Structural Models Of Wage And Employment Dynamics

STRUCTURAL MODELS OF WAGE AND EMPLOYMENT DYNAMICS

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CONTRIBUTIONS TO ECONOMIC ANALYSIS 275

Honorary Editors: D.W. JORGENSON J. TINBERGENy Editors: B. Baltagi E. Sadka D. Wildasin

Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo ii

STRUCTURAL MODELS OF WAGE AND EMPLOYMENT DYNAMICS

Henning Bunzel and Bent J. Christensen University of Aarhus, Denmark George R. Neumann University of Iowa, USA Jean-Marc Robin Universite´ de Paris 1, France

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INTRODUCTION TO THE SERIES This series consists of a number of hitherto unpublished studies, which are introduced by the editors in the belief that they represent fresh contributions to economic science. The term ‘economic analysis’ as used in the title of the series has been adopted because it covers both the activities of the theoretical economist and the research worker. Although the analytical methods used by the various contributors are not the same, they are nevertheless conditioned by the common origin of their studies, namely theoretical problems encountered in practical research. Since for this reason, business cycle research and national accounting, research work on behalf of economic policy, and problems of planning are the main sources of the subjects dealt with, they necessarily determine the manner of approach adopted by the authors. Their methods tend to be ‘practical’ in the sense of not being too far remote from application to actual economic conditions. In addition they are quantitative. It is the hope of the editors that the publication of these studies will help to stimulate the exchange of scientific information and to reinforce international cooperation in the field of economics. The Editors

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CONTENTS PREFACE

xix

LIST OF CONTRIBUTORS

xxi

I. INTRODUCTION SUMMARY OF CONFERENCE PAPERS Dale T. Mortensen

3

II. THEORY PAPERS CHAPTER 1

1. 2.

3. 4.

5.

Introduction The model 2.1. Firms 2.2. Workers Bargaining Market equilibria 4.1. Market Equilibrium with On-the-Job Search 4.2. Market equilibrium with no on-the-job search Discussion Acknowledgment References Appendix: Proof of Proposition 5

CHAPTER 2

1. 2.

BARGAINING, ON-THE-JOB SEARCH AND LABOR MARKET EQUILIBRIUM Ken Burdett and Roberto Bonilla

ON-THE-JOB SEARCH AND STRATEGIC BARGAINING Robert Shimer

Introduction Model 2.1. Preferences and technology vii

15

15 18 19 19 22 26 28 30 31 33 33 34

37

37 40 40

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3.

4.

5.

6. 7.

2.2. Wage bargaining and equilibrium concept 2.3. Nonconvexity of the set of feasible payoffs Market Equilibria with Wage Dispersion 3.1. Bellman values 3.2. Subgame perfect equilibria of the bargaining game 3.3. Other market equilibria 3.4. Wage lotteries Degenerate Market Equilibria 4.1. Single wage market equilibrium 4.2. Many-wage market equilibria Heterogeneous Firms 5.1. Definition of equilibrium 5.2. Testable implications 5.3. Comparison with Burlett and Mortensen (1998) Discussion Conclusion Acknowledgment References

CHAPTER 3

1. 2.

3.

4.

5. 6.

Introduction Ex Ante heterogeneity 2.1. A simple model 2.2. Alternative assumptions 2.3. Discussion Ex post heterogeneity 3.1. Permanent shocks 3.2. The law of two wages Other models 4.1. Transitory shocks 4.2. A crime model Ex ante and ex post combined Conclusion Acknowledgments References

CHAPTER 4

1.

ALTERNATIVE THEORIES OF WAGE DISPERSION Damien Gaumont, Martin Schindler and Randall Wright

WAGE DIFFERENTIALS, DISCRIMINATION AND EFFICIENCY Shouyong Shi

Introduction

41 43 44 44 45 46 48 49 49 50 52 52 54 55 56 57 58 58

61

61 64 64 69 70 71 71 73 75 75 77 79 80 80 81

83

83

ix

2.

3.

4. 5.

6.

The model 2.1. Workers and firms 2.2. A candidate equilibrium The candidate is the unique equilibrium 3.1. Separation is not an equilibrium 3.2. Firms rank the two types of workers strictly 3.3. High-productivity workers have the priority Properties of the equilibrium and the social optimum Extension to many types of workers 5.1. The equilibrium and its properties 5.2. Numerical examples Conclusion Acknowledgment References Appendix A: Proof of Proposition 3 Appendix B: Proof of Proposition 4

CHAPTER 5

1. 2.

3. 4. 5. 6.

LABOR MARKET SEARCH WITH TWO-SIDED HETEROGENEITY: HIERARCHICAL VERSUS CIRCULAR MODELS Pieter A. Gautier, Coen N. Teulings and Aico van Vuuren

Introduction The circular job search model 2.1. Production 2.2. Labor supply 2.3. Labor demand 2.4. Job search technology 2.5. Wage setting 2.6. Free entry condition Equilibrium conditions Characterization of the equilibrium The cost of search Final remarks Acknowledgment References Appendix

87 87 89 91 92 94 95 97 100 100 104 106 108 108 109 111

117

117 121 121 122 122 122 123 123 123 125 126 129 129 129 130

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CHAPTER 6

1. 2.

3.

4.

Introduction Model framework 2.1. The Burdett–Mortensen equilibrium search model with heterogeneous productivity across firms 2.2. Remuneration of firm owners The right-hand tail of the income distribution 3.1. Finite maximum wage 3.2. Tail weight indicators 3.3. Mandelbrot’s weak pareto law Conclusions Acknowledgement References Appendix: Proof of Proposition 1

CHAPTER 7

1. 2.

3.

4.

5.

6. 7. 8.

THE WEAK PARETO LAW IN BURDETT–MORTENSEN EQUILIBRIUM SEARCH MODELS Gerard J. van den Berg

COMPETITIVE AUCTIONS: THEORY AND APPLICATION John Kennes

Introduction Competitive auctions 2.1. The model 2.2. The bidding game 2.3. Buyers choice of seller to bid for 2.4. Sellers’ reserve price choice 2.5. A large market Dynamics 3.1. The model 3.2. Equilibrium The mortensen rule 4.1. Efficient entry 4.2. Efficient technology dispersion 4.3. Efficient job creation On-the-job search 5.1. The model 5.2. Equilibrium What makes a seller? Wages Price posting

133

133 134 134 137 138 138 138 140 142 142 142 143

145

145 149 149 149 150 150 151 151 151 153 153 153 155 156 156 156 158 158 159 160

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9. 10. 11.

Imperfect information 9.1. The model Further applications Conclusions Acknowledgment References

CHAPTER 8

1. 2. 3. 4. 5.

Introduction Model Optimal dividing values Worker and employer behavior Conclusions Acknowledgment References

CHAPTER 9

1. 2.

3.

4.

5. 6. 7.

BLOCK ASSIGNMENTS Michael Sattinger

SOCIAL SECURITY AND INTERGENERATIONAL REDISTRIBUTION Joydeep Bhattacharya and Robert R. Reed

Introduction The model 2.1. Environment 2.2. Time line 2.3. The labor market 2.4. Costs 2.5. Specification of labor market policies 2.6. Workers’ payoffs 2.7. Payoffs to firms 2.8. Matching Bargaining and wage determination 3.1. Wage functions 3.2. Discussion of the wage function for the young Equilibrium 4.1. Definition and existence 4.2. Labor market participation conditions 4.3. Equilibrium entry condition 4.4. Partial equilibrium effects of increasing the generosity of benefits The absence of policy Are pension programs welfare enhancing? Conclusion

161 161 163 165 166 166 169

169 171 173 174 180 181 181

183

183 186 186 186 188 189 190 191 192 192 193 193 194 196 196 196 198 199 201 203 207

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Acknowledgments References Appendix A: Proof of Lemma 1 Appendix B: Proof of Lemma 2 Appendix C: Proof of Lemma 3 Appendix D: Proof of Lemma 4 Appendix E: Proof of Proposition 5

208 209 210 210 211 212 213

III. MICROECONOMETRIC PAPERS CHAPTER 10

1. 2. 3. 4. 5. 6. 7.

THE JOB LADDER Audra J. Bowlus and George R. Neumann

217

Introduction Earnings growth in U.S. census data The Burdett–Mortensen equilibrium search model Data description Fitting the earnings distribution 10 years in the future Downward wage mobility Conclusions Acknowledgments References Appendix

217 219 220 221 222 227 231 233 233 235

CHAPTER 11

1. 2. 3.

4. 5.

6.

HETEROGENEITY IN FIRMS’ WAGES AND MOBILITY POLICIES J.M. Abowd, F. Kramarz and S. Roux

Introduction A simple theory of wages, productivity, and mobility A general set of wage and mobility equations 3.1. Starting-wage equation 3.2. The firm-specific model for wages and mobility Data description 4.1. The DADS Estimation results 5.1. Starting wages 5.2. The firm-specific wage and mobility equations Conclusion

237

237 239 244 244 246 248 248 249 249 249 261

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Acknowledgment References Appendix A: The likelihood function for the firm-specific model of wages and mobility Appendix B: Starting-wage equation estimates Appendix C: Estimation of the corrected covariance matrix CHAPTER 12

1. 2.

3.

4.

5.

6.

7.

THE EMPIRICAL CONTENT OF THE JOB SEARCH MODEL: LABOR MOBILITY AND WAGE DISTRIBUTIONS IN EUROPE AND THE U.S. Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

Introduction Facts about worker turnover and wages 2.1. A brief description of the sample 2.2. Worker turnover 2.3. Wages 2.4. Summary A simple model of worker turnover 3.1. The environment 3.2. Individual labor market transitions 3.3. Stationary worker flows and stocks 3.4. Discussion Structural estimation 4.1. Estimation procedure 4.2. Results Fit 5.1. Transitions across employment states 5.2. Job durations 5.3. The sampling distributions of wage offers 5.4. Wage mobility Identification and specification analysis 6.1. Inference from transition data 6.2. Inference from both transition and duration data 6.3. Inference from cross-sectional wage data 6.4. Inference from wage mobility Conclusion Acknowledgment References Appendix: Data

262 262 263 266 266

269

270 271 272 273 276 278 278 278 280 281 282 283 283 285 288 288 291 293 296 296 297 298 300 302 304 304 305 306

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CHAPTER 13

1. 2. 3. 4. 5. 6. 7.

Motivation The literature The descriptive model Data Estimation Discussion Conclusions Acknowledgment References

CHAPTER 14

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

IDENTIFICATION AND INFERENCE IN DYNAMIC PROGRAMMING MODELS Bent J. Christensen and Nicholas M. Kiefer

Introduction Dynamic programming: the marketing example Discrete states and controls Identification: a preview Likelihood functions Measurement error Imperfect control Random utility models A continuously distributed Utility shock Continuous state and optimal stopping: the search Model Conclusion Acknowledgments References

CHAPTER 15

1. 2. 3. 4.

JOB CHANGES AND WAGE GROWTH OVER THE CAREERS OF PRIVATE SECTOR WORKERS IN DENMARK Paul Bingley and Niels Westergaard-Nielsen

ON ESTIMATION OF A TWO-SIDED MATCHING MODEL Linda Y. Wong

Introduction The model Data Estimation strategy 4.1. Specification 4.2. The likelihood function and solution method

309

310 311 313 315 321 324 327 327 327

331

331 334 335 337 339 346 350 354 356 358 361 362 362

365

365 367 368 370 370 372

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5.

6.

Results 5.1. A low interest rate 5.2. A high interest rate 5.3. Subsamples 5.4. Characteristics of the baseline model 5.5. Regime 1 5.6. Regime 2 Conclusion References

CHAPTER 16

1. 2. 3.

4. 5.

6.

Introduction The data The empirical implementation of a structural nonstationary job search model 3.1. The model 3.2. Parameterization 3.3. Likelihood of the sample Results Simulation of different economic policy changes 5.1. Reform A: 14% increase in insurance benefits, keeping unchanged the declining time sequence 5.2. Reform B: The replacement of the declining time sequence of UI benefits by a constant sequence 5.3. Reform C: The reform B combined with the imposition of punitive sanctions if two job offers are refused 5.4. Reform D: A 3-month increase in the duration of UI entitlement Conclusion Acknowledgment References

CHAPTER 17

1. 2. 3.

A STRUCTURAL NONSTATIONARY MODEL OF JOB SEARCH: STIGMATIZATION OF THE UNEMPLOYED BY JOB OFFERS OR WAGE OFFERS? Ste´fan Lollivier and Laurence Rioux

CAN RENT SHARING EXPLAIN THE BELGIAN GENDER WAGE GAP? Francois Rycx and Ilan Tojerow

Introduction Theoretical framework Description of the data

374 374 374 375 375 376 379 379 380

381

382 385 389 389 392 393 397 405 405 407 409 410 410 410 411

413

414 416 418

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4.

5.

Empirical analysis 4.1. Benchmark specification 4.2. Group effects 4.3. Individual and firm characteristics 4.4. Industry wage differentials 4.5. Endogeneity of profits 4.6. Rent sharing for men and women Conclusion Acknowledgment References Appendix

CHAPER 18

1. 2.

3.

4.

5.

6.

MODELING INDIVIDUAL EARNINGS TRAJECTORIES USING COPULAS: FRANCE, 1990–2002 Ste´phane Bonhomme and Jean-Marc Robin

Introduction Copula models for earnings dynamics 2.1. Copulas 2.2. Transition probability matrices as copula approximations 2.3. Linear dynamics of levels and dynamics of ranks 2.4. Parametric copulas Estimation of mixtures of copula models 3.1. Heterogeneity 3.2. A sequential EM algorithm for discrete mixtures of copulas An empirical model of French individual earnings and employment dynamics, 1990–2002 4.1. The French Labour Force Survey, 1990–2002 4.2. Model specification 4.3. Estimation methodology 4.4. Estimation results 4.5. Model fit Relative earnings mobility 5.1. Spearman rho 5.2. Results Conclusion Reference Appendix A: Parametric copulas Appendix B: The plackett coupla Appendix C: Asymptotic properties of the sequential Em algorithm for copula model Appendix D: Detailed estimation procedure

419 420 422 422 423 424 425 427 428 428 431

441

441 443 443 444 445 446 448 448 449 451 451 452 454 454 463 465 465 466 468 468 470 471 472 475

xvii

IV. MACROECONOMETRIC PAPERS CHAPTER 19

1. 2. 3.

4.

5.

Introduction Environment Equilibrium 3.1. Definition of stationary equilibrium 3.2. Worker side 3.3. Firm side 3.4. Equilibrium distribution of workers 3.5. Characterization of equilibrium Calibration 4.1. Comparative statics – aggregate productivity 4.2. Comparative statics – destruction rate Conclusion References Appendix

CHAPTER 20

1. 2.

3.

AMPLIFICATION OF PRODUCTIVITY SHOCKS: WHY DON’T VACANCIES LIKE TO HIRE THE UNEMPLOYED? E´va Nagypa´l

EVALUATING THE PERFORMANCE OF THE SEARCH AND MATCHING MODEL Eran Yashiv

Introduction The search and matching model 2.1. The basic set-up 2.2. Matching 2.3. Firms 2.4. Wages 2.5. Equilibrium 2.6. The dynamics and the steady state 2.7. The contribution of Dale Mortensen U.S. labor market data 3.1. The relevant pool of unemployment 3.2. Job vacancies 3.3. The flow of hires and the worker job-finding rate 3.4. Wages 3.5. Other data series 3.6. Data properties

481

482 485 488 488 489 492 493 495 496 498 502 503 504 506

509

510 512 512 513 514 515 517 517 520 521 521 525 529 530 531 532

xviii

4.

5. 6.

Calibration and model-data fit 4.1. Calibration 4.2. Model-data fit The underlying mechanism Conclusions Acknowledgment References Appendix: Data (Source and definitions)

CHAPTER 21

1. 2.

3.

4.

5.

6.

PRODUCTIVITY GROWTH AND WORKER REALLOCATION: THEORY AND EVIDENCE Rasmus Lentz and Dale T. Mortensen

Introduction A model of creative destruction 2.1. Household preferences 2.2. The value of a firm Deterministic productive heterogeneity 3.1. Product creation 3.2. The distribution of firm size Market equilibrium 4.1. Firm entry and labor market clearing 4.2. Existence Evidence and estimation 5.1. Danish firm data 5.2. Model estimation 5.3. Model simulation 5.4. Identification 5.5. Estimation results Concluding remarks References

SUBJECT INDEX

536 536 538 542 545 546 546 549

551 552 554 554 555 558 558 559 560 561 562 565 565 567 569 570 571 575 576 579

PREFACE

This volume contains papers presented at the Conference on Labor Market Models and Matched Employer–Employee Data held at Sandbjerg Manor in Sønderborg, Denmark, August 15–18, 2004. The conference was held in honor of Dale Mortensen upon the occasion of his 65th birthday. He has served as the Chair of the Department of Economics and as the Ida C. Cook Professor of Economics and the Director of the Mathematical Methods in the Social Sciences Program at Northwestern University. His research and teaching interests are in labor economics, macroeconomics, and economic theory. Professor Mortensen received his B.A. in Economics from Willamette University in 1961 and his Ph.D. in Economics from Carnegie-Mellon University in 1967. Although he has been on the faculty of Northwestern University since 1965, he has also held visiting appointments at the University of Essex, Hebrew University, New York University, California Institute of Technology, and Cornell University as well as visiting research appointments at the Central Institute of Mathematics–Economics in Moscow, Russia, the Center for Labor and Social Research in Aarhus, Denmark, Research School of Social Sciences, The Australian National University, and Research Fellow at the Institute for the Study of Labor (IZA), Bonn, Germany. He presented the Zeuthen Lectures at the University of Copenhagen in November 2000, the Mackintosh Lecture at Queens University in March, 2002, and the Schumpeter Lecture Series at Humboldt University, Berlin, in June 2002, and the Marshall Lecture at the 2005 meeting of the European Economic Association. Mortensen pioneered the theory of job search and search unemployment and extended this approach to study labor turnover, research and development, personal relationships, and labor reallocation. His insight, that friction is equivalent to the random arrival of trading partners, has become the leading technique for the analysis of labor markets and the effects of labor market policy. The development of equilibrium dynamic models designed to account for wage dispersion and the time-series behavior of job and worker flows are the principal topics of his current research. His publications include over 50 scientific articles and contributions to books. His new book, Wage Dispersion: Why are Similar Workers Paid Differently, was published in 2003 by MIT Press. Mortensen is a past president of the Society of Economics Dynamics, one of the founding editors of the Review of Economic Dynamics, a Fellow of the xix

xx

Econometric Society, a Fellow of the American Academy of Arts and Sciences, and a Fellow of the Society of Labor Economists. It was entirely appropriate that this conference, the sixth in a series, be held at Sandbjerg, where it began in 1982. From the beginning there has been close interplay among economic theorists, econometricians, and applied economists in analyzing what can be learned about the dynamics of employment and wage formation, and Dale Mortensen has had an important influence on all three areas. These conferences have had significant influence on how we think about public policy in the labor market, and what kinds of data would be needed to answer questions about these policies. The scientific program of the conference was prepared by a committee consisting of the editors and Dale Mortensen. After the conference, all papers have been subjected to a refereeing process, with papers being sent to one or two referees chosen from the participants. The conference has received valuable financial support from the Aarhus University Research Foundation, Danmarks Nationalbank, The Carlsberg Foundation, the Danish National Research Foundation, and from the George Daly Research Fund in the Tippie College of Business at the University of Iowa. We especially want to thank Mrs. Kirsten Stentoft for her work on organizing the conference, for her patience in dealing with slow authors and editors. Henning Bunzel University of Aarhus Bent J. Christensen University of Aarhus George R. Neumann University of Iowa Jean Marc Robin Universite´ de Paris I

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List of Contributors Abowd, M. John

ILR Labor Economics, Cornell University, 259 Ives Hall, 38 Beckett Way, Ithaca, NY 14850, USA. e-mail: [email protected]

Bhattacharya, Joydeep

Department of Economics, Iowa State University, Ames IA 50011-1070, USA. e-mail: [email protected]

Bingley, Paul

National Centre for Register-based Research, University of Aarhus, Taasingegade 1, DK-8000 Aarhus C, Denmark. e-mail: [email protected]

Bonhomme, Ste´phane

CREST-INSEE, 15 Boulevard Gabriel Pe´ri, 92245 Malkoff Cedex, France. e-mail: [email protected]

Bonilla, Roberto

Department of Economics, University of Newcastle upon Tyne Business School, Ridley Building, Newcastle upon Tyne, NE1 7RU, UK. e-mail: [email protected]

Bowlus, Audra J.

Department of Economics, University of Western Ontario, London, Ontario N6A 5C2, Canada. e-mail: [email protected]

Bunzel, Henning

School of Economics and Management, University of Aarhus, Building 322, 8000 Aarhus C, Denmark. e-mail: [email protected]

Burdett, Kenneth

Department of Economics, University of Pennsylvania, 439 McNeil, 3718 Locust Walk, Philadelphia, PA 19104-6297, USA. e-mail: [email protected]

Christensen, Bent J.

School of Economics and Management, University of Aarhus, Building 322, 8000 Aarhus C, Denmark. e-mail: [email protected] xxi

xxii

Gaumont, Damien

Universite´ Paris II Pantheon-Assas and ERMES, 12 Place du Panthe´on, F-75231 Paris Cedex 05, France. e-mail: [email protected]

Gautier, Pieter A.

Tinbergen Institute, Free University of Amsterdam, Roetersstraat 31, 1018 WB Amsterdam, The Netherlands. e-mail: [email protected]

Jolivet, Gregory

CREST-INSEE, 15 Boulevard Gabriel Pe´ri, 92245 Malakoff Cedex, France. e-mail: [email protected]

Kennes, John

University of Copenhagen, Centre for Applied Microeconometrics (CAM), Studiestræde 6, DK-1455 Copenhagen K, Denmark. e-mail: [email protected]

Kiefer, Nicholas M.

Department of Economics, Cornell University, 476 Uris Hall, Ithaca, NY 14853-7601, USA. e-mail: [email protected]

Kramarz, Francis

De´partement de la recherche, INSEE, 15 bd Gabriel Pe´ri, 92245 Malakoff Cedex, France. e-mail: [email protected]

Lentz, Rasmus

Department of Economics, University of Wisconsin–Madison, 1180 Observatory Drive, room 6432, Madison WI 3706–1393, USA. e-mail: [email protected]

Lolliver, Ste´fan

INSEE, 18 Boulevard A. Pinard, Timbre F001, 75675 Paris Cedex 14, France. e-mail: [email protected]

Mortensen, Dale T.

Department of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL. 60208, USA. e-mail: [email protected]

Nagypa´l, E´va

Department of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL. 60208, USA. e-mail: [email protected]

Neuman, George R.

Department of Economics, University of Iowa, Iowa City, IA 52242, USA. e-mail: [email protected]

xxiii

Postel-Vinay, Fabien

PSE, Ecole Normale Supe´rieure, 48 boulevard Jourdan, 75014 Paris, France. e-mail: [email protected]

Reed, Robert R.

Gatton College of Business and Economics, Department of Economics, University of Kentuckey, Lexington, KY 40506, USA. e-mail: [email protected]

Rioux, Laurence

CREST-INSEE, 15 Bd Gabriel Pe´ri, 92245 Malakoff Cedex, France. e-mail: [email protected]

Robin, Jean-Marc

Universite´ de Paris 1 - Panthe´on – Sorbonne, Maison des Sciences Economiques, EUREQua, 106/112 bd de l’Hoˆpital, 75647 Paris Cedex 13, France. e-mail: [email protected]

Roux, Se´bastien

CREST-INSEE, 15 Boulevard Gabriel Pe´ri, 92245 Malakoff Cedex, France. e-mail: [email protected]

Rycx, Francois

Department of Applied Economics, Universite´ Libre de Bruxelles, CP140, Avenue F.D. Roosevelt 50, 1050 Bruxelles, Belgium. e-mail: [email protected]

Sattinger, Michael

Department of Economics, University at Albany, Albany NY 12222, USA. e-mail: [email protected]

Schindler, Martin

Research Department, International Monetary Fund, 700, 19th Street NW, Washington, DC 20431, USA. e-mail: [email protected]

Shi, Shouyong

Department of Economics, University of Toronto, 150 St. George Street M5S 3G7, Canada. e-mail: [email protected]

Shimer, Robert

Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637, USA. e-mail: [email protected]

Teulings, Coen N.

SEO, University of Amsterdam, Roetersstraat 29, 1018 WB Amsterdam, The Netherlands. e-mail: [email protected]

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Tojerow, Ilan

Department of Applied Economics, Universite´ Libre de Bruxelles, CP140, Avenue F.D. Roosevelt 50, 1050 Bruxelles, Belgium. e-mail: [email protected]

van den Berg, Gerard J.

Department of Economics, Free University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. e-mail: [email protected]

van Vuuren, Aico

Department of Economics, Free University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. e-mail: [email protected]

Westergaard-Nielsen, Niels

Department of Economics, Aarhus School of Business, Prismet, Silkeborgvej 20, 8000 Aarhus C, Denmark. e-mail: [email protected]

Wong, Linda Y.

Department of Economics, Binghamton University, PO Box 6000, Binghamton, NY 13902, USA. e-mail: [email protected]

Wright, Randall

Department of Economics, University of Pennsylvania, 439 McNeil, 3718 Locust Walk, Philadelphia, PA 19104-6297, USA. e-mail: [email protected]

Yashiv, Eran

Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK. e-mail: [email protected]

I. INTRODUCTION

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Summary of the conference papers Dale T. Mortensen It is a privilege to summarize and comment on the papers included in this volume. I am also honored that the conference participants came to present them at Sandbjerg Manor at least in part to acknowledge my advancing age. The conference documented in this volume is the latest in a series held in Denmark and elsewhere on the theoretical and empirical topics in labor market dynamics organized by Henning Bunzel with the able assistance of Kirsten Stentoft. The contributions presented at three previous conferences were also published in the North-Holland ‘‘Contributions to Economic Analysis’’ series. This collection provides an interesting record of the history of the application of search and matching theory to the analysis of panel data. On behalf of myself and others in this general field of economic research, I thank my friends Henning and Kirsten for their tireless organizational efforts. I also thank them and the editor of the volume for organizing this conference in my honor. Allowing for search on-the-job has both enriched and complicated the study of wage and employment dynamics in recent years. The problems that arise do so because on-the-job search by an employee adversely affects the value of the match to the employer. Because the decision to search or not depends on the expected future wages earned on the current job, there is considerable room for interesting strategic interaction in the wage determination process when on-thejob search is a possibility. In what some refer to as the ‘‘bench mark’’ paper by Burdett and Mortensen (1998), wage dispersion, defined as different wages paid to observable equivalent workers, is the outcome when employer’s post wages that trade off a higher wage against the benefit of a lower turnover rate. Two of the theory papers in this volume look at bargaining as an alternative form of wage determination in a similar environment. Roberta Bonilla and Ken Burdett in ‘‘Bargaining, On-the-Job Search, and Labor Market Equilibrium’’ study a model in which the initial wage of a worker when transiting from unemployment to employment is determined as the outcome of a particular bilateral bargaining game. Given that all agents are risk neutral and there is a strictly positive cost-of-search effort, the authors show that an employed worker does not search on-the-job if the wage exceeds a specific value, which the authors choose to call the ‘‘search wage.’’ As the employer 3

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Dale T. Mortensen

knows the value of the search wage even if the worker’s search behavior is not observed, she can either agree to pay the search wage, and thereby enjoy the benefit of a longer match duration, or pay a lower but acceptable wage, and let the worker look for a better alternative while employed. In their formulation, all employers and workers are respectively identical. The authors also assume that once found, an alternative employer and the current employer will bid up the wage to the level that equates the value of a match to that of a vacancy. The authors show that a steady-state equilibrium solution to the model will always be characterized by search on-the-job when the cost is low enough and by no search when the cost is sufficiently high. Furthermore, for intermediate values of the cost of search on-the-job, multiple equilibria, one with and another without search on-the-job, coexist. The multiplicity reflects the complementarity of the strategic interaction that arises when the search and wage setting decisions are simultaneously determined. In a related paper, ‘‘On-The-Job Search and Strategic Bargaining,’’ Rob Shimer also studies a model of wage determination through bargaining in the same Burdett–Mortensen environment. He begins by noting that Nash’s axiomatic solution to the bargaining problem does not generally apply when an increase in the wage induces an increase in match duration. Instead, he formulates and solves a strategic alternating offer bargaining game of the Rubinstein type under the assumption that the worker must choose to negotiate with only one employer among the set contacted. (In other words, an alternative and current employer does not engage in Bertrand competition as assumed by Bonilla and Burdett and others in the literature.) Shimer characterizes the set of sub-game perfect equilibria when search on-the-job is costless. As in the Burdett–Mortensen model, the most plausible among these implies wage dispersion. Specifically, a unique continuous wage distribution exists such that the Nash product of employer’s and employee’s match values is maximal and the same for all wages in its support. Shimer also characterizes the nature of equilibrium wage dispersion when productivity differs across employers. Finally, he concludes that his bargaining model is more tractable and has more plausible empirical implications than the Burdett–Mortensen wage posting model. As its title suggests, ‘‘Alternative Theories of Wage Dispersion’’ by Damien Gaumont, Martin Schindler, and Randall Wright is also concerned with modeling wage dispersion. However, the theories considered are more in spirit to the classic model of Albrecht and Axell (1984) rather than the Burdett–Mortensen model in the sense that workers value equivalent jobs differently. In addition to making several suggestions about directions for the study of wage and price dispersion, the paper contains two notable contributions. First, the authors show that when worker preferences over job are different ex ante and search is costly, only no trade is an equilibrium. The reason is that the set of equilibrium wages is contained in the set of reservation wages. Hence, the highest wage in any equilibrium is the reservation wage of the pickiest type. But, given a positive search cost, a worker of this type will not participate because he or she rationally

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anticipates no rents ex post. Second, models in which heterogeneity of worker preference are generated ex post, after a contact has been made, do not suffer from this problem and generally have equilibria with wage dispersion. However, the authors also show that the support of the equilibrium distribution in this case generically includes at most two points. That a positive relationship should hold between worker ability and wages received is virtually axiomatic. In ‘‘Wage Differentials, Discrimination, and Efficiency,’’ Shouyong Shi develops a model of wage determination with the property that less able workers can be paid more. Workers differ by skill and skill is observable. The model is one of ‘‘directed search’’ in which each employer has but one job to fill that can be occupied by a worker of any skill. Employers announce which skills they are willing to hire, the wage that will be paid to each skill type, and the skill rank order used to fill the job from among the applications. With this information, workers must decide where to apply. An employer’s strategy maximizes expected profit while each worker selects a probability distribution over employers that determines where he or she will apply, one that maximizes the worker’s expected wage. This environment suffers from a coordination problem in the sense that every worker must take into account the application strategy of every other while every employer chooses a hiring strategy that takes account of its effect on the application decisions of workers. In the case of two skill types and identical employers, Shi demonstrates that a unique symmetric equilibrium exists with the property that all firms are willing to hire workers of both types, all employers offer a common wage to each worker type, and a more-skilled worker is placed in the job ex post if and only if one is available. Because in equilibrium, the expected wage is the same across all firms for each worker type but a more-skilled worker ranks above a less-skilled worker in job placement, the more-skilled is compensated in part by a higher probability of employment. As a consequence, the wage paid to a more-skilled worker is less than that of a less-skilled worker for all sufficiently small differences in productivity. Furthermore, this ‘‘perverse’’ wage differential is constrained efficient because the ‘‘crowding out’’ effect on each worker’s application decision on other workers is internalized as in directed search models more generally. Finally, Shi shows that his result generalizes to any finite number of skill types and that the model can produce wage distractions of the kind observed in the data. Pieter Gautier, Coel Teulings, and Aico van Vuuren in their paper ‘‘Labor Market Search with Two-Sided Heterogeneity: Hierarchical versus Circular Models’’ derive and characterize search equilibrium (without search on-the-job) in an environment in which the matching of worker and job types takes place with friction. The crucial problem is the determination of the set of acceptable types for each individual. A symmetric equilibrium is a non-cooperative reservation strategy for each agent that maximizes expected value given the choices of all other worker and employer types. The authors assume that worker types and

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job types can each be represented as points on a circle that match output declines with the difference between the actual and first best pairing. Although this specification differs from the usual assumption that match output increases with both type indices, they prove that the equilibrium for the circular model shares most of the properties derived in previous work by the authors for the hierarchical model. Moreover, because there is no ‘‘corner problem’’ in the circular model they are able to establish that a unique equilibrium exists. Furthermore, the equilibrium is essentially equivalent to one in which match output is stochastic. Gerard van den Berg begins his paper, ‘‘The Weak Pareto Law in Burdett–Mortensen Equilibrium Search Models,’’ with the well-known observation that income distributions have Pareto-like right tails. Given mild restrictions on the underlying distribution of productivity, he shows that the right tail of the income distribution implied by the Burdett–Mortensen model with firmspecific productivity heterogeneity has this property if the earnings of firm owners are those that receive high incomes, as suggested by empirical evidence. This property follows from the fact that the employer’s profit share of output, as well as the wage, increases with productivity. Intuitively, profit income is a convex function of productivity because the firm’s ‘‘monopsony power’’ increases as the relative wage paid rises. The result is a Pareto tail. John Kennes in his paper ‘‘Competitive Auctions: Theory and Application’’ makes a case for a specification in which wages are determined as the outcome of a decentralized ‘‘competitive auction’’ rather than either bargaining or wage posting, the mechanisms typically assumed in search and matching models. For example, in a model in which employer’s sample worker, the worker receives a wage equal to the value of the ‘‘second best alternative,’’ which is either his or her productivity in the next best job, if two or more offers are received or the value of the worker’s outside option, if only one offer arrives. As he notes, I observed in Mortensen (1982) that the outcome of a broad class of matching games is constrained efficient if (1) matches with positive surplus form and (2) the individual agent responsible for any particular match receives the entire match surplus. The reason is that the social return to the creation of the match is generally the product of the agent’s contribution to the probability of match creation and the surplus value of the match. A problem with the rule is implementation; how is the initiator of the match determined? In the context of a search for a single product or service, one in which each buyer demands a single unit, in which buyers and sellers are respectively identical, Kennes shows that outcome of the auction by the seller described satisfies the Mortensen Rule, the two conditions stated above. Furthermore, efficiency is obtained with any finite number of agents of the two types even though the ‘‘matching function’’ is not linearly homogenous as required by the well-known Hosios condition. He also provides a second example with endogenous heterogeneity, one in which workers invest in skill prior to matching, for which the Mortensen Rule supports efficient investment decisions. Because of the heterogeneity of types, the Hosios condition

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cannot be applied in this case either. It is of interest to note that, price dispersion in the sense that identical agents can receive different prices ex post occurs in both of these examples. Indeed, dispersion in this sense is an obvious corollary of the Mortensen Rule. Other applications considered include search on-the-job. In the paper, ‘‘Block Assignments,’’ Michael Sattinger asks whether separate markets better serve the matching needs of employers and workers than would a market in which workers and employers of all types search and match in the same market. For example, suppose worker skill and employer skill are complements so that the first best-matching equilibrium is strictly assortative. Given search friction, he shows that two markets perform better than one if ‘‘blocks’’ of high-skill workers and employers participate in one while low-skill workers and employers match in the other. Unlike the case of a ‘‘marriage market’’ with non-transferable utility where this pattern can be an equilibrium outcome as demonstrated by Burdett and Coles (1997), he argues that the arrangement will not arise as a consequence of reservation choices made by individual agents when utility is transferable. Hence, market institutions of these types are not supported by the decentralized search decisions of individuals. In the final theory paper, Joydeep Bhattacharya and Robert Reed give a novel efficiency rationale for the cross-country prevalence of government pension plans funded by ‘‘pay-as-you-go’’ tax systems that redistribute income from the young to the old in their paper ‘‘Social Security and Intergenerational Redistribution.’’ They first note that these systems seem to be designed to induce retirement by the elderly in order to free up jobs for the young. They argue that search and matching friction create congestion and hold-up problems such that younger employees may be preferred to older ones. Under this circumstance, reducing the supply of older workers in the model can encourage job creation and higher wages. As a consequence, they argue, pension programs that provide incentives for earlier retirement for those workers with long employment histories foster job creation and raise welfare. This is a theoretical paper that uses a relatively standard search model set in an overlapping generation structure to generate formal proofs of the authors assertions. The remaining papers in the volume are organized in groups of contributions to micro- and macroeconomics, respectively. As one who has never felt that this distinction was particularly meaningful, I am somewhat reluctant to endorse it here. However, for the purpose of exposition I will follow along. Since the original paper on search on-the-job by Ken Burdett (1978), the empirical question has been the extent to which job changes explain the growth in wages of individual workers rather than improvements in earning directly attributable to tenure on a job and years of work experience. In their paper, ‘‘The Job Ladder,’’ Audra Bowlus and George Neumann use the structure of my equilibrium version of that model, Mortensen (1990), to answer the question. Specifically, using the parameters of the model of search on-the-job previously estimated from National Longitudinal Survey of Youth (NLSY) data, they show that the wage distributions obtained by simulating individual wage trajectories

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fit the experience contingent distributions found in the data quite well, particularly if one allows for a common trend that they attribute to productivity growth. However, because the model does not allow for the wage reductions that characterize actual wage trajectories, it fails in some important dimensions. In ‘‘Heterogeneity in Firms’ Wages and Mobility Policies,’’ John Abowd, Francis Kramarz, and S. Roux estimate a model that allows for firm wage policy and mobility differences using French longitudinal matched employer–employee data. Firm differentials in both the level of the wage and its rate of growth with tenure are allowed. They find substantial heterogeneity across firms. Although there is no average seniority effect on wages in the data, the strongest association suggested by the parameter estimates is that a continuum of firms exists with a group that pay low wages, offer a high return to seniority, and experience high turnover located at one end, and another group that pay high wages, a low return to seniority, and experience low turnover at the other end. These results suggest that the wage-turnover trade-off exists and plays an important role in a firm’s choice of wage and mobility policy. The paper ‘‘The Empirical Content of the Job Search Model: Labor Mobility and Wage Distributions in Europe and the U.S.’’ by Gregory Jolivet, Fabien Postel-Vinay, and Jean-Marc Robin uses comparable panel data from 10 European countries and the US to test the ‘‘very tight correspondence between the determinants of labor turnover and individual wage dynamics on one hand, and the determinants of wage dispersion on the other’’ implied by the Burdett– Mortensen model. Specifically, they use the turnover parameters estimated from the data to derive the implied difference between the steady-state distributions of wages offered and earned, and compare these differences with those observed in the data. They find that the model does a very good job explaining the extent to which the distribution of wages earned stochastically dominates the wage offer distribution in almost all of the 10 country cases. In the model estimated, the authors allow for job-to-job transitions that need not result in wage gains in their empirical model. They attribute these ‘‘involuntary’’ job-to-job transitions to ‘‘reallocation shocks.’’ In the extended Burdett–Mortensen model estimated, such an event is equivalent to a layoff followed by a transition to a new job without an intervening unemployment spell. Interestingly, the authors find that turnover is primarily ‘‘voluntary’’ in low turnover countries and is largely ‘‘involuntary’’ in countries that experience high turnover rates. Paul Bingley and Niels Westergaard-Nielsen take a more direct approach to documenting the role of wage differentials in explaining job-to-job movements in their paper ‘‘Job Changes and Wage Growth over the Careers of Private Sector Workers in Denmark.’’ Using the Danish Integrated Database for Labor Market Research (IDA) data, they first estimate firm wage level and growth equations that include firm fixed effects. They then show that firms with higher wage levels and growth rates experience lower separation rate as suggested by models of search on-the-job. They also show that workers are more likely to

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separate if alternative competing employers pay more and/or offer faster wage growth. These results confirm in a less-restrictive model those of Christensen, Lentz, Mortensen, Neumann, and Werwatz (2005) who use the same data. They are also consistent with the finding for France reported in this volume by Abowd, Kramarz, and Roux. B.J. Christensen and Nick Kiefer address a number of econometric issues that arise when estimating dynamic stochastic decision models in ‘‘Identification and Inference in Dynamic Programming Models.’’ In the paper, they take the reader step by step through the empirical implications of a simple discrete state and control dynamic programming problem as a means of illustrating that only interval identification of many parameters of interest can be guaranteed. The authors go on to discuss the dual curses of ‘‘degeneracy’’ and ‘‘determinacy’’ and to develop the implications of alternative ways of dealing with them. They sum up by showing how these issues arose and were handled in their prior work on estimating the parameters of search models. In her paper, ‘‘On Estimation of a Two-Sided Matching Model,’’ Linda Wong proposes a method of estimating ‘‘marriage model’’ using data on marital status. Except for the assumption of non-transferable utility, the model is related to that analyzed by Gautier, Teulings, and van Vuuren in this volume. As in that paper, meeting occurs at Poisson arrival rates and match utility depends on the types of the two agents matched. The essential decision is the determination of the set of types of the opposite sex that are acceptable mates for each individual given the choices of all the others in the ‘‘marriage market.’’ The principal indicators of type are wage and education and the parameter of interest is the relative weight put on these two factors as determinates of ‘‘relative attractiveness.’’ Given the specification as a function of attractiveness, the equilibrium of the model is characterized by stratification into partitions of the type space into a set of marriage classes as in Burdett and Coles (1997). Ste´fan Lollivier and Laurence Rioux study the determinants and effects of job security perceptions in their paper ‘‘A Structural Non-Stationary Model of Job Search: Stigmatization of the Unemployed by Job Offers or Wage Offers?’’ Their data is from the European Community Household (ECHP) panel, which includes survey questions that inquire about the worker’s expectations regarding job loss. Whether perceived job security improves the prediction of the subsequent duration of the current job is one of the issues they address. In their study, ‘‘Can Rent Sharing Explain the Belgian Gender Wage Gap?’’, Francois Rycx and Ilan Tojerow begin with the observation that workers employed by more profitable firms earn more holding worker and other firm characteristics constant. In this paper, they ask whether there are differences in the relationship by sex. Based on a matched employer–employee data set for Belgium, they find strong evidence for profit sharing in the private sector but no significant difference between the effect of profitability on wages for men and women. However, their results do suggest that about half of the gap can be attributed to the fact that women generally work for less-profitable firms.

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Ste´phane Bonhomme and Jean-Marc Robin develop a new methodology for the analysis of earnings inequality in a dynamic context in ‘‘Modeling Individual Earnings Trajectories Using Copulas: France, 1990–2002.’’ The technique uses copula representations of multidimensional densities to decompose a sequence of earning distributions into the product of two components: the product of an initial marginal – or cross-section – density and the likelihood of the sequence of individual ranks in the marginal distributions. They then apply the technique using French Labor Force Survey data. They find that 20% of the variance of intertemporal log-earnings means is transitory in nature while 60% of the longitudinal variance is permanent, and 33% is persistent. They also establish that earnings mobility slows down when cross-section inequality increases at business-cycle frequencies. In his recent critique of the search and matching model of the unemployment as codified in Mortensen and Pissarides (1994), Shimer (2005) shows that productivity shocks in the most rudimentary version of the model with reasonable parameter values do not induce movements in vacancies and unemployment in the model of the magnitudes observed in US post-WWII time series data. Indeed, the empirical elasticity of ‘‘market tightness’’ with respect to productivity is an order of magnitude larger than that implied by the model. The paper by Eva Nagypa´l and Eran Yashiv addresses this issue in different ways. In her paper, ‘‘Amplification of Productivity Shocks: Why Don’t Vacancies Like to Hire the Unemployed?’’ Nagypa´l develops an extension of the standard search and matching model that allows for search on-the-job. By adding a fixed cost of vacancy creation and endogenous search effort, she shows that employers can expect a higher profit from an employed worker than an unemployed worker. Because search effort by the employed and vacancies are complements in the matching process, productivity shocks are amplified in this case. Specifically, a positive shock to productivity induces an initial increase in vacancies, which stimulates the search intensity of employed workers by more than the unemployed. Given that employers prefer the employed, a further increase in vacancy creation takes place. After calibrating the model in ways that make it comparable with Shimer’s analysis, she finds that this multiplier process can explain nearly 70% of the observed variation in the US vacancy–unemployment ratio. In his paper, ‘‘Evaluating the Performance of the Search and Matching Model’’ Yashiv claims his more complex version of the basic search and matching model is fully consistent with the high persistence and volatility in macro labor variables observed as well as the negative correlation between vacancies and unemployment. In his analysis, the exogenous processes that induce employment fluctuations are estimated from the data using a structural VAR rather than assumed to be shocks to labor productivity. He specifies and estimates a convex costs of vacancy creation and hiring rather than making the usual linear assumption. His operational definition of unemployed search is more expansive than that used by Shimer. Finally, counter cyclic movements in job separation

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rates are also important in his model. All of these differences seem to contribute to the different result. In my joint paper with Rasmus Lentz, ‘‘Productivity Growth and Worker Reallocation: Theory and Evidence,’’ we develop and estimate a structural model of aggregate endogenous growth. The model is one of growth through new product creation and destruction in the Schumpeterian tradition. In the model, firms differ with respect to innovative ability, which reflects itself in differential rates of firm growth. The estimated model implies that a large share of productivity growth is associated with the reallocation of employment from slow-growing and dying firms to more innovative fast-growing firms. We plan to extend the model in ways that allow for search friction in future research. I thank the authors for providing this very stimulating collection of new and original research on some of my favorite topics. References Albrecht, J. and B. Axell (1984), ‘‘An equilibrium model of search unemployment’’, Journal of Political Economy, Vol. 92, pp. 824–840. Burdett, K. (1978), ‘‘A theory of employee search and quits’’, American Economic Review, Vol. 68, pp. 212–220. Burdett, K. and M. Coles (1997), ‘‘Marriage and class’’, Quarterly Journal of Economics, Vol. 112, pp. 141–168. Burdett, K. and D.T. Mortensen (1998), ‘‘Wage differentials, employer size and unemployment’’, International Economic Review, Vol. 39, pp. 257–273. Christensen, B.J., R. Lentz, D.T. Mortensen, G. Neumann and A. Werwatz (2005), ‘‘Job separations and the distribution of wages’’, Journal of Labor Economics, Vol. 23, pp. 31–58. Mortensen, D.T. (1982), ‘‘Property rights and efficiency in mating, racing and related games’’, American Economic Review, Vol. 72, pp. 968–979. Mortensen, D.T. (1990), ‘‘Equilibrium wage distributions: a synthesis’’, in: J. Hartog, G. Ridder and J. Theeuwes, editors, Panel Data and Labor Market Studies, Amsterdam: North-Holland. Mortensen, D.T. and C.A. Pissarides (1994), ‘‘Job creation and job destruction in the theory of unemployment’’, Review of Economic Studies, Vol. 61, pp. 397–415. Shimer, R. (2005), ‘‘The cyclical behavior of equilibrium unemployment and vacancies’’, American Economic Review, Vol. 95(1), pp. 25–49.

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II. THEORY PAPERS

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CHAPTER 1

Bargaining, On-the-Job Search and Labor Market Equilibrium Ken Burdett and Roberto Bonilla Abstract The objective of this paper is to investigate the equilibrium consequences of assuming workers can search when employed, but only at a cost. When an unemployed worker contacts a firm with a vacancy, they use a finite strategic bargaining game to establish the wage to be paid. Although search is unobservable both are aware the worker may search while employed if the wage is low enough. If an employee does contact another firm, both firms are assumed to bid for the services of the worker. This, of course, implies wage dispersion exists as an equilibrium characteristic. Further, it will be shown that multiple equilibria can exist – an equilibrium where employees do not search, and another where employees do search.

Keywords: search, bargaining, equilibrium JEL classifications: J31, J64 1. Introduction A worker may choose to search for another job while employed. Indeed, many of us choose to search on-the-job at times. Of course, searching for another job is costly at least in terms of time and effort. The object of this study is to

Corresponding author. r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75001-X

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investigate equilibrium in a labor market where employed workers can choose to search for another job but only at a cost. There are few studies that have focussed on this decision in an equilibrium context. Moscarini (2003) and Burdett et al. (2004) provide the only two examples known to the authors. Moscarini studies such a decision in the context of the worker and firm learning about the quality of the match. Burdett et al. investigate the decision to search while matched within the context of a marriage market model. In this case a husband’s desire to look for a new partner depends, in part, on whether his wife is looking for a new partner, or not. As in this study, they assume throughout the search decision is a discrete one. Varying search intensity may well be difficult in many situations. To investigate the equilibrium consequences of the above decision a model is constructed with two major features. First, the wage paid to an unemployed worker who contacts a firm is established by a strategic finite stage random offer game. Second, if an employed worker does search and contacts a firm with a vacancy, the two firms bid for the services of the worker. Using a simple model of this type where all firms and all workers are homogeneous, it will be shown that the decision to search, or not, by employed workers has a profound impact on equilibrium and this leads to several new insights into the equilibrium workings of labor markets that have not been explored before. For example, it will be shown that even in the simple labor market considered, the wage bargains in equilibrium may imply that employed workers pay to search while employed. This, of course, implies wage dispersion exists as an equilibrium characteristic. Further, it will be shown that multiple equilibria can exist – an equilibrium where employees do not search, and another where employees do search. There is no doubt most employees search every now and then even though such behavior is costly to the individual. Indeed, Fallick and Fleischman (2004) estimate that 50 percent of job changes involve no interim unemployment. This and other work has led to the recognition that on-the-job search plays an important role in the labor market. All this implied a recent explosion of work on this topic (see, for example, Burdett, 1978; Pissarides, 1994; Burdett and Mortensen, 1998; Postel-Vinay and Robin, 2002a,b; Cahuc, et al., 2003; Shimer, 2004). In these studies workers search while employed by assumption. There is no search decision by the worker. In contrast, the search decision of employed workers, and the consequences of this decision on wages and employment, is the focus of this study. In the literature on labor market with frictions, two approaches have been used to determine how wages are established. The first approach assumes that firms post wages at which they hire workers (see, for example, Burdett and Mortensen, 1998; Mortensen, 2003). In the second approach, it is assumed that when a worker and firm make contact they bargain over the wage paid to the worker if he or she becomes an employee (see, for example, Pissarides, 2000). In the vast majority of studies to date based on the bargaining approach it has been

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assumed that workers and firms reach a Nash bargain in establishing the wage. Given employees must also decide whether to search or not when employed, we consider a strategic finite random offer bargaining game.1 As the worker’s search behavior while employed is assumed not to be observable to the firm, the bargain cannot condition on search behavior. Given the worker does search on the job, we need to describe what happens when a firm with a vacancy is contacted by an employed worker. Several approaches are possible. Here we follow Postel-Vinay and Robin (2002a) and assume the two firms bid for the services of the worker. Given all firms are the same, both firms will bid the same wage. This wage will make each firm indifferent between hiring the worker, or posting a vacancy. For simplicity we assume the worker stays with his or her original employer. Let z denote the wage received by a worker after this bidding process. Clearly, as all firms are homogeneous, no other firm in the market can offer a greater wage than z and therefore a worker receiving such a wage will not search. Suppose now a firm with a vacancy contacts an unemployed worker and they bargain over the wage to be paid. Both realize, given z, and the cost of search, c, that if the wage bargained is less than some wage Q – the search wage – then the worker will search on-the-job if employment is accepted. If the wage negotiated is greater than the search wage, then the worker will not search while employed. Even though search behavior cannot be observed, in equilibrium the firm will have correct beliefs about whether the worker will search, or not while employed. Of course, given a wage, the firm prefers its employees not to search as such employees yield a greater expected return. Hence, under certain parameter values, a firm is willing to offer a worker a greater wage so the worker will not search. As will be shown, there is a limit to the firm’s willingness to stop its employees from searching. In particular, if the worker’s cost of search is small, it will be shown that the bargained wage implies the worker will search on-the-job. In this case employed workers are paid one of two wages – the wage bargained between an unemployed worker and firm, or the wage bid by two firms. Note that, when employees do search, there is a social loss as there is no output gain and search is costly. On-the-job search in this case is purely for private gain. It is assumed that firms only hire one worker and the number of firms is fixed and equal to the fixed number of workers, it follows that the number of vacancies equals the number of unemployed. Throughout it is assumed the labor market is in steady-state. There are two steady-states to consider – one where employees do not search, and another where some employees search.

1 Some have argued that Nash bargaining does not appear appropriate as the bargaining set is not convex. Bonilla and Burdett (2005) consider a Nash bargaining model where workers may search while employed.

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The paper can be outlined as follows. First, a simple labor market model with frictions is outlined. We then consider the decision problems faced by firms and workers. Then we consider the bargaining process between a worker and firm given the outside options of both the worker and firm, i.e., the worker’s expected return when unemployed, U, and the firm’s expected return when posting a vacancy, V, are taken as given. The bargaining process implies the wage offered is a function of U and V (as well as the other parameters of the model). Next, given employed workers do not search while employed, it is shown that expected e and a firm’s expected steadysteady-state payoff to an unemployed worker, U, state payoff when posting a vacancy, Ve , can be expressed as a function of the bargained wage (as well of the other parameters). A similar result can be established when we assume that the market is in a steady-state where employed workers search on-the-job. A market equilibrium is essentially a fixed point. That is an equilibrium where workers (at least initially) search on-the-job exists if the U and V are taken as given when a firm with a vacancy and an unemployed worker bargain a wage, wf, turns out to be the U and V generated in a steadystate where employees search given that the wage paid is wf. A similar logic establishes the existence of an equilibrium where workers do not search while employed. It will be shown that given the parameters, market equilibria can be fully described. Finally, the implication of the results established are discussed and an example is provided.

2. The model Suppose there is a unit mass of both workers and firms. Both workers and firms are homogenous. In particular, any employed worker generates revenue p per unit of time when working at any firm. Independent of employment status, if a worker pays flow cost c, a firm with a vacancy is contacted at Poisson rate a. An unemployed worker obtains b per unit of time. Any firm employs at most one worker. If a worker is an employee of a firm, the partnership breaks up at an exogenous rate d. If such job destruction occurs, the firm costlessly posts a vacancy and seeks a new employee, whereas the worker becomes unemployed and looks for a job. A firm with a vacancy contacts a searching worker at Poisson rate af. If a firm with a vacancy contacts an unemployed worker the wage paid is determined by a bargaining process that is discussed in detail later. Essentially, we assume a worker and firm play a finite time random offer game to establish the wage paid. Whether a firm’s employee searches, or not, is not observable to that firm. Hence, when bargaining, the worker and firm cannot condition on the worker’s search behavior. Suppose a firm’s employee does search on-the-job. Further, assume this worker contacts another firm with a vacancy. Here, the newly contacted firm and the worker’s current employee are assumed to enter a wage bidding competition for the worker services.

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All discount the future at rate r40. Each worker maximizes expected discounted income, whereas each firm maximizes its expected discounted profit. To reduce descriptions, in what follows we shall say workers and firms maximize their expected payoffs. We now describe the behavior of firms and workers. 2.1. Firms Given the model described above, let V denote the firm’s expected payoff when it posts a vacancy. The object here is to specify a firm’s expected return when it hires a worker, taking V as given. Let J(w, 0) denote a firm’s expected payoff when currently employing a worker at wage w given the worker does not search while employed. It follows rJðw; 0Þ ¼ p

w þ d½V

Jðw; 0ފ

Now let J(w, 1) denote a firm’s expected return that employs a worker at wage w and the worker searches on-the-job. In this case rJðw; 1Þ ¼ p

w þ d½V

Jðw; 1ފ þ a½J H

Jðw; 1ފ

where JH denotes the firm’s expected payoff after its employee contacts another firm and they bid for the worker’s services. Suppose for a moment that a firm’s employee contacts another firm with a vacancy. In this case, by assumption, the two firms bid for the worker’s services. As firms are homogeneous, each firm is willing to bid up to z, such that if the worker accepts the firm’s expected return is the same as if the firm posts a vacancy, i.e., J H ¼ V . Without loss of generality, assume the worker stays at his/ her current employer. Further, as no firm is willing to pay a worker more than this, at wage z, the employee does not search, i.e., J H ¼ V ¼ Jðz; 0Þ Using the logic described above it follows Jðw; 0Þ ¼

p

and Jðw; 1Þ ¼

p

w þ dV rþd w þ ðd þ aÞV rþdþa

It is now simple to establish that Jðz; 1Þ ¼ Jðz; 0Þ ¼ V ¼ ðp z¼p

ð1Þ

ð2Þ zÞ=r, which implies

rV

2.2. Workers Let U0 denote an unemployed worker’s expected return. Here a worker’s expected payoff when employed is derived taking U0 as given. Suppose for the

20

Ken Burdett and Roberto Bonilla

moment a worker is employed at wage w and does not search on-the-job. This worker’s expected return, U(w, 0), can be written as rUðw; 0Þ ¼ w þ d½U 0

Uðw; 0ފ

ð3Þ

Let U(w, 1) denote a worker’s expected payoff when employed at wage w and searching while employed. It follows rUðw; 1Þ ¼ w þ d½U 0

Uðw; 1ފ þ a½Uðz; 0Þ

Uðw; 1ފ

c

ð4Þ

where U(z, 0) denotes the worker’s expected return after the worker has contacted another firm and these two firms have bid for his/her services (as was described previously). Taking U0 as given, it follows that Uðw; 0Þ ¼ and Uðw; 1Þ ¼

w þ dU 0 rþd

ð5Þ

wðr þ dÞ þ ðr þ d þ aÞdU 0 þ az ðr þ d þ aÞðd þ rÞ

cðr þ dÞ

ð6Þ

Note, 0o@Uðw; 1Þ=@wo@Uðw; 0Þ=@w: We are now in a position to specify Q – the search wage of a worker, and R – the reservation wage of an unemployed worker. As shown below, a worker will search on-the-job if the wage paid when first employed is less than Q. At any wage above Q, an employee will not search. The reservation wage of an unemployed worker is the minimum wage an unemployed worker will accept, given U0. The relationship between these two objects is presented in the following Proposition. Proposition 1. (a) For any fixed U0, Uðw; 1Þ_Uðw; 0Þ if and only if wwQ, where Q¼z

cðr þ dÞ a

ð7Þ

(b) For fixed U0, Uðw; 1Þ_U 0 as w_R1 , where R1 ¼

rðd þ a þ rÞU 0 þ cðd þ rÞ ðr þ dÞ

az

ð8Þ

(c) For fixed U0, Uðw; 0Þ_U 0 as w_R0 , where R0 ¼ rU 0 (d) R1wQ as cwc1 , where c1 ¼

a½z rU 0 Š rþd

ð9Þ

Bargaining, On-the-Job Search and Labor Market Equilibrium

21

Similarly, R0 _Q as cwc1 . (e) The reservation wage of an unemployed worker, R, is such that R ¼ R1 , if c  c1 , and R ¼ R0 , if c4c1 .

Proof. The results follow directly from manipulation of (3)–(6). Given a wage paid when employed, Proposition 1 specifies the three options a worker can select – work and not search, work and search, and unemployment. Suppose for the moment coc1 , where c1 is defined by (9) – this implies R1 oQ. At wage R1 a worker is indifferent between work and search, and unemployment, i.e., UðR1 ; 1Þ ¼ U 0 . At any wage greater than R1 a worker strictly prefers to work and search rather than be unemployed. Further, as R1 oQ, at any wage w 2 ðR1 ; QÞ, a worker prefers work and search to work and not search. At wage R0 a worker is indifferent between unemployment, and work and not search, i.e., UðR0 ; 0Þ ¼ U 0 . However, as coc1 , by assumption, at wage R0 oQ, the worker strictly prefers work and search over work and not search, and unemployment, i.e., UðR0 ; 1Þ4UðR0 ; 0Þ ¼ U 0 . Hence, if w 2 ðR1 ; QÞ, the worker’s desired option is to search and work. At any wage w4Q, a worker strictly prefers to not search while working over searching while employed, and unemployment. The relationships discussed above are illustrated in Figures 1 and 2. Note, either R1oR0oQ, as shown in Figure 1, or QoR0oR1, as shown in Figure 2. Assume now that c4c1 in this case QoR0oR1 as shown in Figure 2. At wage Q, a worker is indifferent between work and search, and work and not search. The worker in this case, however, prefers unemployment rather than the two employment options at wage Q. At any wage greater than R0 the worker prefers work and not search over the other two options. Indeed, given c4c1 , as shown

Figure 1.

R1oR0oQ

U(w,0)

U(w,1)

U0 R1

R0

Q

w

22

Ken Burdett and Roberto Bonilla

Figure 2.

QoR0oR1

U(w,0)

U(w,1)

U0 Q

R0

R1

w

in Figure 1, a worker selects the work and search option. If c4c1 , a worker strictly prefers unemployment if facing a wage less than R0. Assume for the moment that coc1 and therefore R1 oQ. At wage Q a worker is indifferent between work and search, and work and no search. Further, a worker prefers both these options to unemployment. In what follows it is assumed that if a worker is indifferent between searching or not searching while employed, the worker accepts the option selected by his/her employer. This relatively harmless restriction makes easy the analysis in what follows. This restriction implies if the firm ever offers wage Q, it will always tell the worker not to search while employed.

3. Bargaining Bargaining between a firm with a vacancy and an unemployed worker is assumed to work as follows. First, the firm makes a wage offer to the worker. If the worker accepts this offer, then employment starts immediately at the offered wage. If the worker rejects the offer, they wait till time period D. Then with the probability of a half the worker (the firm) makes a take-it-or-leave-it wage offer to the firm (worker). If this offer is rejected they separate for good. If this offer is accepted employment starts immediately at the offered wage. Both take V and U as given. An equilibrium to such a game specifies the best responses of both parties given they have correct beliefs. Throughout we assume the bargaining set is non-empty. To achieve this goal we assume rV þ rU 0 op. Consider first the last offers made by both parties. As shown before, the highest wage a firm will pay is z, where V ¼ Jðz; 1Þ ¼ Jðz; 0Þ. Therefore, the worker’s final offer will always be z.

23

Bargaining, On-the-Job Search and Labor Market Equilibrium

Assume for the moment coc1 , where c1 is defined in (9). In this case R ¼ R1 oQ. The firm’s final offer in this case is R if JðR; 1Þ  JðQ; 0Þ, and Q if JðR; 1ÞoJðQ; 0Þ. It will be found useful to define w(Q) by JðwðQÞ; 1Þ ¼ JðQ; 0Þ Note, using z ¼ p wðQÞ ¼

az

rV , the above equality yields

cðd þ a þ rÞ oQ a

ð10Þ

The interpretation of w(Q) is simple. A worker employed at wage Q, who does not search while employed, yields the same expected discounted profit to a firm as employing a worker at wage w(Q) who will search while employed. At any wage wow(Q), a firm strictly prefers to hire a worker at this wage (realizing the worker will search while employed) rather than pay the worker Q. This implies the firm’s final offer will be: (a) R ¼ R1 if Row(Q) (b) Q if wðQÞoRoQ, (c) R ¼ R0 if R>Q These results are illustrated in Figures 3 and 4. In Figure 3, RowðQÞ, whereas R>Q in Figure 4. In any case, the worker’s expected return before the final offer is made, Ew, can be written as E w ¼ ð1=2Þ½U 0 þ Uðz; 0ފ Figure 3.

Row(Q)oQ

J(w,0)

J(w,1)

V R

w(Q)

Q

z

w

24

Ken Burdett and Roberto Bonilla

Figure 4.

w(Q)oRoQ

J(w,0)

J(w,1)

V w(Q)

Q

R

z

w

Now define w1 by Uðw1 ; 1Þ ¼ E w . Given R ¼ R1 oQ, w1 ¼

U 0 rðr þ d þ aÞ þ ðr þ d aÞz þ 2cðr þ dÞ 2ðr þ dÞ

ð11Þ

Note, w1>R1. Accepting a job at the wage w1 and searching while employed yields the worker the same expected payoff as Ew. Note, the wage w1 may, or may not, be less than Q. It is now possible to establish three simple but important claims. Proposition 2. Given coc1, (a) R>w(Q) if and only if c>c2, where ða þ d þ rÞ aðz rU 0 Þ c2 ¼ ð2a þ d þ rÞ ðd þ rÞ (b) w1>Q, if and only if c>c3 1 aðz UrÞ c3 ¼ 2 ðr þ dÞ (c) w1>w(Q), if and only if c>c4, where ðr þ d þ aÞ aðz UrÞ c4 ¼ 2ð2a þ d þ rÞ ðr þ dÞ Proof. From Proposition 1 it follows that if coc1 then R ¼ R1 oQ. Claim (a) now follows directly from (8) and (10). In the same way Claim (b) follows from (7) and (11), whereas (10) and (11) are used to establish (c). This completes the proof.

Bargaining, On-the-Job Search and Labor Market Equilibrium

25

Note, c1 4c2 4c3 4c4 40. Using these costs it is possible to specify the unique equilibrium to the bargaining game. Clearly, given the final offers made, the firm’s first wage offer, wf, will maximize its expected payoff given the offer yields the worker at least Ew. The next Claim specifies the details. Proposition 3. Case A: If cA(0,c4] then the unique bargaining equilibrium implies the firm’s last offer is R ¼ R1 , whereas the worker’s final offer is z. The firm’s first offer is wf ¼ w1 owðQÞ, where w1 satisfies (11). This offer is accepted by the worker who searches while employed. Case B: If cA(c4,c3], the unique bargaining equilibrium implies the firm’s last offer is R ¼ R1 , whereas the worker’s final offer is z. The firm’s first offer is wf ¼ Q, i.e., cðr þ dÞ ð12Þ wf ¼ p rV a which is accepted by the worker who does not search while employed. Case C: If cA(c3,c2], the unique bargaining equilibrium implies the firm’s last offer is R ¼ R1 , whereas the worker’s last offer is z. The firm’s first offer is wf>Q, where U(wf,0) ¼ (1/2)[U0+U(z,0)], i.e., p rV þ rU ð13Þ wf ¼ 2 This is accepted by the worker who does not search while employed. Case D: If cA(c2,c1], the unique bargaining equilibrium implies the firm’s last offer is wf>Q, whereas the worker’s last offer is z. The firm’s first offer is wf, where U(wf,0) ¼ (1/2)[U(Q,0)+U(z,0)], i.e., 2ðp rV Þ cðr þ dÞ=a ð14Þ wf ¼ 2 This is accepted by the worker who does not search while employed. Case E: If c>c1, the firm’s last offer is R ¼ R0 , whereas the workers final offer is z. The firm’s first offer is wf such that wf ¼ (1/2)[U(R0,0)+U(z,0)], i.e., ½rU 0 þ ðp rV ފ 2 This is accepted by the worker who does not search while employed. wf ¼

ð15Þ

Proof. From Proposition 2 it follows that: (a) If cA(0,c4], then R ¼ R1 ow1 owðQÞoQ. This implies the firm’s last offer is R ¼ R1 , and its first offer is w1.

26

Ken Burdett and Roberto Bonilla

(b) If cA(c4,c3], then R ¼ R1 owðQÞow1 oQ. This implies the firm’s last offer is R1, and its first offer is Q. (c) If cA(c3,c2], then R ¼ R1 owðQÞoQow1 : This implies the firm’s last offer is R1, and its first offer is wf>Q. (d) If cA(c2,c1], then wðQÞoR ¼ R1 oQ: This implies the firm’s last offer is Q, and its first offer is wf>Q. (e) If c>c1, then R ¼ R0 4Q: In this case the firm’s last offer is R0, and its first offer is wf>R0. The Claims now follow from the logic described above. To illustrate essentials, Figure 5 presents Case B. Clearly, the firm’s last offer will be R ¼ R1 . As w1>w(Q), Jðw1 ; 1ÞoJðQ; 1Þ: Hence the firm’s first offer is wf ¼ Q. Note, in this case the worker’s expected payoff UðQ; 0Þ4E w , i.e., the wage that maximizes the firm’s expected payoff given the offer implies the worker’s expected payoff is at least Ew, implies the worker does better than Ew.

4. Market equilibria To make further progress, we need next to specify an encounter function. This specifies the number of encounters (e) per unit of time between number of searching workers (s) and the number firms with a vacancy (v), i.e., e ¼ e(s,v). For simplicity, assume e ¼ sv This implies a ¼ v, and af ¼ s. This in turn implies that the number of searchers who make contact with a firm, as, equals the number of encounters between workers and firms, sv, which equals the number of firms that contact a worker, af v. Hence, under all circumstances a ¼ v ¼ u and af ¼ s

ð16Þ

As an unemployed worker is hired when contacting a firm, it follows in any steady-state ua ¼ ð1 uÞd: Therefore, the steady-state number unemployed is always u¼

d dþa

As shown above, however, a ¼ u and therefore pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ðd þ 4Þ d u¼ ð17Þ 2 If employees do not search, s ¼ u, p ¼ 1, and af ¼ u ¼ a. If new employees search, af ¼ s, p ¼ u/s, and af p ¼ u ¼ a.

Bargaining, On-the-Job Search and Labor Market Equilibrium

Figure 5.

27

Case B

J(w,0)

J(w,1)

w

V R

W(Q)

w1

Q

z

U(w,0)

U(w,1)

U w1

R

Q

Given either of the two steady-states – one where employees search and one where they do not – it is possible to write the expected payoff to an unemployed worker, and the expected payoff to a firm posting a vacancy as a function of the wages paid in the market. Now let wf be the wage bargained between an unemployed worker and a firm with a vacancy. Suppose for the moment that employed workers do not search, then the two expected returns are rU 0 ¼ b þ a½Uðwf ; 0Þ

U 0Š

c

28

Ken Burdett and Roberto Bonilla

and rV ¼ af ½Jðwf ; 0Þ

VŠ

Hence, given wf and assuming employed workers do not search when employed, using (1) and (5) in the above implies the expected payoff to an unemployed worker is U0 ¼

ðb

cÞðr þ dÞ þ awf rðr þ a þ dÞ

ð18Þ

and the expected return to a firm posting a vacancy is V¼

aðp wf Þ rðr þ d þ aÞ

ð19Þ

Note, the above two expected returns depend on the given wage bargained between an unemployed worker and a firm with a vacancy, wf. Suppose now that employed workers search when first employed. In this case, a worker’s expected return when unemployed can be written as rU 0 ¼ b þ a½Uðwf ; 1Þ

U 0Š

c

whereas in this case a firm’s expected return when posting a vacancy is rV ¼ af p½Jðwf ; 1Þ

VŠ

where p is the probability an unemployed worker is contacted given a worker contacted. Using (2) and (6) in the above equation it is straightforward to establish these two expected returns: U0 ¼

ðd þ rÞðr þ d þ aÞb

and V¼

cðd þ rÞð2a þ d þ rÞ þ a½az þ wf ðd þ rފ rðr þ d þ aÞ2

a½p wf Š rðr þ d þ 2aÞ

ð20Þ

ð21Þ

Note, while V still depends only on wf , U0 now depends on z as well as on wf. However, z ¼ p rV . 4.1. Market Equilibrium with On-the-Job Search In the bargaining section, it was demonstrated that the wages paid in the market are functions of U0 and V. For example, if the cost of search is low enough, the unique bargaining equilibrium is where workers search while employed and wf ¼ wf ðU 0 ; V Þ, as given in (11), and z ¼ zðV Þ (as z ¼ p rV ) are the two wages offered. In the Section 3 it was shown that if the market is in a particular steadystate, then U0 and V can be written as functions of the given wages offered, i.e., if the market is in a steady-state where employees search (when facing wage wf,

Bargaining, On-the-Job Search and Labor Market Equilibrium

29

then U ¼ U 0 ðwf ; zÞ and V ¼ V ðwf Þ as shown in (20) and (21). A steady-state market equilibrium exists when, given V and U0, the wages established by the bargaining process, wf and z, turn out to be the same wages that generate the given V and U0 in the relevant steady-state. This implies for a steady-state equilibrium with on-the-job search we require that, (20), (21), wf ¼ w1 (as given by (11)) and z ¼ p rV be simultaneously satisfied. Solving those four equation’s yields U0 ¼

bð2d þ a þ 2rÞða þ d þ rÞ þ paða þ d þ rÞ 2rða þ d þ rÞ2

cðr þ dÞð2d þ 3a þ 2rÞ ð22Þ

and V¼

a½ðp

bÞða þ d þ rÞ cðr þ dފ 2rða þ d þ rÞ2

ð23Þ

The resulting two wages can be written as wf ¼

bða þ d þ rÞð2a þ d þ rÞ þ ðd þ rÞ½cð2a þ d þ rÞ þ pða þ d þ rފ 2ða þ d þ rÞ2

ð24Þ

and z¼

pðr þ d þ aÞð2d þ a þ 2rÞ þ baðr þ d þ aÞ þ acðr þ dÞ 2ðr þ d þ aÞ2

ð25Þ

Further, for such an equilibrium to exist we require wfow(Q). It is straightforward to show wfow(Q) if and only if coq5, where q5 ¼

aðr þ d þ aÞðp bÞ ða þ 2d þ 2rÞð2a þ r þ dÞ

ð26Þ

Finally, we need to check that the expected returns to an unemployed worker and a firm with a vacancy are not negative. From (22) it follows U0Z0 if crk(A), where kðAÞ ¼

ða þ d þ rÞ½bða þ 2ðd þ rÞÞ þ apŠ ðd þ rÞð3a þ 2ðd þ rÞÞ

It follows immediately that VZ0 in any equilibrium of this type as p b>0. It is now simple to check that if coq5, then coK(A). The following claim summarizes the result obtained above. Proposition 4. If (26) is satisfied there exists a steady-state equilibrium where new employees search and (24) and (25) denote the wages paid.

30

Ken Burdett and Roberto Bonilla

4.2. Market equilibrium with no on-the-job search As shown in Proposition 3, there are four cases to deal with here (Cases B, C, D, and E). Intuitively, in each case k, the bargaining section implied we solve for the wage paid as a function of the parameters and the given U0 and V, i.e., wk ¼ wk ðU 0 ; V Þ. In a no on-the-job search steady-state, however, (18) and (19) must be satisfied which specify U 0 ¼ U 0 ðwk Þ and V ¼ V ðwk Þ The solution of these equations establishes the existence of a market equilibrium with no on-the-job search. This is relegated to an Appendix and the relevant results are stated below. Essentially, the various types of steady-state market equilibria where employees do not search depend upon the given cost of search in a straightforward way. Difficulties arise, however, as some types of equilibria do not exist as a worker’s utility of leisure, b, is small enough that U0o0 in a candidate equilibrium. Below we assume p4b4bn , where a p ð27Þ bn ¼ a þ 2ðd þ rÞ

which rules out this problem. More complex statements are required if bobn . Proposition 5. Assume p4b4bn . (a) Given cA(q4,q3], where q4 ¼

aðp bÞ aðp bÞ and q3 ¼ ð3a þ 2ðd þ rÞÞ ða þ 2ðd þ rÞÞ

there exists a steady-state equilibrium where employees do not search and are paid wf, where cða þ d þ rÞ a Further, in equilibrium ap

wf ¼

U0 ¼ and

ap þ bðr þ dÞ cða þ 2ðd þ rÞÞ rða þ d þ rÞ

c V ¼ 40 r

ð28Þ

ð29Þ

ð30Þ

(b) Given cA(q3,q2], where q2 ¼

aðp bÞ aþdþr

a steady-state equilibrium exists where employees do not search and are paid wf ¼

pþb 2

c

ð31Þ

Bargaining, On-the-Job Search and Labor Market Equilibrium

31

(c) Given cA(q1,q2], where aðp bÞ dþr a steady-state equilibrium exists where employees do not search and are paid q1 ¼

wf ¼

2ap

cða þ d þ rÞ 2a

ð32Þ

(d) If cA(q1,qk], where ap qk ¼ b þ ða þ 2ðd þ rÞÞ

there exists a steady-state equilibrium where employees do not search and are paid wf ¼

pþb 2

c

ð33Þ

Proof. See the Appendix.

5. Discussion Note, q5, defined by (26), is strictly larger than q4, defined in Proposition 5(a). This implies that q1 4q2 4q3 4q5 4q4 40 This, of course implies there is multiple equilibria when cA(q4,q5). We will return to this below. To illustrate the situation specified formally above suppose the parameters of the model are given by r ¼ 0:10; d ¼ 0:25; b ¼ 20; p ¼ 50

It follows from (17) that a ¼ 0:39 in this case. Further, using the relevant equations it also follows that q1 ¼ 33:46; q2 ¼ 15:81; q3 ¼ 10:74; q4 ¼ 6:25; q5 ¼ 7:03

and

b ¼ 17:90 The market wage bargained between a firm with a vacancy and an unemployed worker is indicated by wf in Figure 6 which illustrates how the equilibrium wage changes as the given cost of search changes. A couple of general conclusions are illustrated in this figure. First, equilibrium wages increase as the cost of search

32

Ken Burdett and Roberto Bonilla

Figure 6. wages

q4

q5

Market equilibrium wages

q3

q2

q1

qk

50

Z 40 wf 30 wf

wf

wf

20 wf

c

0 0

5

10

15

20

25

30

35

40

Search Equilibrium No-search Equilibrium

increases when some employees search while employed. This illustrates (24) and (25). Note also that the equilibrium wage ‘‘jumps up’’ as the given cost of search increases. First, where the market equilibrium becomes one where employees do not search from one where they do. This follows as it can be established that wf in a search equilibrium is smaller than the wage in a no search equilibrium where cA(q4,q5). Second, the wage ‘‘jumps up’’ at q2. When the cost of search is a little smaller, the firm’s last offer is R, and therefore the firm’s first offer, wf satisfies Uðwf ; 0Þ ¼ ð1=2Þ½UðR; 1Þ þ Uðz; 0ފ If the cost of search is a little greater than q2, the firm’s final offer is Q>R, and therefore its first offer, wf must satisfy Uðwf ; 0Þ ¼ ð1=2Þ½UðQ; 0Þ þ Uðz; 0ފ Hence, the equilibrium wage ‘‘jumps up’’ at q2. To see why there is multiple equilibrium in the region where cA(q4, q5), first note from (23) and (30) that the equilibrium expected payoff to a firm with a vacancy when workers search on-the-job (denoted by V s) is greater than the equilibrium payoff to a firm if employees do not search (denoted by V ns), i.e., V s>V ns if cA(q4, q5) and therefore firms prefer the equilibrium where employees search to one where they do not. Further, from (22) and (29) it can be

Bargaining, On-the-Job Search and Labor Market Equilibrium ns

33

s

established that U >U when cA(q4, q5), i.e., unemployed workers prefer a no search equilibrium to one where employees search. Note V s>V ns implies from (7) that Q ns>Q s, i.e., there is a larger search wage in the no search equilibrium than in the equilibrium where employees search. Using (10) it also follows that w ns ðQ ns Þ ¼

aðp

rV ns Þ

a

cðd þ a þ rÞ

4

aðp

rV s Þ

a

cðd þ a þ rÞ

¼ w s ðQ s Þ

when cA(q4, q5). Substituting (22) and (23) into (11) yields wage w1s , whereas substituting (29) and (30) into (11) yields the wage w1ns . It is now simple to establish that if all expect employees to search on-the-job w1s ow s ðQ s Þ, whereas if all expect employees not to search, then w1ns 4w ns ðQ ns Þ; given cA(q4, q5). Hence, if all expect employees to search, in equilibrium a firm’s first wage offer is w1s ow s ðQ s Þ, which is accepted by the worker. On the other hand, if all expect employees not to search, then (in equilibrium) the firm’s first offer is Qns, as w1ns 4w ns ðQ ns Þ: This implies two equilibria can be sustained when cA(q4, q5) – one where employees search on-the-job, the other where employees do not search. Acknowledgment We would like to thank R. Wright for providing helpful comments. References Bonilla, R. and K. Burdett (2005), ‘‘Nash bargaining and costly on-the-job search’’, Mimeo, University of Pennsylvania. Burdett, K. (1978), ‘‘A theory of employee search and quits’’, American Economic. Review, Vol. 68, pp. 212–220. Burdett, K., R. Imai and R. Wright (2004), ‘‘Unstable relationships’’, B.E. Journals in Macro, Frontiers in Macroeconomics, Vol. 1(1), Article 1. Burdett, K. and D.T. Mortensen (1998), ‘‘Wage differentials, employer size, and unemployment’’, International Economic Review, Vol. 39, pp. 257–273. Cahuc, P., F. Postel-Vinay and J.-M. Robin (2003), ‘‘Wage bargaining with onthe-job search: theory and evidence’’, Mimeo, Universite´ de Paris I, September. Fallick, B. and C. Fleischman, (2004), ‘‘Employer-to-employer flows in the U.S. labor market: the complete picture of gross worker flows’’, Federal Reserve Board, Finance and Economics Discussion Series Working Paper, 2004–2034. Mortensen, D.T. (2003), Wage Dispersion: Why are Similar Workers Paid Differently, Cambridge, MA: MIT Press. Moscarini, G. (2003), ‘‘Job matching and the wage distribution’’, Yale Working Paper, Yale University.

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Pissarides, C. (1994), ‘‘Search unemployment with on-the-job search’’, The Review of Economic Studies, Vol. 61(3), pp. 457–475. Pissarides, C. (2000), Equilibrium Unemployment Theory, 2nd edition, Cambridge, MA: MIT Press. Postel-Vinay, F. and J.-M. Robin, (2002a), ‘‘The distribution of earnings in an equilibrium search model with state-dependent offers and counter-offers’’, forthcoming in: International Economic Review, Vol. 43(4), pp. 989–1371. Postel-Vinay, F. and J.-M. Robin (2002b), ‘‘Equilibrium wage dispersion with worker and employer heterogeneity’’, Econometrica, Vol. 70(6), pp. 2295–2350. Shimer, R. (2004). ‘‘On the job search, bargaining, and wage dispersion’’, Mimeo, University of Chicago. Appendix: Proof of Proposition 5 We address (a) first. Using the relevant equations for Case B and solving implies (29) and (30) in the Proposition. Substituting these two equations in (12) yields (28) in the Proposition. Note, VZ0 always holds and U0Z0, when p4b4bn . This completes the proof of (a). We now address (b). Using the relevant equations for Case C and solving implies U0 ¼

ðb

cÞða þ 2ðd þ rÞÞ þ ap 2rða þ d þ rÞ

and V¼

aðp þ c bÞ 40 2rða þ d þ rÞ

Substituting these two equations in (13) yields (31) in the Proposition. It is simple to check U0Z0, and VZ0 is guaranteed if p4b4bn , when cA(c3,c2]. This completes the proof of (b). We now address (c). Using the relevant equations for Case D and solving implies U0 ¼ and

2ap þ 2bðd þ rÞ cða þ 3ðd þ rÞÞ 2rða þ d þ rÞ

c 40 2r Substituting these two equations in (14) yields (32) in the Proposition. Again, it is simple to check that U0Z0 and VZ0 if p4b4bn , when cA(c1,c2]. This completes the proof of (c). V¼

Bargaining, On-the-Job Search and Labor Market Equilibrium

35

We now address (d). Using the relevant equations for Case E and solving implies U0 ¼ and

ðb

cÞða þ 2ðd þ rÞÞ þ ap 2rða þ d þ rÞ

ðp þ c bÞ 40 2 Substituting these two equations in (15) yields (32) in the Proposition. Note, U0Z0 if and only if coqk, where V¼

qk ¼

bða þ 2ðd þ rÞÞ þ ap ða þ 2ðd þ rÞÞ

Further, qk>q1 if p4b4bn : This completes the proof.

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36

CHAPTER 2

On-the-Job Search and Strategic Bargaining Robert Shimer Abstract This paper studies wage bargaining in a simple economy in which both employed and unemployed workers search for better jobs. The axiomatic Nash bargaining solution and standard strategic bargaining solutions are inapplicable because the set of feasible payoffs is nonconvex. I instead develop a strategic model of wage bargaining between a single worker and firm that is applicable to such an environment. I show that if workers and firms are homogeneous, there are market equilibria with a continuous wage distribution in which identical firms bargain to different wages, each of which is a subgame perfect equilibrium of the bargaining game. If firms are heterogeneous, I characterize market equilibria in which more productive firms pay higher wages. I compare the quantitative predictions of this model with Burdett and Mortensen’s (1998) wage posting model and argue that the bargaining model is theoretically more appealing along important dimensions.

Keywords: job-to-job transitions, wage bargaining, nash bargaining, non-convex payoffs JEL Classifications: C78, J31, J64

1. Introduction In recent years, search theorists have grown increasingly aware of the need to incorporate on-the-job search into their models. In part this is because job-to job transitions are pervasive in the United States economy. According to conservative r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75002-1

37

38

Robert Shimer

estimates, job-to-job transitions are about half as common as unemployment-toemployment transitions (Blanchard and Diamond, 1989). Using evidence from a newer data set, Fallick and Fleischman (2004) argue that half of all new employment relationships results from a job-to-job transition rather than a movement from unemployment or out of the labor force into employment. But the interest in on-the-job search models is also a consequence of the novel theoretical results that they generate. Burdett and Mortensen (1998) develop a wage posting model in which firms offer high wages to attract workers from other firms and to reduce worker turnover. They show that the unique equilibrium of the labor market is characterized by a continuous wage distribution, even if all workers and firms are identical. If firms are heterogeneous, higher productivity firms pay higher wages. This paper has spawned a number of extensions. Stevens (2004) and Burdett and Coles (2003) allow firms to post wage contracts rather than just a single wage. The latter paper shows that if workers are risk averse, equilibrium involves a distribution of contracts, each with an upward-sloping wage profile. Postel-Vinay and Robin (2002) allow firms to match outside offers and show that workers may voluntarily take a wage cut in order to move to a firm that is likely to be more aggressive in matching outside offers in the future. Cahuc et al. (2003) explicitly model the bargaining game between a worker and one or more potential employers. Moreover, many of these models have been tested using matched worker-firm data sets; Mortensen (2003) is a prominent example. At the same time, there is a substantial gap between this model and the ‘standard’ labor market model of search, summarized in Pissarides’(2000) textbook. In the simplest version of that model, only unemployed workers search for jobs. When a worker and firm meet, the wage is set in accordance with the Axiomatic Nash (1953) bargaining solution. Pissarides shows that this results in the worker and firm splitting the gains from trade, with the worker’s share determined by her (exogenous) bargaining power. There have been some attempts to introduce on-the-job search into the bargaining model. Pissarides (1994) assumes that a worker and firm split the surplus from matching. The equilibrium of the resulting model is qualitatively different from the equilibrium of the Burdett and Mortensen (1998) model: if workers and firms are homogeneous, then all workers earn the same wage at all jobs, so there is no wage dispersion. The natural conclusion is that whether there is wage dispersion in a homogeneous agent economy with on-the-job search depends critically on whether firms post wages or wages are bargained. This paper revisits this conclusion. The first finding is that the Axiomatic Nash bargaining solution is inapplicable in this environment. Nash (1953, p. 129) writes ‘‘The only important thing is the set of those pairs ðu1 ; u2 Þ of utilities which can be realized by the players if they cooperatey. It should be a compact convex set in the ðu1 ; u2 Þ plane.’’ I find that in the model with on-the-job search, the set of feasible payoffs is typically nonconvex because an increase in the wage raises the duration of an employment relationship. This possibility is

On-the-Job Search and Strategic Bargaining

39

absent from models without on-the-job search, but is central to wage setting in the environment of interest to this paper. This leads me to focus on strategic bargaining games. I assume that when a worker and firm first meet, they bargain over the wage for the duration of the employment relationship, taking as given the wage bargained by other workers and firms, the ‘‘wage distribution.’’ I model bargaining as an infinite horizon alternating offers game with a small risk that bargaining breaks down between offers. I require that any wage w that is paid in a market equilibrium be a subgame perfect equilibrium of the strategic bargaining game when the risk of breakdown is sufficiently small. The existing literature on such games, including Rubinstein (1982), Shaked and Sutton (1984), and Binmore et al. (1986) shows that under some conditions there is a unique subgame perfect equilibrium in this strategic bargaining game. Unfortunately, these results are also inapplicable to my environment because all of these papers also assume that the set of feasible payoffs is convex. When I extend their approach to handle models with nonconvex payoffs, I find that the subgame perfect equilibrium of the bargaining game with a given wage distribution is no longer unique. Instead, I get a precise characterization of the set of subgame perfect equilibria. In a market equilibrium, each wage in the support of the wage distribution corresponds to one of these subgame perfect equilibria. In an environment with homogeneous firms and on-the-job search, I find there are many market equilibria. There is a continuum of market equilibria each characterized by a different continuous wage distribution. In each market equilibrium every wage in the support of the distribution is a subgame perfect equilibrium of the bargaining game. Depending on how employed workers behave when they encounter a firm paying their current wage, there may also be a continuum of market equilibria with a degenerate wage distribution and more generally a continuum of market equilibria with an n-point wage distribution for arbitrary n. I then extend the model to have heterogeneous firms, with a continuous distribution of productivity x across firms. I provide a simple characterization of market equilibria in which more productive firms pay strictly higher wages: there is a function fx ðyÞ such that for each firm type x, fx ðxÞ  fx ðyÞ for all y in a neighborhood of x. This is a generalization of a naı¨ ve application of the Axiomatic Nash bargaining solution to this model (see Mortensen 2003, Section 4.3.4), which imposes the stronger condition that fx ðxÞ  fx ðyÞ for all firm types x and y. This paper proceeds as follows. Section 2 lays out the basic model with homogeneous workers and firms and discusses convexity of the set of feasible payoffs. Section 3 characterizes the set of market equilibria with a continuous wage distribution, while Section 4 shows that, if workers never switch employers when they are indifferent, the model has many market equilibria characterized by a mass of firms paying the same wage. I argue that such market equilibria seem contrived compared to the ones with a continuous wage distribution, since they are broken if firms are concerned that workers might sometimes accept equal

40

Robert Shimer

outside offers. Section 5 explores the model with heterogeneous firms. I provide a concise definition of a market equilibrium when more productive firms pay higher wages. I then show that, like the Burdett and Mortensen (1998) model, the strategic bargaining model of on-the-job search predicts the productivity of each worker conditional on her wage and the entire wage distribution. Moreover, the model implies that some wage distributions cannot be produced by this model regardless of the distribution of productivity. Section 6 discusses the connection between this paper and existing attempts to use the Axiomatic Nash (1953) bargaining solution to set wages in models with on-the-job search. Finally, the paper concludes in Section 7 by evaluating the advantages and disadvantages of bargaining and wage posting models of on-the-job search. 2. Model 2.1. Preferences and technology I consider a continuous time, infinite horizon economy. There are two types of economic agents, firms and workers. All agents are risk-neutral, infinitely lived, and discount future income at rate r40. Let v denote the measure of firms in the economy, indexed by j 2 ½0; vŠ. Firms are ex-ante identical but may pay different wages. Let W ðjÞ denote the wage that  denote the support of the wage distribution, and firm j pays its workers, ½w; wŠ F ðwÞ denote the fraction of firms paying a wage strictly less than w. This is a critical object determined in the market equilibrium of this economy but taken as given by each individual agent. Each firm is endowed with a constant returns to scale production technology using only labor. More precisely, each employee produces output x and hence yields a flow profit x W ðjÞ to firm j. Each firm contacts a worker at a constant rate, regardless of the firm’s bargained wage or how many filled jobs it has. This means that there is no opportunity cost of hiring a worker.1 Employment relationships end exogenously at rate s40, leaving the worker unemployed and the firm with nothing. Normalize the measure of workers to 1. Each worker may be employed or unemployed. An unemployed worker gets flow utility zox from leisure and unemployment income, while a worker employed by firm j earns the wage W ðjÞ. All workers search for jobs, contacting a randomly selected firm at rate l40. A worker’s optimal search behavior is simple: she takes any job that raises the present value of her income. There is one subtle but important tie-breaking assumption: I look at market equilibria in which a worker switches jobs when indifferent. Relaxing this behavioral restriction enlarges the set of market equilibria, a possibility I explore in Section 4.

1

In a market equilibrium, a firm’s profit depends on its bargained wage, unlike in Burdett and Mortensen (1998).

On-the-Job Search and Strategic Bargaining

41

For a given wage distribution F , I can characterize the equilibrium through a series of Bellman equations. Let EðwÞ denote the expected value of income for a worker currently employed at wage w and U denote the expected value of income for an unemployed worker. These satisfy Z w rEðwÞ ¼ w þ s ðU EðwÞÞ þ l maxfEðw0 Þ EðwÞ; 0g dF ðw0 Þ ð1Þ w

rU ¼ z þ l

Z

w

maxfEðw0 Þ

U; 0g dF ðw0 Þ

ð2Þ

w

An employed worker earns a wage w; the match ends at rate s, leaving the worker unemployed; and the worker gets another wage offer at rate l, leading to a capital gain Eðw0 Þ EðwÞ if Eðw0 Þ  EðwÞ and zero otherwise. An unemployed worker earns income z and finds a firm at rate l as well. It is useful to simplify these expressions to obtain an expression for the surplus a worker gets from a match. Observe from (1) and (2) that EðzÞ ¼ U, so a worker is indifferent between unemployment and working at a wage equal to her unemployment income. Then differentiate (1) to prove E 0 ðw0 Þ ¼ 1=ðr þ s þ l ð1 F ðw0 ÞÞÞ 40. Integrate this using the terminal condition EðzÞ ¼ U to get Z w 1 dw0 EðwÞ U ¼ ð3Þ 0 ÞÞ r þ s þ lð1 F ðw z In particular, workers prefer higher wages and move whenever they find a job that pays a higher wage, at rate lð1 F ðwÞÞ. Next, consider a firm paying a wage w. Since it has constant returns to scale production technology, one can evaluate each of its filled jobs in isolation. Their value is defined recursively by rJðwÞ ¼ x

w

ðs þ lð1

F ðwÞÞÞJðwÞ

ð4Þ

The job produces flow profit x w, but ends either exogenously at rate s or endogenously when the worker finds at least as good a wage offer, at rate lð1 F ðwÞÞ. When a job ends, the firm loses the full value of the job JðwÞ since its opportunity cost is zero. 2.2. Wage bargaining and equilibrium concept A critical issue in this environment is how wages are set. A worker will take any job paying at least her value of unemployment z, EðwÞ  U if w  z. Similarly, a firm will hire any worker if the wage is no more than x, JðwÞ  0 if w  x. This introduces a bilateral monopoly problem in wage setting. Following Diamond (1982) and Mortensen (1982), I assume that the worker and firm settle on a wage by bargaining. Wage bargaining is complicated in this environment. Standard axiomatic bargaining solutions Nash (1953) and strategic bargaining games (Rubinstein,

42

Robert Shimer

1982, Shaked and Sutton 1984, Binmore et al., 1986) assume that the set of feasible allocations is convex, a restriction that I show below may be violated in this model. So rather than apply an out-of-the-box bargaining solution, I am forced to return to the foundations of two person bargaining theory and analyze the subgame perfect equilibria of a particular extensive form game. Before discussing a particular bargaining game, I mention some important features of wage setting in this environment. I assume that when an unemployed worker meets a firm, the pair bargains over a wage. The wage subsequently remains fixed for the duration of the match.2 At some later date, the worker may meet another firm. At this juncture, the worker must choose an employer and then, if she switches employers, bargain with the new employer with no possibility of recalling her old job. If she stays at her old job, her wage is unchanged. I make this assumption to parallel Pissarides (1994) as closely as possible. This rules out the possibility that a worker can exploit multiple job opportunities to raise her wage. Postel-Vinay and Robin (2002) examine what happens if a worker who has multiple job opportunities can get her employers to bid for her labor and Moscarini (2004) looks at firms’ incentive not to match outside job offers. Another important assumption is that a worker who contacts a firm can observe the firm’s index j and rationally anticipate the bargained wage W ðjÞ. Although this assumption might seem extreme in an environment with homogeneous firms, it is more plausible when firms are heterogeneous. I show in Section 4 that with heterogeneous firms, there is a market equilibrium in which more productive firms pay higher wages and workers move whenever they contact a more productive firm. Finally, I do not permit bargaining over wage lotteries rather than simply over wages. I show in Section 3.4 that, because the set of feasible allocations may be nonconvex, wage lotteries could play a nontrivial role. I now describe how a worker and a firm set the wage. I consider an alternating offers game, based closely on Binmore et al. (1986) ‘‘strategic model with exogenous risk of breakdown.’’ For simplicity, assume that a worker and firm bargain in artificial time, so wage bargaining does not delay production. Denote time in the bargaining game by t ¼ 1, 2, 3, y, with an infinite horizon. In each ‘odd’ stage t ¼ 1, 3, 5, y, the firm makes a wage offer w, which the worker can accept or reject.3 If the worker accepts the offer, bargaining ends, production starts, and the worker is paid the negotiated wage for the remainder of the match,

2

This rules out the possibility that the pair bargains over a wage contract. If they could, the optimal contract would be simple to describe: the firm would pay the worker her full marginal product x for the duration of the match and the pair would bargain over an initial transfer from the worker to the firm. Since the worker receives her entire marginal product, she has no incentive to switch employers at a later date. Thus this contract eliminates all job-to-job transitions, which are inefficient from the perspective of a particular worker and firm. This is reminiscent of Stevens’ (2004) findings in the Burdett and Mortensen (1998) wage posting model of on-the-job search. 3 This notation suggests that the firm makes the first wage offer. The results are unchanged if the worker makes the first wage offer or if the first mover is determined by a coin flip.

On-the-Job Search and Strategic Bargaining

43

giving the worker and firm expected values EðwÞ and JðwÞ, respectively. If the worker rejects the offer, negotiations break down with probability d, leaving the worker unemployed with expected income U and the firm with nothing. Otherwise, the game proceeds to stage t+1 ¼ 2, 4, 6, y, an ‘even’ stage. Now, the worker makes a wage demand to the firm, which again may be accepted or rejected. Rejection leads to a probability d that negotiations break down; otherwise the game proceeds to stage t+2, another ‘odd’ stage that is identical to stage t. The worker and firm treat the wage distribution F, and hence the Bellman values J, E, and U, as fixed when bargaining. In a market equilibrium, some firms must be willing to offer each wage in the support of the wage distribution F. To be precise, market equilibrium imposes that for all
40 and all firms j, there is a sufficiently small d40 such that there is a subgame perfect equilibrium of the strategic bargaining model in which the bargained wage lies within e of W ðjÞ. The wages W ðjÞ must integrate up to the distribution F. 2.3. Nonconvexity of the set of feasible payoffs Nash (1953) examined two-person-bargaining problems in which the feasible set of payoffs is convex. He argued that there are four reasonable restrictions on the outcome of a bargaining game: it should be invariant to equivalent representations of players’ von Neumann–Morgenstern utility functions; it should be independent of irrelevant alternatives;4 it should be Pareto efficient; and it should be symmetric if the underlying problem is symmetric. Nash proved that if these four axioms hold, there is a unique solution to the bargaining problem and it maximizes ðEðwÞ UÞJðwÞ. Similarly, uniqueness theorems in the literature on alternating offers bargaining games (Rubinstein, 1982; Shaked and Sutton 1984; Binmore et al., 1986) assume that the set of feasible payoffs is convex. Unfortunately, the set of feasible payoffs is typically nonconvex in this environment and so these results are inapplicable. Consider the simplest wage distribution, F ðwÞ degenerate at some wage w 2 ðz; xÞ. Then the value functions (3) and (4) reduce to

EðwÞ

4

8 w z > < rþsþl U¼ w w > w z : þ rþs rþsþl

if w  w ð5Þ if w4w

More precisely, suppose some outcome x is the bargaining solution in one problem. We now eliminate some feasible payoffs but x remains feasible. Then, it should still be the outcome of the restricted problem.

44

Robert Shimer

and 8 x w > if w  w < rþsþl JðwÞ ¼ x w > : if w4w rþs

ð6Þ

 while JðwÞ is disNotably EðwÞ U is continuous but not differentiable at w,  For sufficiently small e, both the worker and firm prefer a fair continuous at w. lottery between w
and w þ
to a wage of w for sure. The worker prefers the lottery because EðwÞ has a convex kink at w while the firm prefers it because JðwÞ  In the next section, I show that this nonconvexity jumps up discontinuously at w. carries over to many other wage distributions, including any wage distribution associated with a market equilibrium.

3. Market Equilibria with Wage Dispersion This section considers market equilibria in which the wage distribution is continuous. I prove that there is a family of such market equilibria, parameterized
 by the lower bound of the wage distribution w 2 12 ðx þ zÞ; x : rþsþl 1 F w ðwÞ ¼ l

 x x

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi! w x w x w 1 2 log w w z x w

ð7Þ

 Clearly and differentiable on its support. with support ðw; wÞ.
F w is continuous  The restriction that w 2 12 ðx þ zÞ; x ensures that F w is increasing. Moreover,  ¼ 1 pins down the upper bound of the support of F w ; F w ðwÞ ¼ 0 while F w ðwÞ since F w ðxÞ41, w 2 ðw; xÞ. I start in Sections 3.1 and 3.2 by proving that each of these wage distributions is consistent with market equilibrium. Section 3.3 proves that there are no other market equilibria with a continuous wage distribution while Section 3.4 considers wage lotteries.

3.1. Bellman values The first step is to characterize the Bellman values when wages satisfy (7). Sub stitute (7) into Equation (3) to show that for w 2 ½w; wŠ,

ðw EðwÞ

U¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi x w x w zÞ 1 2 log x w w z rþsþl

ð8Þ

On-the-Job Search and Strategic Bargaining

45

 This is strictly increasing in w. One can also solve for E(w) U when we½w; wŠ and confirm that E is globally increasing. Similarly, substituting (7) into Equa earns expected profit tion (4) shows that a firm paying a wage w 2 ½w; wŠ JðwÞ ¼

x w sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi x w x w ðr þ s þ lÞ 1 2 log w z x w

ð9Þ

 and This is strictly decreasing in w. Again, one can solve for J(w) when we½w; wŠ confirm that J is globally decreasing. The set of feasible payoffs is nonconvex when the wage distribution F satisfies Equation (7). In particular, Equations (8) and (9) imply ðEðwÞ

UÞJðwÞ ¼

ðx wÞðw zÞ ðr þ s þ lÞ2

ð10Þ

 so this region of the Pareto frontier of the bargaining is constant for w 2 ½w; wŠ, set is convex. That the Pareto frontier is convex under the wage distribution F w is not an accident. Section 3.3 shows that in any market equilibrium with a continuous wage distribution, ðEðwÞ UÞJðwÞ is constant on the support of the wage distribution. 3.2. Subgame perfect equilibria of the bargaining game To prove that the wage distribution (7) is consistent with a market equilibrium, I  there is a d40 such that there is a must show that for any
40 and all w 2 ðw; wÞ, subgame perfect equilibrium of the strategic bargaining model in which the bargained wage lies within e of w. Consider the following strategies: in each odd period, the firm proposes a low  The worker accepts any offer w  wf and refuses lower wage wage wf 2 ½w; wÞ.  The offers. In each even period, the worker proposes a high wage ww 2 ðwf ; wŠ. firm accepts w  ww and rejects higher wage demands. To prove that this is a subgame perfect equilibrium, I must show that no one-stage deviation is profitable, which puts strong restrictions on the relationship between wf and ww . First, suppose the firm considers offering a wage different from wf . Any higher wage is accepted, but so is wf , and since J is decreasing (Equation (9)), a wage increase reduces the firm’s payoff. Any lower wage is rejected, in which event the firm accepts the higher wage ww in the next period if negotiations do not break down first. Again, this reduces firm’s payoff from Jðwf Þ to ð1 dÞJðww Þ. Similarly, offering ww is the worker’s best response to the firm’s strategy since E is increasing (Equation (8)). Next turn to the acceptance thresholds. Monotonicity of JðwÞ and EðwÞ U in w ensure that threshold rules are optimal. The threshold must be at the point where the respondent is indifferent. A firm is indifferent between accepting ww now or facing the risk that negotiations break down but otherwise having wf

46

Robert Shimer

accepted next period if Jðww Þ ¼ ð1

dÞJðw f Þ

ð11Þ

Worker’s analogous indifference condition is Eðw f Þ ¼ ð1

dÞEðww Þ þ dU

ð12Þ f

w

This is a pair of equations in w and w . But substituting from (8) and (9)  both Equations (11) and (12) imply indicates that for any w  wf oww  w, ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi         x w x wf x w x ww 1 2 log log ¼ ð1 dÞ 1 2 x w x w w z w z ð13Þ f

w

That the two equations imply the same relationship between w and w is because of the constancy of ðEðwÞ UÞJðwÞ; this would not be true for an arbitrary distribution F.  2 satisfying Equation (13) is a subgame perfect Any pair fwf ; ww g 2 ½w; wŠ equilibrium of the bargaining game with fixed d. In particular, take an arbitrary  Let wf ¼ w and select ww using (13). For sufficiently small d, this w 2 ðw; wÞ.  \ ðw
; w þ
Þ. Since ww 2 ðwf ; wÞ,  this is a subgame perfect defines ww 2 ðwf ; wÞ equilibrium of the bargaining game. And since w 2 ðw
; w þ
Þ, the bargaining outcome lies with e of w. All that remains is to assign wages to firms. One possibility is to let firm j 2 ð0; vÞ pay a wage W ðjÞ solving F w ðW ðjÞÞ ¼ j=v, which gives rise to the desired wage distribution. But any reshuffling of the wages paid is also consistent with market equilibrium, as long as the correct density of firms pay the correct wage and workers know which firm pays what wage. In summary, when a worker encounters firm j, she rationally anticipates that the bargaining game will conclude at some wage W ðjÞ. The worker prefers to meet a firm j 0 with W ðj 0 Þ4W ðjÞ, but if firm j always offers the wage W ðjÞ and refuses any higher offer when playing the bargaining game, the worker’s best response is to accept the low-wage offer. Conversely, firms that bargain to lower wages earn more profits per worker, but it is impossible for a firm that is expected to offer a high wage W ðj 0 Þ to get away with paying its worker a lowwage W ðjÞ. To the readers accustomed to the logic of wage posting models like Burdett and Mortensen (1998), this might seem perverse: why cannot firms unilaterally lower the wage they pay if this is in their interest? Here the possibility of doing so is limited by workers’ expectations and their associated strategies in the bargaining game. 3.3. Other market equilibria There is no market equilibrium in which a positive measure of firms pay the same wage. To prove this, suppose to the contrary that a positive of measure of

47

On-the-Job Search and Strategic Bargaining 5

firms pays wox. Suppose one of those firms considers offering its worker a slightly higher wage w þ
in the odd stage of the bargaining game. Workers prefer higher wages and so the worker will naturally accept the offer. For sufficiently small e, the firm also benefits from the higher wage offer: by assumption, workers switch employers whenever they are indifferent, so by raising its wage offer slightly above the mass point, the firm discretely reduces its turnover, increasing JðwÞ. Thus, this is not a market equilibrium. Now, consider an arbitrary market equilibrium with a continuous wage dis so each wage in the support is a subgame perfect tribution F with support ðw; wÞ, equilibrium of the bargaining game. First note that the worker’s value E must be increasing and firm’s value J must be decreasing on the support of the wage distribution. That E is increasing follows immediately from Equation (3). Equation (4) allows for the possibility that J is increasing in w, but this cannot happen in a market equilibrium: if J is increasing at w, a worker and firm would not agree on a wage of w since both would prefer a higher wage, i.e. w is not in the support of the wage distribution; but if w is not on the support of the wage distribution, Equation (4) indicates that J is decreasing at w. It follows that Equations (11) and (12) carry over to this environment and jointly imply ðEðww Þ

UÞJðww Þ ¼ ðEðwf Þ

UÞJðwf Þ

 In other words, the product of the surplus that the worker for all ww , wf 2 ðw; wÞ. gets from matching and the surplus that the firm gets from matching must be constant on the support of the wage distribution. This is a strong restriction on the wage distribution. To see how strong it is,  note from Equations (3) and (4) that for w 2 ðw; wÞ, ðEðwÞ

Z

1 UÞJðwÞ ¼ dw0 F ðw0 ÞÞ z r þ s þ lð1   x w  r þ s þ lð1 F ðwÞÞ

In particular, ðEðwÞ 

5

w

w z rþsþl



UÞJðwÞ ¼ ðEðwÞ

x w rþsþl



Z

 ð14Þ

UÞJðwÞ for all w or

w

1 dw0 ¼ 0 ÞÞ r þ s þ lð1 F ðw z  x w  r þ s þ lð1 F ðwÞÞ

It is easy to rule out market equilibria in which the wage is w  x.



48

Robert Shimer

Differentiating this yields a first order nonlinear differential equation for F: F 0 ðwÞ ¼

r þ s þ lð1 F ðwÞÞ lðx wÞ

ðw

ðx wÞðr þ s þ lÞ2 zÞðx wÞlðr þ s þ lð1

F ðwÞÞÞ

ð15Þ

 Any continuous wage distribution must satisfy this condition for w 2 ðw; wÞ. Integrating (15) with the terminal condition F ðwÞ ¼ 0 gives Equation (7) for F w . Finally, if woðx þ zÞ=2, F 0 ðwÞo0, which is inconsistent with F being a cumulative distribution function. Thus the only possible market equilibria are those already analyzed.

3.4. Wage lotteries I have so far assumed that a worker or firm can offer its counterpart a wage in the bargaining game, but it cannot offer a wage lottery. To understand why this restriction may be important, consider a firm j that is supposed to offer a worker  when the wage distribution is F w . Since workers’ value a wage W ð jÞ 2 ðw; wÞ function E is increasing, this gives the worker a weighted average of her utility from the lowest possible and highest possible wages: EðW ð jÞÞ  aEðwÞ þ ð1

 aÞEðwÞ

ð16Þ

for some a 2 ð0; 1Þ. Suppose instead the firm offers the worker a lottery. If the worker accepts the lottery, the wage is w with probability a and w with probability 1 a. Since the worker is indifferent about accepting W ðjÞ, she is also indifferent about accepting this lottery and strictly prefers any more generous lottery. But the firm’s payoff is higher under this lottery. To prove this, recall that ðEðW ð jÞÞ UÞJðW ð jÞÞ ¼   Substitute this into Equation (16) to get UÞJðwÞ. ðEðwÞ UÞJðwÞ ¼ ðEðwÞ EðW ð jÞÞ

U ¼a

ðEðW ð jÞÞ

UÞJðW ð jÞÞ þ ð1 JðwÞ

aÞ

ðEðW ð jÞÞ

UÞJðW ð jÞÞ  JðwÞ

or 1 1 ¼a þ ð1 JðW ðjÞÞ JðwÞ

aÞ

1  JðwÞ

 Lotteries enable the so Jensen’s inequality implies JðW ð jÞÞoaJðwÞ þ ð1 aÞJðwÞ. worker and firm to convexify the feasible set of payoffs, raising the possibilities for both. I have ruled out lotteries by fiat, but one reason that lotteries might not be possible is if there is no third party who can verify their outcome. Future research should explore how allowing for lotteries affects the conclusions of this paper.

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4. Degenerate Market Equilibria This section modifies the restriction that workers switch from firms j to j 0 even if they are indifferent. Instead, I consider the opposite tie-breaking assumption: a worker moves only when she encounters a firm paying a strictly higher wage. It is straightforward to see that the continuous wage distributions F w found in the previous section remain market equilibria under this alternative restriction, since workers never encounter a firm paying their current wage, but I show that this change in behavior introduces many additional market equilibria, each with a discrete wage distribution. I start with the simplest type of market equilibrium. 4.1. Single wage market equilibrium Suppose all firms offer a common wage w 2 ðz; xÞ. Equation (3) is unaffected by the change in the workers’ behavior since they are indifferent about whether they move when they encounter a firm offering the same wage. Specializing it to this case gives 8 w z > if wow < rþsþl EðwÞ U ¼ ð17Þ > : w z þ w w if w  w rþs rþsþl

 This is continuous in w even at w. Since workers do not switch when they are indifferent, a firm paying w or higher does not suffer any turnover, while a lower paying firm loses a worker whenever she encounters another firm. Adapting Equation (4) to this environment gives 8 x w > if wow < rþsþl JðwÞ ¼ x w ð18Þ > : if w  w rþs

Notably this jumps up discontinuously at w To see whether this is a market equilibrium, I again examine the bargaining game with a small risk d of a breakdown in negotiations. I look for a subgame perfect equilibrium in which the firm always offers w and the worker always  These wages have the property that the firm is strictly indemands ww 4w. different between accepting ww this period or taking its chances that negotiations break down and offering w next period. On the other hand, subgame perfect equilibrium requires only that the worker weakly prefers accepting w this period rather than waiting until next period to offer ww . To understand why, note that  the firm might even if the worker strictly prefers to accept the firm’s offer w,

50

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choose not to cut its wage, knowing that if it did so its profits would fall discretely from the increase in turnover.6 The first step is to solve for the worker’s offer ww as a function of the firm’s  The worker’s offer must be acceptable but must leave the firm indifferent, offer w. for otherwise the worker would benefit from demanding a higher wage: Jðww Þ ¼  or from (18) ð1 dÞJðwÞ ww ¼ ð1

dÞw þ dx

ð19Þ

  ð1 dÞ Next, the worker must be willing to accept the firm’s offer: EðwÞ Eðww Þ þ dU. Using (17) to solve for EðwÞ U and (19) to eliminate ww gives w 

ð1

dÞðr þ s þ lÞx þ ðr þ sÞz ð2 dÞðr þ sÞ þ ð1 dÞl

Note that the right-hand side is continuously decreasing in d. In particular, for any  w4

ðr þ s þ lÞx þ ðr þ sÞz  wn 2ðr þ sÞ þ l

ð20Þ

 it is a subgame perfect equilibrium of the there is a d40 such that if all firms pay w, bargaining game for a firm to offer w and a worker to offer ww ¼ ð1 dÞw þ dx. In summary, any single wage w 2 ðwn ; xÞ is associated with a market equilibrium of this model when workers stay at their employer if indifferent. One interesting feature that all these market equilibria share is that the  worker’s surplus EðwÞ U ¼ ðw zÞ=ðr þ s þ lÞ exceeds the firms’ surplus  ¼ ðx wÞ=ðr  JðwÞ þ sÞ, although in the limiting case of w ¼ wn , the two terms are equal. It is straightforward to show that without on-the-job search, the worker and firm would divide the match surplus equally, giving rise to a unique equilibrium wage wn . The possibility of on-the-job search therefore raises the wage if there is a degenerate wage in the market equilibrium.

4.2. Many-wage market equilibria Using the same logic, one can construct other market equilibrium wage distributions. For example, there can be a market equilibrium with N wages, w1ow2o?owNox  wN+1, in which a fraction pi of firms pay a wage wi,

6

Since the firm’s value function is nonmonotone in the wage, one also has to verify that a large reduction in the wage is unacceptable to the worker. In particular, the firm earns the same profit from   JðwÞ ¼ JðwÞ.  If the probability of the low wage w ¼ w lðx wÞ=ðr þ sÞ as from the high wage w, breakdown is sufficiently small, the worker will refuse this or any lower wage offer, preferring to wait one period and receive ww .

On-the-Job Search and Strategic Bargaining

51

PN

i¼1 pi ¼ 1. Workers move to higher wage firms whenever presented with the possibility. In the bargaining game, a ‘type i’ firm offers a wage wi and a worker bargaining with such a firm responds with a slightly higher wage wwi ¼ ð1 dÞwi þ dx. If d is sufficiently small, wwi owiþ1 . The firm is indifferent about accepting an offer, while the worker weakly prefers to accept wi . This is a market equilibrium for sufficiently small d. Firms are indifferent about accepting wwi or waiting one period to have wi accepted. Workers are willing to accept wi when Eðwi Þ U  ð1 dÞðEðwwi Þ UÞ, which holds for small d if

wi

zþl

P wj Þ jk¼1 pk ðr þ s þ lÞðx wi Þ 4 P P rþsþl N rþsþl N k¼iþ1 pk k¼jþ1 pk

i 1 X ðwjþ1 j¼1

for all i. In the special case N ¼ 1, this reduces to condition (20), while for N ¼ 2, two wages are a market equilibrium if w1

z4

ðr þ s þ lÞðx w1 Þ r þ s þ lð1 p1 Þ

zþ

lðw2 w1 Þp1 ðr þ s þ lÞðx 4 rþs r þ s þ lð1 p1 Þ

and w2

w2 Þ

For a given p, the first condition places a lower bound on w1 while the second condition places a lower bound on w2 that is increasing in w1 . Both conditions hold when w1 and w2 are sufficiently close to productivity x. To summarize, this simple model of on-the-job search admits a plethora of market equilibria with mass points in the wage distribution. These equilibria hinge on workers’ willingness not to switch employers when they are indifferent. It is unclear whether that assumption is more reasonable than the extreme alternative that workers always switch when indifferent. What happens if one looks for a middle ground? I can think of at least two reasonable ‘refinements’: (1) When a worker at firm j contacts firm j 0 , she moves if W ð j 0 Þ4W ð jÞ or W ð j 0 Þ ¼ W ð jÞ and j 0 4j. Otherwise she remains at firm j. (2) When a worker at firm j contacts firm j 0 she moves with probability 1 if W ð j 0 Þ4W ð jÞ and with probability p40 if W ð j 0 Þ ¼ W ð jÞ. Either refinement eliminates the possibility of a mass in the wage distribution, since, at least for some firms, an arbitrarily small increase in the wage above the mass point leads to a discrete increase in the duration of the match and hence in

52

Robert Shimer

the firm’s value. The only market equilibria that are robust to this refinement are the ones with a continuous wage distributions stressed in Section 3. 5. Heterogeneous Firms I now extend the basic model to introduce firm heterogeneity. I assume thatproductivity x is distributed across firms according to a cumulative distribution  Each firm function H(x), continuously differentiable with convex support ½x; xŠ. contacts a worker at the same constant rate, regardless of the firm’s bargained wage or how many filled jobs it has. Put differently, I treat the distribution H(x) as a primitive of the model and do not ask why both high- and low-productivity firms recruit workers. I also maintain the assumption that the opportunity cost of hiring a worker is zero, independent of x. This follows if all firms costlessly contact workers at a constant rate. I abuse notation slightly to allow for firm heterogeneity. I refer to a firm by its productivity x rather than its index j and let W(x) denote the wage paid by firm x. I also let Jx(w) denote the expected present value of a match for firm x if the worker receives a wage w. To simplify the exposition, I assume that the lower bound of the productivity distribution is workers’ value of leisure, x ¼ z. This ensures that W ðzÞ ¼ z, since that is the only wage that both the worker and firm are willing to accept. Finally, I look only at market equilibria in which the wage function is increasing and continuously differentiable, W 0 ðxÞ40 for all x. This implies that the fraction of firms with productivity less than x is equal to the fraction of firms that pay a wage less than W ðxÞ; HðxÞ ¼ F ðW ðxÞÞ, and that F inherits the continuous differentiability of H and W. It seems likely that for some parameterizations of the model, other market equilibria exist, but I do not characterize them here.

5.1. Definition of equilibrium To characterize a market equilibrium, start again with the Bellman values. The worker’s surplus from a match is unchanged from Equation (3) and is continuously differentiable. The value of a match to a firm is a trivial generalization of Equation (4), x w J x ðwÞ ¼ ð21Þ r þ s þ lð1 F ðwÞÞ Since F is assumed continuously differentiable, Jx(w) is also a continuously differentiable function of w. Assume that F(w) is such that Jx(w) is a decreasing function of w, at least for w  x. Now consider an alternating offers wage bargaining game between a worker and a type x firm, taking the wage distribution F(w) as given. Let d denote the

On-the-Job Search and Strategic Bargaining

53

probability that negotiations break down following each rejected offer and let ww and wf denote the worker’s and firm’s wage offers, respectively. The bargaining problem is analogous to the one in Section 3 since both value functions are monotone. In particular, these offers are part of a subgame perfect equilibrium if the firm is indifferent about accepting ww and the worker is indifferent about accepting wf : J x ðww Þ ¼ ð1

dÞJ x ðwf Þ and

Eðwf Þ ¼ ð1

dÞEðww Þ þ dU

We are interested in characterizing the solution when d is small, so ww and wf converge to W(x). To do so, first differentiate the preceding expressions with respect to d: dww dwf ¼ J x ðwf Þ þ ð1 dÞJ 0x ðwf Þ and dd dd dwf dww E 0 ðwf Þ ¼ Eðww Þ þ U þ ð1 dÞE 0 ðww Þ dd dd J 0x ðww Þ

In the limit as d converges to zero, ww ¼ wf ¼ W ðxÞ. Since Jx and E are continuously differentiate, these expressions reduce to  w  dw dwf  J x ðW ðxÞÞ EðW ðxÞÞ U ¼ ¼ 0 J x ðW ðxÞÞ E 0 ðW ðxÞÞ dd dd d!0

The last equation delivers the critical result E 0 ðW ðxÞÞ J 0 ðW ðxÞÞ ¼0 þ x EðW ðxÞÞ U J x ðW ðxÞÞ

ð22Þ

Equation (22) generalizes the results from the model with homogeneous firms in Section 3, where I proved that ðEðwÞ UÞJðwÞ is constant along the support of the wage distribution. With heterogeneous firms, firm x bargains to a wage W(x) only if W(x) is a local extremum of ðEðwÞ UÞJ x ðwÞ, so the wage elasticity of a type x firm’s value of the match Jx(w) plus the wage elasticity of the worker’s value of the match EðwÞ U must sum to zero. To further refine this characterization of a subgame perfect equilibrium wage, differentiate (22) with respect to x:     d E 0 ðW ðxÞÞ J 0 ðW ðxÞÞ d J 0x ðW ðxÞÞ þ x W 0 ðxÞ þ ¼0 dW ðxÞ EðW ðxÞÞ U J x ðW ðxÞÞ dx J x ðW ðxÞÞ One can verify directly from (21) that the second term is 1=ðx W ðxÞÞ2 40 and so the first term must be negative. Since W 0 ðxÞ40, this implies  d2 logððEðwÞ UÞJ x ðwÞÞ o0  dw2 w¼W ðxÞ

54

Robert Shimer

That is, W(x) is a local maximum of logððEðwÞ UÞJ x ðwÞÞ and hence is a local maximum of ðEðwÞ UÞJ x ðwÞ as well. Since W(x) is continuous and increasing, this is equivalent to requiring that x is a local maximum of fx ðyÞ  ðEðW ðyÞÞ

UÞJ x ðW ðyÞÞ

Formally, let B
ðxÞ  ðx
; x þ
Þ be a ball of radius e around a point x. Then in a market equilibrium7 for every x there is an e>0 such that fxg ¼ arg max fx ðyÞ y2B
ðxÞ

Substituting from (3) and (21), this is equivalent to  Z y  W 0 ðy0 Þ x W ðyÞ 0 fxg ¼ arg max dy y2B
ðxÞ Hðy0 ÞÞ r þ s þ lð1 HðyÞÞ z r þ s þ lð1

ð23Þ

ð24Þ

A market equilibrium is a continuously differentiable and increasing-wage function W(
) such that (24) holds. 5.2. Testable implications Mortensen (2003) discusses the empirical content of the Burdett and Mortensen (1998) model. If one has data on the wage offer distribution F(w), the model allows us to infer the productivity of each firm. The same is true in this model. Let X(w) be the inverse of W(x), the productivity of a firm that pays a wage of w. Use (3) and (21) to substitute for the worker’s and firm’s match value in Equation (22) and simplify Rw ðr þ s þ lð1 F ðwÞÞÞ z 1=ðr þ s þ lð1 F ðw0 ÞÞÞdw0 R X ðwÞ ¼ w þ ð25Þ w 1 þ lF 0 ðwÞ z 1=ðr þ s þ lð1 F ðw0 ÞÞÞdw0

Given any wage distribution F, one can back out the implied productivity of each firm. Even if one does not have data on each worker’s productivity, the model is still testable. In the proposed market equilibrium, more productive firms pay higher wages, so X(w) should be an increasing function. Differentiating (25) gives X 0 ðwÞ40 if and only if Z w  2 1 0 þ dw F 0 ðwÞ4 0 ÞÞ l r þ s þ lð1 F ðw z Z w 2 1 0 ðr þ s þ lð1 F ðwÞÞÞ dw F 00 ðwÞ F ðw0 ÞÞ z r þ s þ lð1

7

By the same logic, if there is a market equilibrium with a decreasing wage function, x must be a local minimum of fx ðyÞ.

On-the-Job Search and Strategic Bargaining

55

This condition holds if the cumulative distribution function F is concave, or equivalently if the wage density F 0 is decreasing, but otherwise it may be violated. For example, suppose l ¼ 20ðr þ sÞ and z ¼ 0. Then this model implies the wage distribution F ðwÞ ¼ w5 with support [0, 1] is inconsistent with any market equilibrium in which the wage function W(x) is increasing. 5.3. Comparison with Burdett and Mortensen (1998) It is useful to compare the equilibrium wage function from the bargaining-model with a similar function obtained in the Burdett and Mortensen (1998) wage posting model. My treatment of this model follows Mortensen (2003). When a worker meets a firm, the firm unilaterally offers the worker a wage without knowing the worker’s employment status. If the firm offers the worker a wage w  z, the worker accepts the job if she is unemployed, with probability u ¼ s=ðs þ lÞ, or employed at a lower wage, with probability ð1 uÞGðwÞ, where GðwÞ ¼ sF ðwÞ=ðs þ lð1 F ðwÞÞÞ is the steady-state distribution of wages paid by firms8. In this event, the firm’s expected discounted profit is J x ðwÞ ¼ ðx wÞ=ðr þ s þ lð1 F ðwÞÞÞ. Putting this together, a type x firm chooses its wage to maximize    s lsF ðwÞ x w þ ð26Þ s þ l ðs þ lÞðs þ lð1 F ðwÞÞÞ r þ s þ lð1 F ðwÞÞ The necessary first order condition is that a type x firm posts a wage w if x ¼ X BM ðwÞ defined by X BM ðwÞ ¼ w þ

ðs þ lð1 F ðwÞÞÞðr þ s þ lð1 F ðwÞÞÞ ðr þ 2s þ 2lð1 F ðwÞÞÞlF 0 ðwÞ

ð27Þ

This generalizes equation (3.16) in Mortensen (2003) to the case of r40. Of course, (27) might represent a minimum of (26). It is in fact a maximum if and only if XBM is increasing or equivalently 2lF 0 ðwÞ2 4ðr þ 2s þ 2lð1

F ðwÞÞÞF 00 ðwÞ

ð28Þ

As in the bargaining model, any concave cumulative wage distribution function F can be rationalized by some underlying productivity distribution. For nonconcave distributions, including the example in the previous section, the condition may be violated. Other functions are consistent with one model but not the other; for example, a log-normal wage distribution cannot be justified using the Burdett–Mortensen model but is consistent with the bargaining model.

8 In steady state, the flow of workers into employment is lu and the flow of workers out of employment is s(1 u). Equating these gives u ¼ sX(s+l) The flow of workers into jobs paying less than o is luF(w), the rate at which unemployed workers find such jobs. The flow of workers out of such jobs is (s+l(1 F(w)))(1 u)G(w), the rate at which workers in these jobs either become unemployed or find a better job. Equating these and using u ¼ sX(s+l) delivers the equation for G in the text.

56

Robert Shimer

Despite their apparent similarities, the quantitative predictions of the two models differ substantially. Suppose, for example, that F ðwÞ ¼ 1 expð wÞ with support [0, N]. Also set r ¼ 0:05, s ¼ 0:5, l ¼ 10, and z ¼ 0. Both models can explain this data using some underlying productivity distribution, but the distributions are distinct, particularly in the right tail. For example, according to the Burdett–Mortensen model, the productivity of a firm paying a wage of 10 in this example must be x ¼ 587.4. In the bargaining model, the implied productivity is a much more reasonable x ¼ 16:9. This is not just a theoretical curiosity. Mortensen (2003) makes the same point in his empirical analysis of Danish wage distributions; compare Figures 4.3 and 4.5 in his book.

6. Discussion Some previous authors have attempted to use the Nash (1953) bargaining solution to set wages in models with on-the-job search. For example, after arguing that Danish wage data are inconsistent with the predictions of the Burdett and Mortensen (1998) model. Mortensen (2003, p. 87) examines ‘‘whether the Nash bilateral bargaining model is consistent with the Danish data on the distribution of average wages paid.’’ To implement this, he imposes in section 4.3.4 that the wage function satisfy fxg ¼ arg max fx ðyÞ

ð29Þ

y

This paper shows that while (29) is a sufficient condition for the equilibrium, equilibrium only imposes the weaker restriction (23). It is unclear whether other market equilibria exist in Mortensen’s model. Other authors, notably Pissarides (1994) and (2000), have examined models like this and assumed that the worker and firm simply split the output from a match, EðW ðxÞÞ

U ¼ J x ðW ðxÞÞ

ð30Þ

for all x. From Equation (22), this is consistent with the equilibrium if and only if E 0 ðW ðxÞÞ þ J 0x ðW ðxÞÞ ¼ 0. But one can verify directly from (3) and (21) that E 0 ðW ðxÞÞ þ J 0x ðW ðxÞÞ ¼

lF 0 ðW ðxÞÞ ðr þ s þ lð1 F ðW ðxÞÞÞÞ2

which is never zero if some firm x is supposed to pay W(x). The ‘surplus splitting’ rule (2.30) ignores the fact that by raising the wage, the worker and firm increase the duration of the match, a critical feature for wage bargaining in environments with on-the-job search. In fact, there are situations in which surplus splitting is Pareto inefficient. Consider a firm that is slightly less productive than most of the other firms in the economy. If all firms split the surplus from matching, this firm will pay a slightly lower wage than most others and suffer high turnover. By raising the wage, the

On-the-Job Search and Strategic Bargaining

57

firm increases the worker’s utility and may increase its profit by reducing turnover. One does not need a very extreme parameterization of the model to illustrate this possibility. Let H(x) be uniform on (z, z+l). Then if l43ðr þ sÞ and all firms split the surplus according to (30), one can show that some firms – more precisely, the most productive firms – would gain by unilaterally raising their workers’ wages. 7. Conclusion The Burdett and Mortensen (1998) model has become an important workhorse of theoretically motivated empirical labor economics. This paper introduces a related model of bargaining and on-the-job search that delivers results that are qualitatively, if not quantitatively, similar to the wage posting model. Why might an economist prefer one model to the other? The wage posting model has one undeniable appeal: it has a unique market equilibrium. Even in the simplest model with homogeneous workers and homogeneous firms, and even if one is willing to ignore the less robust market equilibria with mass points in the wage distribution, the bargaining model admits a multiplicity of market equilibria, each characterized by a continuous wage distribution. Future research should explore which of these market equilibria is most plausible. For example, one can prove that there is only one wage dis are local maxima of ðEðwÞ UÞJðwÞ. tribution, F ðxþzÞ=2 , such that all w 2 ½w; wŠ With any other wage distribution F w and w4ðx þ zÞ=2, it is easy to show that ðEðwÞ UÞJðwÞ is a local minimum. The characterization of market equilibrium with heterogeneous firms, condition (23), therefore suggests that only the wage distribution F ðxþzÞ=2 is the limit of market equilibria of heterogeneous agent economies, with wages monotonic in productivity, as heterogeneity grows less important. Along other dimensions, the bargaining model seems more attractive than the posting model. Consider the out-of-steady-state dynamics of the two models. In the wage posting model, the payoff-relevant state of the economy is described by the unemployment rate u and the distribution of wages paid to employed workers G. Burdett and Mortensen prove that if these are at their steady state values, then there is a market equilibrium in which the wage offer distribution F is constant over time. But suppose instead the economy starts off out of steady state. Does it converge to steady state? What do the nonstationary dynamics look like? Although it is possible to answer these questions under special conditions, a general characterization of the nonstationary dynamics remains elusive (Shimer, 2003). In the bargaining model, the characterization of market equilibrium when the economy is away from steady state is trivial – in fact, it was not necessary to mention the unemployment rate u or the distribution of wages paid G anywhere in the paper. Whether a wage distribution F is a market equilibrium is independent of whether u and G are in steady state.

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Allowing for aggregate shocks, e.g. changes in the arrival rate of offers l, further complicates the posting model. First is the question of whether firms should be able to post offers that are contingent on the aggregate shock. If they can, one can show that firms will use the shock in order to artificially create an upward-sloping wage profile, much as in Stevens’ (2004) and Burdett and Coles’ (2003) deterministic wage contracting models. This conclusion seems unappealing, and so one is led to assume that the firms cannot make wage offers contingent on the aggregate state. But in such a model, the payoff relevant state of the economy is the aggregate shock, the unemployment rate, and the wage distribution across workers. Solving for a market equilibrium is complex at best. In this environment, the bargaining model is appealing along two dimensions. First, it is natural to assume that workers and firms continually re-bargain in the face of shocks. Second, the payoff relevant state is again only the aggregate shock, and so it is possible, at least in principle, to find a solution to the model in which the wage offer distribution depends on current and expected future values of the shock. Finally, the bargaining model addresses an important theoretical concern with the wage posting model. In the latter model, wages are time inconsistent, since a firm would like to cut the wage as soon as the worker agrees to take a job. Although reputation concerns might keep firms paying high wages, reputations are complicated to model and usually ignored; a notable exception is Coles (2001). In the wage bargaining model, a worker and firm can re-bargain at any time and the old wage would remain a subgame perfect equilibrium. Acknowledgment I am grateful for insightful comments from an anonymous referee and the seminar audience at the Labour Market and Matched Employer-Employee Data Conference in Sønderborg, Denmark, especially John Kennan, Dale Mortensen, Eric Smith, and Randall Wright. I also thank the National Science Foundation and the Sloan Foundation for financial support. References Binmore, K., A. Rubinstein and A. Wolinsky (1986), ‘‘The Nash bargaining solution in economic modeling’’, Rand Journal of Economics, Vol. 17(2), pp. 176–188. Blanchard, O. and P. Diamond (1989), ‘‘The Beveridge curve’’, Brookings Papers on Economic Activity, Vol. 1, pp. 1–60. Burdett, K. and M. Coles (2003), ‘‘Equilibrium wage tenure contracts’’, Econometrica, Vol. 71(5), pp. 1377–1404. Burdett, K. and D. Mortensen (1998), ‘‘Wage differentials, employer size and unemployment’’, International Economic Review, Vol. 39, pp. 257–273.

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Cahuc, P., F. Postel-Vinay and J.–M. Robin (2003), Wage Bargaining with On-the-Job Search: Theory and Evidence, Universite de Paris I, Mimeo (September). Coles, M. (2001), ‘‘Equilibrium wage dispersion, firm size, and growth’’, Review of Economic Dynamics, Vol. 4(1), pp. 159–187. Diamond, P. (1982), ‘‘Wage determination and efficiency in search equilibrium’’, Review of Economic Studies, Vol. 49(2), pp. 217–227. Fallick, B. and C. Fleischman (2004), ‘‘Employer-to-employer flows in the US labor market: The complete picture of gross worker flows’’, Federal Reserve Board, Finance and Economics Discussion Series Working Paper 2004–34. Mortensen, D. (1982), ‘‘Property rights and efficiency in mating, racing, and related games’’, American Economic Review, Vol. 72(5), pp. 968–979. Mortensen, D. (2003), Wage Dispersion: Why are Similar Workers Paid Differently, Cambridge, MA: MIT Press. Moscarini, G. (2004), Job-to-Job Quits and Corporate Culture. Yale, Mimeo. Nash, J. (1953), ‘‘Two-person cooperative games’’, Econometrica, Vol. 21(1), pp. 128–140. Pissarides, C. (1994), ‘‘Search unemployment with on-the-job search’’, The Review of Economic Studies, Vol. 61(3), pp. 457–475. Pissarides, C. (2000), Equilibrium Unemployment Theory, 2nd edition, Cambridge, MA: MIT Press. Postel-Vinay, F. and J.-M. Robin (2002), ‘‘Equilibrium wage dispersion with worker and employer heterogeneity’’, Econometrica, Vol. 70(6), pp. 2295–2350. Rubinstein, A. (1982), ‘‘Perfect Equilibrium in a Bargaining Model’’, Econometrica, Vol. 50(1), pp. 97–109. Shaked, A. and J. Sutton (1984), ‘‘Involuntary unemployment as a perfect equilibrium in a bargaining model’’, Econometrica, Vol. 52(6), pp. 1351–1364. Shimer, R. (2003), Out-of-Steady-State Behavior of the Burdett-Mortensen Model, Mimeo, University of Chicago. Stevens, M. (2004), ‘‘Wage-tenure contracts in a frictional labour market: firms’ strategies for recruitment and retention’’, Review of Economic Studies, Vol. 71(2), pp. 535–551.

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CHAPTER 3

Alternative Theories of Wage Dispersion Damien Gaumont, Martin Schindler and Randall Wright Abstract We analyze labor market models where the law of one price fails – i.e., models with equilibrium wage dispersion. We begin considering ex ante heterogeneous workers, but highlight a problem with this approach: if search is costly the market shuts down. We then assume homogeneous workers but ex post heterogeneous matches. This model is robust to search costs, and delivers equilibrium wage dispersion. However, we prove that the law of two prices holds: equilibrium implies at most two wages. We explore other models, including one combining ex ante and ex post heterogeneity which is robust and delivers more realistic wage dispersion.

Keywords: search, wage dispersion, search costs, heterogeneous workers JEL classifications: J31, J64 1. Introduction According to Mortensen (2003, p. 9), ‘‘If the law of one price were to hold in the labor market, similar workers would not be paid differently.’’ This observation is both obvious and deep. The fact is, similar workers do appear to be paid differently. As Mortensen (2003, p. 1) reports, ‘‘Although hundreds if not thousands of empirical studies

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CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75003-3

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Damien Gaumont, Martin Schindler and Randall Wright

that estimate so-called human capital wage equations verify that worker characteristics that one could view as indicators of labor productivity are positively related to wages earned, the theory is woefully incomplete in its explanatory power. Observable worker characteristics that are supposed to account for productivity differences typically explain no more than 30 percent of the variation in compensation.’’ What explains the rest? It is clear that one needs a model with some sort of frictions to address the issue. Search theory is ideally suited to the task. In their survey, Eckstein and van den Berg (2005, p. 25) argue that ‘‘equilibrium search models provide a framework to empirically analyze the sources of wage dispersion: (a) workers heterogeneity (observed and unobserved); (b) firm productivity heterogeneity (observed and unobserved); (c) market frictions. The equilibrium framework can [...] empirically measure the quantitative importance of each source.’’ For example, van den Berg and Ridder (1998) estimate that up to 25% of wage variability is attributable to frictions, in the sense that this is what would emerge from a model without heterogeneity. Postel-Vinay and Robin (2002) estimated up to 50%. The benchmark model for studying wage dispersion is the one developed by Burdett and Mortensen (1998), which is based on wage posting and on-the-job search. The goal of this paper is to explore some alternative models where the law of one price does not hold in the labor market. It is not that there is anything especially wrong with Burdett–Mortensen.1 However, it is good to have some alternatives on the table. These alternatives need not be mutually exclusive, of course; there may be several reasons for wage dispersion in the world and it seems interesting to consider various options, perhaps ultimately integrating the different models in one framework and using the data to measure the importance of each. Here we are not that ambitious, and the goal is to develop theoretically several alternative models that each imply the possibility of a nondegenerate wage distribution. We emphasize that it is not easy to get wage dispersion across homogeneous workers in equilibrium, in the sense that the well-known Diamond (1971) model, which seems on the surface the natural framework in which to think about the issues, predicts a single wage even in the presence of search frictions. Of course, there are several well-known ways to get around this result, including Burdett and Mortensen (1998), which introduces on-the-job search, including Burdett and Judd (1983), which introduces the idea that some workers may get multiple offers, and including Albrecht and Axell (1984), which introduces heterogeneous

1

There are some issues with this model, however, such as the fact that the baseline model predicts an unrealistic wage distribution, in the sense that the density is upward sloping (there are more high wage than low-wage workers). This can be fixed by adding firm heterogeneity or ex ante firm investments, e.g. as discussed in Mortensen (2000, 2003), but it would be nice if a simpler version was more in line with the data.

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2

outside options (values of leisure) for workers. Note that in all of these examples there is a sense in which workers are heterogeneous, but the point is that they have the same productivity and yet still can end up receiving different wages in equilibrium. In any case, the goal here is to explore some new models and ideas.3 We begin with models where workers have ex ante heterogeneous outside options in the spirit of Albrecht and Axell (1984), although the details are quite different. We point out a flaw in this class of models: once we introduce positive costs of search, no matter how small, equilibrium unravels. That is, there will always be one type who will drop out, but once they do, another type drops out and so on, until we are back to Diamond (1971). Given this, we introduce a framework where workers are homogeneous but matches are heterogeneous. That is, ex ante all workers are the same, but there are match-specific shocks so that different workers attach different valuations to different jobs even though productivity is still the same. In this model, the market does not shut down for positive search costs, and it does deliver equilibria where the law of one price fails. However, we prove that the law of two prices holds: as shown in Curtis and Wright (2004) in the context of monetary theory, in search models with match-specific shocks one can get more than one price, but generically one cannot get more than two. We also explore some other models. To motivate these, note that in Albrecht–Axell or Burdett–Judd style models, the reason different firms may post different wages is that high-wage firms have a high inflow of workers (they recruit faster). In Burdett–Mortensen, high-wage firms also have a high inflow of workers and additionally have a low outflow (they lose workers more slowly). We present a model where all firms have the same inflow but high-wage firms have a low outflow, and one where high-wage firms get better performance from their workforce.4 In each case, we prove that the law of one price does not hold, but the law of two prices does. We also show how to combine approaches. This is important because the unraveling of models with search costs and the law of two prices can both be overturned when we have ex ante and ex post shocks. Hence, the combination of both delivers a robust model with an empirically more interesting wage distribution.

2

Diamond (1987) presents a consumer search model that is quite similar to Albrecht and Axell’s (1984) labor market model. In addition to work mentioned below, some other related papers include MacMinn (1980), Rob (1985) and Albrecht and Vroman (1992, 2005). See Mortensen (2003) for additional references. 3 In this paper, we are interested in models where firms post wages, as opposed to bargaining with individual workers after they meet. Any bargaining model with heterogeneous agents or heterogeneous matches, such as Mortensen and Pissarides (1994), can generate wage dispersion, but we follow the literature that asks how one can get a nondegenerate distribution of posted wages. 4 The first of these is in the spirit of the Burdett et al. (2003) model of crime, although considerably simpler; the second is in the spirit of the efficiency wage literature, such as Shapiro and Stiglitz (1984). Typical efficiency wage models do not generate endogenous wage dispersion, but see Albrecht and Vroman (1998).

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The rest of the paper is organized as follows. In Section 2, we present some models with ex ante heterogeneity and discuss unraveling. In Section 3, we introduce models with ex post heterogeneity and prove the law of two wages. In Section 4, we discuss some other approaches. In Section 5, we combine ex ante and ex post heterogeneity. In Section 6, we conclude. 2. Ex Ante heterogeneity 2.1. A simple model There is a [0, 1] continuum of firms and a [0, L] continuum of workers. There are K types of workers. A measure Lj of workers are type j, and they have P utility of leisure bj where we order types such that bj>bj 1, j ¼ 2,y, K, and j Lj ¼ mp.5 Firms post wages. Each firm has a constant returns technology with labor as the only input and productivity y>bK (if there are any workers with bj>y, they will drop out). For now we follow Burdett and Mortensen (1998) by assuming firms are interested in maximizing steady-state profit and will hire as many workers as are willing to accept; we consider different models of firm behavior below. All agents are risk neutral and discount at rate r. Unemployed workers contact firms at rate aw, and there is no on-the-job search. Matches end at an exogenous rate d. Given any distribution of posted wages F(w), it should be obvious that each type of worker will have a reservation wage wj such that he accepts wXwj and rejects wowj, with wj+1>wj. It is equally apparent that, in any equilibrium, no firm would post anything other than one of the reservation wages, as a firm posting wA(wj, wj+1) could reduce w down to wj and make more profit per worker without changing the set of workers who accept. A special case of this is the Diamond (1971) result when K ¼ 1: with homogeneous workers, all firms post w1. Moreover, in this case w1 ¼ b1 . To see why, assume all firms are posting w>b1; then as long as r>0, a firm can post w e for some e>0 and still hire every worker it contacts. So in equilibrium, all firms must post w ¼ w1 ¼ b1. Consider the case K ¼ 2. Then there are at most two wages w1 and w2 posted in equilibrium. Let yA[0, 1] be the fraction of firms posting w2 and thus 1 y the fraction posting w1. Let Uj be the value function of an unemployed worker of type j and Wj(w) the value function of a type j worker employed at w. Since we already know the only posted wages are w1 and w2, the relevant flow Bellman equations for unemployed workers are rU 1 ¼ b1 þ aw ð1

yÞ½W 1 ðw1 Þ

rU 2 ¼ b2 þ aw y½W 2 ðw2 Þ

5

U 2 Š;

U 1 Š þ aw y½W 1 ðw2 Þ

U 1Š

Albrecht and Vroman (2005) assume unemployment benefits vary with the duration of unemployment and thus endogenize heterogeneity in b.

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where we use the result that type 2 accepts w2 but not w1, while type 1 accepts both offers. Indeed, the reservation property implies W 1 ðw1 Þ ¼ U 1 and W 2 ðw2 Þ ¼ U 2 , and so the expressions simplify to rU 1 ¼ b1 þ aw y½W 1 ðw2 Þ rU 2 ¼ b2 :

U 1Š

Again using the reservation property, the relevant Bellman equations for employed workers are rW 1 ðw1 Þ ¼ w1

rW 1 ðw2 Þ ¼ w2 þ d½U 1 rW 2 ðw2 Þ ¼ w2 :

W 1 ðw2 ފ

Taken together, these equations imply w2 ¼ b2 and w1 ¼

ðr þ dÞb1 þ aw yb2 . r þ d þ aw y

ð1Þ

Notice w1 is a weighted average of b1 and b2, and w1>b1 if and only if y>0. Type 1 workers would not accept w ¼ b1 if y>0, because there is a chance of getting w2 ¼ b2 . Notice that qw1/qy>0. Now consider firms. For now we follow Burdett–Mortensen and assume each firm is interested in maximizing steady-state profit. To compute this, let rj be the probability a random unemployed worker accepts wj. Then a firm posting wj hires at rate afrj, the rate at which it meets workers times the probability they accept, and expects to earn (y wj)/(r+d) from each worker it hires. Hence, firms care about6 Pj ¼

af rj ðy wj Þ . rþd

For a firm posting w2, r2 ¼ 1, and for a firm posting w1, r1 ¼ L1 u1 = ðL1 u1 þ L2 u2 Þ, where uj is the steady-state unemployment rate for type j workers, with u1 ¼ d=ðd þ aw Þ and u2 ¼ d=ðd þ aw yÞ. Hence, r1 ¼

L1 ðaw y þ dÞ . L1 ðaw y þ dÞ þ L2 ðaw þ dÞ

We are interested in the sign of P2 P1, since this determines the optimal wage posting strategy. This is equal in sign to y w2 r1(y w1), which, after inserting

6

The original Burdett–Mortensen model actually proceeds by noting that in steady state a firm posting wj ends up with a stock afrj/d of workers and is interested in maximizing (afrj/d)(y wj). This yields exactly the same results in the model under consideration.

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r1, w1 and w2 and simplifying, can be shown to be equal in sign to the following linear function of y,
 TðyÞ ¼ ½L1 ðr þ dÞ þ L2 ðaw þ dފð y b2 Þ L1 ðr þ dÞð y b1 Þ aw y þ ½L1 dðr þ dÞ þ L2 ðaw þ dÞðr þ dފð y b2 Þ L1 dðr þ dÞð y

b1 Þ.

ð2Þ

The following best response condition must hold in any equilibrium:7 y¼0

if Tð0Þo0;

y¼1

if Tð1Þ40; and if y 2 ð0; 1Þ then TðyÞ ¼ 0. ð3Þ

When TðyÞ ¼ 0, we can solve explicitly for y¼

rþd L1 dð y b1 Þ ½L1 d þ L2 ðaw þ dފð y b2 Þ aw ½L1 ðr þ dÞ þ L2 ðaw þ dފð y b2 Þ L1 ðr þ dÞð y

b1 Þ

.

ð4Þ

Proposition 1. For all aw, there exists a unique solution to (3), and 0oyo1 iff yoyoy where y ¼ b2 þ

dL1 ðb2 b1 Þ raw L1 ðb2 b1 Þ and y ¼ y þ . ðaw þ dÞL2 ðr þ aw þ dÞðaw þ dÞL2

ð5Þ

Proof. Existence is easy. If T(0)p0 then y ¼ 0 satisfies (3). Suppose T(0)>0. If T(1)X0 then y ¼ 1 satisfies (3). If T(1)o0 then by continuity there exists a yA(0,1) that satisfies (3). Uniqueness follows from the result that T 0 (y)o0 at any y such that TðyÞ ¼ 0, which is easily verified to be true as long as y 2 ðy; yÞ. The thresholds y and y come from checking when T(0)>0 and T(1)o0. When productivity is low, all firms pay w1 ¼ b1 , when it is high, all firms pay w2 ¼ b2 , and when it is in the intermediate region ðy; yÞ, there is wage dispersion. We can now solve explicitly for wages as well as the number of workers earning each wage, the unemployment rate and so on, in the range where yA(0, 1) by inserting (4) into (1). For example, normalizing b1 ¼ 0 without loss of generality, we have w1 ¼

L2 ðd þ aw Þðy b2 Þ rL1

L1 db2

.

Notice w1 ¼ b1 ¼ 0 at y ¼ y and w1 is linearly increasing in y up to w1 ¼ ðaw b2 Þ=ðr þ d þ aw Þ at y ¼ y. 7

When yA(0, 1), one can equivalently interpret the outcome as a Nash equilibrium where symmetric firms use asymmetric pure strategies, or where they all use the same mixed strategy.

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The distribution of wages paid can be calculated easily given the steady-state conditions: w ¼ w1 with probability 1 p and w ¼ w2 with probability p, where p ¼ ‘2 =ð‘1 þ ‘2 Þ and ‘i is the steady-state measure of workers who are employed at wi. This is to be contrasted with the distribution of wages posted, which is given by w ¼ w1 with probability 1 y and w ¼ w2 with probability y. Typically, p is bigger than y since w2-firms have more workers than w1-firms, which is precisely how they can have equal profits. Notice that in this model we can easily have a decreasing density, in the sense that po1/2o1 p. This is in contrast to the basic Burdett–Mortensen model where the density must be increasing, contrary to the data. Of course, in the K ¼ 2 case, our density is not very realistic in another sense – there are only two wages. We show below how to generalize this. It remains to discuss the arrival rates. As we mentioned earlier, the measure of firms is fixed at unity and each firm will hire as many workers as it can get. Suppose we assume a CRS meeting technology m(nu, nf), where nu is the number of unemployed workers and nf ¼ 1 is the number of firms. Then the rate at which workers contact firms is aw ¼ mðnu ; nf Þ=nu , which given nf ¼ 1 and nu ¼ L1 u1 þ L2 u2 ¼ L1 d=ðd þ aw Þ þ L2 d=ðd þ aw yÞ can be written   d d ;1 m L1 þ L2 d þ aw d þ aw y aw ¼ . ð6Þ d d L1 þ L2 d þ aw d þ aw y An equilibrium is then a pair (aw, y) satisfying (3) and (6). Once aw is known, af ¼ mðnu ; nf Þ=nf can be calculated, but notice that af only affects the level of profits and not the sign of P2 P1, and so it does not affect the equilibrium values of y, wj and so on. Consider for the sake of illustration the special case where mðnu ; nf Þ ¼ A minfnu ; nf g ¼ A minfL1 u1 þ L2 u2 ; 1g a matching function that arises in various applications (see, e.g. Lagos, 2000). This implies mðnu ; nf Þ ¼ Anu and hence aw ¼ A as long as L1u1+L2u2p1 which always holds if Lp1, an assumption we are free to make. In this case, the arrival rate for workers is essentially exogenous. Hence, equilibrium is completely characterized by (3), and everything to be said about it is contained in Proposition 1.8 We will not dwell on existence or uniqueness/multiplicity in the case of a general matching technology, but instead we present another, less extreme,

8

It is not that we think these assumptions about the matching function or Lr1 are particularly general or realistic; the point is simply that we can come up with assumptions that generate simple wage-dispersion equilibrium without having to worry about the endogeneity of arrival rates.

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Figure 1.

Multiple equilibria





E


M

w

example. Consider the Cobb–Douglas specification mðnu ; nf Þ ¼ An1u g ngf . We can solve (6) in this case for y¼

L2 d=aw ðA=aw Þ1=g

L1 d=ðaw þ dÞ

d=aw .

ð7Þ

Figure 1 plots (4) and (7) in (aw, y) space, the former labeled yE for ‘‘equal profit’’ and the latter labeled yM for ‘‘matching function.’’ As one can see, there are two solutions for yA(0, 1). Hence, the model is not only capable of generating wage dispersion, it also yields multiple equilibria with wage dispersion.9 The intuition for this result, which seems novel compared to the related literature on multiplicity going back to Diamond (1981, 1982), is simple. Suppose many firms are paying the high wage w2 ¼ b2 , so that y is relatively big. Then from (1) we see that w1 is relatively big. This makes it relatively less profitable to try to get away with posting the low wage w1, and hence more firms end up posting w2.

9

The point is not that we always get multiplicity, but that we can. Also, note that it is not easy to construct examples for realistic parameter values with yA(0, 1), because, as one can see from (5), the interval ðy; yÞ is small when r is small. The example in the figure uses r ¼ 0:1, b1 ¼ 0:1, b2 ¼ 1:8, y ¼ 2:165, L1 ¼ L2 ¼ 0:5, d ¼ 0:05, A ¼ 0:17 and g ¼ 0:5.

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2.2. Alternative assumptions An alternative assumption about firm behavior is that each employer may post at most one vacancy, along the lines of the models in Pissarides (2000), at cost k. Then firms maximize the present discounted value of vacancies (as opposed to steady-state profit). Mortensen (2000) shows that this alternative approach gives similar results in the basic Burdett–Mortensen model, under some conditions, and we want to see how it affects outcomes here. Let Vj denote the value of a firm with a vacancy posting wage wj, and Jj the value of having the job filled. Then we have rV j ¼ af rj ðJ j rJ j ¼ y

V jÞ

wj þ dðV j

k

ð8Þ

J j Þ.

ð9Þ

Solving the system yields rV j ¼

af rj ðy wj þ kÞ r þ d þ af r j

k.

ð10Þ

Compared with Pj, the differences are that k appears, and that afrj shows up in the denominator of Vj. Inserting r1 and w1 into V2 V1, we see that it takes the same sign as    L2 ðd þ aw Þ L2 ðd þ aw Þ TðyÞ ¼ L1 þ k aw y ðy b2 Þ L1 ðy b1 Þ þ r þ d þ af r þ d þ af   L2 ðd þ aw Þðr þ dÞ ðy b2 Þ L1 dðy b1 Þ þ L1 d þ r þ d þ af L2 ðd þ aw Þðr þ dÞ þ k. r þ d þ af Assume first k ¼ 0. As in the previous model, the best response condition (3) must hold in any equilibrium, and for any aw and af there exists a unique solution to this condition, with 0oyo1 iff yoyoy (although the values for y and the thresholds y and y are different). Again we can use   d d ;v m L1 þ L2 d þ aw d þ aw y aw ¼ d d L1 þ L2 d þ aw d þ aw y

to determine aw and then af, except here we need to replace v ¼ 1 with v ¼ 1 Lð1 L1 u1 L2 u2 Þ since, with k ¼ 0; all firms that do not have a worker are recruiting (in the previous model all firms were recruiting, even those that had workers, because there firms want to employ as many people as they can get). Because now both aw and af enter the T function, we cannot use mðnu ; nf Þ ¼ A minfnu ; nf g to eliminate the arrival rates from T, as we did in the previous model: we can eliminate aw ¼ A, but that still leaves af. However, a different

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trick to simplify matters is to assume equal numbers of workers and firms: L ¼ 1. Then, given we are assuming k ¼ 0 so that all firms post vacancies, every filled job takes one worker and one vacancy off the market, leaving the ratio nu/nf unchanged. Hence, constant returns in the matching function implies the arrival rates are again effectively exogenous, and all that one needs to determine is y. Now consider k>0, so not all firms necessarily post vacancies. Free entry implies V j ¼ 0 for any wj that is actually posted. To focus on the more interesting outcomes, consider any equilibrium with y>0. Then some firms post w2 ¼ b2 , so V 2 ¼ 0 and we can solve (10) with j ¼ 2 for af ¼

kðr þ dÞ . y b2

This pins down af, from which we can determine the vacancy-unemployment ratio v/u through af ¼ mðu=v; 1Þ, and then aw ¼ mð1; v=uÞ. Substituting af and aw into T then allows us to determine y, which completes the description of equilibrium.

2.3. Discussion We have illustrated under various assumptions that simple models with ex ante heterogenous workers can generate wage dispersion. As we said above, this is very much in the spirit of the Albrecht–Axell model, although the details of our set up are different. Moreover, this framework generalizes quite easily to the case of K>2 types. There will be K reservation wages w1P ,y, wK, and in equilibrium these are posted with probabilities y1,y, yK where K j yj ¼ 1. Of course, some values of yj may be 0 in equilibrium, but clearly no firm will post anything other than one of the K reservation wages.10 However, this class of models, with any value of K, has a problem: the equilibrium is not robust to the introduction of any search cost e>0. In the case of K ¼ 2, the high reservation wage workers (those with b ¼ b2 ) get zero surplus from search – they reject w1, and while they may accept w2, for them it is no better than unemployment. Hence, they will not search if e>0. But then no firm will post anything other than w1 and we are back to Diamond (1971). Obviously this is true for any K: worker types with the highest bK get no surplus, so they drop out if e>0, and there are K 1 types left, and so on. We cannot get robust wage dispersion with ex ante heterogeneity. Indeed, things are worse than one might think: once all but type 1 workers drop out, given e>0, the type 1 workers will drop out as well and the market shuts down. Based on these observations, we think it is worth considering some alternatives to ex ante heterogeneity.

10

Eckstein and Wolpin (1990) analyze this version of the model empirically.

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71

3. Ex post heterogeneity 3.1. Permanent shocks Consider a model where workers are ex ante homogeneous, but matches are ex post heterogeneous. In particular, when a worker contacts a firm, he draws at random a match-specific c 2 fc1 ; ::: ; cK g, where c is the per period cost to accepting the job. For example, c could be the cost of commuting, working with people you may not like, etc.11 For now c is permanent for the duration of the match; we consider below the case where workers draw a new c each period. As in the previous section, we start with K ¼ 2 and consider K>2 below. Thus, c ¼ c1 with probability l and c ¼ c2 4c1 with probability 1 l. Assume b+c2oy. Again, we begin by assuming that firms post wages to maximize steady-state profit. It should be obvious that each worker now has two reservation wages: he accepts wXw1 if he draws c1 in a match, and accepts wXw2 if he draws c2. For the same reason as in the previous model, there will be at most two wages posted – no firm would post anything other than w1 or w2. We let y be the fraction of firms posting w2 as before, and we now let Wj(w) be the value to having a job with wage w and c ¼ cj and U the value of unemployed search. A key difference from the previous section is that here U is not indexed by type – there are no types, as all workers are ex ante identical. Also, there Wj(w) denoted the value function for a type j worker employed at w, while here it is the value function for worker in a type j match employed at w. The reservation wage conditional on c ¼ cj satisfies W j ðwj Þ ¼ U. The Bellman equation for unemployed workers is rU ¼ b þ aw ly½W 1 ðw2 Þ

UŠ,

ð11Þ

where we have used the facts that a worker who draws c2 does not accept w1, and that a worker who draws cj accepts wj but gets no surplus from doing so. Also, rW j ðwÞ ¼ w

cj þ d½U

W j ðwފ.

Solving these equations, we can derive aw ly ðc2 rþd aw ly ðc2 w2 ¼ b þ c2 þ rþd w1 ¼ b þ c1 þ

11

c1 Þ c1 Þ:

This model is similar to Burdett and Wright (1998), which is also based on nonpecuniary matchspecific shocks, except there the wage is fixed. Burdett and Wright (1993) and Masters (1999) discuss ways to endogenize w in that model, but they do not consider wage dispersion. See also Albrecht and Jovanovic (1986).

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A firm posting w1 hires at rate afl and expects to earn (y w1)/(r+d) from each worker it hires. Similarly, a firm posting w2 hires at rate af and expects to earn ðy w2 Þ=ðr þ dÞ. Therefore, af lðy w1 Þ rþd af ðy w2 Þ . P2 ¼ rþd

P1 ¼

The same methods used above imply that P2 P1 takes the same sign as TðyÞ ¼ ðr þ dÞ½ð1

lÞðy

bÞ

ðc2

lc1 ފ

aw lð1

lÞðc2

c1 Þy.

ð12Þ

An equilibrium still must satisfy the best response condition (3). Proposition 2. For all aw, there exists a unique solution to (3), and 0oyo1 iff yoyoy where y¼bþ

c2 lc1 aw lð c 2 c 1 Þ and y ¼ y þ . rþd 1 l

ð13Þ

Proof. Existence is the same as in Proposition 1. Uniqueness is even easier here since T 0 (y)o0 for all y. Again, the thresholds y and y come from checking when T(0)>0 and T(1)o0. One can again solve TðyÞ ¼ 0 for y¼

r þ d ð1 aw

lÞðy bÞ ðc2 lc1 Þ . lð1 lÞðc2 c1 Þ

ð14Þ

Using this value of y, we can solve for wages in the case where yA(0, 1): c2 lc1 . 1 l When y ¼ 0, the unique posted wage is w1 ¼ b þ c1 , and when y ¼ 1, the unique posted wage is w2 ¼ b þ c2 þ aw l=ðr þ dÞðc2 c1 Þ. Note that w2>b+c2 because a worker who draws c2 and has offer w2 would prefer to turn it down and wait to get c1 and w2.12 We can again consider different assumptions regarding firm behavior. When each firm can hire at most one worker and has to post a vacancy to recruit, the Bellman equations are again given by (8) and (9). If k ¼ 0, we have   1 aw ¼ m 1; ð15Þ uðy; aw Þ wj ¼ y þ cj

12

Also note that in this model y4y even in the limit as r-0, unlike the model in the previous section where y ¼ y at r ¼ 0.

Alternative Theories of Wage Dispersion

73

where uðy; aw Þ ¼

d d þ aw ½y þ ð1

yÞlŠ

is the unemployment rate. In an equilibrium with wage dispersion, the fraction y of firms posting w2 is still determined by (3) where T is now given by (12). An equilibrium is a pair (y, aw) satisfying (3) and (15). Again, in the special case mðu; vÞ ¼ A minfu; vg, we can guarantee aw ¼ A and equilibrium is fully characterized by (3). If k>0, and assuming yA(0, 1), the free entry condition V j ¼ 0 pins down af rj ¼

kðr þ dÞ . y wj

Given af we can determine aw and this can be inserted into the T(y) function, which then pins down y. 3.2. The law of two wages The model with ex post heterogeneity is not fragile with respect to introducing search costs. As long as y>0, it is clear that we can have rU>b, and hence workers would be willing to search even at a cost. It is also clear that we can generalize the analysis to the case where c ¼ c1 ; ::: ; cK with probability l ¼ l1 ; ::: ; lK and what we said goes through. In particular, there will exist K conditional reservation wages w1 ; ::: ; wK such that any worker accepts wXwj if he draws cj in a match. Things do not unravel here the way they did with ex ante heterogeneity because there are no types to drop out, and any worker gets positive gains from search as long as y1o1, since then there is a chance he can get a job at a wage high enough to make him accept even if he draws cj>c1, but he gets lucky and draws c ¼ c1 . Now for something that may be more surprising (if one has not seen a version of it before). The usual Diamond logic guarantees that no firm will post any wage other than one of the K reservation wages w1 ; ::: ; wK , so if we let yj be P the fraction posting wage wj, we know j yj ¼ 1. We now claim that generically there are never more than two wages actually posted. Adapting the language in Curtis and Wright (2004), we call this the law of two wages. Proposition 3. For generic parameter values, we can have yj>0 for at most two values of j. Proof. For any K, workers’ Bellman equations are rU ¼ b þ aw rW j ðwÞ ¼ w

K P

j¼1

lj

K P yi W j ðwi Þ i¼j

cj þ d U

U

W j ðwÞ :



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The reservation property implies W j ðwj Þ ¼ U for all j. This implies wj ¼ cj þ rU.

ð16Þ

Now consider profits, and suppose yi>0, yj>0, yk>0 for distinct i, j, and k. Then Pi ¼ Pj ¼ Pk ¼ maxfP1 ; ::: ; PK g. Using (16) we can write af rj ðy wj Þ rþd af rj ðy cj rUÞ , ¼ rþd

Pj ¼

P where rj ¼ jh¼1 lh is the probability a random worker accepts wj. The condition Pi ¼ Pj ¼ Pk therefore implies gi ðUÞ ¼ gj ðUÞ ¼ gk ðUÞ, where gj ðUÞ  rj ðy

cj Þ

rrj U.

For generic parameter values, there does not exist a solution to gi ðUÞ ¼ gj ðUÞ ¼ gk ðUÞ – i.e., there does not exist a value of U such that the three linear functions are all equal. The result is depicted in Figure 2, which shows profit as a function of the wage, P(w). For wow1 the firm hires no one, so P ¼ 0. At w ¼ w1 , P(w) jumps up because now the firm hires any worker who shows up and draws c ¼ c1 . For wA[w1, w2), P(w) is linearly decreasing in w. And so on. If w1, y, wK were exogenous, then generically P(w) will attain a maximum at a unique wj. But w1, y, wK are not exogenous. Hence, one might think they can adjust endogenously until there are multiple wj maximizing P(w). Indeed, this is precisely what we did in the case K ¼ 2 to get P1 ¼ P2 . However, the reservation wages are all related by (16); hence, it may be possible to adjust one of them so that P(w) is maximized at more than one point, but we cannot independently adjust another one so that P(w) is maximized at more than two points.13

13 Jafarey and Masters (2003) analyze a model related to Curtis–Wright, except the match-specific shock is a uniformly distributed continuous variable. They show this implies exactly one price will be posted in equilibrium (this is not inconsistent with the law of two prices, of course, which says there are no more than two). Intuitively, with a continuum of shocks there is a continuum of reservation wages, and the function P(w) in Figure 2 does not have discrete jumps. Generalizing Jafarey and Masters, one can show there is a single wage posted if we assume a continuous distribution, say G, that satisfies G00  2G02 =ð1 GÞ. See Curtis and Wright (2004) for details and other references. This is not a particularly general condition, however, and one should not expect a single wage to be typical, since we already showed that it is easy to get two.

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Alternative Theories of Wage Dispersion

Figure 2.

Profit as a function of w

Π(w)

w1

w2

w3

w4

w

4. Other models 4.1. Transitory shocks The above model assumes that when a worker and firm meet, the match-specific shock c is kept forever. Suppose now that c is an i.i.d. draw each period in a given match, after the job is accepted. Each period, workers decide whether to come to work or to stay home that day without losing the job.14 Consider the case where K ¼ 2, so c ¼ c1 with probability l and c ¼ c2 with probability 1 l. We continue to assume b+c2oy. All workers have a common reservation wage for accepting a job, say w1. At that wage they will come in only on days when c ¼ c1 . However, a firm may choose to pay w2>w1 to entice workers to come in even on days when they draw c ¼ c2 . Obviously no firm would ever post anything other than w1 or w2, and as always we let y be the fraction posting the latter. The Bellman equations for

14

We assume they do not lose their job if they stay home one day because a threat by firms to fire a worker for not showing up is not credible – the best they could do is replace him after some time with someone identical. This is similar to the efficiency wage model of Shapiro and Stiglitz (1984), e.g. in the sense that workers may shirk and firms may choose to pay them enough so they will not shirk, but here layoffs are not used as a discipline device.

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workers are rU ¼ b þ aw y½W ðw2 Þ U Š þ aw ð1 yÞ½W ðw1 Þ rW ðwÞ ¼ E maxfw c; bg þ d½U W ðwފ;

UŠ

where Emax{w c, b} reflects the fact that, at a given wage, the worker will stay home for realizations of c above w b. It is obvious that w1 c2ob and w2 c2 ¼ b; therefore E maxfw1 c; bg ¼ lðw1 c1 Þ þ ð1 lÞb and E maxfw2 c; bg ¼ w2 ½lc1 þ ð1 lÞc2 Š. The reservation property implies W ðw1 Þ ¼ U. Putting these facts together, we can solve for w1 ¼ b þ

ðr þ dÞc1 þ aw yc2 . r þ d þ aw y

Again notice that the reservation wage has to be more than enough to entice the worker to come in on his best day, as long as y>0, since the worker can always hold out for a job that pays enough to come in on a bad day, which delivers a positive surplus every time he has a good day with c ¼ c1 . For firms we have Pj ¼

af rj ðy wj Þ , rþd

ð17Þ

where rj now is the probability a worker shows up on any given day, given wj: r1 ¼ l and r2 ¼ 1. After inserting the wages, we can show that P2 P1 is equal in sign to TðyÞ ¼ ðr þ dÞ½ð1

lÞðy

bÞ

c2 þ lc1 Š þ ð1

lÞðy

b

c2 Þaw y.

Equilibrium still requires the best response condition (3). Notice that, in contrast to the other models, here we have T 0 >0, and so there is the potential for multiple equilibria, even for a given aw. Proposition 4. For all aw, we have the following: y ¼ 0 is the unique solution to (3) if yoy; y ¼ 1 is the unique solution if y4y; and there are three solutions y ¼ 0, y ¼ 1 and yA(0, 1) if yoyoy where y ¼ b þ c2 þ

ðr þ dÞlðc2 c1 Þ aw lðc2 c1 Þ and y ¼ y þ . ðr þ d þ aw Þð1 lÞ ðr þ d þ aw Þð1 lÞ

Proof. Similar to earlier results. It is easy to generalize the analysis to any K>2 and to verify that the law of two wages holds. Suppose that more than one wage is posted in equilibrium. The lowest possible posted wage w1 is the reservation wage. Every other posted wj will be equal to cj+b for some j. To see this, consider a firm posting wA(b+cj, b+cj+1). It would face the same probability of workers showing up on any given day by posting b+cj. Hence, all wages posted must equal b+cj

Alternative Theories of Wage Dispersion

77

for some j, conditional on exceeding the reservation wage w1. At the reservation wage w1, however, this argument breaks down since if w1>b+c1 we cannot lower it and stay in business; as we saw above, with K ¼ 2, w1>b+c1 when y>0. Having clarified the possible structure of equilibrium wages, consider the possibility P that more than two are posted. Then Pj is still given by (17), but now rj ¼ jh¼1 lh . The point is that Pj is a function of wj and exogenous parameters. Hence, we have the same problem as before: Pi ¼ Pj ¼ Pk constitutes two equations in one unknown. The law of two wages again holds. 4.2. A crime model In Sections 2 and 3, firms paying higher wages recruit at a faster rate. In Burdett–Mortensen, high-wage firms recruit faster and additionally lose workers more slowly. Here we present a model where they lose workers more slowly. Following Burdett et al. (2003), we interpret this as a model of crime. Thus, any employed worker randomly comes across an opportunity to commit crime at rate m, with gross reward R. There is a probability n of getting caught, which means having to leave one’s job and being forced into unemployment. More generally, in Burdett et al., when a worker is caught he is put in jail for a while, which means he obviously cannot keep his job. For simplicity here we assume jail time is 0, but still assume that a worker who gets caught loses his job, since this is what matters for the purpose of generating wage dispersion.15 Workers are ex ante homogeneous, and have a common reservation wage w0. Firms can hire any worker they contact by posting w0. However, a plausible alternative is to pay a wage above w0 to induce a worker to refrain from crime. Firms may find this profitable since, after all, they suffer a capital loss when workers leave. To see how it works, let w1>w0 denote the crime wage at which a worker would refrain from crime rather than risk losing his job, defined by R þ n½U W ðw1 ފ ¼ 0. It is clear that in equilibrium no firm would post anything other than w0 or w1. As above, let y be the fraction of firms posting the higher wage. The Bellman equations for workers are rU ¼ b þ aw y½W ðw1 Þ

rW ðw0 Þ ¼ w0 þ mR rW ðw1 Þ ¼ w1 þ d½U

15

UŠ W ðw1 ފ:

As we mentioned above, in some efficiency wage models firms are supposed to punish workers who get caught engaging in bad behavior by laying them off, even though they have incentive not to lay them off ex post. Having a third party (e.g. the police) take workers away gets around this problem. Also note that for simplicity here we assume the unemployed do not engage in crime, but this is easily generalized.

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Although they accept w0, they get no surplus. Using R þ n½U W ðw0 Þ ¼ U we can solve for

W ðw1 ފ ¼ 0 and

w1 ¼ b þ ðr þ d þ ayÞR=n w0 ¼ b

mR þ ayR=n:

All firms recruit at the same rate af, but those paying w0 lose workers at rate d+mn while those paying w1 lose workers at rate d. Hence, af ðy w0 Þ r þ d þ mn af ðy w1 Þ : P1 ¼ rþd P0 ¼

Following the usual procedure, P1 P0 is proportional to TðyÞ ¼ mnðy

bÞ

ðr þ dÞ2 R=n

mð2r þ 2d þ aw yÞR.

ð18Þ

For any aw, (3) is again an equilibrium condition. There is a unique solution to (3) and y¼

mnðy

bÞ

ðr þ dÞ2 R=n þ 2mðr þ dÞR maw R

is in (0, 1) iff y is in ðy; yÞ where ðr þ dÞ2 R=n þ 2mðr þ dÞR mn y ¼ y þ aw R=m: y¼bþ

Generalizing this model, suppose crime opportunities have potentially K different payoffs: R ¼ Rj with probability mj, for j ¼ 1; ::: ; K. We can also allow the probability of getting caught and hence losing one’s job nj vary with the opportunity. There will be K critical wages at which workers are just indifferent for some j, Rj ¼ nj ½W ðwj Þ UŠ as well as a reservation wage W ðw0 Þ ¼ U. If necessary, we can order the labels so that w1 ow2 o:::owK , and without loss of generality we consider the case where w0ow1 (otherwise we can ignore opportunities for low j). The generalized worker payoffs are rU ¼ b þ aw

K P

j¼0

yj ½W ðwj Þ

rW ðwÞ ¼ w þ d½U

UŠ

W ðwފ þ

K P

i¼jþ1


mi Ri þ ni ½U

 W ðwފ :

Alternative Theories of Wage Dispersion

79

Profit from posting any wj is Pj ¼

af ðy wj Þ , r þ d þ xj

P where xj ¼ K i¼jþ1 mi ni is the probability a worker is forced to leave. We can have Pi ¼ Pj for i6¼j since the higher profit per worker that comes with a lower wage could be offset by a higher rate at which a firm loses workers, as we saw when K ¼ 2. But generically, there is no way to have Pi ¼ Pj ¼ Pk for distinct i, j and k, since the crime wages wi, wj and wk are related through Rj ¼ nj ½W ðwj Þ UŠ. The law of two wages again holds. It seems to be a fairly robust feature of models with ex post heterogeneity. 5. Ex ante and ex post combined We now assume that there are match-specific shocks and that individuals differ permanently in their valuation of unemployment. Let K1 denotes the number of types bi 2 fb1 ; ::: ; bK 1 g and K2 the number of match-specific PKshocks b 1 cP j 2 fc1 ; ::: ; cK 2 g, arranged such that cioci+1 and bjobj+1. Also, i¼1 li ¼ K2 c b c j¼1 lj ¼ 1, where li is the fraction of individuals with b ¼ bi and lj is the probability that any individual draws c ¼ cj . By the usual argument, there will be at most K ¼ K 1 K 2 different reservation wages, and so at most K different posted wages. Let Ui be the value function for an unemployed type i worker and Wij(w) the value function for a type i worker employed at w with c ¼ cj . The Bellman equations are rU i ¼ bi þ aw rW ij ðwÞ ¼ w

K2 X

lcj

j¼1

cj þ d½U i

K X k¼1

yk maxfW ij ðwk Þ

U i ; 0g

W ij ðwފ.

All wages are set so as to satisfy W ij ðwij Þ

Ui ¼ 0

for some pair (i, j).16 Profit is given by Pij ¼

af rij ðy wij Þ rþd

P where we now define rij ¼ z;k2Fij lbz lck and Fij ¼ fz; kjW zk ðwij Þ can now prove the generalized law of two wages. 16

U z X0g. We

The number of reservation wages could be strictly less than K, since it may be that wij ¼ whk for (i, j)6¼(h,k), but this would be a nongeneric case.

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Proposition 5. For generic parameter values, we can have yk>0 for at most 2K1 values of k. Proof. Fix the worker type z and suppose that three of his reservation wages are posted, say wzi, wzj and wzk, where these each satisfy W zt ðwzt Þ

Uz ¼ 0

for t ¼ i; j; k. Then Pzi ¼ Pzj ¼ Pzk . The reservation property implies wzj ¼ cj þ rU z and so the equal profit condition implies gzi ðU z Þ ¼ gzj ðU z Þ ¼ gzk ðU z Þ where gzj ðU z Þ ¼ rzj ðy

cj

rU z Þ.

Once again, this is a system of two equations in which one is unknown, Uz, which cannot generically be satisfied. Therefore, at most two different wages are posted for any given z, and hence the maximum number is 2K1. Combining ex ante and ex post heterogeneity is interesting for the following reason. Having ex ante heterogeneity with K1 types delivers a potentially rich wage distribution, but with only ex ante heterogeneity the equilibrium is not robust – it unravels with any e>0 search costs. Having ex post heterogeneity is robust, but it delivers limited dispersion because even with K2 values of the shock, we can have at most two posted wages. Combining ex ante and ex post heterogeneity remedies the shortcomings of both models – the resulting equilibria are robust, and the model can deliver more interesting distributions, with up to 2K1 wages. 6. Conclusion We have studied some alternative models of the labor market in which the law of one price does not hold. With ex ante heterogeneity, we can generate wage dispersion, but the equilibrium is not robust to the introduction of search costs. With ex post heterogeneity, equilibrium is robust, but dispersion is repressed by the law of two prices. A combination of the approaches generates robust equilibrium and is not subject to the law of two wages. Acknowledgments We thank Dale Mortensen, Jim Albrecht, Ken Burdett, Guillaume Rocheteau, Rob Shimer, Chris Waller and Ken Wolpin for input. We also thank the Federal Reserve Bank of Cleveland, ERMES at Paris II and the National Science Foundation for research support. The views expressed in this paper are those of the authors and not necessarily those of the IMF.

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References Albrecht, J. and B. Axell (1984), ‘‘An equilibrium model of search unemployment’’, Journal of Political Economy, Vol. 92, pp. 824–840. Albrecht, J. and B. Jovanovic (1986), ‘‘The efficiency of search under competition and monopsony’’, Journal of Political Economy, Vol. 94, pp. 1246–1257. Albrecht, J. and S. Vroman (1992), ‘‘Non-existence of single-wage equilibria in search models with adverse selection’’, Review of Economic Studies, Vol. 59, pp. 617–624. Albrecht, J. and S. Vroman (1998), ‘‘Nash equilibrium efficiency wage distributions’’, International Economic Review, Vol. 39, pp. 183–203. Albrecht, J. and S. Vroman (2005), ‘‘Equilibrium search with time-varying unemployment benefits’’, Economic Journal, Vol. 115, pp. 627–644. Burdett, K. and K.L. Judd (1983), ‘‘Equilibrium price dispersion’’, Econometrica, Vol. 51, pp. 955–970. Burdett, K., R. Lagos and R. Wright (2003), ‘‘Crime, inequality, and unemployment’’, American Economic Review, Vol. 93, pp. 1764–1777. Burdett, K. and D.T. Mortensen (1998), ‘‘Wage differentials, employer size, and unemployment’’, International Economic Review, Vol. 39, pp. 257–273. Burdett, K. and R. Wright (1993), ‘‘Search, matching and unions’’, pp. 411–426 in: H. Bunzel, P. Jensen and N. Westergard-Nielsen, editors, Panel Data and Labor Market Dynamics, Amsterdam: North-Holland. Burdett, K. and R. Wright (1998), ‘‘Two-sided search with nontransferable utility’’, Review of Economic Dynamics, Vol. 1, pp. 220–245. Curtis, E. and R. Wright (2004), ‘‘Price setting, price dispersion, and the value of money: or, the law of two prices’’, Journal of Monetary Economics, Vol. 51, pp. 1599–1621. Diamond, P.A. (1971), ‘‘A model of price adjustment’’, Journal of Economic Theory, Vol. 3, pp. 156–168. Diamond, P.A. (1981), ‘‘Mobility costs, frictional unemployment, and efficiency’’, Journal of Political Economy, Vol. 89, pp. 798–812. Diamond, P.A. (1982), ‘‘Aggregate demand management in search equilibrium’’, Journal of Political Economy, Vol. 90, pp. 881–894. Diamond, P.A. (1987), ‘‘Consumer differences and prices in a search model’’, Quarterly Journal of Economics, Vol. 102, pp. 429–436. Eckstein, Z. and G. van den Berg (2005), ‘‘Empirical labor search: a survey’’, Journal of Econometrics forthcoming. Eckstein, Z. and K.I. Wolpin (1990), ‘‘Estimating a market equilibrium search model from panel data on individuals’’, Econometrica, Vol. 58, pp. 783–808. Jafarey, S. and A. Masters (2003), ‘‘Output, prices and the velocity of money in search equilibrium’’, Journal of Money, Credit and Banking, Vol. 35, pp. 871–888.

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Lagos, R. (2000), ‘‘An alternative approach to search frictions’’, Journal of Political Economy, Vol. 108, pp. 851–873. MacMinn, R.D. (1980), ‘‘Job search and the labor dropout problem reconsidered’’, Quarterly Journal of Economics, Vol. 95, pp. 69–87. Masters, A.M. (1999), ‘‘Wage posting in two-sided search and the minimum wage’’, International Economic Review, Vol. 40, pp. 809–826. Mortensen, D.T. (2000), ‘‘Equilibrium unemployment with wage posting: Burdett–Mortensen meets Pissarides’’, in: H. Bunzel, B.J. Christensen, N.M. Kiefer and D.T. Mortensen, editors, Panel Data and Structural Labor Market Models, Amsterdam: Elsevier. Mortensen, D.T. (2003), Wage Dispersion, Cambridge, MA: Zeuthen Lecture Book Series. Mortensen, D.T. and C.A. Pissarides (1994), ‘‘Job creation and job destruction in the theory of unemployment’’, Review of Economic Studies, Vol. 61, pp. 397–415. Pissarides, C.A. (2000), Equilibrium Unemployment Theory, 2nd edition, Cambridge, MA: MIT Press. Postel-Vinay, F. and J.-M. Robin (2002), ‘‘Equilibrium wage dispersion with worker and employer heterogeneity’’, Econometrica, Vol. 70, pp. 2295–2350. Rob, R. (1985), ‘‘Equilibrium price distributions’’, Review of Economic Studies, Vol. 52, pp. 487–504. Shapiro, C. and J.E. Stiglitz (1984), ‘‘Equilibrium unemployment as a worker discipline device’’, American Economic Review, Vol. 74, pp. 433–444. van den Berg, G.J. and G. Ridder (1998), ‘‘An empirical equilibrium search of the labor market’’, Econometrica, Vol. 66, pp. 1183–1221.

CHAPTER 4

Wage Differentials, Discrimination and Efficiency Shouyong Shi Abstract In this paper I analyze a large labor market where workers are heterogeneous in productivity and where (homogeneous) firms post wages and a ranking of workers to direct workers’ search. I show that there is a unique equilibrium, which has the following properties. First, the wage differential is reversely related to productivity when the productivity differential is small. Second, as the productivity differential decreases to zero, the reverse wage differential increases, and so it remains strictly positive in the limit. Third, high-productivity workers are not discriminated against even when they have lower wage, because they always receive higher employment priority and higher expected wage. Fourth, the equilibrium is socially efficient, and so the reverse wage differential is part of the efficient mechanism. Finally, I provide numerical examples to illustrate the wage distribution.

Keywords: search, wage differential, discrimination JEL Classifications: J3, J6, J7 1. Introduction Standard economic theories view wage differentials as a compensation for workers’ human capital or productivity. These theories have encountered great difficulties in explaining the large wage differentials in the US data. For example, Juhn et al. (1993) have found that all the observable characteristics of workers’ productivity, such as education, experience and age, can explain only one-third of the differential between the 90th and the 10th percentile of the wage distribution between 1963 and 1989. On the other hand, wage differentials seem to CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75004-5

83

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

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Shouyong Shi

depend statistically on seemingly irrelevant background characteristics, such as workers’ race, gender and height. To allow this anomalous dependence, the standard theory attributes it to discrimination.1 Given these difficulties, it is useful to explore alternative theories of the labor market. In a seminal paper, Mortensen (1982) provided one such alternative that emphasized search frictions in the labor market. He characterized an efficient compensation scheme in a class of frictional markets and showed that prices (or wages) may depart significantly from Walrasian prices in order to achieve efficiency. Now, there is a large and still growing literature that explores the importance of search frictions in the labor market (see Mortensen, 2002, for reference). The current paper follows this line of research. The main purpose is to illustrate that wage differentials are sometimes a bad indicator of productivity differentials and discrimination. Instead, the difference in expected lifetime earnings is a much better indicator. I will establish the following results. First, a wage differential can be reversely related to the productivity differential and the size of this reverse wage differential increases when workers become more and more similar in productivity. Thus, even when the productivity differential approaches zero, a wage differential still exists in equilibrium. Second, in contrast to the standard interpretation, the reverse wage differential is not discrimination. On the contrary, higher productivity is always rewarded with a higher expected wage, which takes workers’ employment probability into account. Third, the equilibrium is socially efficient, and so the differentials in wages and employment probabilities are part of the efficient mechanism. The model is one with directed search. It can be best described for the case where there are only two types of workers. A worker’s type is determined by one observable skill, which I call productivity. The difference in productivity among workers can be very small. All firms are identical. They simultaneously post wages and ranking schemes for the workers. Each firm can post different wages for different types of workers, but is restricted to give identical workers the same ranking and the same wage. After observing firms’ announcements, workers decide which firm to apply and they cannot coordinate their applications. After receiving the applicants, each firm selects one worker according to the ranking scheme and pays the posted wage. Search is directed because, when choosing the wage and the ranking scheme, each firm takes into account the effect of its

1

See Altonji and Blank (1999) for a survey of the facts and the literature on discrimination in the labor market. A popular spin-off is the theory of statistical discrimination. It argues that when firms are uncertain about workers’ fundamental characteristics, discrimination can be an equilibrium outcome, either because the characteristics on which discrimination is based are correlated with workers’ fundamental characteristics, or because discrimination leads to self-fulfilling separation of worker types.

Wage Differentials, Discrimination and Efficiency

85

announcements on the matching probability. However, because strategies do not depend on agents’ identities; strategies here are still anonymous. I show that there is a unique symmetric equilibrium where identical workers use the same application strategies. Every firm attracts both types of workers with positive probability, and so separation of the two types is not an equilibrium. Moreover, every firm gives high-productivity workers the priority in employment whenever the firm receives both types of workers. However, when the productivity differential is small, this employment advantage comes with a lower wage. The reverse wage differential arises from the trade-off between the employment probability and wage. Workers care about the expected wage, which is the employment probability times the wage. By giving high-productivity workers a higher ranking, a firm can lower the actual wage by a discrete amount for these workers and yet still be able to attract them. The combination of a high ranking and low wage is optimal for a firm, because it enables the firm to increase the utilization of high-productivity workers. The combination is also attractive to high-productivity workers, provided that the combination yields higher expected wage. Indeed, the expected wage is always higher for high-productivity workers than for low-productivity workers. Despite the non-standard features of actual wages, the equilibrium is socially efficient in the following sense: If a social planner tries to maximize expected aggregate output under the constraint of the same matching function as the one generated in the equilibrium, then the planner will choose the same allocation between workers and firms as in the equilibrium. The efficiency result is in accordance with Mortensen’s (1982) general result on efficiency, in the sense that each worker’s expected wage in the current model takes into account the expected crowding-out that the particular worker creates on other workers (see also Hosios, 1990). Note that efficiency entails both the ranking of workers and the wage differential. I extend the model to a market where there are many types of workers, characterize the equilibrium and use numerical examples to illustrate the equilibrium wage distribution. The numerical examples show that even a minuscule difference in productivity can lead to large differences in wage and employment probability. It would be misleading to interpret the wage differential here as an obvious consequence of the trade-off between wage and employment probability. This interpretation would miss two important points of the analysis. First, the wage differential does not always exist in directed search models with homogeneous workers (cited below). Because an equilibrium with zero wage differential also exists there, it is not clear why the equilibrium with a positive wage differential should be selected. Here, I regard an economy with homogeneous workers as the limit of economies with heterogeneous workers. Because the equilibrium is unique when the two types of workers differ in productivity, and because the wage differential remains positive even when this difference in productivity

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approaches zero, the process selects the equilibrium with a positive wage differential in the economy with homogeneous workers. Second, the intuitive trade-off between wage and employment probability is not able to predict the direction in which a wage differential should exist. When workers’ type does not affect their productivity and preferences, the equilibrium can give either type of workers a high wage and a low employment probability. In contrast, I show that only one direction of the wage differential is consistent with the described process of equilibrium selection. This paper belongs to the search literature (see Mortensen, 2002). Most of the papers in this literature assume that search is not directed. For models of directed search see Peters (1991), Moen (1997), Acemoglu and Shimer (1999), Burdett et al. (2001) Julien et al. (2000) and Shi (2001, 2002). These models have either homogeneous agents on both sides of the market, or heterogeneous agents on both sides of the market who are complementary with each other in production. The model in the current paper lies somewhere in between–it has heterogeneous workers and identical firms.2 Search models are used to examine discrimination by Black (1995) and Bowlus and Eckstein (2002). They show that if some firms have prejudice against a subset of workers, then the search cost will support a wage differential in the steady state. In contrast to my model, search in these models is not directed. More importantly, my model does not rely on the exogenous prejudice to generate a large wage differential among similar workers. On the issue of discrimination, the model most closely related to mine is Lang et al. (2002). An important difference is that they restrict each firm to post only one wage for all workers whom the firm tries to attract; i.e., the firm is not allowed to post different wages for different workers. As a result, the two types of workers are completely separated in their model. I show that such separation is neither an equilibrium nor efficient when each firm can post a different wage for each type of workers. I will organize the paper as follows: in Section 2, I will describe the simple model with two types of workers and propose a candidate equilibrium. Section 3 will show that the candidate equilibrium is indeed the unique equilibrium. In Section 4, I will examine the properties of the equilibrium and show that the equilibrium is socially efficient. Section 5 will extend the model to incorporate many types of workers. I will then conclude in Section 6.

2

Shimer (1997) constructs a model similar to mine with two types of workers. Although our results overlap to some extent, his focus is to contrast the effects of different mechanics of wage determination on the division of the match surplus. In particular, he does not emphasize the reverse wage differential that can arise in the directed search environment.

Wage Differentials, Discrimination and Efficiency

87

2. The model 2.1. Workers and firms Consider a labor market with a large number of workers, N. There are two types of workers, types T (for ‘‘tall’’) and S (for ‘‘short’’). Type T workers are a fraction g of all workers and type S a fraction (1 g), where g A (0,1). Sometimes, I use the notation gT ¼ g and gS ¼ 1 g. A worker’s type is observed by the firm immediately upon applying to a job. A type S worker produces y units of output and a type T worker produces (1+d)y, where d>0. In a large part of the analysis in this paper, I will restrict d to be sufficiently small. The purpose is to examine whether a small productivity difference can generate a large wage differential. There are also a large number of firms, M, all of which are identical. For the moment, this number is fixed. Competitive entry of firms can be introduced easily and will be briefly discussed at the end of Section 5.1. Each firm wants to hire only one worker. Denote the tightness of the market as y ¼ N=M. For simplicity, assume that the economy lasts for one period. The recruiting game is as follows. First, all firms make their announcements simultaneously. Each firm i announces two wages, wiT for type T workers and wiS for types S workers, together with a rule that ranks the two types of applicants. Let Ri 2 f1; 0; Fg denote this ranking or priority rule, where F ¼ ½0; 1Š. The firm selects type T workers first if it sets Ri ¼ 1 and type S workers first if it sets Ri ¼ 0; If Ri ¼ F, the firm is indifferent between the two types of workers.3 Once a firm posts the wages and the selection rule, it is committed to them. All workers observe all announcements and then decide which firm to apply to. This application decision can possibly be mixed strategies over the jobs. Let aij denote the probability with which a type j worker applies to firm i. After receiving the applicants, a firm selects a worker according to the announced ranking and pays the corresponding wage. The worker produces immediately, obtains the wage and the game ends. Note that a worker’s strategy does not depend on the worker’s identity. Thus, all workers of the same type must use the same strategy. This symmetry requirement on the workers’ side reflects the realistic feature of the labor market that workers cannot coordinate their application decisions.4 However, the coordination failure would be eliminated if firms could identify each worker and make

3

The ranking scheme is included in a firm’s announcement to ease the description. However, firms do not have to post the ranking literally. For the model to work, all that is needed is that workers expect the firms to use the ranking scheme to select workers after workers apply. This expectation will be fulfilled because the ranking is optimal for the firms, ex post. 4 The set of asymmetric equilibria is large. In a model with two identical agents on each side of the market, Burdett et al. (2001) have shown that there is a continuum of asymmetric equilibria while there is a unique symmetric equilibrium.

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an offer specifically to that worker. To preserve the coordination failure, I assume that each firm’s offer and, in particular, the ranking rule should not depend on the identities of the workers whom the firm receives. In this sense, strategies are anonymous. However, a firm’s strategy is allowed to depend on the firm’s identity. This allowance is necessary for examining the possibility of a separating equilibrium where two groups of firms each attract a distinct type of workers. I will focus on the limit of the economy where N and M approach infinity while their ratio, y, lies in the interior of (0, N). The equilibrium in this limit is significantly easier to characterize than the finite economy, because a single firm’s deviation does not affect workers’ payoff from applying to other firms. More precisely, let pij be the probability with which a type j worker who applies to a firm i gets the job. Define the ‘‘market wage’’ of a type j worker as follows: ðpi0 j wi0 j Þ. E j ¼ max 0

ð1Þ

i ai

In the limit described above, the effect of firm i’s strategy on pi0 j approaches zero (see Burdett et al., 2001). Thus, each firm takes Ej as given. A worker maximizes the expected wage that he can obtain from applying to a job. Given Ej, a type j worker’s strategy is to choose aij ¼ 1 if pij wij 4E j , aij ¼ 0 if pij wij oE j , and aij 2 ð0; 1Þ if pij wij ¼ E j . In the limit economy, it is convenient to express this strategy with a new variable qij ¼ gj Naij . Then, 8 if pij wij 4E j > <¼ 1 if pij wij oE j ð2Þ qij ¼ 0 > : 2 ð0; 1Þ; if p w ¼ E : j ij ij Since the sum of aij over i is one, then qij must satisfy M 1 X q ¼ gj y; M i¼1 ij

j ¼ T; S.

ð3Þ

When all type j workers use the strategy qij, the expected number of type j applicants received by firm i is gj Naij ¼ qij . For this reason, I call qij the queue length of type j workers for firm i. Despite the coincidence between a worker’s strategy and the queue length, it would be incorrect to interpret an individual choice of q as the worker’s ability to influence other workers’ or firms’ decisions. When qij>0, I say that the firm i attracts type j workers. In the limit where the economy becomes infinitely large, the probability with which firm i attracts one or more type j workers is 1 ð1 gj aij ÞN ! 1 e qij . If firm i gives type j workers the selection priority, then the employment probability of a type j worker at the firm is 1

ð1 gaij ÞN 1 e ! qij Ngaij

qij

 Gðqij Þ.

Wage Differentials, Discrimination and Efficiency

89

0

On the other hand, if firm i gives the other type j aj the priority, then the firm will consider a type j worker only if the firm receives no type j 0 applicants. This event occurs with probability e qij0 , in which case each type j worker applying to the firm is chosen with probability G(qij). Thus, for a general priority rule Ri, a worker’s employment probability at firm i is piT piS

¼ ¼

½Ri þ ð1 Ri Þe qiS ŠGðqiT Þ; ½1 Ri þ Ri e qiT ŠGðqiS Þ:

ð4Þ

Note that the function G is continuous and decreasing. Thus, when there are slightly more workers applying to a firm, each applicant is chosen by the firm with a slightly lower probability. Now, consider a firm i’s choices of (wiT, wiS, Ri). The firm’s expected profit is pi ¼ ð1 e qiT Þ½Ri þ ð1 Ri Þe qiS Š½ð1 þ dÞy wiT Š þ ð1 e qiS Þ½1 Ri þ Ri e qiT Šðy wiS Þ.

ð5Þ

The firm’s optimal choices solve the following problem: max pi

subject to ð2Þ for j ¼ T; S.

The constraint (2) reflects the fact that the firm takes into account the effect of its choices on workers’ decisions. Finally, the firm’s ranking rule must be optimal, ex post. That is, the following condition must hold 8 if wiT wiS ody > <1 Ri ¼ 0 if wiT wiS 4dy ð6Þ > : F; if wiT wiS ¼ dy:

A (symmetric) equilibrium consists of firms’ strategies ðwiT ; wiS ; Ri ÞM i¼1 ; workers’ strategies ðqiT ; qiS ÞM and the numbers (E ,E ) such that the following requireT S i¼1 ments are met: (i) given the firms’ strategies and the numbers (ET, ES), each worker’s strategy is given by (2); (ii) given the numbers (ET, ES) and anticipating workers’ responses, each firm’s strategy is optimal and the ranking is ex post optimal; and (iii) the numbers (ET,ES) obey (1). 2.2. A candidate equilibrium Let me construct a candidate equilibrium now and show later that it is the only equilibrium. In this equilibrium, every firm ranks type T workers first and every firm attracts both types of workers. That is, Ri ¼ 1, qiT>0 and qiS>0 for all i. Then, a firm’s maximization problem simplifies to ðPÞ maxðwiT ;wiS Þ pi ¼ ð1

þe

e

qiT

qiT

ð1

Þ½ð1 þ dÞy e

qiS

Þðy

wiT Š

wiS Þ,

ð7Þ

subject to G(qiT)wiTZET and e qiT GðqiS ÞwiS  E S . These constraints are necessary for qiT>0 and qiS>0 (see (2)), which are stipulated for the candidate

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equilibrium. Note that both constraints must hold with equality. If one were to hold with strict inequality ‘‘>’’, then qi ! 1 and Gðqi Þ ! 0 which would violate the corresponding constraint. Using the constraints to substitute (wiT, wiS), I can write the firm’s expected profit as follows: e

p ¼ ð1

qiT

Þð1 þ dÞy þ e

qiT

e

ð1

qiS

Þy

The first-order conditions for (qiT, qiS) yield ye dye

ðqiT þqiS Þ qiT

ðqiT E T þ qiS E S Þ

ð8Þ

¼ ES ,

ð9Þ

þ ES ¼ ET .

ð10Þ

Since the solution to these equations does not depend on the firm’s index i, all firms use the same strategy in the candidate equilibrium. In this case, aiT ¼ aiS ¼ 1=M, which implies the following queue lengths: qT ¼ gy;

qS ¼ ð1

ð11Þ

gÞy.

Finally, the first-order conditions and the constraints in problem (P) become E S ¼ ye

y

E T ¼ E s þ dye wT ¼ y

ð12Þ gy

gy½d þ e ð1 egy 1

¼ ye y ½1 þ deð1 gÞy

Š

;

wS ¼ y

gÞy

Š,

ð1 gÞy . eð1 gÞy 1

ð13Þ ð14Þ

Proposition 1. The market has a unique (symmetric) equilibrium, which satisfies: (E1) All firms have the same strategy and rank type T workers above type S workers (i.e., R ¼ 1); (E2) Each firm posts wT for type T workers and wS for type S workers, as given by (14); (E3) Each worker applies to every firm with the same probability, which yields the queue lengths (qT,qS) in (11) for every firm; and (E4) Workers’ expected wages satisfy (12) and (13). Let me discuss (8), (12) and (13), which will be useful later. Because all firms use the same strategy, I will suppress the firm’s index i in this discussion. First, the condition (8) says that a firm’s expected profit is equal to the difference between expected output and expected wage cost. Note that, although a firm hires only one worker, the expected wage cost on type j workers is qjEj, as if the firm hires a number qj of such workers at a wage rate Ej. Second, the conditions (9) and (10) state that a worker’s expected contribution to output, after subtracting the amount of other workers’ expected output crowded out by this worker, is equal to the worker’s expected market wage. Since adding a type S worker contributes to a firm’s output only when the firm did not receive any other applicant, which occurs with probability e ðqT þqS Þ , a type S worker’s contribution to expected output is ye ðqT þqS Þ . Similarly, a type T worker contributes to a firm’s output by an amount (1+d)y if the firm did not receive

Wage Differentials, Discrimination and Efficiency

91

any other applicant, by dy if the firm received some type S applicants but no type T applicant, and by nothing if the firm received other type T applicants. Since the first case occurs with probability e ðqT þqS Þ and the second case occurs with probability e qT ð1 e qS Þ then the expected contribution of a type T worker to output is e qT ðdy þ ye qS Þ. Under (9), this equals ðE S þ dye qT Þ as (10) states. The equilibrium in Proposition 1 features complete mixing of the two types of workers, in the sense that every firm attracts both types of applicants. However, after workers’ mixed strategy is played out, a firm may or may not receive both types of applicants. If the firm does receive both types of workers, it chooses a high-productivity worker. If the firm receives only low-productivity workers, it selects one of them. Note that the wages in (14) satisfy wT wS ody. That is, the ranking R ¼ 1 is optimal for the firms, ex post. To verify that the strategies described in Proposition 1 indeed constitute an equilibrium, it suffices to show that the following deviations by a single firm are not profitable: D1. A deviation that intends to attract only type T workers. D2. A deviation that intends to attract only type S workers. D3. A deviation that attracts both types of workers but ranks type S workers first. D4. A deviation that has no selection priority. In the next section, I will accomplish this task, and more, by proving that the described equilibrium is the unique (symmetric) equilibrium. 3. The candidate is the unique equilibrium Proposition 1 states that there is no equilibrium other than the one described in the proposition. To establish this result, I need to show that no other possible configuration of strategies forms an equilibrium. The other possibilities are as follows: N1. N2. N3. N4. N5.

Complete separation of the two types of workers; Partial separation of type T workers; Partial separation of type S workers; No priority in a subset or all of the firms; No separation and strict ranking that is different from the one in the equilibrium.

For each possibility, I will use the same procedure to show that it is not an equilibrium. First, supposing that one of the above possibilities is an equilibrium, I will compute the wages posted by each firm, the expected number of applicants of each type whom a firm attracts, and expected wages in the market. Second, I will construct a single firm’s deviation from this supposed equilibrium and toward the one described in Proposition 1. I will show that this deviation is profitable, and so the possibility is not an equilibrium.

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I can take the size of a subset of firms to zero. In this limit, the above possibilities become a single firm’s deviations. In particular, the deviation D1 described before is a limit case of N2, D2 of N3, D3 of N5 and D4 of N4. Thus, by showing each of the above possibilities can be improved upon by making the strategies closer to the ones described in Proposition 1, I also succeed in proving that the deviations D1 through D4 are not profitable against the equilibrium. 3.1. Separation is not an equilibrium Consider first the case of complete separation, i.e., Case N1. Divide the firms into two groups, A and B. Suppose that the firms in group A attract only type T workers and the firms in group B attract only type S workers. This possibility is an equilibrium in a similar model by Lang et al. (2002) who assume that each firm can post only one wage. When each firm can condition the wage on the type of the hired worker, complete separation is no longer an equilibrium. The intuition is as follows. Suppose that complete separation is an equilibrium, as in Case N1. A firm in group A can maintain the same wage for type T workers and rank such workers first as in the supposed equilibrium, but choose a wage to attract type S workers as well. Type T workers will not change their strategy of applying to this firm, and so the expected profit from hiring a type T worker does not change. In the case where no type T worker shows up at the firm, the deviating firm can hire a type S worker and obtain additional profit. Thus, the deviation is profitable, provided that it is feasible and that it attracts type S workers. To verify this intuition, let wA be the wage posted by a group A firm (for type T workers) and wB be the wage posted by a group B firm (for type S workers). Let a be the fraction of firms that are in group A. Then, the expected number of applicants is qA ¼ gy/a for a group A firm and qB ¼ (1 g)y/(1 a) for a group B firm. Let E^ T and E^ S be expected wages of these two types of workers, respectively, in the supposed equilibrium of separation. Then, GðqA ÞwA ¼ E^ T ;

GðqB ÞwB ¼ E^ S .

Because the two types of workers are completely separated, there is no crowding-out between them. Thus, the expected wage of each type of workers is equal to the worker’s expected marginal contribution to output. That is,5 E^ T ¼ ð1 þ dÞye

5

qA

;

E^ S ¼ ye

qB

.

ð15Þ

To verify this, consider a type A firm’s deviation to a wage wA for type T workers and suppose that this deviating firm still attracts only type T workers (e.g., the firm sets zero wage for type S workers). d Then, the first-order condition for this deviation yields ð1 þ dÞye qA ¼ E^ T ; where qdA is the expected number of type T workers whom the deviating firm will attract. For this deviation to be not profitable against the supposed equilibrium, we must have qdA ¼ qA ¼ gy=a. Thus, E^ T ¼ ð1 þ dÞye qA . The expression for E^ S can be obtained similarly by considering a group B firm’s deviation.

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Wage Differentials, Discrimination and Efficiency

Thus, the expected profit of a firm in the two groups is, respectively, as follows: pA ¼ ð1 e qA Þ½ð1 þ dÞy wA Š ¼ ð1 þ dÞy½1 ð1 þ qA Þe qA Š, e

pB ¼ ½1

qB

Šðy

wB Þ ¼ y½1

ð1 þ qB Þe

qB

Š.

For a to be in (0,1), a firm must be indifferent between being in the two groups. Thus, pA ¼ pB . This requirement yields qAoqB, i.e., a> g, for all d>0. If d ¼ 0, then a ¼ g. Now, consider the following deviation by a single firm in group A. The firm maintains the wage wA for type T workers and still ranks these workers first. In contrast to the supposed equilibrium, the deviating firm posts wdAS for type S workers and ranks them below type T workers. It is clear that a type T worker will apply to the deviating firm with the same probability as in the supposed equilibrium, and so the expected number of type T workers whom the firm will receive is qA ¼ gy=a. Let qdAS be the expected number of type S workers whom the deviating firm will receive. Then, the probability with which an individual type S worker who applies to the deviating firm will be selected is e qA GðqdAS Þ. Let wdAS and the associated queue length qdAS satisfy the following conditions: d e qA Gðqd Þwd ¼ E^ S ; ye ½qA þqAS Š ¼ E^ S . ð16Þ AS

AS

The first condition requires the deviation to give a type S applicant the same expected wage as in the market, and the second condition requires the deviation to be the best of its kind so that a type S worker’s expected output is equal to the expected wage in the market. The deviation has the following features. First, the deviation indeed attracts type S workers. To see this, substituting E^ S from (15) into the second equation in (16) yields qdAS ¼ qB qA , which is positive as shown earlier. Second, the deviation is feasible in the sense that wdAS oy: To verify this, combine the two equations in (16) to obtain qd wdAS ¼ y qd AS oy. e AS 1 Note that the strict inequality implies that the deviating firm obtains a positive profit from hiring a type S worker when no type T worker shows up at the firm. Finally, the deviator’s ranking of the two types of workers is optimal for the firm, ex post; i.e., the deviation satisfies wA wdAS ody. To verify this, temporarily denote D ¼ qB qA ð40Þ. Substituting ðwdAS ; wA Þ from the above, I can rewrite the compatibility condition as 1 ð1 þ D þ qA Þe ðDþqA Þ 1 ð1 þ DÞe D 4 . 1 e qA 1 e D Because 1 þ DoeD , then ð1 þ D þ qA Þe ðDþqA Þ oe qA þ qA e ðDþqA Þ . Also, because qA o1o 1 De D , then eqA 1 ð1 þ D þ qA Þe ðDþqA Þ qA e D 41 41 1 e qA eq A 1 That is, the required condition holds. 1

D

De 1

e

D

¼

1

ð1 þ DÞe 1 e D

D

.

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Shouyong Shi

Because the deviating firm’s expected profit from type T workers is the same as in the supposed equilibrium and the additional expected profit from type S workers is strictly positive, the deviation is profitable. Thus, complete separation is not an equilibrium. Similarly, partial separation as in Cases N2 and N3 cannot be an equilibrium. A firm that attracts only one type of workers can increase its expected profit by deviating to a strategy that attracts both types of workers. 3.2. Firms rank the two types of workers strictly In all the cases examined so far, firms rank the two types of workers strictly. Now, I show that Case N4, where some or all firms give no priority, is not an equilibrium. Thus, it is optimal for all firms to rank the two types of workers. Again, divide the firms into groups A and B. Suppose, to the contrary, that there is an equilibrium in which one or both groups of firms use no priority. Because complete separation is not an equilibrium, at least one group of firms must attract both types of workers. Without loss of generality, let group A firms attract both types of workers. Let a be the fraction of firms in group A. For each firm in group A, let pA be the firm’s expected profit, wAj the wage for type j workers and qAj the expected number of type j workers for the firm, where j ¼ T; S. Let E^ j be the expected market wage of a type j worker. Because a type A firm does not rank the workers, the queue length of workers for the firm is qAT+qAS, which is temporarily denoted Q. For both types of workers to apply to the firm, the following conditions must hold: GðQÞwAj ¼ E^ j ;

j ¼ T; S.

A group A firm must be indifferent between the two types of workers after both show up at the firm. That is, wAT wAS ¼ dy. The two conditions together solve: E^ T E^ S E^ T E^ S ; wAS ¼ ; GðQÞ ¼ . GðQÞ GðQÞ dy Moreover, ðE^ T ; E^ S ; wAT ; wAS Þ are all positive and the relationship E^ S  ye Q holds.6 A group A firm can profit by deviating to a strategy that ranks type T workers first. Let ðwdAT ; wdAS Þ be the wages for the two types of workers posted wAT ¼

6 To show E^ S 40, suppose to the contrary that E^ S ¼ 0. Then wAS ¼ 0 and pA ¼ ð1 e Q Þy. In this case, a single firm in group A can deviate to wdAS ¼
and wdAT ¼ wAT þ
; where e>0 is a sufficiently small number. It can be shown that this deviation is profitable. Because Q>0, then E^ S 40 implies that ðE^ T ; wAT wAS Þ are all positive. To show E^ S  ye Q , suppose E^ S oye Q , instead. An individual firm in group A firm can deviate to wdAT ¼ 0 and wdAS 40. This deviation attracts only type S workers. d Let wdAS and the associated queue length qdAS satisfy GðqdAS ÞwdAS ¼ E^ S ¼ ye qAS . Then, this deviation can be shown to be profitable.

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Wage Differentials, Discrimination and Efficiency

by the deviating firm and deviation satisfy GðqdAT ÞwdAT ¼ E^ T ; ½qdAT þqdAS Š

qdAT

qdAS Þ

be the corresponding queue lengths. Let the

GðqdAS ÞwdAS ¼ E^ S ,

qdAT

¼ E^ T E^ S . Denote Q ¼ þ Because E^ S  ye Q , then Qd  Q: Because dyGðQÞ ¼ d q E^ T E^ S ; then e AT ¼ GðQÞ: Since e zoG(z) for all z>0, then GðQÞoGðqdAT Þ; which implies Q4qdAT . The deviator’s expected profit is ye

d

pdA ¼ ð1 ¼ ð1

¼ E^ S ;

e

ðqdAT ;

qdAT

e

dye

qdAS :

qdAT

Þð1 þ dÞy

ð1 þ qdAT

dÞy

qdAT E^ T þ ðe ÞE^ T qd E^ S

qdAT

e

Qd

Þy

qdAS E^ S

AS

d The second equality is obtained from substituting e qAT ¼ ðE^ T E^ S Þ=ðdyÞ and d e Q ¼ E^ S =y. The expected profit from not deviating is pA ¼ ð1 e Q Þð1 þ dÞy QE^ T . The gain from the deviation is

pdA

pA ¼ ye

Q

½1 þ d z

dð1 þ qdAT

QÞeðQ

qdAT Þ

The function (1 z)e is decreasing for all z>0. Since pdA pA 40. That is, the deviation is profitable.

ð1 þ Qd

Q4qdAT

QÞeðQ

Qd Þ

Š.

and Q  Qd ;then

3.3. High-productivity workers have the priority Since Cases N1 through N4 are not an equilibrium, all firms in an equilibrium must attract both types of workers and have a strict ranking of the two types. Now I show that the ranking must be R ¼ 1; that is, it is optimal for firms to rank the workers according to productivity. This is done by excluding the possibility N5 as equilibrium. Dividing the firms into groups A and B, the possibility has two subcases: (N5a) group A firms give priority to type S workers and group B to type T workers; (N5b) all firms rank type S workers first. Consider Case N5a first, where group A firms rank type S workers first and group B firms rank type T workers first. In this case, a firm’s expected profit is higher in group B than in group A, and so the case is not an equilibrium. To see this, let (wiT, wiS) be the wages posted by a firm in group i, where i ¼ A; B and (qiT, qiS) be the corresponding queue lengths. Temporarily denote the total expected number of workers attracted by a group i firm as Qi ¼ qiT þ qiS , and the firm’s expected profit as pi. Denote a type j worker’s expected market wages as E^ j . Then, the maximization problem of a firm in each group yields ð1 þ dÞye ye

QB

QA

¼ E^ S ;

¼ E^ T ; dye

dye qBT

qAS

¼ E^ T

¼ E^ T

E^ S ,

E^ S .

These equations imply qBT ¼ qAS . Also, a group B firm attracts more workers than a group A firm. More precisely, with the temporary notation

96

Shouyong Shi

D ¼ QB QB

qBT ð40Þ, I can derive the following result from the above equations:   1 þ deD QA ¼ ln 40. 1þd

After computing a firm’s expected firm in each group, I can obtain pB pA ¼ d½1 ð1 þ qBT Þe qBT Š þ e QB ½ðQA qBT ÞdeðQB qBT Þ ðQB y

QA ފ.

The first term of this difference is positive. So is the second term.7 Thus, pB 4pA . Finally, suppose that case N5b is an equilibrium, where each firm attracts both types of workers and ranks type S workers first. Let the wages posted by each firm be (wT, wS) and the corresponding queue lengths be (qT, qS). Denote Q ¼ qT þ qS and denote a firm’s expected profit as p. Then, ð1 þ dÞye

Q

¼ E^ T ;

p ¼ y þ dy½1 þ qS Še

¼ E^ T

E^ S .

dye

qS

qS

ð1 þ QÞe

Q

Moreover, since all firms attract the same expected number and composition of workers, qT ¼ gy and qS ¼ ð1 gÞy. Consider a deviation by an individual firm to wages ðwdAT ; wdAS Þ that are intended to attract both types of workers and that give type T workers priority. Let ðqdAT ; qdAS Þ be the corresponding queue lengths and let the deviation be the best of its kind (i.e., let it satisfy the corresponding first-order conditions). Let ðqdAT ; qdAS Þ serve the roles of (qAT, qAS) in the above proof for case N5a, and let (qT, qS) serve the roles of (qBT,qBS). Then, the same proof shows that the deviation increases the firm’s expected profit. In addition, it can be shown that the condition, ð1 þ dÞy wdAT 4wdAS ; holds so that the ranking in the deviation is optimal for the firm, ex post.8 Therefore, it is profitable to

7

To verify this, substitute QA to obtain: ðQA

qBT ÞdeðQB

qBT Þ

ðQB

QA Þ ¼ dDeD

ð1 þ deD Þ ln

1 þ deD . 1þd

This expression is an increasing function of D for all D>0 and, at D ¼ 0, it is equal to 0. Thus, the expression is positive for all D>0. 8 To verify this condition, use the first-order conditions to solve the queue lengths induced by the deviation as qdAT ¼ ð1 gÞy and qdAS ¼ gy lnð1 þ d degy Þ. The wages are wdAT ¼ E^ T =GðqdAT Þ and d wdAS ¼ E^ S eqAT =GðqdAS Þ: Then the required condition holds if and only if the following condition holds: 0od

ð1 þ dÞ

qdAS ð1 gÞye gy þ ð1 gÞy ð1 þ dÞðegy e 1

1Þ

.

Using the solution for qAS obtained earlier, I can show that the right-hand side of the above inequality is increasing in d, and so it is greater than its value at d ¼ 0 which is proportional to ½gey egy þ 1 gŠ. The last expression is a concave function of g, and it is equal to 0 at both g ¼ 0 and g ¼ 1. Thus, it is positive for all g 2 ð0; 1Þ.

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deviate from the strategies in N5b to one that ranks type T workers above type S workers. 4. Properties of the equilibrium and the social optimum In this section, I examine the properties of the equilibrium and show that the equilibrium is socially efficient. The following proposition can be readily confirmed from (11) through (14). Proposition 2. The equilibrium described in Proposition 1 has the following properties. (i) A firm’s ex post profit from a type T worker is higher than that from a type S worker. (ii) A type T worker has a higher employment probability than a type S worker. (iii) ET>ES, and (ET ES) is of the same order of magnitude as d. (iv) 9d1 4d0 40 such that wTowS for all dA(0,d0) and wT>wS for all d>d1. (v) Define D ¼ wS =wT 1: 9d2 40 such that, if 0odo minfd0 ; d2 g, then ðdD=ddÞ o0; ðdD=dyÞ40; ðdD=dgÞ40. The properties (i), (ii) and (iii) are intuitive. Property (i) repeats an earlier result that it is optimal for all firms to rank the workers according to productivity. Such ranking gives each high-productivity worker a higher employment probability and higher expected wage than a low-productivity worker. Moreover, high-productivity workers get a higher expected wage, and the differential in the expected wage is of the same order of magnitude as the productivity differential. However, the actual wage is not always higher for high-productivity workers. As stated in property (iv), only when the productivity differential is sufficiently large do high-productivity workers get a higher actual wage than low-productivity workers. When the productivity differential is small, workers with higher productivity get lower wages. This reverse wage differential arises from the ranking of workers. An increase in the ranking increases a worker’s employment probability, and hence the expected wage, by a discrete amount. So, when a firm rewards slightly more productive workers with a higher ranking, it can cut the wage for these workers by a discrete amount and yet still be able to attract them. By doing so, the firm can increase expected profit. Moreover, the size of the reverse wage differential increases as the productivity differential (d) decreases. Although unconventional, this result is quite intuitive. When the productivity advantage of one set of workers shrinks relative to other workers, maintaining the employment priority for these workers is optimal for a firm only if their wage is reduced. An implication of this result is that the reverse wage differential remains strictly positive as the productivity differential approaches zero. However, if the economy literally has d ¼ 0, there is another equilibrium in which all workers are paid the same wage. This equilibrium with a

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uniform wage is not selected as the limit outcome of a sequence of economies in which the productivity differential is positive. The above results suggest that actual wages can sometimes be a bad indicator of workers’ productivity. Thus, the standard practice in labor economics that attributes wage differentials to productivity differentials should be taken with caution. First, a large residual wage differential might be attributed to a statistically insignificant differential in productivity, as it is the case here when d is small. In this case, it is futile trying to explain the wage differential by ever expanding the list of workers’ characteristics. Second, there is nothing abnormal about a residual wage differential; rather, it is part of the equilibrium with fully rational players in a frictional labor market. Also, the residual wage differential is socially efficient, as shown below. Third, when similar workers get different wages, the ones who receive lower wage are not necessarily discriminated against. For example, where the productivity differential is small, the reverse wage differential is just a compensation to low-productivity workers for being ranked low in the selection. Despite the lower wage in this case, high-productivity workers are not discriminated against, because they are ranked the first for the job and they obtain higher expected wage. The wage differential in this model also responds to the market condition in an interesting way. When there is a reverse wage differential as a result of a small productivity differential, an increase in the overall ratio of workers to firms, y, increases the reverse wage differential. The explanation is as follows. When jobs become more scarce, workers value the employment probability more than the wage. Since high-productivity workers are given a higher employment probability through the ranking scheme, they are willing to take a larger wage cut to maintain this difference in the employment probability. I now turn to the efficiency of the equilibrium. Since the equilibrium has unconventional features in wages, it is interesting to see whether the equilibrium is efficient under the constraint of the matching frictions. To examine efficiency, let me take aggregate output as the measure of social welfare. This measure is appropriate here because all agents are risk neutral. Suppose that a fictional social planner tries to maximize aggregate output, subject to the same restrictions that matching frictions generate in the equilibrium. One of these restrictions is that the firms cannot separate two identical workers. This restriction requires that the planner must treat all workers in the same group in the same way in the matching process. Thus, the planner can divide the firms into at most two groups, with each group potentially targeting a different group of workers. Another restriction is that the matching function in each group of workers and firms must be the same as in the equilibrium. To describe the matching function, let the two groups of firms be indexed by i, where i ¼ A, B. Let group i firms be a fraction ai of all firms, where aA+aB ¼ 1. Let qij be the expected number of type j workers for each firm in group i, where j ¼ T, S. Let Ri be the ranking of the workers by a firm in group i, where Ri A {1,0,F}. As in the market equilibrium, the matching function facing the social

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planner is such that, in each group i, a firm receives one or more type j workers with probability ð1 e qij Þ. A planner’s allocation is ðai ; qiT ; qiS ; Ri Þi¼A;B . Expected output of a firm in group i is Ri ½ð1

þ ð1

e

qiT

Þð1 þ dÞy þ e

Ri Þ½ð1

e

qiS

qiT

ð1

Þy þ e

qiS

e

qiS

ð1

ÞyŠ

e

qiT

Þð1 þ dÞyŠ.

Re-arranging terms and weighting each group’s output by the group’s size, I can express expected output per firm in the economy as follows: X ai fy½1 e ðqiT þqiS Þ Š þ dyð1 e qiT Þ½Ri þ ð1 Ri Þe qiS Šg: i¼A;B

The planner chooses ðai ; qiT ; qiS ; Ri Þi¼A;B to maximize this output, subject to qiT  0; qiS  0 and the following (resource) constraints: aA þ aB ¼ 1;

aA ; aB 2 ½0; 1Š;

aA qAj þ aB qBj  gj y; for

j ¼ T; S:

ð17Þ

Here gT ¼ g and gS ¼ 1 g. The first constraint is self-explanatory. The second constraint is the adding-up constraint (3) in the current context. The following proposition holds and a proof is supplied in Appendix A. Proposition 3. The efficient allocation coincides with the equilibrium allocation described in Proposition 1. The reason why the equilibrium is efficient is similar to that in other directed search models, e.g. Moen (1997), Acemoglu and Shimer (1999) and Shi (2001, 2002). In particular, the directed search framework allows the firms to internalize the matching externalities. One way to see this is to recall the expressions for workers’ expected market wages, ET and ES, which are given by (12) and (13). For each type of workers, the expected market wage is equal to the worker’s marginal contribution to expected output which takes into account the worker’s crowding-out on other workers’ expected output. As a result, a worker’s expected wage in the equilibrium is equal to the worker’s social marginal value. To express this equality more formally, note that a type j worker’s social marginal value is equal to the Lagrangian multiplier of (17) in the planner’s problem, denoted lj. Then, it can be verified that lj ¼ E j , for both j ¼ T and S. Because expected wages take into account each worker’s crowding-out on other workers, they share the property of the efficient mechanisms described by Mortensen (1982, pp. 968–969) and Hosios (1990). Let me make a few remarks on the efficiency result. First, it is expected wages, not actual wages, which serve the role of the efficient compensation scheme. To achieve efficiency, the market mechanism requires the use of not only prices (wages) but also the ranking of the workers. Second, separating the two types of workers in the matching process, completely or partially, is inefficient. Third,

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because the equilibrium is the only equilibrium, the wage differential (some times a reverse one) and the ranking are efficient.9 Now, it is useful to contrast the results in my model with those in Lang et al. (2002). As mentioned before, the main difference is that Lang et al. restricts each firm to post only one wage. This restriction generates several important differences in results. First, the complete separation is an equilibrium in Lang et al. but not in my model. Second, the workers who have a lower employment probability in my model can get a higher wage than other workers when the productivity differential is small. This does not occur in Lang et al. Third, the equilibrium in my model is efficient, but not in Lang et al. In particular, the separation of the two types of workers, the wage differential and the employment difference between workers in the model of Lang et al. are all inefficient. 5. Extension to many types of workers In this section, I extend the model to incorporate many types of workers. One purpose of this analysis is to check whether the results in the simple model are robust. The other purpose is to examine how wages vary when the workers’ productivity varies in a wide range. 5.1. The equilibrium and its properties Let there be j  2 types of workers, which are indexed by j ¼ 1; 2; . . . ; j. The productivity of a type j worker is yj and each worker is of type j with probability gj, where y1 4y2 4


4yj and SJj¼1 gj ¼ 1. As before, I examine only the symmetric equilibrium, where all workers of the same type use the same strategy. In the presence of a large number of types of workers, it is difficult to establish the uniqueness of the equilibrium. Thus, I will only establish the existence of an equilibrium resembling the one in the simple model, i.e., an equilibrium in which all firms attract all types of workers and rank the workers according to productivity. To examine this equilibrium, I need to examine deviations that do not attract all types of workers or do not rank the workers according to productivity. To do so, I introduce the following general notation. Use the notation i  i0 to mean that a firm ranks type i workers above type i0 workers in the hiring process. For any integer K with 1  K  j, I use K to stand for both the number and the ordered set f1; 2; . . . ; Kg, where the ordering is 1  2 


 K. As in the simple model, it is not optimal for a firm to give the same ranking to two types of workers. Thus, I will restrict attention to strategies, including deviations, that have strict ranking over the types of workers. A strategy is defined by three characteristics: a strictly ordered set of types C, a vector of wages w and the

9

Although I have not examined entry by firms, it can be done easily (see the end of Section 5.1).

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10

associated vector of queue lengths q. The set C contains the types of workers whom a firm attracts and the ordering of these types. It is useful to express C as C ¼ ðj k ÞK j k 2 J; K  J, k¼1 ; so that the ranking in C is according to the index k, with j1 being ranked the first. The vectors w and q are sorted according to the ordering in C. Given (C,w,q), it is convenient to define k X qj i . ð18Þ Qj k ¼ i¼1

Then, Qj k is the expected number of workers whom the firm attracts and whose ranking is higher than or equal to k in C. Set Qj0 ¼ 0. An equilibrium consists of a strategy, (C,w,q), where the ordering in C is strict, and a vector of expected market wages, E, such that (i) given E, the strategy is optimal for each firm, and (ii) the strategy induces E. The particular equilibrium which resembles the unique equilibrium in the simple model has the following additional features: (i) All firms use the same strategy; (ii) Every firm attracts all types of workers; and (iii) C ¼ J, i.e., every firm ranks the workers according to productivity. With these features, each worker applies to all firms with the same probability, and so qj ¼ gj y for all j 2 j. To characterize this equilibrium, suppose that each firm attracts all types of workers and ranks the workers according to the ordered set J. Then, with probability e Qj 1 , a firm receives no applicant whose productivity is higher than yj. Thus, the probability with which the firm successfully hires a type j worker is e Qj 1 ð1 e qj Þ ¼ e Qj 1 e Qj . Similarly, a type j worker who applies to the firm will be selected with the following probability: pj ¼ e

Qj

1

Gðqj Þ ¼ ½e

Qj

1

e

Qj

Š=qj .

ð19Þ

Taking the vector E as given, each firm chooses (w, q) to solve the following problem: J
P max p ¼ e Qj 1 e Qj ðyj wj Þ j¼1

s:t: pj wj  E j for all j:

ð20Þ

Because the firm attracts all types of workers in the conjectured equilibrium, then qj>0 for all j and (20) holds with equality. Thus, wj ¼ E j =pj for all j.

10

ð21Þ

Strictly speaking, q should be treated as the workers’ strategy, rather than the firm’s. However, the use of q as a firm’s choice is convenient and it does not change the analysis. As it is clear from the simple model, a firm that intends to attract a type of workers must offer a combination of a wage and a queue length that gives the worker the expected market wage. Under this constraint, the firm’s choice of w effectively determines q.

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In fact, as mentioned earlier, q is given as qj ¼ gj y for all j.

ð22Þ

Because qj>0, the first-order conditions of the above problem yield: E j ¼ yj e

Qj

J X

i¼jþ1

yi ðe

Qi

1

e

Qi

Þ; all j.

ð23Þ

Note that, since G(q)>e q for all q>0, then pj 4e Qj from (19) and pjwj ¼ Ejopjyj from (23). That is, the wage rate specified by (21) is feasible for all j. This compensation scheme generalizes (12) and (13). The right-hand side is a type j worker’s expected contribution to output. The first term is a type j worker’s contribution to output when the firm does not receive any applicant whose productivity is higher than or equal to yj. The second term is expected output that a type j worker crowds out on the workers of lower productivity. Thus, the compensation scheme requires a worker’s expected wage to be equal to the worker’s expected output minus the worker’s expected crowding-out on other workers’ output. For this reason, the compensation scheme is socially efficient. Proposition 4. The strategy (J,w,q) and the vector E form an equilibrium if the ranking in J is strict and if (21), (22) and (23) are satisfied. In this equilibrium, pj 4pjþ1 ; E j 4E jþ1 ; wj oyj and wj yj owjþ1 yjþ1 for all j. However, wj owjþ1 if ðyj yjþ1 Þ is small. A proof of this proposition appears in Appendix B. To see why the particular (J,w,q) and E form an equilibrium, consider a deviation by a single firm to (C,qd,wd), where C ¼ ðj k ÞK k¼1 for 1  K  J. First, any deviation that does not rank the workers according to productivity can be improved upon. If there are two types of workers in C, say types js and js+1, whose relative ranking is opposite to productivity, then I can construct a further deviation that switches the ranking of these two types while maintaining the same wages and queue lengths for other types in C. Relative to the original deviation, the further deviation attracts more type js+1 workers and fewer type js workers. Since type js+1 workers are more productive than type js workers, the gain from the further deviation outweighs the loss, and so this further deviation improves upon the original deviation. Second, any deviation that does not attract all types of workers can be improved upon. If the deviation does not attract type j workers, where jsojojs+1, then I can construct a further deviation that attracts type j workers and ranks type j between js and js+1. This further deviation gives the same wages and queue lengths to all the workers in the set C, except type js+1. In addition, the sum of the queue lengths of type j and type js+1 workers in the further deviation is equal to the queue length of type js+1 workers in the original deviation. Again, the gain from the higher productivity workers (type j)

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outweighs the loss from the lower productivity workers (type js+1), and so this further deviation improves upon the original deviation. Note that the further deviations in the above construction move the strategies toward the equilibrium strategy described in Proposition 4. They share with the equilibrium strategy the feature that every firm attracts all types of workers and ranks the workers according to productivity. By construction, however, all strategies with this feature are dominated by the strategy in the described equilibrium. Thus, the equilibrium described in Proposition 4 is indeed an equilibrium. Because the equilibrium in this extended economy is of the same type as the one in the simple model, it is not surprising that it has similar properties. In particular, higher productivity workers have a higher employment probability and higher expected wage. Also, actual wages exhibit a differential opposite to the productivity differential when the productivity differential is small.11 The dependence of the actual wage on productivity can exhibit a number of different patterns. One is that the actual wage decreases in productivity, which occurs when the productivity differential (yj yj+1) is small for all j. The opposite pattern is that the actual wage increases in productivity, which occurs when (yj yj+1) is large for all j. Non-monotonic patterns can also arise. For example, if (yj yj+1) is large for large j and decreases sharply as j decreases, then the actual wage increases first when productivity increases from the lowest level but decreases when productivity is sufficiently high. The equilibrium wage distribution also depends on the pattern of productivity and the distribution of workers. To compute the wage distribution, recall that each type j worker is employed with probability pj. Since the size of type j workers in the labor force is gj, the expected number of type j workers who are employed (at wage wj) is gj p j ¼

1 e y

yGj

1

e

yGj




;

where Gj ¼

j X

gi .

i¼1

The total number of employed workers of all types is (1 e y)/y, which is independent of the distribution of types. Therefore, the density of workers employed at wage wj is12 gj p j g j pj y . ¼ f j  PJ 1 e y j¼1 gj pj

ð24Þ

Because pj>pj+1, then fj/gj>fj+1/gj+1. That is, the higher the productivity of a type of workers, the larger the fraction of these workers will be employed. If all

11

As in the simple model, expected wages in the equilibrium internalize the matching externalities. Thus, the equilibrium allocation is efficient. 12 Obviously, there is only a finite number of wage levels in the equilibrium distribution and so the phrase ‘‘density function’’ really means the frequency function.

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types of workers have the same size in the labor force, then the density of wages is an increasing function of productivity. However, because wages are not necessarily an increasing function of productivity, the density of wages can be increasing, decreasing, or non-monotonic in wages. In the next subsection, I will provide some numerical examples to illustrate this distribution. Before providing the numerical examples, let me introduce firms’ entry into the market. Suppose that firms compete in setting up vacancies and the cost of setting up a vacancy is c. After the vacancies are set up, the firms play the recruiting game as described before. The expected profit of each firm from recruiting will depend on the overall market tightness, y, and so let me denote it as p(y). In the equilibrium, y is determined by pðyÞ ¼ c. 5.2. Numerical examples I use two examples to illustrate the wage distribution. In particular, these examples show that the model can generate a hump-shaped density of the wage distribution as the one documented in the literature (e.g., Mortensen, 2002). Since the model is not calibrated to the data, these examples are only illustrative. In both examples, I will set J ¼ 20, y1 ¼ 100, c ¼ 0:2yJ and yj ¼ y1 ð1 þ jDÞ1 j for all j, where D will be determined by the choice of yJ through the requirement yj ¼ y1 ð1 þ jDÞ1 j . The two examples differ from each other in D and the distribution of workers in the labor force. Of course, the equilibrium tightness and wage distribution also differ in the two examples. .P J 5 0:3 and Example 1. yJ ¼ 98 and gj ¼ j 0:3 j¼1 j . These imply D ¼ 5.319  10 y ¼ 0:819. In this example, the difference between the highest and the lowest productivity is small, and the distribution of workers’ productivity in the labor force is a decreasing function. Figure 1 depicts the equilibrium in this example. In the upper panel, the variable on the horizontal axis is an increasing function of productivity. This panel shows that the employment probability is an increasing function of productivity, as the theory predicts. Also, because the productivity differential is small, the wage rate is a decreasing function of productivity. The magnitude of this reverse wage differential is worth noting. Although the lowest productivity is only 2% lower than the highest productivity, the wage of the least productive workers is twice as much as the wage of the most productive workers! Despite this large reverse wage differential, high-productivity workers are rewarded properly as they receive higher expected wage. This is made possible by a positive differential in the employment probability that outweighs the reverse wage differential. In Figure 1, I depict the wage density and the distributional density of workers’ productivity. The wage density is hump-shaped. Because the wage rate is a decreasing function of productivity in this example, the shape of the wage density implies that more workers with medium productivity are employed than

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Figure 1. The equlibrium and the wage density in Example 1. j: worker’s type, with a lower j corresponding to higher productivity; wj/y1: wage of type j workers, normalized by y1; pj: employment probability of a type j worker; gj: fraction of type j workers in the labor force; sw: wages sorted in an ascending order; f: wage density 1

1 0.8

wj y1

0.6

pj

0.4

γj

0.2 0

0

5 1

10 (J + 1 - j)

15

20 J

0.1 0.08

fk

0.06

γk

0.04 0.02 0

0

50 48.86

55

60 swk

63.52

workers with either very high-productivity or very low-productivity. There are very few low-wage workers in the equilibrium because these workers are highproductivity workers who are a small fraction of the labor force. There are also very few high-wage workers because these workers are low-productivity workers, who have a very low ranking and hence a very low employment probability. The shape of the wage density sharply contrasts with the distribution of workers in the labor force, g. In the lower panel of Figure 1, the plot of g is the hypothetical density of wages when all workers of each type are employed. In contrast to the humped shape of the actual density, this hypothetical density is increasing. For any wage level, if the actual density exceeds the hypothetical

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density, the workers at that wage are employed at a rate higher than the average employment rate; if the actual density falls below the hypothetical density, the workers at that wage are employed at a rate lower than the average employment rate. Thus, in this example, low-wage workers are employed at a rate higher than the average rate while high-wage workers are employed at a rate lower than the average rate. In the above example, the hump-shaped wage density comes with a reverse wage differential. However, this is not an inevitable prediction of the model, as the following example shows. .P J Example 2. yJ ¼ 50 and gj ¼ ½ jðJ þ 11 jފ4 jފ4 . These imply j¼1 ½ jðJ þ 11 3 D ¼ 1:858  10 and y ¼ 0:696. In contrast to Example 1, this example has a much larger difference between the highest and the lowest productivity. As a result, the wage rate is an increasing function of productivity, as depicted in the upper panel of Figure 2. Also, the distribution of workers in the labor force has a single peak at an intermediate level of productivity, rather than being a decreasing function of productivity. Notice that the wage differential is small, in contrast with the large difference in productivity. The large difference in productivity induces a large difference in the employment probability, and hence in the expected wage. The density of the wage distribution is still hump-shaped, as depicted in the lower panel of Figure 2. The hypothetical wage density when all workers are employed, g, is also hump-shaped. In comparison with this hypothetical density, the equilibrium wage density peaks at a higher wage and a larger mass is distributed at higher wages. Thus, low-wage workers are employed at a rate lower than the average employment rate while high-wage workers are employed at a rate higher than the average rate. This result reflects the fact that high-wage workers in this example are high-productivity workers who are employed with a higher probability. 6. Conclusion In this paper, I construct a search model of a large labor market in which workers are heterogeneous in productivity and (homogeneous) firms post wages to direct workers’ search. Each firm can rank the applicants and post different wages for different types of workers. Workers cannot coordinate their applications. I show that there is a unique (symmetric) equilibrium. In this equilibrium, high-productivity workers have a higher priority in employment and higher expected wages than low-productivity workers. The equilibrium is socially efficient. However, actual wages are higher for high-productivity workers than for low-productivity workers only when the productivity differential is large. When the productivity differential is small, workers of higher productivity are paid lower wages. Moreover, this reverse wage differential increases as the

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Figure 2. The equlibrium and the wage density in Example 2. j: worker’s type, with a lower j corresponding to higher productivity; wj/y1: wage of type j workers, normalized by y1; pj: employment probability of a type j worker; gj: fraction of type j workers in the labor force; sw: wages sorted in an ascending order; f: wage density 1

1 0.8

wj y1

0.6

pj

0.4

γj

0.2 0

0

5 1

10 (J + 1 - j)

15

20 J

0.1 0.08

fk

0.06

γk

0.04 0.02 0

0

50 48.86

55

60 swk

63.52

productivity differential decreases. Thus, the wage differential is strictly positive even when the productivity differential approaches zero. These results suggest that actual wages may not be a good indicator of productivity; rather, the expected lifetime earnings are a better indicator. They also show that wage differentials among similar workers should not always be construed as discrimination, as they have often been viewed in the literature. Rather, the difference in workers’ expected payoffs is a more reliable measure of discrimination. For example, when the productivity differential is so small that it appears statistically insignificant to an econometrician (but observable to the

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employers), the current model implies that the workers with slightly higher productivity obtain lower wage than other workers. Guided by the convention, the econometrician would conclude that high-productivity workers are discriminated against. This would be misleading, because high-productivity workers enjoy a higher ranking and higher expected wage. A natural question is whether the wage differential can persist in the long run. In a dynamic environment, the search model will have exogenous separation between matched firms and workers. A wage differential will continue to exist in the steady state, but it may not be reversely related to productivity when the productivity differential is small. For the reverse wage differential to survive in the steady state, exogenous job separation must be high. Finding out how high a job separation rate is needed in a quantitative exercise. An extension to a dynamic economy will also generate some interesting predictions on the time path of wages. For example, consider the model with only two types of workers and suppose that the productivity differential between the two is small. Then, employed workers of high productivity are more likely to search on the job than employed workers of low productivity. This implies that high-productivity workers will have a steeper wage path, even though there is no learning-by-doing or human capital accumulation. Analyzing such on-the-job search is difficult in an environment with directed search. The reason is that, when offering a wage, a firm must take into account not only workers’ trade-off between the wage and the current employment probability, but also the trade-off between the current wage and the probability of getting higher wage in the future. Perhaps a quantitative analysis can be conducted. Acknowledgment This paper has been presented at the Conference on Labour Market Models and Matched Employer–Employee Data (Denmark, 2004) and at University of British Columbia. I have benefited from conversations with Mike Peters, Rob Shimer, Audra Bowlus and Michael Baker. I gratefully acknowledge the financial support from the Bank of Canada Fellowship and from the Social Sciences and Humanities Research Council of Canada (SSHRCC). The opinion expressed here is my own and it does not represent the view of the Bank of Canada. References Acemoglu, D. and R. Shimer (1999), ‘‘Holdups and efficiency with search frictions’’, International Economic Review, Vol. 40, pp. 827–850. Altonji, J.G. and R.M. Blank (1999), ‘‘Race and gender in the labor market’’, pp. 3143–3259. In: O. Ashenfelter and D. Card, editors, Handbook of Labor Economics, Vol. 3c, Amsterdam: Elsevier. Black, D.A. (1995), ‘‘Discrimination in an equilibrium search model’’, Journal of Labor Economics, Vol. 13, pp. 309–334.

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Bowlus, A.J. and Z. Eckstein (2002), ‘‘Discrimination and skill differences in an equilibrium search model’’, International Economic Review, Vol. 43, pp. 1309–1345. Burdett, K., S. Shi and R. Wright (2001), ‘‘Pricing and matching with frictions’’, Journal of Political Economy, Vol. 109, pp. 1060–1085. Hosios, A. (1990), ‘‘On the efficiency of matching and related models of search and unemployment’’, Review of Economic Studies, Vol. 57, pp. 279–298. Juhn, C., K.M. Murphy and B. Pierce (1993), ‘‘Wage inequality and the rise in returns to skill’’, Journal of Political Economy, Vol. 101, pp. 410–442. Julien, B., J. Kennes and I. King (2000), ‘‘Bidding for labor’’, Review of Economic Dynamics, Vol. 3, pp. 619–649. Lang, K., M. Manove and W. Dickens (2002), Racial discrimination in labor markets with posted wage offers, manuscript, Boston University. Moen, E.R. (1997), ‘‘Competitive search equilibrium’’, Journal of Political Economy, Vol. 105, pp. 385–411. Mortensen, D. (1982), ‘‘Property rights and efficiency in mating, racing, and related games’’, American Economic Review, Vol. 72, pp. 968–979. Mortensen, D. (2002). Wage dispersion: why are similar workers paid differently? manuscript, Northwestern University. Peters, M. (1991), ‘‘Ex ante price offers in matching games: non-steady state’’, Econometrica, Vol. 59, pp. 1425–1454. Shi, S. (2001), ‘‘Frictional assignment. i. efficiency’’, Journal of Economic Theory, Vol. 98, pp. 232–260. Shi, S. (2002), ‘‘A directed search model of inequality with heterogeneous skills and skill-biased technology’’, Review of Economic Studies, Vol. 69, pp. 467–491. Shimer, R. (1997), Do good guys come in first? How wage determination affects the ranking of job applicants, unpublished manuscript, Princeton University. Appendix A: Proof of Proposition 3 I can assume aA>0 and aB>0 without loss of generality. To see this, suppose that an allocation has aA ¼ 1, aB ¼ 0 and (qiT, qiS, Ri)i ¼ A,B. Then, an alternative allocation ðani ; qniT ; qniS ; Rni Þi¼A;B can be constructed as follows: anA ¼ anB ¼ 12 ; qnAT ¼ qnBT ¼ qAT ; qnAS ¼ qnBS ¼ qAS and RnA ¼ RnB ¼ RA . This alternative allocation is equivalent to the original allocation, except that it re-labels half of the firms in the original group A as group B. Let lj be the Lagrangian multiplier of (17) in the planner’s maximization problem. Then, the first-order conditions of qiT and qiS are as follows: lT =y  e

ðqiT þqiS Þ

lS =y  e

ðqiT þqiS Þ

þ de dð1

qiT

½Ri þ ð1 e

qiT

Þð1

Ri Þe

qiS

Ri Þe

qiS

Š;

;

‘‘ ¼ ’’ if qiT 40, ‘‘ ¼ ’’ if qiS 40.

ðA:1Þ ðA:2Þ

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Shouyong Shi

The first-order condition of aA (taking aB ¼ 1 0 ¼ dð1 e qAT Þ½RA þ ð1 RA Þe qAS Š dð1 e qBT Þ½RB þ ð1 RB Þe qBS Š þ ½e

ðqBT þqBS Þ

e

ðqAT þqAS Þ

Š

ðqAT

aA into account) is

qBT ÞlT =y

ðqAS

qBS ÞlS =y. ðA:3Þ

The following features can be easily deduced from the planner’s problem. First, from inspecting the objective function, I can infer that the choice Ri ¼ 1 is efficient whenever qiT>0 and qiS>0. Second, it is efficient to utilize both types of workers, in the sense that qAj+qBj>0 for both j ¼ T; S. To see this, suppose qAS ¼ qBS ¼ 0, to the contrary. Then, (17) does not bind and so lS ¼ 0. Because type S workers are not assigned to match with any firm, the ranking R is irrelevant for the allocation, and so Ri can be set to 1. In this case, (A.2) implies 0  e qiT for i ¼ A; B; which is a contradiction. Similarly, the choices qAT ¼ qBT ¼ 0 are not efficient. With the above features, the efficient allocation must be one of the following cases: (i) qiT>0 and qiS>0 for both i ¼ A and B; (ii) qAT>0 and qAS>0 but qBT ¼ 0oqBS; (iii) qAT>0 and qAS>0 but qBT>0 ¼ qBS; (iv) qAT>0 ¼ qAS and qBT ¼ 0oqBS. (All other cases are equivalent to these four cases, with the labels A and B being switched.) In the remainder of the proof, I will first show that case (i) is a solution to the planner’s problem and it yields the same allocation as the equilibrium described in Proposition 1. Then I will show that cases (ii) through (iv) are not efficient. Consider case (i) first. In this case, Ri ¼ 1 for both i ¼ A and B. Also, the first-order conditions for qiT and qiS hold as equality for both i ¼ A and B. These first-order conditions and constraint (17) together yield qAT ¼ qBT ¼ gy and qAS ¼ qBS ¼ ð1 gÞy. Thus, the two groups are identical, and the first-order condition for a is trivially satisfied. This allocation is the same as in the equilibrium described in Proposition 1. In case (ii), RA ¼ 1. Also, (A.2) holds as equality for both i ¼ A and B, which yields qAT þ qAS ¼ qBT þ qBS . In addition, (A.1) holds with equality for i ¼ A. Combining the conditions (A.1) and (A.2) for i ¼ A yields lT lS ¼ dye qAT . Substituting these results into (A.3), I have 0 ¼ d½1

ð1 þ qAT Þe

qAT

z

Š.

Because 1>(1+z)e for all z>0, the right-hand side of the above condition is strictly positive. This is a contradiction, and so case (ii) is not a solution to the planner’s problem. In case (iii), RA ¼ 1, (A.1) holds as equality for both i ¼ A and B, and (A.2) holds as equality for i ¼ A. Thus, lT ¼ ye

qAT

qBT ¼ qAT

ðe

qAS

þ dÞ;

lS ¼ ye   d þ e qAS ln . 1þd

ðqAT þqAS Þ

,

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Wage Differentials, Discrimination and Efficiency

Substituting these conditions into (A.3) and simplifying, I have:   d þ e qAS qAS . 0 ¼ ln 1 þ deqAS 1þd The right-hand side of the above condition is an increasing function of qAS and, at qAS ¼ 0, its value is 0. Because qAS>0, then the above condition is violated. Thus, case (iii) is not a solution to the planner’s problem. Finally, consider case (iv). In this case, (A.1) holds as equality for i ¼ A and (A.2) holds as equality for i ¼ B. Thus, lT ¼ ð1 þ dÞye qAT and lS ¼ ye qBS . Substituting these results into (A.3) yields 0 ¼ ð1 þ qBS Þe

qBS

þd

ð1 þ dÞð1 þ qAT Þe

qAT

.

ðA:4Þ

Because no type S worker applies to a group A firm, RA can be set to 1. Also, (A.2) holds as inequality for i ¼ A. Substituting lS ¼ e qBS and RA ¼ 1, this inequality implies qBS  qAT . Since the function ð1 þ zÞe z is decreasing for all z>0, the result qBS  qAT and the condition (A.4) imply 0  d½1 ð1 þ qAT Þe qAT Š. This cannot hold for qAT>0. Thus, case (iv) is not a solution to the planner’s problem. QED Appendix B: Proof of Proposition 4 Before proving that the strategy described in the proposition is an equilibrium, let me verify the properties of the equilibrium stated in the proposition. Using (19) I have ! qj qjþ1 e 1 1 e pj pjþ1 ¼ e Qj 40. ðB:1Þ qj qjþ1 The inequality follows from the facts that ez 1>z>1 e (23), I have Ej

E jþ1 ¼ ðyj

yjþ1 Þe

Qj

wjþ1 ¼ ðyj

yjþ1 Þ

qj eq j

for all z>0. From

40.

Using this result, (21) implies wj

z

1

E jþ1

1 pjþ1

! 1 . pj

ðB:2Þ

Because qj oeqj 1 and pj>pj+1, the above condition implies wj wjþ1 oyj yjþ1 for an j. Thus, wj yj owjþ1 yjþ1 for all j. In turn, this implies that wjoyj for all j if and only if wJoyJ. The latter condition indeed holds, because (21) implies qj Ej oyj . ¼ yj q j wj ¼ e 1 pj The final property of the equilibrium is that wjowj+1 if (yj yj+1) is small. To show this, note from (B.1) that (pj pj+1) is bounded strictly above zero even

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Shouyong Shi

when (yj yj+1) approaches zero. Thus, (B.2) implies that wjowj+1 when (yj yj+1) is sufficiently small. Now, I prove that there is no profitable deviation from the strategy described in the proposition. Consider a deviation by a single firm to the strategy (C, qd,wd), where C ¼ ðj k ÞK k¼1 and 1  K  J. Although the ranking in C may not necessarily be strict, I will consider only the case of strict ranking; that is, j 1  j 2 


 j K . A similar proof can be constructed to show that a deviation to having no priority between some elements in C is not profitable. With strict ranking in C, the deviation must satisfy the following properties for all k 2 K: e

Qdj

e

k 1

Qdj

k

qdjk yj k e

Qdj

k

K X

wdjk ¼ E jk ,

i¼kþ1

 yj i e

Qdj

i 1

ðB:3Þ e

Qdj

i



¼ E jk

ðB:4Þ

where Qdjk is defined similarly to (18), with qd replacing q. The first condition is type jk workers’ participation constraint, and the second condition comes from the first-order condition of wdjk . If one of these conditions is violated, then the deviation can be improved upon by a further deviation to the strategy that satisfies these conditions. The deviation can differ from the equilibrium strategy in two ways: the ranking in C may not be according to workers’ productivity and C may not include all elements of J. In Lemmas 5 and 6 below, I will show that if the deviation has either of these differences, then it can be improved upon by a further deviation that eliminates the difference. Thus, a profitable deviation must attract all types of workers and rank them according to productivity. By construction, the best strategy of this kind is the strategy proposed in the equilibrium. Lemma 5. A deviation that does not rank workers according to productivity can be improved upon by another deviation that attracts the same types of workers as in the original deviation but that ranks the workers according to productivity. Proof. Consider the deviation (C,wd,qd) described above and suppose that the ranking in C is not according to productivity. Then, there exists a number s ¼ maxfk 2 K : yjk oyjkþ1 g. That is, type js and type js+1 are the lowest ranked pair which exhibits a relative ranking  opposite to  productivity. Consider a furn n n , where C n ¼ ðnk ÞK ther deviation from this deviation C ; qnk ; wnk k¼1 . This k2K further deviation attracts the same types of workers as those contained in C but it reverses the ranking between type js and type js+1 workers. That is, nk ¼ jk for all kefs; s þ 1g; ns ¼ j sþ1 and nsþ1 ¼ j s . Use q to construct Q similarly to (18). Furthermore, let this further deviation have the following properties: n (i) For all kefs; s þ 1g; wnnk ¼ wdjk and (B.3)  is satisfied  with ðqnnk ; Qnk ; wnnk Þ n d n d d n replacing ðqjk ; Qj k ; wj k Þ. (ii) For k 2 fs; s þ 1g; qnk ; Qnk ; wnk solve the following

Wage Differentials, Discrimination and Efficiency

113

problem: max

sþ1  X

e

n

Qnn

e Q nk

k 1

k¼s

 yn k

wnnk



subject to e

n

Qnn

k 1

e Q nk

qnnk

wnnk ¼ E nk ;

k ¼ s; s þ 1,

qnns þ qnnsþ1 ¼ qdjs þ qdjsþ1

ðB:5Þ ðB:6Þ

For k  s 1, the two deviations have the same wages and both satisfy (B.3). Comparing the condition (B.3) for the two deviations and working from k ¼ 1, I obtain Qnnk ¼ Qdjk for all k  s 1, which implies qnnk ¼ qdjk for all k  s 1. With (B.6), I have Qnnsþ1 ¼ Qdjsþ1 . Using this result and again comparing the condition (B.3) for the two deviations, I have Qnnk ¼ Qdjk and hence qnnk ¼ qdjk for all k  s þ 2. Thus, the only difference between the two deviations lies in the employment and wages of type js and type js+1 workers. To show that the further deviation improves upon the original deviation, substitute (B.5) and (B.6) for ðwnns ; wnnsþ1 ; qnnsþ1 Þ and use the facts that ns ¼

j sþ1 ; nsþ1 ¼ j s ; Qnns 1 ¼ Qdjs 1 and Qnnsþ1 ¼ Qdjsþ1 . Then, the objective function of the above maximization problem becomes       d n n Qd e Qjs 1 Q Qns yj sþ1 qnns E jsþ1 þ e Qns Q jsþ1 yj s qdjs þ qdjsþ1 qnns E j s . The first-order condition for qnns yields  
n 0 ¼ yjsþ1 yj s e Qns þ E j s E j sþ1 .

From (B.4), I can derive the following relationship:   d E js E j sþ1 ¼ yj sþ1 yjs e Qjs .

ðB:7Þ

The above two conditions imply Qnns ¼ Qdjs . Then, Qdns

qnns ¼ Qnns

1

¼ Qdjs

Qdjs ¼ qdjs .

The constraint (B.6) then yields qnnsþ1 ¼ qdjsþ1 . Let pd be the firm’s expected profit under the original deviation and pn under the further deviation. Then, pd ¼

K  X k¼1

e

Qdj

k 1

e

Qdj

k



yj k

 wdjk ,

114

Shouyong Shi n

and p is given similarly. Subtract the two: pn

pd ¼

sþ1 h X k¼s

e

Qnn

k 1

n

e Q nk

 yn k

wnnk



 e

Qnn

k 1

n

e Q nk



yn k

wnnk

i

.

Substituting the above results for (Q, q), the constraint (B.5) for the wages, and the relationship (B.7), I have     h i d qd pn pd ¼ yjsþ1 yjs e Qjs eqjs 2 þ qdjs qdjsþ1 þ e jsþ1 .

Since ez>1+z and e z>1 z for all z>0, the expression in [
] above is positive. Because yjsþ1 4yjs , then pn 4pd : QED

Lemma 6. A deviation that does not attract all types of workers can be improved upon by a further deviation that attracts all types. Proof. Let the original deviation be (C, wd, qd), where C ¼ ðj k ÞK k¼1 and j k 2 J. Because the deviation does not attract all types, then KoJ. With Lemma 5, I can assume that C ranks the workers by productivity, i.e., j 1 oj 2 o


oj K . Also, since any deviation that does not satisfy (B.3) and (B.4) can be improved upon, the deviation must satisfy these conditions. Let j n ¼ maxfj 2 J\Kg. There are two cases to consider: j*>jK and j*oJK. If j>jK, construct the further deviation (C,w,q) as follows. Set  C ¼ {j1,j2, y,jK,jK+1}, where j Kþ1 ¼ j n . For every k  K, set the wage wnjk ¼ wdj k and let the constraint (B.3) hold. For k ¼ K+1, let (B.3) and (B.4) hold. Then, qnjk ¼ qdjk for all k  K. This further deviation increases the firm’s expected profit, provided wnjkþ1 oyj kþ1 . The latter condition can be verified. Now examine the case jojK. Let soK be such that j s 1 oj n oj s . From the original deviation, I construct a further deviation (C, w,q), where C n ¼ ðnk ÞKþ1 k¼1 is given as 8 if k  s 1 > < jk ; n nk ¼ j ; if k ¼ s > : j k 1 ; if k  s þ 1:

 That is, Pk then type j is inserted between the types js 1 and js. Define n Qnk ¼ i¼1 qni . Let the strategy in the further deviation satisfy the following properties:

(i) For all k  s 1; wnnk ¼ wdjk and (B.3) is satisfied with ðqnnk ; Qnnk ; wnnk ; K þ 1Þ replacing ðqdjk ; Qdjk ; wdjk ; KÞ. (ii) For all k 2 fs; s þ 1; . . . ; K þ 1g, (B.3) and (B.4) are satisfied with ðqnnk ; Qnnk ; wnnk ; K þ 1Þ replacing ðqdjk ; Qdjk ; wdjk ; KÞ. As in the proof of Lemma 5, property (i) implies Qnnk ¼ Qdjk ; qnnk ¼ qnj k and wnk ¼ wdjk for all k  s 1. For k  s, recall that nkþ1 ¼ j k . Subtract (B.4) for jk n

115

Wage Differentials, Discrimination and Efficiency

in the original deviation and for nk+1 in the further deviation, I have 0 ¼ yj k D j k

K X

yj i Dj i

i¼kþ1 Qnn

Dj i

1

1

Qd




;

all k  s

where Dj k ¼ e kþ1 e jk . Changing the index k to k+1 and subtracting the resulted equation from the above equation, I obtain yj k yjkþ1 Djk ¼ 0. Since yj k 4yjkþ1 , then Djk ¼ 0 for all k  s. That is, Qnnkþ1 ¼ Qdjk ; all k  s.

In particular, Qnnkþ1 ¼ Qdjk . Then, Qnnkþ1

Qnnkþ1 ¼ Qdjk

Qdjk ; all k  s.

Working from k ¼ K 1 to k ¼ s, the above equation yields qnnkþ1 ¼ qdjk ;

all k  s þ 1.

qnns þ qnnsþ1 ¼ Qnnsþ1

Qnns

1

¼ Qdjs

Qdjs

1

¼ qdjs .

Since (B.3) holds in both deviations, the above equalities imply wnnkþ1 ¼ wdjk ;

all k  s þ 1

Thus, the only difference between the two deviations lies in type ns and type ns+1 workers. Let pd be the firm’s expected profit with the original deviation and pn with the further deviation. Then pn

pd ¼



e

Qdj

s 1

e

Qdjs

 yj s

sþ1   X e wdjs þ k¼s

Qnn

k 1

e

Qnn

k

 yn k

 wnnk .

Substitute (B.3) for wd and its counterpart for w, substitute the relationship (B.7) and use the above relationships between (Q,q) and (Qd,qd). Then,     ni h n pn pd ¼ yns ynsþ1 e Qns 1 1 1 þ qnns e qns 40.

Thus, the original deviation can be improved upon by including the type ns ¼ j n in the set of workers to attract. This completes the proof of Lemma 6 and hence of Proposition 4. QED

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116

CHAPTER 5

Labor Market Search with Two-Sided Heterogeneity: Hierarchical versus Circular Models Pieter A. Gautier, Coen N. Teulings and Aico van Vuuren Abstract Search models with two-sided heterogeneity and hierarchical sorting are difficult to solve analytically. Teulings and Gautier (2004) apply Taylor expansions, which are only valid close to the Walrasian equilibrium. This paper applies a simpler circular model instead of the more realistic hierarchical model. Workers and firms are located on a circle. We derive the unique analytical equilibrium of this model and show that it has the same characteristics as the Taylor expansion of the hierarchical model. This raises hope that the former can be used as a proxy for the latter in more complex settings.

Keywords: equilibrium search, heterogeneity, bargaining JEL classifications: J41, J64 1. Introduction Recently, there has been increased attention for search models of the labor market with two-sided heterogeneity and hierarchical sorting due to the comparative advantage of high-skilled workers in complex jobs (see e.g. Shimer and Smith, 2000). Those papers combine the classical literature on Walrasian

Corresponding author. r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75005-7

117

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Pieter A. Gautier, Coen N. Teulings and Aico van Vuuren

assignment and hedonic prices of Becker (1973), Rosen (1974), Sattinger (1975), and Teulings (1995) on the one hand and the matching models of Diamond (1982), Mortensen (1982), and Pissarides (2000) on the other hand. In the Walrasian version of these models, the equilibrium is a one-to-one mapping of worker to job types, where the highest worker type is assigned to the highest job type, and vice versa. We therefore refer to them as hierarchical search model. In the presence of search frictions, agents cannot afford to wait forever till the optimal match comes along. Hence, the equilibrium is not characterized by a one-to-one mapping, but by a band around this mapping, that is, matching sets are convex.1 A serious drawback of these hierarchical search models is that they do not allow for an analytical characterization of the equilibrium even for the most convenient specification of tastes and technologies. As a substitute for an exact characterization of the equilibrium, Teulings and Gautier (2004) provide a Taylor expansion of the equilibrium, which gives a second-order approximation of the effect of search frictions on the economy. However, extending this approach to models that also allow for on-the-job search turned out to be a bridge too far.2 The fundamental reason why analytical equilibria cannot be obtained in random search models is that for sufficiently small search frictions the matching sets of intermediate worker and job types will have an interior upper and lower bound, while there will always be some high types for which the upper bound of the matching set is the upper support and some low types for which the lower bound is the lower support.3 This can easily be seen by realizing that in an hierarchical search model both the index for the worker and the job type can be transformed into a uniformly distributed index without loss of generality.4 Matching sets can then be represented graphically in a unit square reflecting the worker-job type space. The Walrasian assignment is a one-to-one correspondence from the lower left (0, 0) corner of the square to the upper right (1,1) corner, while the search equilibrium is a band around this locus, see Figure 1a

1

Shimer and Smith (2000) show that the production technology should be log supermodular for this statement to be true. Otherwise, matching sets need not even be connected. 2 An important reason for this problem is that the closer one gets to the Walrasian equilibrium by raising the efficiency of the search process, the more important on-the-job search becomes relative to search while unemployed, simply because the cost of accepting a job goes down by the greater efficiency of on-the-job search. Hence, the equilibrium without on-the-job search is not a useful starting point for Taylor expansions. 3 In the directed search models of Shi (2002) and Shimer (2004), matching sets are convex and increasing in worker types. Those models are also constraint efficient. Albrecht et al. (2004) show that this efficiency result breaks down if workers can apply to multiple jobs. Directed search models do require workers to have a lot more information than random search models; i.e. on the locations of firms and the wages that each firm offers at each point in time. 4 Let s be the index for the worker type and let F(s) be its distribution function. Then, F(s) is a transformed worker index that is uniformly distributed.

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Labor Market Search with Two-Sided Heterogeneity

Figure 1. (a) Illustration of an equilibrium as a band arround the Walrasian assignment; (b) Illustration of an equilibrium with complete segmentation 1

S

(a)

0 C 1

S

(b)

0

C

120

Pieter A. Gautier, Coen N. Teulings and Aico van Vuuren

where worker types are denoted by s and job types by c. Obviously, matching sets for the extreme types cannot have both an interior upper and lower bound. Moreover, the vertical (respectively horizontal) distance measures the probability that a random job offer (applicant) is in a worker’s (firm’s) matching set. So, we can never have an assignment where both the workers probability to receive an acceptable job offer and the firm’s probability to receive an acceptable applicant are constant across all worker and job types, respectively, because in the corners the acceptance probability has to go down, either for workers, or for jobs, or for both. We refer to this phenomenon as the corner problem. Figure 1b depicts the only exception: a complete segmentation of the market in a number of disjoint segments, where within each segment contacts are always accepted and workers never match with jobs outside of their segment. This is the type of equilibrium analyzed by Burdett and Coles (1997). The critical condition for this type of equilibrium to exist is that utility is non-transferable, which might be an appropriate assumption for the marriage market, as in Burdett and Coles (1997), but not for the labor market. This paper investigates for the case without on-the-job search whether circular models can be a useful alternative. In these models, worker and job types are locations, not on a line, but on the circumference of a circle, as in Marimon and Zilibotti (1999), Moscarini (2001) and Barlevy (2002). The obvious advantage of this circular set up above hierarchical search model is that there are no corners in the type space, so all upper and lower bounds of matching sets can be interior. This advantage of circular models has to be traded off against the realism of hierarchical search model, which makes them easier to implement empirically, i.e. it is hard to think of a worker characteristic that can be meaningfully displayed as a location on a circle while many characteristics can be hierarchically ordered, like IQ or years of schooling. This paper analyzes how the exact equilibrium in a circular model compares to the approximate characterization of the equilibrium of the hierarchical search model of Teulings and Gautier (2004). We assume that worker types are distributed uniformly over the circumference of the circle. Furthermore, there is free entry of vacancies for each type. We first prove that in that case the vacancy distribution must be uniform. Marimon and Zilibotti (1999) provide a similar proof for their model, but that proof assumes unemployment to be distributed uniformly, while unemployment is endogenous in our model. Given the uniform vacancy distribution, the model is equivalent to the standard stochastic job search model where heterogeneity is revealed ex-post, i.e. after the worker and firm meet (see Pissarides, 2000). In the circular model, the agents must determine the maximum distance of a match while in the stochastic job match model, the agents must decide upon the minimum productivity. Since there is a one-to-one mapping between reservation distance and reservation productivity the equilibrium of our model makes the same predictions as a stochastic job match model. Our analysis shows that the characteristics of the equilibria of the circular and the hierarchical search model are also very similar. However, our interpretation of the stochastic job search

Labor Market Search with Two-Sided Heterogeneity

121

model as reflecting an approximation of the equilibrium of an assignment model with search frictions yields one important additional constraint on the shape of the distribution function of the stochastic job quality. Since the optimal match is interpreted as an interior maximum of a differentiable production function (no further functional form assumptions are needed), the derivative of this function is zero at the optimum. Hence, wages of jobs that are located close to the optimum are almost as high as the highest wage and the equilibrium wage distribution has a heavy right tail, just as in Burdett and Mortensen (1998). This feature has usually been considered as a burden from an empirical point of view but Gautier and Teulings (2005) have shown that this feature fits the data quite well and that log wages for given worker types are decreasing in the distance to the optimal assignment, i.e. they are concave in the job type. Finally, we prove uniqueness of the equilibrium in the circular model, something we were unable to prove in the hierarchical search model. Given the quadratic specification of the contact technology, which implies strongly increasing returns to scale, this is an all but obvious conclusion. The paper is organized as follows. Section 2 starts with the assumptions. We derive the equilibrium conditions in Section 3. In Section 4 we characterize the equilibrium. The costs of search are derived and discussed in Section 5 and Section 6 concludes. 2. The circular job search model The economy that we consider has the following properties. 2.1. Production There is a continuum of worker types s and job types c; s and c are locations on a circle with circumference 1, so that s ¼ 1 is equivalent to s ¼ 0, and the same for c. Workers can only produce output when matched to a job. Let k be an arbitrary constant, then: xðs; cÞ ¼ minjs k2Z

c þ kj

This implies that the range of the function x is ½0; 12Š. We assume the output, Y, only to depend on the value of x (s, c). Hence, for every pair (s, c) the distance between s and c is the only relevant characteristic that determines the level of output. For convenience, we drop the arguments of x in the remainder of this subsection. We assume that Y is a twice differentiable concave function of x with a unique maximum at x ¼ 0. It is symmetric around this maximum, so that Y(x) ¼ Y( x). Without loss of generality, the maximum can be normalized to unity: Y(0) ¼ 1. Hence, x can be interpreted as the distance from the optimal assignment, x ¼ 0, or as a mismatch indicator. Furthermore, these assumptions imply Y0 (0) ¼ 0. This is an important observation that follows directly from our

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Pieter A. Gautier, Coen N. Teulings and Aico van Vuuren

interpretation of the model where the optimal assignment is an interior maximum of a differentiable production function. The simplest possible production function that meets those criteria is: 1 2 gx 2 which we will use to characterize the equilibrium. Our main results like Proposition 2, do not depend on this specific functional form. We make the additional assumption that g>8 to rule out trivial situations where workers accept all jobs, even when the value of leisure is equal to zero (since then Y ð12Þo0). The parameter g is analogous to the complexity dispersion parameter discussed in Teulings and Gautier (2004, p. 558), the lower g, the better substitutes different worker types are. Y ðxÞ ¼ 1

2.2. Labor supply We assume that labor supply per s type is uniformly distributed over the circumference of the circle and, without loss of generality, normalize total labor supply to one. Hence, the density of type s is also equal to one. Unemployed workers receive a value of leisure B. Employed workers supply a fixed amount of labor and their pay off is equal to the wage they receive. 2.3. Labor demand There is free entry of vacancies for all c types. The flow cost of maintaining a vacancy is equal to K per period. After a vacancy is filled, the firm only pays for the wage of the worker. 2.4. Job search technology We assume a quadratic contact technology, where the number of contacts is proportional to the product of the number of job seekers and vacancies. The quadratic contact technology implies increasing returns to scale (IRS) and the absence of congestion externalities: the number of unemployed workers per unit of s, u (s), does not enter into the contact rate for unemployed workers and mutatis mutandis the same applies for the number of vacancies per unit of c, v (c) in the contact rate of vacancies. Hence, the contact rate of an unemployed worker at location s with a vacancy at location c equals: ls!c ¼ lvðcÞ

where l>0. Since every contact of a particular type of worker with a particular job is at the same time a contact of a particular job with a particular type of worker, the contact rate of job types must satisfy: lc!s ¼ luðsÞ

where u(s) denotes the density of unemployed workers of type s. This is the same contact technology that Shimer and Smith (2000) use. We can interpret the

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efficiency parameter l as the scale of the labor market. The larger the scale of the market, the more efficient the search process. The main reason for using IRS is that it avoids congestion effects between rocket engineers and hamburger flippers. Another motivation is that it simplifies the model. The limiting case, l-N, yields the Walrasian equilibrium. Matches between workers and jobs are broken up at a rate d>0. 2.5. Wage setting We assume that wages are determined by Nash bargaining over the match surplus between the worker and the firm. 2.6. Free entry condition We assume that firms can enter without restrictions and hence the asset value of opening a vacancy is equal to zero in equilibrium. 3. Equilibrium conditions Let VE(s, W) be the asset value of employment and let VJ(s, c, W) be the asset value, for a firm at location c that employs a worker at location s, where the firm pays the worker a wage W. These asset values satisfy the Bellman equations: V E ðs; W Þ ¼

W þ dV U ðsÞ rþd

V J ðs; c; W Þ ¼

ð1Þ

Y ½xðs; cފ W rþd

ð2Þ

where VU(s) is the asset value of unemployment of a worker at location s. Wages maximize the Nash product:
W ðs; cÞ ¼ arg max V E ðs; W Þ W

V U ðsÞ

b


1 V J ðs; c; W Þ

b

The first factor in square brackets is the increase in expected wealth for the worker relative to the status of unemployment and the second factor is the increase in wealth for the firm. W(s,c) satisfies the first-order condition:
 bV J ½s; c; W ðs; cފ ¼ ð1 bÞ V E ½s; W ðs; cފ V U ðsÞ

where we use V EW ðs; W Þ ¼ V JW ðs; c; W Þ. This condition states that the gain from a marginal wage increase for the worker must equal the cost of that increase for the firm, both weighted by their respective bargaining power b and (1 b). Substitution of (1) and (2) into this relation yields: W ðs; cÞ ¼ bY ½xðs; cފ þ ð1

bÞrV U ðsÞ

ð3Þ

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Pieter A. Gautier, Coen N. Teulings and Aico van Vuuren

Since the option value of a vacancy must be equal to zero by the free entry condition, we have: Z 1   K ¼l uðsÞ max V J ðs; c; W Þ; 0 ds 0

¼ ð1

bÞ

l rþd

Z

0

1

 uðsÞ max Y ½xðs; cފ

rV U ðsÞ; 0



ds

ð4Þ

where the second line follows from substituting (2) and (3) into (4). The asset value for an unemployed worker of type s equals the utility of leisure, B, plus the probability that she finds a job that improves her expected discounted life time wealth times the expected wealth change: Z 1   U rV ðsÞ ¼ B þ l vðcÞ max V E ðs; W Þ V U ðsÞ; 0 dc 0

l ¼Bþb rþd

Z

0

1

 vðcÞ max Y ½xðs; cފ

rV U ðsÞ; 0



dc

ð5Þ

where the second line follows from substituting (1) and (3) into (5). Note that both (4) and (5) imply that only matches for which output exceeds the flow value of unemployment, Y[x(s,c)]ZrVU(s), are acceptable. For the marginally acceptable job, for which Y[x(s,c)] ¼ rVU(s), Equation (3) implies: W(s,c) ¼ Y[x(s,c)].  be the value of x that satisfies this constraint. Hence: Let xðsÞ   Y ½xðs; s  xðsÞÞ ¼ rV U ðsÞ ¼ W ðs; s  xðsÞÞ

ð6Þ

By the symmetry of Y(x) around x ¼ 0, the equality applies both to the left and to the right of x ¼ 0. Finally, the equilibrium flow condition for unemployment is equal to:     luðsÞ V ½s þ xðsÞ
V ½s
xðsÞ ¼ d½1
uðsÞ Z c V ðcÞ  vðxÞ dx 0

The left-hand side is the number of people finding a job, the right-hand side is the number of people losing their job. Solving for u(s) yields: uðsÞ ¼

d    d þ l V ½s þ xðsÞ
V ½s
xðsÞ 

ð7Þ

We can now define a labor market equilibrium as:  satisfying Definition 1. An equilibrium consists of the set {rVU(s), u(s), v(c), xðsÞ} the Equations (4)–(7).

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4. Characterization of the equilibrium The following proposition simplifies the analysis considerably: Proposition 1. Only a uniform vacancy distribution: 8cA[0,1]:v(c) ¼ v, is consistent with free entry.5 Proof. See the Appendix. The intuition behind Proposition 1 is that unemployed workers located at points with relatively many vacancies in their neighborhood have high outside options. In addition, there are relatively few unemployed workers at these locations because they find acceptable offers quickly. From Equation (4) it then directly follows that the good outside option as well as the low number of unemployed workers implies that the value of a vacancy at a crowded location must be relatively low. This violates the assumption of free entry and zero expected profits for all firms. Hence, a situation where some locations have more vacancies than others cannot be an equilibrium. Proposition 1 simplifies the notation of our analysis considerably since it implies that everything can be stated in terms of differences from the optimal allocation where workers and jobs are located at the same point. The functions W as well as the (other) equilibrium conditions can be stated in terms of this difference between s and c. In addition, x and VU can be expressed without any 2l argument. Let k ¼ rþd , then the equilibrium conditions simplify to:  Z x  1 2 U U gx rV 1 dx ð8Þ rV ¼ B þ bkv 2 0 kð1 bÞ K¼ 1 þ kvx

Z x  0

 ¼ rV U ¼ 1 W ðxÞ

1

1 2 gx 2

rV

U



dx

gx 2

ð9Þ

ð10Þ

The unemployment rate u can be solved as a post-endogenous variable: u¼

1 1 þ kvx

ð11Þ

5 Marimon and Zilibotti (1999) prove that given u(s) is uniform, v(c) must be uniform. As can be seen in the Appendix, a part of the proof is also related to this issue (i.e. Lemma A.3). Here, we prove that given that l(s) is uniform, both u(s) and v(c) must be uniform.

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Note that these relations depend on the composite parameter k, not on its separate components, r, d, and l. Substitution of (10) into the right-hand side of (8) yields: Z x 2

1 rV U ¼ B þ gbkv x2 dx x 2 0 1 ¼ B þ bgkvx 3 3 This is a similar expression as the Taylor expansion in Teulings and Gautier (2004, Proposition 2). However, in this model the relationship is exact and not an approximation. There are two assumptions made in this model that lie behind this result. First, the supply of labor is uniform and, as we showed, this implies that the vacancy distribution is uniform as well while it can take any form in Teulings and Gautier (2004). Second, Y(x) is a quadratic function. Further substitution of (10) in the left-hand side of (8) and solving for v yields: v¼

1

1 2 2gx

B

ð12Þ

1 3 3bgk x

Substitution of (10) into (9) gives: Z 1 kgð1 bÞ x 2 K¼ ðx x2 Þ dx 2 1 þ kvx 0 1 ð1 bÞgk 3 ¼ x 3 1 þ kvx Substitution of (12) in this equation and some rearrangement of terms yields:     1 1 1 b gx 2 K ¼ ð1 bÞbg2 kx 5 ð13Þ 1 B 2 3 9 Now we can state: pffiffiffi Proposition 2. For Ko23 2ð1 bÞkð1 rium with positive vacancy supply.

BÞ3=2 g

1=2

there exists a unique equilib-

The proof of this proposition follows directly from Equation (13), which has a qffiffiffiffiffiffiffiffiffiffiffi 2ð1 BÞ  For x4  single positive solution for x. this equilibrium is characterized by g

full unemployment and no vacancies. For larger K we get the trivial equilibrium where no vacancies are opened and where everybody is unemployed. 5. The cost of search

Teulings and Gautier (2004) introduce the concept of the cost of search X: the relative difference between output in the optimal job type and the flow value of

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127

unemployment:
 X  Y ð0Þ
rV U =Y ð0Þ

X equals foregone output due to unemployment, mismatch, and the cost of vacancy creation. Since output in the optimal assignment is normalized to one and since the flow value of unemployment is equal to the reservation wage, which is equal to the flow value of output in the marginal job type,  the costs of search takes a very simple form: rV U ¼ W ðxÞ, 1 X ¼ 1
rV U ¼ gx 2 2

ð14Þ

It is convenient to study the economy close to the Walrasian equilibrium, that is, for a large scale of the labor market, k. Then, the cost of search X are small, so that we can ignore higher order terms. Substitution of the expression for X in the previous equation and ignoring higher order terms in X yields:6 pffiffiffi 4 2 ð1
bÞbk 5=2 ð1
BÞK ffi X ð15Þ 9 g1=2 The elasticity of the cost of search X with respect to the scale of the labor market k is
5/2. The cost of search are minimal for an even distribution of bargaining power, b ¼ 1/2, and the cost of search are increasing in the cost of holding vacancies, K, and increasing in one minus the replacement rate, 
B=W ðxÞ  ffi 1
B.7 ½W ðxÞ The private cost of unemployment are equal to the probability of unemployment u times the loss in pay off during that period, 1–B. Similarly, the cost of vacancies are vK. The cost of suboptimal assignment are equal to the expected loss in output compared to the optimal assignment, conditional on employment, E ½Y ð0Þ
Y ðxÞjx  x . For all three components, we can derive (approximate) expressions: 8 2 bX 3 2 vK ffi ð1
bÞX 3 Z 1 x 1 2 1 gx dx ¼ X E ½Y ð0Þ
Y ðxÞjx  x  ¼ 3 x 0 2 uð1
BÞ ffi

ð16Þ

6 Here and in the sequel the symbol ffi means that we ignore relative deviations of order X, that is, factors 1+O[X]. 7


B 1
X
B W ðxÞ ¼ ¼ ð1
BÞð1 þ O½X Þ  W ðxÞ 1
X 8

2 2  ½W ðxÞ
B bgx

1 bX ð1
X
BÞ

3 
B ¼ ¼ 3 Ignoring factors of order (1+O[X]) yields the first equau½W ðxÞ 2 1 2  1
B
ð1
3bÞX ½W ðxÞ
Bþ 3bgx tion.

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Teulings and Gautier (2004) arrive at similar expressions as (15) and (16). For b ¼ 1/2 , all three components account for one-third of the cost of search. From an empirical point of view, these equations provide two ways for estimating the cost of search X. First, they can be estimated from the Equation (1). As an  illustration, setting the value of leisure B equal to zero (so that W ðxÞ B ffi 1), the bargaining power of workers b equal to 1/2, and the unemployment rate u equal to the natural rate, e.g. 5%, implies that X ¼ 15%. Second, the cost of search can be estimated from wage data, 1 E ½W ð0Þ
W ðxÞjx x  ¼ bX 3 This equality follows directly from the wage setting relationship: W ð0Þ
W ðxÞ ¼ b½Y ð0Þ
Y ðxÞ. One can use standard human capital variables and occupation and industry dummies to obtain an estimate of the mismatch indicator x. This mismatch indicator can be used to estimate  Setting b equal to 1/2, we can combine these figures to E½W ð0Þ
W ðxÞjx  x. get an estimate for X. This estimate of X is 25%, which is higher than the one derived from the first method, see Gautier and Teulings (2005). Wages satisfy:  1
W ðxÞ ¼ 1
g ð1
bÞx 2 þ bx2 2

Hence, wages are concave in the mismatch indicator x, i.e., taking the worker index s as given, they are concave in the job index c. Since x is distributed uniformly on the support ½0; 12 and since W0 (x)o0 for x>0, the wage offer distribution satisfies: Pr½W oW ðxÞ ¼ 1
2xðW Þ where x(W) is the inverse function of W(x) for x>0. Hence, the density function f(W) of the wage distribution reads: f ðW Þ ¼
2

dxðW Þ 2 ¼ dW gbxðW Þ

Since x(W) ¼ 0 in the optimal assignment, the wage offer distribution has a fat right tail, just as in Burdett and Mortensen (1998). This feature fits the data quite well. Finally, we derive an explicit expression for the unemployment rate by the substitution of Equation (16) in (15): pffiffiffi 2 6

u5=2 ð1
bÞ

½3u þ 2ð1
uÞb3=2

¼

pffiffiffi gK

kð1
BÞ3=2

We can conclude that all crucial relations of the hierarchical search model carry over to the circular model.

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6. Final remarks In this paper we made three contributions. First, we showed that the equilibrium properties of the more realistic hierarchical assignment model with search frictions carry over to the simple circular model with search frictions. Second, we showed that if skills are distributed uniformly over the population, then the vacancy distribution must be uniform. In that case, the circular model makes the same predictions as the standard stochastic job search model, with an additional constraint on the shape of the offer distribution, namely a fat upper tail, since the derivative of wages with respect to the job quality distribution is zero in the optimal job type. Hence, search in a hierarchical assignment model is similar to a specific form of stochastic job search model. Third, we showed that the resulting equilibrium is unique. This is all but obvious, since we use a quadratic contact technology, and this could therefore not be proven in the hierarchical assignment model of Teulings and Gautier (2004). The similarity of the equilibria for the circular and the hierarchical assignment model without on-the-job search provides hope that the circular model can also be a useful proxy for the hierarchical assignment model in a world with on-the-job search. That is what we do in a subsequent paper, Gautier et al. (2005). There, we show that on-the-job search reduces both the cost of search and the reservation wage.

Acknowledgment We thank an anonymous referee and participants at the Sandbjerg (2004) conference in honor of Dale Mortensen for useful comments.

References Albrecht, J.W., P.A. Gautier and S.B. Vroman (2004), ‘‘Equilibrium directed search with multiple applications’’, Mimeo, Tinbergen Institute, Amsterdam. Barlevy, G. (2002), ‘‘The sullying effect of recessions’’, Review of Economic Studies, Vol. 69, pp. 65–96. Becker, G. (1973), ‘‘A theory of marriage: part I’’, Journal of Political Economy, Vol. 81, pp. 813–846. Burdett, K. and M. Coles (1997), ‘‘Marriage and class’’, Quarterly Journal of Economics, Vol. 112, pp. 141–168. Burdett, K. and D. Mortensen (1998), ‘‘Wage differentials, employer size, and unemployment’’, International Economic Review, Vol. 39,pp. 257–273. Diamond, P.A. (1982), ‘‘Wage determination and efficiency in search equilibrium’’, Review of Economic Studies, Vol. 49, pp. 217–227.

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Gautier, P.A. and C.N. Teulings (2005), ‘‘How large are search frictions?’’, Journal of the European Economic Association. Forthcoming. Gautier, P.A., C.N. Teulings and A. van Vuuren (2005), ‘‘On-the-job search and sorting’’, Mimeo, Tinbergen Institute, Amsterdam. Marimon, R. and F. Zilibotti (1999), ‘‘Unemployment vs. mismatch of talents: reconsidering unemployment benefits’’, The Economic Journal, Vol. 109, pp. 266–291. Moscarini, G. (2001), ‘‘Excess worker reallocation’’, Review of Economic Studies, Vol. 68, pp. 593–612. Mortensen, D. (1982), ‘‘The matching process as a non-cooperative bargaining game’’, in: J.J. McCall, editor, The Economics of Uncertainty, Chicago: University of Chicago Press. Pissarides, C.A. (2000), Equilibrium Unemployment Theory, Cambridge: MIT Press. Rosen, S. (1974), ‘‘Hedonic prices and implicit markets: differentiation in pure competition’’, Journal of Political Economy, Vol. 82, pp. 34–55. Sattinger, M. (1975), ‘‘Comparative advantage and the distribution of earnings and abilities’’, Econometrica, Vol. 43, pp. 455–468. Shi, S. (2002), ‘‘A directed search model of inequality with heterogeneous skills and skill based technology’’, Review of Economic Studies, Vol. 69, pp. 467–491. Shimer, R. (2004), ‘‘The assignment of workers to jobs in an economy with coordination frictions’’, Mimeo, University of Chicago. Shimer, R. and L. Smith (2000), ‘‘Assortative matching and search’’, Econometrica, Vol. 68, pp. 371–398. Teulings, C.N. (1995), ‘‘The wage distribution in a model of the assignment of skills to jobs’’, Journal of Political Economy, Vol. 103, pp. 280–315. Teulings, C.N. and P.A. Gautier (2004), ‘‘The right man for the job’’, Review of Economic Studies, Vol. 71, pp. 553–580.

Appendix Proof of Proposition 1. Define Cðs; DÞ ¼

Z

sþD s D

Y ½xðs; cފvðcÞ dc

We first prove the following lemmas:9

9

For the proofs in the Appendix we assume that v is continuous and differentiable. Although this is a restrictive assumption, similar but more lengthy proofs can be derived for the non-continuous and non-differentiable case. These proofs can be provided by the authors upon request.

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Labor Market Search with Two-Sided Heterogeneity 0

Lemma A.1. C(s, D) is only constant on s A [0,1] when v (c) ¼ 0 for every cA[0,1]. Proof. Taking derivatives of C(s, D) with respect to s gives the following condition for C(s, D) to be constant: Cs ðs; DÞ ¼ Y ½xðs; s þ Dފvðs þ DÞ Y ½xðs; s Z sþD þ Y 0 ½xðs; cފx0 ðs; cÞvðcÞ dc ¼

Z

Dފvðs

DÞ

s D sþD

s D

Y 0 ½xðs; cފv0 ðcÞ dc ¼ 0

The second line is obtained by using partial integration. Since Y(s, c) is symmetric around s for every s, v(c) must be symmetric around s for every s as well. This implies that v0 (c) must be constant. Since by the properties of our circle, v(c) ¼ v(c+k); k 2 N this can only be realized when v0 (c) ¼ 0. Lemma A.2. (i) The reservation wages are constant for every sA[0,1] only if  0 ÞÞ4Cðs1 ; xðs  1 ÞÞ v0 (c) ¼ 0 for every cA[0,1]. (ii) 8s0, s1A[0,1] such that Cðs0 ; xðs it must be that rVU(s0)>rVU(s1). Proof. Assume that v0 (c)6¼0 and that rVU(s) is constant for every sA[0,1]. Hence  is constant as well. According to Lemma A.1 there must be s0, s1A[0,1] such xðsÞ  0 ÞÞ4Cðs1 ; xðs  1 ÞÞ. This implies that that Cðs0 ; xðs Z s0 þxðs  0Þ  lb Y ½xðs0 ; cފ rV U ðs0 Þ vðcÞ dc rV U ðs0 Þ ¼ B þ r þ d s0 xðs  0Þ Z s0 þxðs  1Þ  lb ¼Bþ Y ½xðs0 ; cފ rV U ðs0 Þ vðcÞ dc r þ d s0 xðs  1Þ Z s0 þxðs  1Þ  lb 4B þ Y ½xðs1 ; cފ rV U ðs0 Þ vðcÞ dc r þ d s0 xðs  1Þ ¼ rV U ðs1 Þ

This is in contradiction with the assumption that rVU(s) is constant. The second part of this lemma follows directly. Lemma A.3. The unemployment rate is constant for every sA[0,1] if and only if v0 (c) ¼ 0, cA[0,1]. In addition when C(s0, D)>C(s1, D) then u(s0)ou(s1). Proof. This is obtained by taking derivatives of u(s)     us ðsÞ ¼ vðs þ xðsÞÞ vðs xðsÞÞ þ x 0 ðsÞ½vðs þ xðsÞÞ þ vðs Z s0 þxðsÞ  1  uðsÞð1 uðsÞÞ vðcÞ dc  s0 xðsÞ

 Š xðsÞÞ



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Pieter A. Gautier, Coen N. Teulings and Aico van Vuuren

We know that x 0 ðsÞ ¼ 0 if and only if v(c) ¼ 0 for all cA[0,1]. Hence, when v(c) ¼ 0 for all c we have that us(s) ¼ 0. It remains to prove that the unemployment rate cannot be constant when v0 (c)6¼0 for some c. From the above equation we have x 0 ðsÞ ¼

 vðs þ xðsÞÞ vðs  vðs þ xðsÞÞ þ vðs

 xðsÞÞ  xðsÞÞ

  This can only be satisfied when vðs þ xðsÞÞ ¼ vðs xðsÞÞ, excluding the event that v(c)6¼0 for some cA[0,1]. The second part follows on similar lines. Proof of Proposition 2. Let v0 (c)6¼0 for some cA[0,1]. Then it must be possible to construct two values c0 and c1 such that C½c0 ; Dðc0 ފ4C½c1 ; Dðc1 ފ

Using this and Lemmas A.2 and A.3 gives: Z 1   l K ¼ ð1 bÞ uðsÞ max Y ½xðs; c0 ފ rV U ðsÞ; 0 ds rþd 0 Z 1   l oð1 bÞ uðsÞ max Y ½xðs; c1 ފ rV U ðsÞ; 0 ds rþd 0 ¼K

This is a contradiction and hence v0 (c) ¼ 0, cA[0, 1].

CHAPTER 6

The Weak Pareto Law in Burdett–Mortensen Equilibrium Search Models Gerard J. van den Berg Abstract Empirical income distributions almost universally satisfy the weak Pareto law. We show that equilibrium search models based on the Burdett–Mortensen model also give rise to such income distributions. We explicitly take account of the remuneration for firm owners. Remarkably, the admissible set of values for the index parameter of the Pareto tail corresponds exactly to the range of values (1,2] found in empirical studies on income distributions.

Keywords: income distribution, productivity, regular variation, wages, job search JEL classifications: J3, D83, J42, J6, C72 1. Introduction Weak Pareto laws state that the right-hand tail of a distribution behaves in the limit as a simple power function (see e.g. Kra¨mer and Ziebach, 2003, for an overview). It is well known that empirical income distributions almost universally satisfy such laws (see e.g. Aoyama et al., 2000, for a survey). Theoretical explanations usually focus on markets where agents’ incomes follow simple stochastic processes, which may be justified by kinetic models of simple selforganizing markets where agents engage in random trading opportunities with each other (see e.g. Cordier et al., 2004). In this paper, we show that equilibrium search models based on the model by Burdett and Mortensen (1998) also give rise to income distributions satisfying a CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75006-9

133

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

134

Gerard J. Van Den Berg

weak Pareto law. Our main model builds on the Burdett–Mortensen model in which productivity is heterogeneous across firms. This model has been extensively studied by Bontemps et al. (2000). We extend this model by introducing remuneration for the owners of the firms. The remuneration is based on the steady-state profit flow of the firm. It can be thought of as the earnings of the CEO of the firm or its entrepreneur, or the dividend yield of the major shareholder, or the income of the star employee (like in the entertainment sector; see De Vany and Walls, 2002). In practice such incomes are related to the profit flow of the firm, and it has been acknowledged in the empirical literature on income distributions that these incomes drive the right-hand tail of these distributions (see e.g. Atkinson, 2003). For simplicity, we refer to the recipients of such incomes as firms owners, and to their incomes as earnings. The income distribution in the model is a mixture of the distribution of wages across employees and the distribution of earnings across firm owners. Since we are only concerned with the right-hand tail of the income distribution, we can easily generalize our results to more general models where, for example, workers are heterogeneous in terms of their value of leisure or unemployment benefits. The derivation of the Pareto law is fundamentally different from that in the above-mentioned literature on kinetic models. First, in our model, workers and firms engage in long-lasting matches, a firm may be simultaneously matched with multiple workers, and workers may jump from lower-paying jobs to higherpaying jobs. The model feature that workers can adopt a strategy that systematically moves them up on the wage ladder is a distinct advantage from an empirical point of view. Second, we require a restriction on the set of equilibrium wage distributions in order to narrow down the set of income distributions to the distributions satisfying the Pareto law. Specifically, we do not allow the density of the wage distribution to be vertical at the maximum of its support. This restriction is verified by an abundant empirical literature. Indeed, the density is generally found to be horizontal at this point. From this innocent restriction we do not only obtain the Pareto law for the income distributions, but we simultaneously obtain that the coefficient of the Pareto tail is exactly in the narrow range [ 2, 1) that is generally found in the empirical literature. In Section 2 we present the model. Section 3 contains the main results, and Section 4 concludes and outlines some avenues for further generalizations. 2. Model framework 2.1. The Burdett– Mortensen equilibrium search model with heterogeneous productivity across firms We start with a description of the Burdett–Mortensen model in which productivity is heterogeneous across firms. Since this model has been described in numerous studies (see e.g. Bontemps et al., 2000; Mortensen, 2003), the present

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exposition can be very brief. We are not concerned with multiplicity or existence of equilibria (see Bontemps et al., 2000 and Van den Berg, 2003, for details). The model considers a labor market consisting of fixed continuums of workers and firms. The supply side of the model is equivalent to a standard partial job search model with on-the-job search. Workers obtain wage offers, which are random drawings from the endogenous wage offer distribution F(w), at a rate l0 when unemployed and l1 when employed. Whenever an offer arrives, the decision has to be made whether to accept it or to reject it and search further for a better offer. Firms post wage offers and they do not bargain over the wage. Layoffs accrue at the constant exogenous rate d. The opportunity cost of employment is denoted by b and is assumed to be constant across individuals and to be inclusive of unemployment benefits and search costs. We take 0ol0, l1, d oN and bZ0. For expositional reasons, we restrict attention to the limiting case in which the discount rate is infinitesimally small. The results are robust with respect to this. The optimal strategy of the workers has the properties that unemployed workers accept any wage offer exceeding their reservation wage f, and employed workers accept any wage offer exceeding their current wage. An active firm does not offer a wage below f, so that all wage offers will be acceptable for the unemployed. Note that in this paper we are only concerned with the right-hand tail of the income distribution. We can easily generalize our results to more general models where for example, workers are heterogeneous in terms of their value of b. Bontemps et al. (1999) show that this does not qualitatively affect the right-hand tail if the support of b does not stretch out too far to the right. In fact, many model aspects that only concern wages and job search behavior in unemployment and at middle and lower wage ranges are irrelevant for our analysis. We therefore make the convenient assumption that there is a fixed minimum wage level or wage floor w with w  f. Let the distribution of wages paid to a cross-section of employees (or the wage distribution) have distribution function G. By imposing that the steadystate flows into and out of this cross-section are equal it follows that GðwÞ ¼

F ðwÞ , 1 þ kF ðwÞ

ð1Þ

where k :¼ l1 =d and F ðwÞ :¼ 1 F ðwÞ. Next, we derive the steady-state supply of labor l(w) to an employer setting a wage w. Somewhat loosely, one may say that this must equal the number of workers earning w in a steady state, divided by the number of firms paying w in the steady state. Note that it is assumed that a firm pays the same wage to all of its employees. As a result, lðwÞ ¼ A

1 2

ð1 þ kF ðwÞÞ

,

ð2Þ

where m and n denote the measures of workers and firms in the market, A :¼ mk0 ð1 þ kÞ=ðnð1 þ k0 ÞÞ, and k0 :¼ l0 =d. The equation above only holds if

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F does not have mass points, which is an equilibrium property. Note that l(w) increases in w. Also note that the firm size of a firm paying w only depends on w by way of the rank order F(w) of w. This is because employed workers’ strategies are myopic: they are only concerned with whether the offered wage exceeds the current wage or not. Now consider a firm with a flow p of marginal revenue product generated by employing one worker. For convenience, we assume that p does not depend on the number of employees, i.e. we assume that the production function is linear in employment. We refer to this firm as a firm of type p and to p as the (labor) productivity of this firm. Each firm sets a wage w so as to maximize its steadystate profit flow pðp; wÞ ¼ ðp

wÞlðwÞ,

given F and given the behavior of workers. We assume that p is continuously distributed across firms in the market, with distribution function G. The lower bound of the support of G is denoted by p. For convenience we take p  w. Following Bontemps et al. (2000), we make the following regularity assumptions on G: Assumption 1. The distribution G of p across firms has a density g that is continuous and positive on its connected support ðp; pÞ. The mean of p is finite. In equilibrium, the profit-maximizing wage for a firm of type p defines a mapping w ¼ KðpÞ, with K increasing and continuous and K(p)op. Consequently, the distribution function of wage offers is F ðwÞ ¼ GðK 1 ðwÞÞ, and F and G have no mass points and have a connected support that can be denoted as ðw; wÞ.1 It is useful to note that for a given firm of type p, the first-order condition @pðp; wÞ=@w ¼ 0 gives
 ð3Þ 1 þ kF ðwÞ þ 2kf ðwÞðp wÞ ¼ 0,

under the restriction that w  w, where w ¼ KðpÞ and f(w) is the density associated with F(w). This can be rewritten as K 0 ðpÞ ¼

2kgðpÞ ðp 1 þ kGðpÞ

KðpÞÞ,

ð4Þ

with G :¼ 1 G. Solving for K yields w ¼ KðpÞ ¼ p

1

2

ð1 þ kGðpÞÞ

"Z

w

p

dx

#


2
1 þ kGðxÞ

ð5Þ

As shown by Van den Berg and Van Vuuren (2002), these results imply that thepsteady-state ffiffiffiffi firm size distribution has a density proportional to l 3=2 . Specifically, it equals 12 Ak1l 3=2 on 2 A=ð1 þ kÞ oloA.

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1

By invoking F ðwÞ ¼ GðK ðwÞÞ, we obtain the expressions for F and G in terms of the primitives of the model. In general, it is not possible to obtain closed-form expressions for K(p), F(w) or G(w). The profit flow pðpÞ :¼ pðp; KðpÞÞ of a firm with productivity p offering K(p) is Z p dx pðpÞ ¼ A ð6Þ
2 . w 1 þ kGðxÞ Note that p0 ðpÞ ¼ lðKðpÞÞ. Together with the facts that l and K are increasing, this implies that the profit flow p(p) is convex in p, indicating that highproductivity firms have disproportionally more monopsony power in the labor market than low-productivity firms. We return to this later on. 2.2. Remuneration of firm owners We now extend the model by incorporating remuneration based on the firms’ profits, like the earnings of the CEO of the firm or its entrepreneur, or the dividend yield of the major shareholder, or the income of the star employee. As a start, let us assume that each firm has one owner. Further, he/she receives a remuneration (or earnings) flow z that equals a fraction b of the steady-state profit flow. So, the owner of the type-p firm receives an earnings flow OðpÞ ¼ bðp KðpÞÞlðKðpÞÞ. We focus on the steady-state distribution of the earnings, ignoring short-term fluctuations in the profit flow originating from short-term fluctuations in the firm size. This seems reasonable, given that the income data used to empirically verify Pareto laws are always time-aggregated into yearly intervals. As can be seen from the results below, the number of owners or individuals benefiting from the firms’ profits can be changed to any fixed positive number. Indeed, since we are only concerned with the right-hand tail of the income distribution, we may specify the number of owners or star employees, or the remuneration fraction b, to be a smooth function of p. The distribution H of income y across individuals in the population is a mixture of the distribution G of wages across employed workers, the distribution J of earnings z across firm owners, and the distribution of benefits b across unemployed workers. Clearly, J satisfies JðzÞ ¼ GðO 1 ðzÞÞ. We assumed that b is constant, but even if we generalize this toward restrictions that the support of its distribution is not stretched out too far to the right, then the income distribution across unemployed workers is irrelevant for the right-hand tail of H. Therefore, we ignore it throughout the remainder of the paper. Hence, HðyÞ ¼ ¼

mk0 n m JðyÞ þ GðyÞ þ Iðy  bÞ nþm ðn þ mÞð1 þ k0 Þ ðn þ mÞð1 þ k0 Þ mk0 GðK 1 ðyÞÞ ðn þ mÞð1 þ k0 Þð1 þ kGðK

1

ðyÞÞÞ

þ

nGðO 1 ðyÞÞ mIðy  bÞ þ
ð7Þ nþm ðn þ mÞð1 þ k0 Þ

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3. The right-hand tail of the income distribution 3.1. Finite maximum wage Bontemps et al. (2000) provide some results on the behavior of G at the upper bound of its support. First of all, the upper bound w is finite and equal to ! Z p 1 dxo1, ð8Þ 1 w¼wþ 2 w ½1 þ kGðxފ so wage distributions with support extending to N are not supported by the model. This confirms that firms with a high productivity have very high monopsony power. Indeed, if p ¼ 1 then the monopsony power index (p w)/p tends to its maximum value 1 for high values of p. This is in sharp contrast to the position of low-productivity firms. If p ¼ w then the firms with the lowest productivity have a monopsony power index of 0. The latter is obviously due to the fact that firms must pay more than the wage floor. For firms at the lower end of the market, the wage floor destroys monopsony power almost completely. Let us return to the wages set by high-p firms. The values of p may be widely dispersed among high-p firms, but this does not result in the wages being widely dispersed as well. Since pðp; wÞ ¼ ðp wÞlðwÞ and l only depends on w by way of the rank order of p, there is an incentive for a high-p firm to set a wage that is only marginally higher than that of the firms with a somewhat lower p. Formally, Equations (4) and (5) imply that the value of K0 (p), given g on (0, p), varies linearly in the value of the density g at p, so that a locally wide dispersion of p (i.e. a low value of g(p)) does not lead to a locally wide dispersion of wages. Note that when search is undirected, it takes on average a very long time for an employee of a high-productivity firm to find a job at a firm with even higher productivity. Similarly, further increasing the wage does lead to a significant increase in the inflow of workers from elsewhere. Of course, there is a countervailing force due to the competition between firms for workers. This can be most clearly seen when we tilt the productivity distribution toward higher values of p. Specifically, if G is replaced by a distribution that is first-order stochastically dominating, with a heavier right-hand tail, then w increases.

3.2. Tail weight indicators It is useful to define sets of distribution functions with right-hand tails that are well behaved in the sense that they allow for certain limits to exist. We denote the distribution (function), survivor function and density of a generic random variable by H; H :¼ 1 H, and h, respectively.

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Definition 1. Regular Tail and Rapid Tail. A distribution of a non-negative random variable has a regular tail with index aA( N, 0) if there is a cA(0, N) such that  lim x a HðxÞ ¼ c.

x!1

We denote this as H 2 RT a . A distribution H : ½x0 ; 1Þ ! ð0; 1Þ of a non-negative random variable has a rapid tail if for all a 2 ð 1; 0Þ,  lim x a HðxÞ ¼ 0.

x!1

We denote this as H 2 RT 1 . The extension to the definition of a regular or rapid tail of a density (or indeed general functions) is straightforward. For example, a density of a non-negative random variable has a regular tail with index aA( N, 1) if there is a cA(0,N) such that limx!1 x a hðxÞ ¼ c. We denote this as h 2 fRT a : 1oao 1g. These concepts are closely related to the familiar concepts of regular variation and rapid variation, which are defined as Definition 2. Regular Variation and Rapid Variation. A distribution of a nonnegative random variable is regularly varying with index aA( N, 0) if  HðtxÞ ¼ ta  x!1 HðxÞ lim

8t40.

We denote this as H 2 RV a . A distribution of a non-negative random variable is rapidly varying if  HðtxÞ ¼0  x!1 HðxÞ lim

8t41.

We denote this as H 2 RV 1 . Again, the extension to the definition of a regularly or rapidly varying density is straightforward. See, for example, Feller (1971), Mikosch (1999) and Andrade and O’Hagan (2004), for overviews of these concepts. Distributions with H 2 RV 1 (or, simply stated, distributions in RV 1 ) are also called light-tailed distributions, while distributions in fRV a : 1oao0g are also called heavy-tailed distributions. See Andrade and O’Hagan (2004) for the close relation to the ‘‘credence’’ measure of tail weight. The sets RRT :¼ fRT a : 1  ao0g and RRV :¼ fRV a : 1  ao0g are large in the sense that they contain virtually every well-known family of

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distributions for non-negative random variables with support stretching to infinity. Many families are in RT 1 and RV 1 , like the exponential, lognormal, Weibull and gamma families and the family of normal distributions truncated from below at non-negative values. The sets fRT a : 1oao0g and fRV a : 1oao0g contain the Pareto, Student-t and Singh–Maddala families and certain families of distributions with infinite means and variances. It can be shown that H 2 RT a ) H 2 RV a , but the reverse does not hold. Loggamma distributions are rare examples of distributions that are in RVa but not in RTa. Also, if the density h 2 RT a 1 then H 2 RT a , and if h 2 RV a 1 then H 2 RV a . The reverse of these statements does not hold in general, but it does for the above-mentioned families. 3.3. Mandelbrot’s weak Pareto law Mandelbrot’s weak Pareto law states that the income distribution H(y) satisfies the following: there are aA( N,0) and cA(0,N) such that lim y a HðyÞ ¼ c.

y!1

In other words, H 2 RT a for some aA( N, 0). Note that this is a substantial restriction compared to H 2 RRT . In the context of the Pareto law, a is called the Pareto tail index. We now proceed toward a justification of this law in the context of our model. First, observe that because the highest wage is finite, we need the highest productivity to be infinite, otherwise the income distribution has a finite support, violating the Pareto law. Because of this, the right-hand tail of the income distribution equals the right-hand tail of the earnings distribution of firm owners. It has been acknowledged in the empirical literature on income distributions that these earnings drive the right-hand tail of the income distribution (see e.g. Atkinson, 2003). Incidentally, from Bontemps et al. (2000) we know that p ¼ 1 implies that the density g of G at w satisfies lim gðwÞ ¼ 0. w"w

In fact, this result immediately follows from Equations (3) and (2). Moreover, it can also be reversed, so lim gðwÞ ¼ 0 w"w

iff p ¼ 1,

lim gðwÞ 2 ð0; 1Þ w"w

iff po1.

The requirement that p ¼ 1 is thus aligned with the property that the wage density has a right-hand tail that decreases toward zero at the upper bound of its support. The latter is a typical feature of empirical wage densities.

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We now impose some smoothness on the set of admissible productivity distributions G(p). In particular, we take Assumption 2. The density g of p across firms has a rapid tail or a regular tail, so gARRT. (which we take to imply that p ¼ 1). Recall that this is a very weak restriction as it includes virtually all families of distributions of non-negative random variables with support stretching to infinity. The main result of the paper is: Proposition 1. Given the regularity Assumptions 1 and 2, the income distribution satisfies Mandelbrot’s weak Pareto law iff limw"w g0 ðwÞ4 1. In that case, the Pareto tail index a of the income distribution satisfies 2rao 1, and all of these values are admissible. Proof. see the Appendix. We must emphasize two aspects. First, the proposition states that the Pareto law is aligned with the empirical fact that the wage density has a right-hand tail that is not vertically sloped at the upper bound of its support. The latter is a universal feature of empirical wage densities, even if one conditions on worker and firm characteristics. Indeed, wage densities typically have a horizontal righthand tail, leading to limw"w g0 ðwÞ ¼ 0. So, by requiring the resulting equilibrium wage density to have a property that is innocent from an empirical point of view, we obtain that the set of possible income distributions satisfies the Pareto law. From the proof we obtain some intuition for this. One needs a productivity distribution G with a heavy right-hand tail to obtain a wage distribution with an empirically sensible right-hand tail. Without the former, the highest wages are so compressed that the wage density is vertical. If G has a thin tail then the high-p firms are relatively close in terms of their productivity, and this makes it less attractive for any of them to offer a wage above the range offered by the others. The required heavy tail of the productivity distribution also gives rise to a heavy-tailed profit distribution, in turn giving rise to an empirically sensible income distribution. The second aspect of the proposition that we need to emphasize is that it shrinks the set of possible equilibrium income distributions in two steps. Mandelbrot’s law limits the set of possible H to RTa with aA( N,0). But the imposition of the restriction that delivers the validity of this law simultaneously achieves that the tail index a is restricted to the set [ 2, 1). Remarkably, the set [ 2, 1) for a corresponds exactly to the range of values found in empirical studies on income distributions (see the references in Section 1). The tail-index values aZ 1 are ruled out because of the assumed finiteness of the mean of p.2 2

It is relatively straightforward to extend the results to Kakwani’s weak Pareto law, which is defined as follows (see Kra¨mer and Ziebach, 2003): there is an a 2 ð0; 1Þ such that limy!1 yhðyÞ=HðyÞ ¼ a. This is equivalent to h 2 RV a 1 (see e.g. Mikosch, 1999).

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4. Conclusions We demonstrated that a version of the Burdett–Mortensen equilibrium search model gives rise to income distributions satisfying the weak Pareto law. In accordance to the empirical evidence, the upper tail of the income distribution is not based on wages but on incomes of firm owners or star employees. Remarkably, the admissible set of values for the index parameter of the Pareto tail corresponds exactly to the range of values found in empirical studies on income distributions. The analysis has also provided some insight into competition at the high end of a market with repeated search. One may say that, among the very productive firms, competition is much less severe than among the other firms. This is because, with undirected search, the relative rather than the absolute wage level drives jobto-job mobility. As a consequence, one needs a productivity distribution with a very heavy right-hand tail to obtain a wage distribution with an empirically sensible right-hand tail. And, as we have shown, the required tail of the productivity distribution is such that it also gives an empirically sensible income distribution. The above intuition gives rise to some issues for further research. First, it would be interesting to investigate the implications of more directed search of high-wage firms by workers. Second, it may be interesting to relax the assumption that the mean productivity across firms is finite, in order to allow for even heavier tails of the productivity distribution. Finally, it may be interesting to endogenize the productivity distribution. Acknowledgement I thank Dale Mortensen for the inspiration that his work has given me to study equilibrium search models. Also, thanks to an anonymous referee for useful comments. References Andrade, J.A.A. and A. O’Hagan (2004), ‘‘Bayesian robustness modelling of scale and location parameters using regularly varying distributions’’, Working paper, University of Sheffield, Sheffield. Aoyama, H., Y. Nagahara, M.P. Okazaki, W. Souma, H. Takayasu, and M. Takayasu (2000), ‘‘Pareto’s law for income of individuals and debt of bankrupt companies’’, Kyoto: Working paper, Kyoto University. Atkinson, A.B. (2003), ‘‘Income inequality in OECD countries: data and explanations’’, CESifo Economic Studies, Vol. 49, pp. 479–513. Bontemps, C., J.-M. Robin and G.J. van den Berg (1999), ‘‘An empirical equilibrium job search model with search on the job and heterogeneous workers and firms’’, International Economic Review, Vol. 40, pp. 1039–1074. Bontemps, C., J.M. Robin and G.J. van den Berg (2000), ‘‘Equilibrium search with continuous productivity dispersion: theory and non-parametric estimation’’, International Economic Review, Vol. 41, pp. 305–358.

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Burdett, K. and D.T. Mortensen (1998), ‘‘Wage differentials, employer size, and unemployment’’, International Economic Review, Vol. 39, pp. 257–273. Cordier, S., L. Pareschi and G. Toscani (2004), On a kinetic model for a simple market economy, Orle´ans: Working paper, University of Orle´ans. De Vany, A.S. and W.D. Walls (2002), Momentum, motion picture profit, and the curse of the superstar, Irvine: Working paper, University of California. Feller, W. (1971), An Introduction to Probability Theory and Its Applications II, New York: Wiley. Kra¨mer, W. and T. Ziebach (2003), The weak Pareto law and regular variation in the tails, Dortmund: Working paper, Universita¨t Dortmund. Mikosch, T. (1999), Regular variation, subexponentiality and their applications in probability theory, Groningen: Working paper, Groningen University. Mortensen, D.T. (2003), Wage Dispersion, Cambridge: MIT Press. Van den Berg, G.J. (2003), ‘‘Multiple equilibria and minimum wages in labor markets with informational frictions and heterogeneous production technologies’’, International Economic Review, Vol. 44, pp. 1337–1357. Van den Berg, G.J. and A. van Vuuren (2002), ‘‘Using firm data to assess the performance of equilibrium search models of the labor market’’, Annales d’E´conomie et de Statistique, Vol. 67/68, pp. 227–256. Appendix: Proof of Proposition 1 We first characterize the shape of the right-hand tail of G in terms of the shape of the right-hand tail of G. Lemma 1. Let G satisfy the regularity Assumptions 1 and 2. In equilibrium, the wage density gðwÞ satisfies limw"w g0 ðwÞ ¼ 0 if and only if limp!1 p3 gðpÞ ¼ 1. If limp!1 p3 gðpÞ ¼ 0 then limw"w g0 ðwÞ ¼ 1. If limp!1 p3 gðpÞ 2 ð0; 1Þ then limw"w g0 ðwÞ 2 ð 1; 0Þ. This lemma is similar to Bontemps et al. (2000)’s Proposition 8, and the proof is along their lines. By differentiating Equation (3) with respect to w (keeping in mind that p is related to w by way of w ¼ KðpÞ; note that the differentiation is allowed in the interior of G), we obtain f 0 ðwÞ ¼

f ðwÞ 2ðp wÞ

ð1 þ kF ðwÞÞ

ðdp=dw 1Þ . 2kðp wÞ2

From Equation (4) we obtain dp=dw ¼

1 þ kF ðwÞ . 2kgðpÞðp wÞ

By substituting this into the above expression for f 0 ðwÞ we obtain f 0 ðwÞ ¼

f ðwÞ 2ðp wÞ

ð1 þ kF ðwÞÞ2 ð1 þ kF ðwÞÞ . þ 2kðp wÞ2 4k2 gðpÞðp wÞ3

ðA:1Þ

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Now let w " w (so p " p ¼ 1). The first and third terms on the right-hand side both converge to zero, since wo1 and limw"w f ðwÞ ¼ 0. Thus, the limiting behavior of f 0 ðwÞ is determined by the limiting behavior of the second term on the right-hand side. The latter is determined by the limiting behavior of p3 gðpÞ. Under Assumption 3.3, the limits of the terms in Equation (9) as p ! 1 are well defined. This immediately leads to the text of the lemma with g0 replaced by f0 . Next, it is straightforward to show that these results on the limiting behavior of f 0 ðwÞ are also valid for g0 ðwÞ. This completes the proof of Lemma 1. With g 2 RRT (so g 2 RT 1 or g 2 fRT a : 1oao 1g), the limit limp!1 p3 gðpÞ depends on a. It is infinite iff 3oao 1, it equals a finite positive value iff a ¼ 3, and it is zero otherwise. Note that, given Assumption 2, g 2 fRT a : 3  ao 1g is equivalent to G 2 fRT a : 2  ao0g. In fact, in the latter set we can rule out values 1rao0 because they give rise to distributions with infinite means which are ruled out by Assumption 1. In sum, we have demonstrated that, under some regularity conditions on the productivity distribution, limw"w g0 ðwÞ4 1 is equivalent to G 2 fRT a : 2  ao 1g, where all of these values of a are admissible. Now let us consider the income distribution H. We aim to prove that for any a 2 ½ 2; 1Þ the statement that G 2 RT a is equivalent to the statement that H satisfies Mandelbrot’s weak Pareto law, with H 2 RT a . n GðO 1 ðyÞÞ. Consequently, For y  w, Equation (7) implies that HðyÞ ¼ nþm a a y y lim ¼ lim n 1 y!1 HðyÞ y!1 nþmGðO ðyÞÞ  a y n þ m ðO 1 ðyÞÞa
¼ lim y!1 O 1 ðyÞ n GðO 1 ðyÞÞ Oð:Þ is continuous and strictly increasing, and limp!1 OðpÞ ¼ 1 due to the fact that both K(p) and l(K(p)) converge to a finite number as p-N. So, if the limit in the above equation exists then it can be expressed as   OðpÞ a n þ m pa ¼ lim
ðA:2Þ y!1 p n GðpÞ Consider the first ratio on the right-hand side. This equals   bðp KðpÞÞlðKðpÞÞ a , p which converges to (bA)a. Now consider the third ratio on the right-hand side of (10). This converges to some positive constant iff G 2 RT a . Consequently, for any aA[ 2, 1] the statement that G 2 RT a is equivalent to the statement that H satisfies Mandelbrot’s weak Pareto law, with H 2 RT a .

CHAPTER 7

Competitive Auctions: Theory and Application John Kennes Abstract The theory of competitive auctions offers a coherent framework for modeling search frictions as a non-cooperative game. The theory represents an advancement over cooperative approaches that make exogenous assumptions about how output is divided between buyers and sellers and about the forces that bring buyers and sellers into local markets. Moreover, unlike price-posting models, which fix the terms of trade prior to matching, competitive auction models have a bidding process that allocates the good (or service) to the highest valuation bidder at a price equal to the second highest valuation. Therefore, the competing auction model is more robust to problems in which there are heterogenous valuations. This paper offers an introduction to the theory and application of competing auction theory.

Keywords: auctions, search, competition JEL classifications: E24, J31, J41, J64, D44 1. Introduction At the heart of every economic theory is a description of how people exchange. Economists generally choose between one of the two possible extremes: Walrasian or random matching. In the Walrasian extreme, the cost of communication between buyers and sellers is assumed to be zero. Therefore, buyers and sellers in a Walrasian economy need only report their characteristics to a central

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mechanism designer and a set of transfers are then carried out using this information.1 At the opposite extreme is random matching. In random-matching models, the cost of communication is assumed to be almost insurmountable. Instead, buyers and sellers are brought together by an exogenous matching technology and only then can they communicate with their potential trading partner(s). Recent research has developed models in which trade is neither Walrasian nor random matching. These so-called directed search models allow a range of communications that falls within the extremes of Walrasian and randommatching environments. For example, in a directed search model we might assume that sellers can easily communicate their locations to buyers, but we might also assume that buyers cannot communicate with each other over which seller to visit. This type of model has equilibrium mixed strategies for buyers concerning their location decision over sellers. Moreover, the mixed strategy equilibrium is a useful method to describe the difficulties of coordination in a large decentralized market. Thus, the early applications of these models have been to the theory of unemployment. Directed search models confront a number of modeling issues that are not found in either Walrasian or random-matching environments. The basic premise is that the selling mechanism of each seller matters for both its selling price and probability of trade with buyers. Moreover, this demand function is not assumed, as in models of monopolistic competition, but is instead derived as the outcome of a market with coordination frictions. In this paper, for example, I derive a simple static model of competitive auctions in which sellers set reserve prices. The model illustrates how competition affects pricing. I demonstrate that sellers choose positive reserve prices in small markets but in the limit, as the market is made large, the sellers choose a reserve price of zero. In other words, intense price competition between sellers leads to the very simple selling mechanism of an auction without reserve price. The rest of this paper is dedicated to showing that this basic model has many applications. Perhaps, the most fundamental application of competing auction theory is the development of a fully operational, dynamic model of the labor market (see Julien et al., 2000). These models offer a very tractable alternative to closely related undirected search models, which are described in Pissarides (2000). The advantage of the competing auction model is it avoids two basic assumptions foreign to conventional general equilibrium theory: (1) wages determined by exogenous Nash bargaining rules and (2) meeting arrival rates determined by an exogenous matching technology. A surprisingly robust feature of competitive auction equilibrium is efficiency. The basic premise – that auctions generate efficient outcomes in matching games – was first advanced by Mortensen (1982). The Mortensen rule, roughly

1

This central figure is often referred to as the Walrasian auctioneer.

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stated, is that the surplus of a match should go to the initiator of a match. I discuss how the Mortensen rule can be stated as a set of axioms that give outcomes equivalent to competitive auctions without reserve prices. I also show that this axiomatic rule gives efficiency in a set of matching games for which the well-known Hosios rule fails. These games include markets of finite size, where the matching technology displays decreasing returns to scale and markets with heterogenous buyers where the matching function contains more than two arguments. The examples also illustrate that a competitive auction model of ex ante investments is not subject to a holdup problem and that such a model can lead to efficient technology dispersion. Competitive auction theory offers a very simple framework in which to study endogenous job destruction. This problem is difficult to study in an alternative framework such as a wage-posting model with coordination frictions, because the posted wage never leaves the worker completely satisfied – they are always looking for more. The auction mechanism gives them what they want – everything – but only if they have multiple offers in the type of job they are currently in. Consequently, the competitive auction model of the labor market offers a much more tractable theory of on-the-job search. To illustrate, I extend the basic dynamic model to have (i) heterogenous jobs that are distinguished by their productivity and capital cost and (ii) heterogenous job searchers who are distinguished by their employment status. On-the-job search leads to wage changes as workers move into higher productivity jobs. The equilibrium is constrained efficient even though employed workers sometimes search. Many models of competitive auction treat the identity of buyers and sellers as exogenous. This is not an innocuous assumption. Suppose that there is large number of red and blue agents who are to be matched together as red–blue pairs. The question arises: what characteristics of red and blue agents cause one type to be the seller? I discuss two separate causal elements. First, I show red agents are likely to be buyers if they are more numerous than blue agents. Second, I show that red agents are likely to be buyers if they are heterogenous and blue agents are homogenous. I also evaluate some closely related models of coordination frictions with price posting. I argue that these model works well if (i) buyers are homogenous and (ii) buyer–seller relationships are stable by assumption. In this case, the auction and price-posting models are equivalent. However, many difficulties arise if buyers are heterogenous. In particular, even in a static model with heterogenous buyers, in order to gain equivalence with efficient competitive auctions, a very complicated discriminatory price-posting structure must be adopted by sellers (see Shi, 2004). Moreover, if valuations are determined ex post by nature, then the predetermined terms of trade dictated by posted prices are inevitably inefficient. I also consider the wage implications of competitive auction models. The solution of a competitive auction model yields an expression for the asset value of a worker at a job (see Julien et al., 2000). However, it is also straightforward to

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derive the implications of this asset equation for wages. Therefore, it is easy to show that a model without on-the-job search does not yield much wage dispersion. Moreover, it can also be shown that the model can explain much wage dispersion if on-the-job search is permitted. The other issue about wages is the outcome of competitive auctions in markets with small frictions. I show that removing frictions from the model leads to a Walrasian outcome in which workers are paid their marginal product. This exercise of comparing the limiting properties of models with frictions to Walrasian out comes was advanced by Rubinstein and Wolinsky (1985) and Gale (1986). Gale argues that this exercise is an important test of how reasonable the assumptions are of the matching game specified. Competing auction models may become more used in the industrial organization literature. In particular, the theory of competitive auctions implies a shift in focus from imperfect information about buyers to imperfect information about sellers. On the one hand, the problem of informational inefficiency concerning buyers is solved by the auction and thus the high valuation buyer is always rewarded the good for sale at a price equal to the second highest valuation. On the other hand, the sellers’ problem of attracting buyers gives them much incentive to advertise themselves as good – a classic lemons problem. Therefore, I consider the effects of exogenously lifting the veil of ignorance about the quality of sellers. This application shows that third party information is used efficiently but that there are important distributional considerations. I show that buyers are always made worse off by small additions to their information set, but can be benefited by a sufficiently large addition. There are a number of other applications of competitive auctions. I attempt to briefly summarize a number of these applications. The list is by no means exhaustive and is meant to offer suggestions about how the theory might be applied. Finally, I caution that the present paper does not attempt to advance the theory of competitive search by Moen (1997) and Shimer (1996a). The theory of competitive search derives a number of very similar results as competitive auction theory, but for a very different reason. In particular, the theory of competitive search obtains efficiency in matching by the assumption of middlemen who oversee frictional submarkets and set the terms of trade in each. This concept runs into difficulties if buyers are heterogenous. For example, in a priceposting model, Shi (2004) shows that, if buyers are heterogenous, the optimal equilibrium decision of sellers requires a vector of ex ante prices – with each price corresponding to the unique valuation of each type of buyer. Moreover, he shows that these pricing announcements need not be monotonic. In particular, sellers may post lower prices for high-valuation buyers than for low-valuation buyers. The competitive auction model is a much simpler framework, because prices are determined ex post and there is no need to solve a complicated multidimensional ex ante pricing game. It also seems more realistic, because the sale of unique goods and services, such as specialized labor or specialized jobs, is rarely done by posted price (see Gautier and Moraga-Gonzalas, 2004).

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The paper is organized into a number of analytical sections that demonstrate some basic results. I then provide a section on further applications, where I discuss a number of other contributions to the theory and I attempt to link these results to ideas presented in the preceding sections. The final section offers concluding remarks. 2. Competitive auctions This section considers a simple game of competitive auctions described in Julien et al. (2000).2 This model introduces the optimal auction of each seller as a choice of reserve price. The coordination frictions arise in the mixed strategy equilibrium of this game. The model illustrates an important result that competition among sellers tends to reduce their reserve prices. In the game presented here, the reserve price is driven to zero in the limit as the number of buyers and sellers is large. The fact that prices cease to play a role in a large market is an important simplification, because this implies that decentralized trade with auctions can be modeled as an ex post pricing game. Many of the models of this survey use this simplification. 2.1. The model There are N identical sellers and M identical buyers, all spatially separated. Each seller has one good for sale worth y to any buyer and worth zero to the seller. All agents are risk neutral and maximize expected income. Buyers can choose the location of only one seller. The sequence of events within the period is as follows. First, each seller announces a reserve price to induce visits from buyers. Buyers then decide which seller to approach. Sellers then auction the good to the highest bidder. 2.2. The bidding game We start with the bidding game for, and given each seller’s announced reserve price ri, where iA{1, 2, y , N} is used to denote sellers. Let miA{1, 2, y , M} denote the number of buyers bidding at seller i and let w(ri, mi) denote the equilibrium price obtained by seller i. As is standard in an ascending-bid auction with homogenous buyers and complete information, the seller’s price is given by 8 > < 0 if mi ¼ 0 wðri ; mi Þ ¼

2

> :

ri y

if mi ¼ 1 if mi 41

ð1Þ

The seminal contributions to competitive auctions (competing mechanisms) are Wolinsky (1988) and McAfee (1993). Peters (1984) and Montgomery (1991) introduce the basic problem as a priceposting game. See also Shimer (2005b).

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Clearly, if no buyer approaches the seller, his price is zero. If only one buyer approaches, then the candidate receives his reserve price ri. If more than one buyer approaches, then Bertrand competition between buyers drives the price up to the point where the seller receives all the gains that the buyer makes from owning the good, y. 2.3. Buyers choice of seller to bid for Having observed the sellers’ reserve price announcement vector, buyers decide which seller to bid for. Since we will be focusing on symmetric equilibria, for notational convenience we will assume that all sellers other than i choose the same reserve price r. Let pi(ri, r) denote the probability that a particular buyer bids for seller i. Thus, for any buyer, the probabilities must sum to one, and given mi sellers at seller i, the probability that seller i will accept any offer is given by Pr{i accepts} ¼ 1/mi. Once the buyer decides to locate at seller i, given mi, the expected payoff to this buyer is Ri ¼ ðy2wðri ; mi ÞÞ Pr{i accepts w(ri, mi)}, which is y–ri if mi ¼ 1 and 0 if mi>1. In a symmetric equilibrium, (1–Pi(ri, r))M 1 is the probability that the buyer will be alone in his offer to candidate i, and [1–(1–pi(ri, r))M 1] the probability that at least one other will make this seller an offer. Hence, before knowing mi, the buyer’s expected payoff if she makes an offer to seller i is Pi ðri ; rÞ ¼ ð1

pi ðri ; rÞÞM

1

ðy

ri Þ

ð2Þ

In a symmetric mixed strategy equilibrium, each buyer chooses pi(ri, r), i ¼ 1; 2; . . . ; N, so that Pi ðri ; rÞ ¼ P. Let p(ri, r) denote the symmetric mixed strategy probability assigned to all other sellers, the constraint that the location probabilities sum to one implies pðri ; rÞ ¼ ð1 pi ðri ; rÞÞ=ðN 1Þ. Using this constraint and (2), one obtains pi ðri ; rÞ ¼ 1

1 þ ðN

N 1
1Þ ðy ri Þ=ðy

rÞ

1=M

ð3Þ

1

2.4. Sellers’ reserve price choice Sellers choose their reserve price to maximize their expected payoffs in a simultaneous moves game with other sellers. Let qi0 ðri ; rÞ ¼ ð1 pi ðri ; rÞÞM and qi1 ðri ; rÞ ¼ Mpi ðri ; rÞð1 pi ðri ; rÞÞM 1 denote the probabilities that seller i will receive zero offers and one offer, respectively. The expected payoff function for seller i is therefore given by V i ðri ; rÞ ¼ qi1 ðri ; rÞri þ ð1

qi0 ðri ; rÞ

qi1 ðri ; rÞÞy

ð4Þ

Since sellers choose their reserve prices simultaneously, an equilibrium array of reserve prices is found by a standard Nash argument, ri ¼ arg max V i ðri ; r Þ. ri

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The symmetric Nash Equilibrium is ri ¼ r ¼

ðM

ðM 1Þ 1Þ þ ðN

1Þ2

y

ð5Þ

2.5. A large market We now consider the properties of this equilibrium, as the scale of the market become large. To do this, we hold the ratio of buyers to sellers constant, f ¼ M/ N, and examine the case where N is very large, but finite number. In this type of environment, the economy can be closely approximated by the limit economy, here N-N. In the large economy, the reserve price becomes zero.3 r ¼ 0

ð6Þ

The matching technology in a large market is given by xðN; MÞ ¼ Nð1

e

f

Þ

ð7Þ

which is the familiar urn ball matching function. The expected payoff to the representative buyer is PðfÞ ¼ e f y, which states that the buyer is paid y if she is alone in her offer, which occurs with frequency e f. Likewise, the expected payoff of the representative seller is V ðfÞ ¼ ð1 e f ðfÞe f Þy, which states that the seller gets y if he has multiple offers, which occurs with frequency 1 e f fe f. 3. Dynamics This section derives a dynamic model of competitive auctions in which workers auction their services to firms. Here, we assume that the reserve price of each worker’s auction is equal to their outside option, even though this assumption can be derived explicitly as an equilibrium outcome as is done in Julien et al. (2000). The model is similar to the basic model of Pissarides (2000). However, we do not need to specify exogenous ‘sharing rules’ or ‘matching functions’. These aspects of the model are derived endogenously. The only friction in the competitive auction model is the length of time between offer rounds. 3.1. The model Consider a simple economy in which N identical workers face an infinite horizon, perfect capital markets, and a common discount factor b. At the start of each period t ¼ 0, 1, 2, y , there are N Et displaced workers of productivity y0 ¼ 0 and Et workers employed in jobs of productivity y1 ¼ y40. The ratio of

3

McAfee and McMillan (1985) derive a similar result in a model of endogenous buyer entry with a monopoly auction.

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Mt job vacancies to job searchers (i.e. displaced workers) is ft ¼

Mt ðN E t Þ

ð8Þ

Each of the vacant jobs carries a capital cost k per job per period. Each worker can work at most one job and one job can employ at most one worker. Furthermore, any match in the current period may dissolve in the subsequent period with probability r. The job vacancies are randomly assigned to job searchers. Therefore, the net addition of Ht workers is given by the following matching technology: H t ¼ ðN E t Þð1 e ft Þ, where 1 e ft is the probability that the worker obtains at least one offer. The exogenous separations at the end of the period imply that the supply employed workers at the start of next period is given by E tþ1 ¼ ð1 rÞðE t þ H t Þ. The labor market is decentralized with each worker using a second price auction for their labor services. Within each period, the order of play is as follows. At the beginning of the period, given the state, new vacancies enter. Once the number of entrants has been established, vacancies choose which workers to approach. Once new vacancies have been assigned to candidates, wages are determined through the auction mechanism. Let Li denote the expected discounted surplus of a match between an unemployed worker and a job of productivity yi at the start of the period (where a job of productivity y0 ¼ 0 is of course the unemployed state – home production). A second price auction implies that the workers share W ji of the total surplus Li which is equal to the surplus Lj of the worker’s second best available job offer. Thus W ji ¼ Lj

ð9Þ

The randomness of the number of jobs at each displaced worker implies that the present value of a displaced worker is given by V t ¼ p0t L0t þ p1t L1t p01t

ft

ft

ð10Þ

¼e þ ft e is the probability the worker has either one or no where job offers in the current period and p1t ¼ 1 e ft ft e ft the probability of multiple offers in which case the workers second best offer is a job of productivity y1. The supply of firms is determined by free entry such that firms earn zero profits in equilibrium. Thus, the expected profit Pt of a firm opening a job is given by   Pt ¼ max q0t ðL1t L0t Þ k; 0 ð11Þ

where q0t ¼ e ft is the probability the buyer of labor (the firm) is alone in its offer to the worker in the current period. The value of a displaced worker that does not find a job is given by L0t ¼ bV tþ1

ð12Þ

and the value of a worker that is employed is given by L1t ¼ y þ b½rV t þ ð1 rÞyŠ þ b2 ð1 rÞ½rV tþ1 þ ð1 rÞyŠ þ . . . , which in a

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steady state is given by L1 ¼

1

y þ brV bð1 rÞ

ð13Þ

3.2. Equilibrium In equilibrium, the supply of job vacancies is given by y k¼ e j 1 p0 ð1 rÞb

ð14Þ

This equation states that the vacancy cost must be offset by the probability that the seller is alone in its offer times an appropriately discounted flow of returns equal to
y each period. The equilibrium unemployment rate is given by u ¼ re f = 1 ð1 rÞe f .

4. The mortensen rule Julien et al. (2005a) apply an axiomatic approach to the efficiency of markets in which the participants in local markets are determined by coordination frictions. This approach is taken from Mortensen (1982), who uses it in a different context. Mortensen’s axioms are useful here, because the outcome of these axioms is an auction without reserve price. Therefore, we can apply these axioms to the game presented previously and to its natural extensions to heterogenous buyers. In both cases, we will show that the Mortensen rule gives efficiency. Moreover, in these games, the well-known axioms of Hosios (1990) do not apply.4 This section will also illustrate why the competitive auction framework is not subject to a holdup problem and why endogenous technology dispersion – the endogenous heterogenous valuation of buyers – is efficient. 4.1. Efficient entry There are N identical sellers and M identical buyers, all spatially separated. Each seller has one good for sale worth y to any buyer and worth 0 to the seller. All agents are risk neutral and maximize expected income. Buyers are randomly allocated to sellers. Therefore, the expected number of matches is given by xðN; MÞ ¼ Nð1

ð1

1=NÞM Þ

ð15Þ

The matching function has decreasing returns to scale, but in the limit where N and M are large, it has the function form given by Equation (7), which displays constant returns to scale. Let mA{0, 1, 2, y , M} denote the number of buyers bidding at a seller. A local market contains one seller S and a set of identical buyers, B ¼ (B1, B2, y , 4

See Davis (2001) for a discussion.

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John Kennes

Bm) if m>1 and no buyers if m ¼ 0.5 The surplus of a match between the seller and any particular buyer Bi is given by LðS; Bi Þ ¼ V ðS; Bi Þ

d s ðBÞ

d i ðBÞ

ð16Þ

where V(S, Bi) is the total value of the match, ds(B) the threat point of the seller, and di(B) the threat point of the buyer. The total valuation of a match is given by V ðS; Bi Þ ¼ y. The disagreement point of the buyer is zero, because once inside the local market, the buyer can trade only with the seller. The disagreement point of the seller is given by max V(S, B i) – the maximum total value of the good to the seller and the set of other buyers. This definition of the seller’s threat point assumes that each player has a conservative assessment about how well their opponent will be rewarded in the event of a disagreement. The surplus of a match is divided by the Mortensen rule. The axioms are as follows:  Axiom 1 (local efficiency). The pair of local market participants with the

highest V(S, Bi) form a match if the surplus L(S, Bi) is positive.  Axiom 2 (initiator of the match). The surplus of the match L(S, Bi) is re-

warded to the initiator of the match – i.e., the buyer. Axiom 1 of the Mortensen rule is also common to Nash’s solution concept. However, the second axiom is simpler than Nash’s other axiom, because it presumes that the identity of the match initiator is known. The Mortensen bargaining rule is equivalent to an auction without reserve price by the seller. In particular, the seller obtains a price y if there are multiple buyers at his local market and a price of zero otherwise. The Mortensen rule has important implications for efficiency in matching. Suppose that the number of buyers is determined by free entry, with each additional buyer to the market paying a capital cost, k. The marginal social benefit of an extra buyer is the extra number of matches created minus this capital cost. It is easy to verify that   N 1 M 1 xðN; MÞ xðN; M 1Þ ¼ ð17Þ N where the right-hand side is the probability the buyer is the sole buyer at his chosen local market. Therefore, the marginal social benefit of the extra buyer is equal to the private return to the extra buyer. Thus, the Mortensen rule gives efficient entry in this economy even though the matching technology does not display constant returns to scale!

5

The idea of local markets was advanced by Lucas and Prescott (1974). King and Grouge (1997) work out the dynamics of this model.

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4.2. Efficient technology dispersion Consider a simple matching game with a large number of N sellers, M1 bad buyers, and M2 good buyers. The good buyers have valuation of the sellers good equal to y2, which is greater than the valuation y1 of bad buyers. Buyers are randomly allocated to sellers. Therefore, if the best matches are always consummated, the expected value of all the matches is given by   ð18Þ S ¼ N ð1 e f2 Þy2 þ ð1 e f1 Þe f2 y1

where f1 ¼ M 1 =N and f2 ¼ M 2 =N. It should be noted that the Hosios rule cannot be applied to this matching game, because the matching technology has more than two arguments. Let m1 and m2 denote the number of bad and good buyers at the local market of a seller. Let the set of buyers with low valuation be L ¼ fðB1 ; B2 ; . . . ; Bm1 Þ if m1  1; f otherwiseg and the set of buyers with high valuation be H ¼ fðBm1 þ1 ; Bm1 þ2 ; . . . ; Bm1 þm2 Þ if m2  1; f otherwiseg. In a local market defined by L and H, the surplus of a match between the seller S and any particular buyer Bi is given by LðS; Bi Þ ¼ V ðS; Bi Þ d s ðL; HÞ d i ðL; HÞ, where the total value of the match is  y2 if i  m1 V ðS; Bi Þ ¼ ð19Þ y1 if m1 þ 1  i  m1 þ m2 the disagreement point of the buyer di(L, H) is zero and the disagreement point of the seller is max V(S, B i) – the maximum total valuation of the good to the seller and the set of other buyers in the local market. The Mortensen rule can also be applied to this game. Suppose we assume free entry of buyers, where the capital cost of a bad buyer is k1 and the capital cost of a good buyer is k2. If we assume the technological opportunity set displays positive but diminishing returns: y1 =k1 4y2 =k2 and y2 k2 4y1 k1 . The decentralized equilibrium under the Mortensen rule is given by e and e

f1

e

f2

y2 þ e

f1

f1

e

f2

y1 ¼ k 1

ð1

e

f2

Þðy2

y1 Þ ¼ k 2

ð20Þ ð21Þ

are both positive. In other words, the decentralized equilibrium has technology dispersion. It is easy to verify that equilibrium is efficient. A social planner faced with the problem of choosing the number of high- and low-valuation buyers to maximize S less the cost of buyers obtains the same solution as Equations (20) and (21). Therefore, the decentralized economy has efficient technology dispersion.6

6

Acemoglu and Shimer (2000) derive related results using a model with non-sequential search, see also Burdett and Judd (1983). Jansen (1999) considers investments by sellers and shows that there is no holdup problem, but also no technology dispersion.

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John Kennes

4.3. Efficient job creation The matching problem in the dynamic model of Section 3 can be also formulated as a social planning problem. The social planner maximizes X   S ¼ max ð22Þ bt yðE t þ H t Þ kM t E t ;E tþ1

such that H t ¼ ðN E t Þð1 e ft Þ and E tþ1 ¼ ð1 rÞðE t þ H t Þ. It is easy to verify that the solution of this simple dynamic programming problem is the same as the decentralized economy. 5. On-the-job search

The dynamic model in Section 3 has employer–employee relationships of random but exogenous duration. The duration of relationships can be made endogenous in a number of ways: economic progress that improves the quality of new jobs, jobs distinguished by the opportunities to create specific and general skills, good and bad jobs leading to on-the-job search in the latter. The important issue is to incorporate these factors into the asset equations of the dynamic model. This section considers a simple model of on-the-job search by Julien et al. (2006), which extends the discussion of technology dispersion in the last section. This model has directed search, because on-the-job searchers – workers in bad jobs – receive fewer offers than workers that are unemployed.7 5.1. The model A large number of N identical risk neutral workers face an infinite horizon, perfect capital markets, and a common discount factor b. Each worker has one indivisible unit of labor to sell. At the start of each period, t ¼ 0; 1; 2; 3; . . . , there exist E0t unemployed workers, of productivity y0 ¼ 0, and Eit workers in jobs of productivity yi>0, where iA{1, 2}. The ratio of good and bad job vacancies to displaced workers at the start of each period is given by fit ¼

ðN

M it E 1t E 2t Þ

ð23Þ

and the ratio of good job vacancies to on-the-job searchers (i.e. workers in bad jobs) is given by ^ ^ ¼ M 2t f 2t E 1t

7

ð24Þ

Price posting is considered by Delacroix and Shi (2003). See Kennes (2005) for the dynamic adjustment path of unemployment and vacancies in a random-matching model.

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A vacant job has a capital cost of ki such that yiZyj and kiZkj for all iZj. A match in any period may dissolve in the subsequent period with fixed probability rA(0, 1). The job vacancies are randomly assigned to job searchers. Therefore, the number of new hires into good and bad jobs is given by H 2t ¼ ðN E 1t E 2t Þp2t þ E 1t p^ 2t and H 1t ¼ ðN E 1t E 2t Þp1t E 1t p^ 2t , where p2t ¼ ð1 e f2t Þ, p1t ¼ ð1 e f1t Þe f2t , and p^ 2t ¼ ð1 e f2t Þ. The fraction r of all jobs dissolves in the next period, therefore, the supply of worker of each type evolves according to the following transition equation: E itþ1 ¼ ð1 rÞðE it þ H it Þ i 2 f1; 2g The labor market is decentralized with each worker using a second price auction for their labor services. Within each period, the order of play is as follows. At the beginning of the period, given the state, new vacancies enter. Once the number of entrants has been established, vacancies choose which workers to approach. Once new vacancies have been assigned to candidates, wages are determined through the auction mechanism. Let Lit denote the expected discounted value of a match between an unemployed worker and a job of productivity yi. The auction implies that the workers share W jit is equal to the value of the worker’s second best available job offer: W jit ¼ Ljt

ð25Þ

where Ljt is the expected discounted value of a match between an unemployed worker and the workers second best available job. The value of a displaced worker is given by V t ¼ p0t L0t þ p1t L1t þ p2t L2t p0t

f1t

f2t

ð26Þ

where is the probability that a worker has one or e ¼ ð1 þ f1t þ f2t Þe fewer offers, p1t ¼ e f2t ð1 f1t e f1t e f1t Þ þ f2t e f2t e f2t the probability of multiple offers only one of which is possibly good, p2t ¼ 1 e f2t f2t e f2t the probability of multiple good offers. The profits from introducing good and bad vacancies directed at unemployed workers are given by   P1t ¼ max ðL1t L0t Þq0t k1 ; 0 ð27Þ P2t ¼ maxfðL2t þ ðL2t

L0t Þq0t

L1t Þq1t

k2 ; 0g

where q0t ¼ e f1t e f2t is the probability a displaced worker receives no other offer and q1t ¼ ð1 e f1t Þe f2t the probability a good firm faces only a competing bad firm in its offer to an unemployed worker. The profits from introducing good vacancies directed at workers in bad jobs are given by   ^ 2t ¼ max ðL2t L1t Þq^ 1 k2 ; 0 P ð28Þ t

^

where q^ 1t ¼ e f2t is the probability a worker is not raided by another good firm. The assumption of free entry ensures that these values are equal to zero. The value of a worker that does not find a match this period is L0t ¼ bV tþ1 ð29Þ

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John Kennes

The value of a high productivity job   L2t ¼ y2 þ b rV tþ1 þ ð1 rÞy2   þ b2 ð1 pÞ rV tþ1 þ ð1 rÞy2 þ . . .

ð30Þ

The value of a low productivity job L1t ¼ y1 þ b½rV tþ1 þ ð1 2

rÞX tþ1 Š

pÞ½rV tþ1 þ ð1

þ b ð1

rÞX tþ2 Š þ . . .

ð31Þ

where Xt+1 summarizes three possibilities: the employed worker is not recruited, the worker is recruited by one good job, and the worker is recruited by multiple high productivity jobs. 5.2. Equilibrium It is straightforward to show that the equilibrium has the following properties: (1) the equilibrium is unique; (2) vacant good jobs are directed at workers employed in bad jobs, if the present value of the sequence of returns equal to the productivity difference between good and bad jobs discounted by b(1 r) is greater than the cost of a good job vacancy (i.e. k2oy2/(1 b(1 r)); (3) vacant bad jobs are directed at unemployed workers if the cost of bad job vacancies are sufficiently low; (4) unemployed workers receive more good offers on average than workers in bad jobs; and (5) the equilibrium is constrained-efficient. 6. What makes a seller? This section considers the choice of agents to become either buyers or sellers (see Kultti and Virrankoski, 2003; Julien et al., 2005a). Suppose that agents on one side of the market are called reds and agents on the other side of the market are called greens. What factors influence who buys and who sells? There are two factors to consider – (i) relative quantities and (ii) relative heterogeneity. Suppose that there are N red agents and M ¼ fN are green agents. The appropriate substitutions into the matching technology given by Equation (7) reveals Nð1

e

f

Þ‘Mð1

e

1=f

Þ

if f‘1

ð32Þ

where the left-hand side of the implication is the number of matches if green agents are sellers and the right side the number of matches if red agents play this role. Thus, other things equal, buyers should be more scarce than sellers for efficiency. A second reason that we can consider is heterogeneity. Consider the case where f ¼ 1. Suppose that red agents are heterogenous – x good red and (1 x) bad red. Using Equation (18), it is easy to show Mð1 4 max Nð1 1z0

e x Þy2 þ Mð1 e

z=x

Þy2 þ Nð1

e x Þe x y1 e

ð1 zÞ=ð1 xÞ

Þy1

ð33Þ

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where y2>y1. Thus, the efficient outcome is heterogenous buyers and homogenous sellers, rather than the other way around. 7. Wages The value of jobs in the dynamic model is given as an asset price. This was done, because the requisite object up for bidding was the present value of the returns to the match. However, it is straightforward to derive equilibrium wages. For example, in the model with one type of job, there are two possible wages given by Lj ¼

wj þ brV 1 bð1 rÞ

ð34Þ

where Lj is the present value of the workers second best offer at the time the wage is negotiated. It is straightforward to extend this exercise to the model of on-the-job search. The model without on-the-job search is unable to produce much wage dispersion, if attention is limited to an economy that has an unemployment rate of 4% or 5% and jobs of significant durability – a separation rate of 4% per month. However, Julien et al. (2006) show that the model with on-the-job search can easily account for the extreme wage disparity in the United States, which is observed by Katz and Autor (1999), while maintaining the extreme assumption that all workers are identical. Moreover, as implied by the theory, this extreme wage dispersion is efficient! There was a major debate about the properties of matching models in markets, where the frictions become infinitesimally small (see Rubinstein and Wolinsky, (1985) and Gale, 1986). This question can be easily studied in the dynamic competitive auction model by reducing the length of time between offer rounds. A consistent shift of the length of offer rounds, Dt, say from two weeks to one, requires us to appropriately scale the other parameters of the model. This scaling is achieved by adjusting the parameters of the model as follows. In each period, the present value of output produced and the capital cost of each job – vacant or filled – are given by Z tþDt yðDtÞ ¼ ye ðsþrÞt dt ð35Þ t

and kðDtÞ ¼

Z

t

tþDt

ke

rt

dt

ð35aÞ

Likewise, the discount rate and separation rates are given by bðDtÞ ¼ e rDt and dðDtÞ ¼ 1 e sDt . Besides being able to illustrate how the Beveridge curve shifts right or left as the size of the friction falls, we can also look at the limiting properties of this economy. In the limit as Dt-0, there is a finite ratio of job searchers to job vacancies, the number of job searchers approach zero.

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Moreover, the wage to a worker is equal to the output of the economy less the capital cost and jobs are paid their capital cost. Therefore, the frictionless economy is Walrasian.8 8. Price posting Hosios (1990) considers a simple model of job entry, based upon Peters model of price posting with capacity constraints. He shows that the equilibrium wage has the following property: w ¼ X 1 =X

ð36Þ

where the function X is the matching technology given in Equation (7). A neat implication is the equivalence of auctions and posted prices (e.g. Kultti, 1999). Coles and Eeckhout (2001) show the use of auctions or price posting is indeterminate in a simple game with homogenous buyers and sellers.9 The price-posting model runs into difficulty with heterogenous buyers. In this case, Shi (2004) shows that if there are two types of buyers then the equilibrium must have each seller posting two posted prices – one for each type of buyer. Besides the uncertain practical relevance of such pricing structures, the model is especially complicated to solve. In other words, the apparent simplicity of price posting is limited by the existence of heterogenous buyers. A similar criticism can be made of model of competitive search (e.g. Moen, 1997). Perhaps the ultimate reason for the existence of markets with competitive auctions is ex post heterogeneity driven by nature. Thus, the early applications of competitive auctions by Wolinsky (1988) and McAfee (1993) have this feature. Ex post uncertainty is conceptually simple to introduce although it can be mathematically taxing, because we must keep track of the statistics for the first and second highest valuations at each local market as a function of the heterogenous number of bidders as is implied by the exponential matching function.10 These results are slightly beyond the scope of the present discussion and serve mainly to connect the results of competitive auctions more closely to the existing auction literature.11 However, the goal of this paper has been to be more focused on the literature on matching, which usually treats heterogeneity quite simply, and which we believe is benefited most by the realism that the competing auction theory provides.

8

De Fraja and Sakovics (2003) consider a slightly different analysis with a matching technology that is not exactly the same as under investigation here. 9 see also Julien et al., 2006. 10 Papers by Peters (1997), Peters and Severinov (1997), and Epstein and Peters (1999) are highly recommended introductions to these topics. 11 A simple approach to the decision problem created by endogenous entry of buyers is given by McAfee and McMillan (1985).

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9. Imperfect information The theory of competitive auctions implies a shift in focus from imperfect information about buyers to imperfect information about sellers. On the one hand, the problem of informational inefficiency concerning buyers is solved by the auction and thus the high-valuation buyer is always rewarded the good for sale at a price equal to the second highest valuation. On the other hand, the sellers’ problem of attracting buyers gives them much incentive to advertise themselves as good – a classic lemons problem. Kennes and Schiff (2004) consider the equilibrium effects of exogenously lifting the veil of ignorance about the quality of sellers. However, there are many applications in which information about sellers is revealed endogenously. Examples of such market mechanisms include reputation systems, accreditation services, and guidebooks. 9.1. The model Suppose that sellers are separated by some mechanism into two quality differentiated submarkets. Let ql and qh denote the expected quality levels of sellers in the two submarkets, and let a denote the fraction of sellers that are allocated to the submarket with expected quality qh. Without loss of generality we assume qh>ql. Buyers are informed of ql, qh, and a. If sellers are separated in such a manner, the average quality of sellers across submarkets cannot change, thus q~ ¼ aqh þ ð1 aÞql . In addition, the number of buyers is fixed, so market tightness for each submarket is related to overall market tightness as follows: F ¼ afh þ ð1 aÞfl , where fl and fh denote the buyer–seller ratios of the two submarkets. The division of sellers into submarkets leads to two basic types of equilibrium distribution of buyers across the submarkets. These distributions depend upon the average quality of sellers in each submarket, their relative numbers, and the overall ratio of buyers to sellers. These conditions are summarized by what we call the exclusion constraint: qh e

F=a

 ql

ð37Þ

The left-hand side of the exclusion constraint is the expected utility of a buyer if all buyers locate in the high-quality submarket. The right-hand side of this constraint is the expected quality of sellers’ products in the low-quality submarket. If (37) is satisfied, a buyer is better off to locate in the high-quality submarket even though if he located in the low-quality submarket, he would not have to compete with any other buyers and could obtain a payoff of ql with certainty. Thus, if the partition of sellers into submarkets satisfies (37), all buyers locate in the high-quality submarket so that fh ¼ F=a and fl ¼ 0. If the partition of sellers into submarkets does not satisfy (37), buyers locate in both the high- and low-quality submarkets. In a mixed strategy equilibrium where (37) is not satisfied, we must have qh e

fh

¼ ql e

fl

ð38Þ

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That is, the expected utility to buyers must be the same from locating in either submarkets. The behavior of buyers can therefore be expressed as a function of the distribution of sellers over the submarkets. This function depends crucially on (37). From F ¼ afh þ ð1 aÞfl and (38), market tightness in the high- and low-quality submarkets are given by ( F=a if EC   fh ¼ ð39Þ F þ ð1 aÞ ln qqh otherwise l

and

fl ¼

(

0 F

a ln

  qh ql

if EC ð40Þ

otherwise

If (37) holds, sellers in the low-quality submarket are excluded, a buyer’s utility in any period is simply U ¼ e F=a qh . If (EC) does not hold, buyers visit both submarkets with strictly positive probability and from (38), a buyer’s utility in any period is U ¼ e fh qh . Substituting in F ¼ afh þ ð1 aÞfl , for any distribution of sellers across submarkets, the expected payoff of a buyer when search is guided is given by ( e F=a qh if EC U¼ ð41Þ F a 1 a e qh q l otherwise The utility of buyers is a linear function of the expected quality of sellers in the high submarket if sellers in the low submarket are excluded. If no sellers are excluded, the utility of buyers is a Cobb–Douglas function of the expected qualities of the sellers’ products in each submarket with the weights being the fraction of sellers in each submarket. This model has some surprising results for the distribution benefits of being better informed. Suppose that buyers are made measurably better informed about seller qualities, but not so well informed as to induce total exclusion of these sellers. In this case, we can compare welfare, with and without guided search as follows: U

U0 ¼ e

¼e o0

F a 1 a q h ql  F a 1 a qh q l

e

F

q~

ðaqh þ ð1 aÞql Þ for all qh aql and a 2 ð0; 1Þ



ð42Þ

To see the inequality, note that in general, xa y1 a oax þ ð1 aÞy for aA(0, 1) and xay. Taking the log of the left-hand side, logðxa y1 a Þ ¼ a log x þ ð1 aÞ log yo logðax þ ð1 aÞyÞ, since log is a concave function, and log monotone implies xa y1 a oax þ ð1 aÞy. Thus, the provision of information about sellers hurts buyers since it forces them to compete more intensely for high-quality products – thus, in equilibrium, the price of high-quality goods increases more than the probability of trade with such

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12

sellers. However, Kennes and Schiff (2004) also show that making buyers better informed about scarce good sellers, such that buyers exclude bad sellers, can make them better off. Another feature of the decentralized economy in this model is that information is used efficiently. This can be seen by setting up the social planning problem and determining the social planner preferred values of fh, fl. This result means that attention can be focused on inefficiencies related to (i) the incomplete marketing of information by a third party and (ii) optimal simple reputation systems – simple algorithms that transfer information across periods (see Kennes and Schiff, 2004, 2005). 10. Further applications This section discusses a number of applications of the theory of competitive auctions. This section is meant to be suggestive of the kind of research that can be pursued. Most of the research described in this section is either very recent, works in progress, or in some cases, speculative. Albrecht et al. (2005) consider a model that combines elements of the wageposting model of Burdett et al. (2001) and the bidding for labor model of Julien et al. (2000). They assume that firms post wages and that workers may apply to more than one job. This model is useful for the purpose of studying two-sided search. An interesting outcome of the model is that the posted wages of firms are driven to zero if they must bid for workers. Therefore, the simple bidding for labor model, which was presented earlier, is the equilibrium outcome of their more elaborate game. The problem of assortative matching is studied by Shimer and Smith (2000) in a random-matching model. The basic idea is that individuals are not always matched to their best jobs. This question can be easily addressed with the model of on-the-job search if we simply assume firms can distinguish between alternative types of labor (e.g. sex, race, schooling, etc.). Shimer (2005b) also considers a related competitive auction model, where heterogenous workers compete for heterogenous jobs.13 Moreover, Coles and Eeckhout (2000) show an interesting limiting property that perfect coordination can be achieved by sufficient heterogeneity on both sides of the market.14 Jovanovic and Rosseau (2004) consider an application in which the assembly of the firm is done by managers who take part in auctions. The model explains

12

Rubinstein and Wolinsky (1987) derive essentially the opposite result, because search in their model is undirected. 13 See Browning et al. (2005) for a model with heterogenous workers and firms, on-the-job search, and aggregate shocks. See also Shi (2002b). 14 This result requires the public observation of all the relevant characteristics of sellers, in addition to their pricing mechanisms.

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movements in product and technology variety since 1990. They argue that rise in the firm specificity of capital explains why (i) labor turnover has declined dramatically, (ii) firms now get a larger fraction of rents, (iii) Tobin’s Q is positively related to the skill premium, (iv) stock prices often rise with no accompanying rise in productivity, and (v) management and venture capitalism play more important roles in running the firm. Burdett et al. (2001) and Shi (2002a) consider the possibility that some firms may have the capacity to hire more than one worker. They show that this feature influences the structure of the equilibrium matching technology. The idea of giving more capacity than the other side of the market – the workers – is closely related to the idea that sellers are more scarce than buyers presented in Section 7. Thus, one explanation for why firms sell jobs is that firms are more scarce than workers. Of course, the fact firms sell multiple units limits the importance of auctions – McAfee and McMillan (1987) suggest that auctions are best used where the seller has a single unit. Therefore, capacity choices could explain why firms do not auction their jobs. However, see Shimer (1996b, 1999) for some implications about worker queues in the alternative market structure. Kennes and Schiff (2004) consider the modeling of reputation systems, such as that found on the auction website, eBay. They consider alternative reputation systems that report only a simple metric (eq. Recommended ¼ 1 or not recommended ¼ 0). One type of reputation system could track whether a seller was honest in the past. Therefore, dishonesty counts as a strike against the seller yielding 0, while honesty would yield 1. An alternative reputation reports something about the quality of the past transaction. For example, a bad transaction – he sold me a bad product – counts as a strike against the seller yielding 0, while the alternative would yield 1. They show that there is a trade-off between these systems, because bad sellers may be ‘excessively’ honest in an honest system. This work provides a basis for designing optimal reputation systems. Kennes and Schiff (2005) consider two methods of information distribution in a competing auction model: guidebooks and accreditation services. They show that a third party will generally sell information through both channels. They show that a monopoly third party will typically overinvest in informationgathering activities, because the punishment of not purchasing information is endogenous. They also show that competing third parties will typically subsidize information distribution on one side of the market in order to charge a higher price for sales of information on the other side of the market. Filges et al. (2004) use a competing auction model to study labor market policy. They observe that cross-country spending on labor market policies is independent of the unemployment rate, but that there are systematic similarities in the combined use of policy instruments. They derive labor market policy in a model with heterogenous workers and opportunities for training. The model is able to explain the strong correlation between active and passive subsidies in the data. The basic mechanism is that increases in passive policies weaken the incentive compatibility constraint on training subsidies. Therefore, higher-quality

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training programs can be implemented without leading advantaged to also choose these programs. The competing auction model can also be used to analyze business fluctuations. For example, the dynamic adjustment of unemployment and vacancies can be modeled by an aggregate Markov process governing the productivity of jobs (see Browning et al., 2005). One goal of this research is to find out whether cyclical changes in the amount of on-the-job search can account for the low variation in vacancies over the business cycle (see Kennes, 2005; Shimer, 2005a). Finally, the sale of a very peculiar asset – money – can also be modeled by competing auctions. For example, Julien et al. (2005b) show that sellers (agents without money) attract buyers (agents with money or agents pursing direct barter transactions) in both greater quantity and quality (measured by the money holdings of each buyer), if they make larger ex ante investments in product quality.15 In other words, the sellers must bid for money. The competing auction model also seems very promising for other questions concerning money holdings. In particular, the theory eliminates the need for assumptions like Nash bargaining and random search, which are made in alternative search-theoretic models of money (see Kiyotaki and Wright, 1993; Trejos and Wright, 1995; Lagos and Wright, 2005). However, it remains to be seen whether the theory can have a comparable impact on research in monetary economics as it is having on research in labor economics.

11. Conclusions Having written this paper in Denmark, I feel at liberty to cite Hans Christian Andersen. Two of his stories come to mind: ‘‘The Emperor’s New Clothes’’ and ‘‘The Ugly Duckling’’. The price-posting model of coordination frictions fails for the simple reason that its main assumption is false. In the real world, the sellers of specialized labor do not make wage demands in their resumes. Nor do the sellers of specialized jobs set wages in their job advertisements. Thus, the emperor has no clothes. Likewise, the theory of competitive auctions has been criticized, or even ignored, because it appears too complex. Therefore, price-posting models have been advanced for their simplicity, despite their questionable assumptions. However, these same models quickly become intractable, or yield questionable results, when they are applied to key problems such as on-the-job search, and worker and firm heterogeneity. By contrast, the theory of competitive auctions offers a tractable alternative with generally sensible results. Thus, this ugly duckling of the matching literature may yet prove to be a beautiful swan.

15 Rocheteau and Wright (2005) consider a model of money that uses Moen’s (1997) notion of competitive search equilibrium. Therefore, buyers – agents with money or agents initiating direct barter transactions – are assumed to be homogenous in each possible submarket.

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Acknowledgment I have benefited from helpful discussions with Benoit Julien, Ian King, Preston McAfee, Dale Mortensen, Aaron Schiff, Torben Tranæs, Yoram Weiss and seminar participants at the 2004 Conference on Labor Market Models, and Match Employer–Employee Data in Honor of Dale Mortensen in Sandbjerg, Denmark.

References Acemoglu, D. and R. Shimer (2000), ‘‘Wage and technology dispersion’’, Review of Economic Studies, Vol. 67, pp. 585–608. Albrecht, J., P. Gautier and S. Vroman (2005), ‘‘Equilibrium directed search with multiple applications’’, Mimeo, Georgetown University. Browning, M., J. Kennes, and A. Schiff (2005), ‘‘Lots of heterogeneity in a matching model’’, Mimeo, University of Copenhagen. Burdett, K. and K. Judd (1983), ‘‘Equilibrium price dispersion’’, Econometrica, Vol. 51, pp. 955–969. Burdett, K., S. Shi and R. Wright (2001), ‘‘Pricing and matching with frictions’’, Journal of Political Economy, Vol. 107, pp. 1060–1085. Coles, M., J. Eeckhout (2000), ‘‘Heterogeneity as a coordination device’’, Mimeo, University of Pennsylvania. Coles, M. and J. Eeckhout (2001), ‘‘Indeterminancy in directed search’’, Journal of Economic Theory, Vol. 111, pp. 265–276. Davis, S. (2001), ‘‘The quality distribution of jobs and the structure of wages in search equilibrium’’, Mimeo, University of Chicago. De Fraja, G. and J. Sakovics (2003), ‘‘Walras retrouved: decentralized trading mechanisms and the competitive price’’, Journal of Political Economy, Vol. 109, pp. 842–863. Delacroix, A. and S. Shi (2003), ‘‘Directed search on-the-job and the wage ladder’’, Mimeo, University of Toronto. Epstein, L. and M. Peters (1999), ‘‘A revelation principle for competing mechanisms’’, Journal of Economic Theory, Vol. 88, pp. 119–161. Filges, T., J. Kennes, B. Larsen and T. Tranæs (2004), ‘‘The equity-efficiency trade-off in a frictional labor market’’, Mimeo, University of Copenhagen. Gale, D. (1986), ‘‘Bargaining and competition part I: characterization’’, Econometrica, Vol. 54, pp. 785–806. Gautier, P. and J.L. Moraga-Gonzalas (2004), ‘‘Strategic wage setting and coordination frictions with multiple applications’’, Mimeo, Tinbergen Institute. Hosios, A. (1990), ‘‘On the efficiency of matching and related models of search and unemployment’’, Review of Economic Studies, Vol. 57, pp. 279–298. Jansen, M. (1999), Job auctions, holdups and efficiency, Manuscript: European University Institute.

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Jovanovic, B. and P. Rousseau (2004), ‘‘Specific capital and division of rents’’, Mimeo, University of Chicago. Julien, B., J. Kennes and I. King (2000), ‘‘Bidding for labor’’, Review of Economic Dynamics, Vol. 3, pp. 619–649. Julien, B., J. Kennes and I. King (2005a), ‘‘The Mortensen rule and efficient coordination unemployment’’, Economics Letters, in press. Julien, B., J. Kennes and I. King (2005b), ‘‘Bidding for money’’, Mimeo, University of Copenhagen. Julien, B., J. Kennes, I. King (2006), ‘‘Residual wage disparity and co-ordination unemployment’’, International Economic Review, in press. Katz, L. and D. Autor (1999), ‘‘Changes in the wage structure and earnings inequality’’, In: Handbook of Labor Economics, Vol. 3, O. Ashenfelter and D. Card, (editors), Elsevier Science B.V., Amsterdam, Ch. 26. Kennes, J. (2005), ‘‘Underemployment, on-the-job search, and the Beveridge curve’’, Economics Letters, in press. Kennes, J. and A. Schiff (2004), ‘‘Simple reputation systems’’, Mimeo, University of Copenhagen. Kennes, J. and A. Schiff (2005), ‘‘Third party information in an equilibrium search model’’, Mimeo, University of Copenhagen. King, I. and R. Grouge (1997), ‘‘A competitive theory of employment dynamics’’, Review of Economic Studies, Vol. 64, pp. 1–22. Kiyotaki, N. and R. Wright (1993), ‘‘A search-theoretic approach to monetary economics’’, American Economic Review, Vol. 83, pp. 63–77. Kultti, K. (1999), ‘‘Equivalence of auctions and posted prices’’, Games and Economic Behaviour, Vol. 27, pp. 106–113. Kultti, K. and J. Virrankoski (2003), ‘‘Physical search’’, Mimeo, University of Helsinki. Lagos, R. and R. Wright (2005), ‘‘A unified framework for monetary economic and policy analysis’’, Journal of Political Economy, Vol. 113, pp. 463–484. Lucas, R.E. and E. Prescott (1974), ‘‘Equilibrium search and unemployment’’, Journal of Economic Theory, Vol. 7, pp. 188–209. McAfee, R.P. (1993), ‘‘Mechanism design by competing sellers’’, Econometrica, Vol. 61, pp. 1281–1312. McAfee, R.P. and J. McMillan (1985), ‘‘Auctions with entry’’, Economics Letters, Vol. 23, pp. 343–347. McAfee, R.P. and J. McMillan (1987), ‘‘Auctions and bidding’’, Journal of Economic Literature, Vol. 25, pp. 699–738. Moen, E. (1997), ‘‘Competitive search equilibrium’’, Journal of Political Economy, Vol. 103, pp. 385–411. Montgomery, J. (1991), ‘‘Equilibrium wage dispersion and interindustry wage differentials’’, Quarterly Journal of Economics, Vol. 105, pp. 163–179. Mortensen, D.T. (1982), ‘‘On the efficiency of mating, racing, and related games’’, American Economic Review, Vol. 72, pp. 968–979.

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Peters, M. (1984), ‘‘Equilibrium with capacity constraints and restricted mobility’’, Econometrica, Vol. 52, pp. 1117–1129. Peters, M. (1997), ‘‘A competitive distribution of auctions’’, Review of Economic Studies, Vol. 64, pp. 97–123. Peters, M. and S. Severinov (1997), ‘‘Competition among sellers who offer auctions instead of prices’’, Journal of Economic Theory, Vol. 75, pp. 141–179. Pissarides, C. (2000), Equilibrium Unemployment Theory, 2nd edition, Oxford: Oxford University Press. Rocheteau, G. and R. Wright (2005), ‘‘Money in search equilibrium, in competitive equilibrium, and in competitive search equilibrium,’’ Econometrica, Vol. 73, pp. 175–202. Rubinstein, A. and A. Wolinsky (1985), ‘‘Equilibrium in a market with sequential bargaining’’, Econometrica, Vol. 53, pp. 1133–1150. Rubinstein, A. and A. Wolinsky (1987), ‘‘Middlemen’’, Quarterly Journal of Economics, Vol. 102(3), pp. 581–594. Shi, S. (2002a), ‘‘Product market and the size-wage differential’’, International Economic Review, Vol. 43, pp. 21–54. Shi, S. (2002b), ‘‘A directed search model of inequality with heterogeneous skills and skill-biased technology’’, Review of Economic Studies, Vol. 69, pp. 467–491. Shi, S. (2004), ‘‘Wage differentials, discrimination and efficiency’’, Mimeo, University of Toronto. Shimer, R. (1996a), ‘‘Contracts in frictional labor market’’, Mimeo, MIT. Shimer, R. (1996b), ‘‘Do nice guys finish last?’’, Mimeo, MIT. Shimer, R. (1999), ‘‘Job auctions’’, Mimeo, Princeton University. Shimer, R. (2005a), ‘‘The cyclical behavior of unemployment and vacancies’’, American Economic Review, Vol. 95, pp. 25–49. Shimer, R. (2005b), ‘‘The assignment of workers to jobs in an economy with coordination frictions’’, Journal of Political Economy, Vol. 113(5), pp. 996–1025. Shimer, R. and L. Smith (2000), ‘‘Assortative matching and search’’, Econometrica, Vol. 68, pp. 343–370. Trejos, A. and R. Wright (1995), ‘‘Search, bargaining, money, and prices’’, Journal of Political Economy, Vol. 103, pp. 118–141. Wolinsky, A. (1988), ‘‘Dynamic markets with competitive bidding’’, Review of Economic Studies, Vol. 55, pp. 71–84.

CHAPTER 8

Block Assignments Michael Sattinger Abstract This paper considers whether decisions of workers and employers would divide a larger labor market into smaller blocks, each a separate labor market, with narrower ranges of worker and job characteristics. This mechanism, if it operated, would bring about a more accurate assingment of workers to jobs in a search context and conserve resources spent searching for matches. Although an efficient division of the larger labor market into two blocks can be found, it is inconsistent with incentives of workers and employers in choosing between blocks in the case considered here.

Keywords: assignment, search, efficiency JEL Classification: J64 1. Introduction This paper reports on computational experiments on the formation of an internal structure for labor markets. Is there one all-encompassing labor market, where all workers compete for all jobs? Are there instead distinct labor markets in which a subset of workers competes for a subset of jobs? Or is the aggregate labor market characterized by some sloppy alternative, where no separate submarkets form? The particular question addressed here is whether a larger labor market can break up into two smaller labor markets (called blocks), within which all workers in a block compete for all jobs in the same block. Formally, the economic context is as follows. Heterogeneous workers are described by a parameter s that covers an interval s0 to s2, and heterogeneous jobs are described by a parameter k that covers an interval k0 to k2. The workers CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75008-2

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

169

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and jobs are assumed to be uniformly distributed over the intervals. Production from a single worker and a single job is f (k, s), an increasing and concave function of the job and worker characteristics. The workers and jobs meet and form matches through a standard search process with bargaining. Now consider whether this labor market could be disaggregated into two submarkets. Let s1 and k1 be values of the worker and job parameters that divide the intervals of workers and jobs, respectively, resulting in two blocks within which workers and jobs match with each other. Suppose workers with values s from s0 to s1 now only match with jobs that have values k between k0 and k1. This will be referred to as Block One. Similarly, suppose workers with values s from s1 to s2 now only match with jobs that have values k between k1 and k2. This is Block Two. Once agents choose blocks, workers in Block One only search in Block One, workers in Block Two only search in Block Two, and similarly for employers with jobs. Figure 1 shows a schematic drawing of the situation. Some relevant questions concerning the disaggregation of the labor market into blocks are as follows. First, are there optimal values of k1 and s1? Second, is this optimal division consistent with the incentives of workers and employers to join one submarket or the other? For example, would workers with s > s1 choose Block Two while workers with s o s1 choose Block One? Is any division of the aggregate labor market into blocks possible using voluntary choice? Block assignments represent a middle ground between two labor market structures that have been previously considered. Sattinger (1995) and Shimer and Smith (2000) analyze markets where heterogeneous workers search randomly among heterogeneous jobs. In this approach, there is no disaggregation of the labor market into smaller markets. In contrast, in an analysis of complete markets, Moen (1997) develops a model in which the labor market breaks up into minimal labor markets as a result of directed search (see the analysis by Mortensen and Pissarides, 1999; Masters, 1999; and Shi, 2001). There is a labor market for each type of job, and in equilibrium only one type of worker chooses to search in each labor market. With block assignments, the aggregate labor Figure 1.

Block Assignment

Worker Characteristics

s2 Block Two s1 Block One s0 k0

k1 Job Characteristics

k2

Block Assignments

171

market breaks into smaller labor markets with narrower ranges of worker and job characteristics as in complete markets, but the labor markets are not so small that heterogeneity in workers and jobs is completely eliminated. Within each block, heterogeneous workers search randomly for jobs at heterogeneous employers. The division of the larger labor market into two blocks brings about a rough assignment of workers to jobs. The upper interval of workers only seek the upper interval of jobs, and the lower interval of workers only seek the lower interval of jobs. Search that may not lead to matches or would lead to inefficient matches (lower interval workers at upper interval jobs, or upper interval workers at lower interval jobs) are eliminated by the ability of workers and employers to choose between blocks. Production could be increased while unemployment and vacancies are reduced. Blocks are a transferable utility analog to the classes described by Burdett and Coles (1997) and Bloch and Ryder (2000) in the marriage case (see also Chade, 2001). In their models, matching suitors form voluntarily into classes, in which they would form unions. The formation of classes arises from the feature of the solution that the reservation utility value of a suitor is the same for all suitors on the other side in a class. This feature does not hold in the block assignments described here, in which wages and profits transfer gains from agents on one side to agents on the other. The conclusion, briefly, is that the optimal division can be found in the case considered here but is not supported by reservation wages and profits. Generally, workers and jobs at the upper ranges in Block One would want to shift into Block Two. Further, there is no equilibrium division even though workers and jobs can costlessly choose blocks, thereby avoiding less productive matches. With threshold workers and jobs in Block One moving into the higher paying Block Two, Block One collapses until there is again only an aggregate labor market. Taxes and transfers that reconcile opportunity costs and reservation wages and profits could possibly support an efficient division. Otherwise the division would require compulsion or restrictive institutional features.

2. Model Let f (k, s) be the production obtained from a worker with parameter s in a job with parameter k. Assume f (k, s) is an increasing and concave function of its arguments. Let m (u, v) be the rate at which matches are formed if there are u unemployed workers and v vacant jobs. Assume that m (u, v) has constant returns to scale, is an increasing, concave function of its arguments, and takes the value 0 if either of its arguments is 0 (see the discussion in Mortensen and Pissarides, 1999; Pissarides, 2000; and Petrongolo and Pissarides, 2001). Suppose matches break up exogenously at the rate g. Suppose there are Nsi workers in block i (employed or unemployed) and Nki jobs (filled or vacant). In a steady- state, the unemployment rate ui and vacancy

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rate vi must satisfy mðui N si ; vi N ki Þ ¼ gð1 ð1

ui ÞN si ¼ ð1

ui ÞN si

vi ÞN ki

ð1Þ ð2Þ

The first condition states that the flows of new matches must equal the number of employed workers that lose their jobs. The second condition states that the number of employed workers must equal the number of filled jobs. For given Nsi and Nki in a block, values of ui and vi that solve 1 and 2 exist and are unique. The argument is as follows. Condition 2 determines a positive relation between ui and vi. If Nsi/Nki r 1, vi Z 0 when ui ¼ 0 Then consider the left and right sides of condition 1 as ui varies between 0 and 1. At ui ¼ 0; mð0; vi N ki Þ ¼ 0, which must be less than the right hand side, gNsi. As ui increases, the left hand side decreases while the right hand side increases. At ui ¼ 1, mðui N si ; vi N ki Þ40 while the right hand side is 0. At a unique value of ui between 0 and 1, both conditions must be satisfied. If Nsi/Nki Z 1, ui Z 0 when vi ¼ 0 and the same argument can be constructed using vi. For some matching functions (including the function used in the next section), it is possible to solve for ui and vi analytically in terms of Nsi and Nki.. In the general case, analytic solutions may not be possible. However, an anonymous referee has proposed the following procedures to establish an analytic relation between the ratio of workers to jobs and the unemployment and vacancy rates. Let ci ¼ N si =N k i. Then 1 and 2 can be rewritten as mðui ci ; vi Þ ¼ gð1 ð1

ui Þci ¼ 1

ui Þci vi

ð3Þ ð4Þ

 be the value of x Assume the matching function m is invertible, and let m1 1 ðv; mÞ  Then 3 and 4 yield such that mðx; vÞ ¼ m, ui ci ¼ m1 1 ðvi ; gð1 ui ci ¼ ci

1 þ vi

vi ÞÞ

ð5Þ ð6Þ

Setting the right hand sides equal yields ci as an analytic function of vi. The unemployment rate can then be found from vi and ci using 6. The remainder of the paper simplifies the derivations by assuming a matching function that yields unemployment and vacancy rates as analytic functions of the numbers of workers and jobs. Assume worker characteristics are uniformly distributed on the interval s0 to s2, and job characteristics are uniformly distributed in the interval k0 to k2. This assumption is not essential to the results but greatly simplifies analysis and computation. In Block One, N s1 ¼ s1 s0 and Nk1 ¼ k1 k0 , while in Block Two N s2 ¼ s2 s1 and N k2 ¼ k2 k1 because of the uniform distributions. Taking s0, k0, s2, and k2 as given, suppose the two conditions 1 and 2 can be solved to yield ui( k1, s1) and vi( k1, s1), i ¼ 1; 2: The resulting unemployment and vacancy rates are constant within blocks.

Block Assignments

173

The transition rate from unemployment to employment in block i is the match rate divided by the number of unemployed workers: li ðk1 ; s1 Þ ¼

mðui ðk1 ; s1 ÞN si ; vi ðk1 ; s1 ÞN ki Þ ðui ðk1 ; s1 ÞN si Þ

ð7Þ

Similarly, the transition rate for jobs in block i from vacancy to filled is the match rate divided by the number of vacancies: di ðk1 ; s1 Þ ¼

mðui ðk1 ; s1 ÞN si ; vi ðk1 ; s1 ÞN ki Þ ðvi ðk1 ; s1 ÞN ki Þ

ð8Þ

These transition rates are constant within a block. Production in Block One can be calculated as the average output for a worker of type s when employed times the proportion of type s employed, integrated over s: R k1 Z s1 k f ðx; yÞdx p1 ðk1 ; s1 Þ ¼ ð1 u1 ðk1 ; s1 ÞÞ 0 dy k1 k0 s0 Z Z 1 u1 ðk1 ; s1 Þ s1 k1 ¼ f ðx; yÞdxdy ð9Þ k1 k0 s0 k0 The same calculation arises using the average output for a job of type k since 1

u1 ðk1 ; s1 Þ 1 v1 ðk1 ; s1 Þ ¼ k1 k0 s1 s0

from 2. Similarly, in Block Two, Z Z 1 u2 ðk1 ; s1 Þ s2 k2 f ðx; yÞdxdy p2 ðk1 ; s1 Þ ¼ k2 k1 s1 k1

ð10Þ

ð11Þ

3. Optimal dividing values In the absence of taxes, transfers, unemployment insurance, or subsidies to employers for vacant jobs, and assuming workers and employers have the same discount rate, the optimal dividing values of k1 and s1 are the values that maximize the sum of p1(k1, s1) and p2 (k1, s1). This occurs because the only source of present value for workers or jobs is production over time. When production is maximized, the sum of present values of worker and job income streams will be maximized. The optimal dividing values can then be found by taking the derivatives of p1 (k1, s1)+p2(k1, s1) with respect to k1 and with respect to s1, setting them equal to 0, and solving. Since the sum of block outputs is maximized, @p1 @p2 @p1 @p2 þ ¼ 0; þ ¼0 @k1 @k1 @s1 @s1

ð12Þ

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Michael Sattinger

Table 1.

Conditions in Blocks One and Two Block One

Production Unemployment rate Vacancy rate @pi/@s1 @pi/@k1 @ui/@s1 @ui/@k1 @vi/@s1 @ui/@k1

0.464 0.152 0.122 0.344 0.640 0.971 1.007 0.309 0.321

Block Two 1.157 0.115 0.139 0.344 0.640 0.607 0.590 0.443 0.431

That is, moving a marginal worker (at the division line) from Block One to Block Two, by lowering s1, raises output in Block Two by as much as output declines in Block One. Second order conditions must also be satisfied.1 Changes in production arise from the production of the marginal worker, diversion of jobs from other workers in Block Two, release of jobs to other workers in Block One, and changes in unemployment and vacancy rates in both blocks. Numerical results have been obtained assuming s0 ¼ k0 ¼ 0:5, s2 ¼ k2 ¼ 2, g ¼ 0:05, a ¼ 1=3 and f ðk; sÞ ¼ ka s1 a ; mðx; yÞ ¼ xy=ð2x þ yÞ

ð13Þ

Then the optimal dividing values are k1 ¼ 1:144 and s1 ¼ 1:167. Values of relevant variables are shown in Table 1. The values for the derivatives for @p2/@s1 and @p2/@k1 are negative because increases in s1 and k1 reduce the number of workers and jobs in Block Two. The maximum output in the two blocks, 0.464 plus 1.157, is greater than the output in the absence of blocks, 1.584.

4. Worker and employer behavior Now consider whether the optimal dividing line could arise from worker and firm behavior. Workers move between employment and unemployment according to a Markov process, with transition rates g and li(k1, s1) in block i. Asset value equations can be defined for the two states and solved to yield the flow of asset value while unemployed: rV ui ðsÞ ¼

1

li ðk1 ; s1 Þ ðwei ðsÞÞ li ðk1 ; s1 Þ þ g þ r

ð14Þ

The second order conditions for a local maximization are that @2 ðp1 þ p2 Þ=@s21 o0; @2 ðp1 þ p2 Þ=@k21 o0, and ð@2 ðp1 þ p2 Þ=@s21 Þð@2 ðp1 þ p2 Þ=@k21 Þ 4 ð@2 ðp1 þ p2 Þ=ð@s1 @k1 ÞÞ2 .

Block Assignments

175

where Vui(s) is the asset value of an unemployed worker with parameter s, discount rate r, and expected wage while employed of wei(s). In general, li(k1, s1) and wei(s) would be functions of the worker’s reservation wage. However, in the block assignments considered here, the worker accepts all offers so the reservation wage is never binding. Nevertheless, it is possible to calculate the hypothetical reservation wage for the worker, the lowest wage rate that would be acceptable to the worker. In a Markov process, this is given by w0i ðsÞ ¼ rV ui ðsÞ ¼

li ðk1 ; s1 Þ ðwei ðsÞÞ li ðk1 ; s1 Þ þ g þ r

ð15Þ

di ðk1 ; s1 Þ ðzei ðkÞÞ di ðk1 ; s1 Þ þ g þ r

ð16Þ

Similarly, jobs move between filled and vacant according to a Markov process, with transition rates di( k1, s1) and g. The reservation profit in block i is given by z0i ðkÞ ¼ rV vi ðkÞ ¼

where r is the same discount rate as for workers, Vvi(k) is the asset value for the vacant job, and zei(k) is the expected profit. Let wi(k, s) and zi(k, s) be the wage and profit in block i for a job with parameter k and a worker with parameter s. Suppose the wage and profit are determined through bargaining in the standard way: wi ðk; sÞ ¼ w0i ðsÞ þ bðf ðk; sÞ zi ðk; sÞ ¼ z0i ðkÞ þ ð1

w0i ðsÞ

bÞðf ðk; sÞ

z0i ðkÞÞ

w0i ðsÞ

z0i ðkÞÞ

ð17Þ ð18Þ

where b represents relative bargaining power of workers. Then wi ðk; sÞ þ zi ðk; sÞ ¼ f ðk; sÞ by construction. The expected wage and profit functions are Z k1 w1 ðx; sÞdx ð19Þ we1 ðsÞ ¼ k0 Þ k0 ðk 1 we2 ðsÞ ¼

Z

ze1 ðkÞ ¼

Z

ze2 ðkÞ ¼

Z

k2 k1 s1

s0 s2

s1

w2 ðx; sÞdx ðk2 k1 Þ

ð20Þ

z1 ðk; yÞdy ðs1 s0 Þ

ð21Þ

z2 ðk; yÞdy ðs2 s1 Þ

ð22Þ

The reservation wage and profit functions can be found analytically in the block assignment case as follows. From 17 and 19, Z k1 ðf ðx; sÞ z01 ðxÞÞdx ð23Þ we1 ðsÞ ¼ ð1 bÞw01 ðsÞ þ b ðk1 k0 Þ k0

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Michael Sattinger

From 15, since l1(k1, s1) is the same for all workers in Block One, we1(s) is a multiple, (l1(k1, s1)+g +r)/l1 (k1, s1), of the reservation wage, w01(s) : we1 ðsÞ ¼

l1 ðk1 ; s1 Þ þ g þ r w01 ðsÞ l1 ðk1 ; s1 Þ

Thus in 23,  l1 ðk1 ; s1 Þ þ g þ rÞ l1 ðk1 ; s1 Þ

ð24Þ

 bl1 ðk1 ; s1 Þ þ g þ r 1 þ b w01 ðsÞ ¼ w01 ðsÞ l1 ðk1 ; s1 Þ Rk b k01 ðf ðx; sÞ z01 ðxÞÞdx ¼ k1 k0

ð25Þ

Then bl1 ðk1 ; s1 Þ þ g þ r 0 @ w01 ðsÞ ¼ b l1 ðk1 ; s1 Þ @s

R k1 k0

ð f ðx; sÞ

z01 ðxÞÞ dx

k0

ð26Þ

f ðx; sÞ dx

ð27Þ

k1

Since z01(x) does not depend on s, w001 ðsÞ

bl1 ðk1 ; s1 Þ=ðk1 k0 Þ @ ¼ bl1 ðk1 ; s1 Þ þ g þ r @s

Z

k1

k0

Thus, the derivative of the reservation wage with respect to s can be found without knowledge of z01(k), yielding the general function bl1 ðk1 ; s1 Þ=ðk1 k0 Þ w01 ðsÞ ¼ bl1 ðk1 ; s1 Þ þ g þ r

Z

k1 k0

f ðx; sÞdx þ C w1

ð28Þ

where Cw1 is a constant of integration. The constant of integration can be determined from a boundary condition. Let w^ 01 be the reservation wage for a worker with s ¼ s1 in Block One. Then  Z k1  Z k1 bl1 ðk1 ; s1 Þ=ðk1 k0 Þ w01 ðsÞ ¼ f ðx; sÞdx f ðx; s1 Þdx þ w^ 01 ð29Þ bl1 ðk1 ; s1 Þ þ g þ r k0 k0 Using the same procedures, the reservation wage functions in Block Two and reservation profit functions in Blocks One and Two can be determined. These functions will depend on constants of integration that can be specified in terms of w^ 02 , z^01 , and z^02 , respectively. Given the dividing values k1 and s1, the reservation wage and profit w^ 01 and z^01 can be determined as follows. Let st and kt be arbitrary values such that

Block Assignments

177

s0pstps1 and k0pktpk1. Then w^ 01 and z^01 must satisfy w01 ðst Þ ¼ ¼

l1 ðk1 ; s1 Þ ðwe1 ðst ÞÞ l1 ðk1 ; s1 Þ þ g þ r R k1 l1 ðk1 ; s1 Þ k0 w1 ðx; st Þdx

ð30Þ

d1 ðk1 ; s1 Þ ðze1 ðkt ÞÞ d1 ðk1 ; s1 Þ þ g þ r R s1 d1 ðk1 ; s1 Þ s0 z1 ðk t ; yÞdy

ð31Þ

l1 ðk1 ; s1 Þ þ g þ r

k1

k0

and z01 ðkt Þ ¼ ¼

d1 ðk1 ; s1 Þ þ g þ r

s1

s0

The conditions 30 and 31 yield two linear relations between w^ 01 and z^01 that can be solved for the values consistent with k1 and s1. Using the same procedure, w^ 02 and w^ 02 can be found for the second block. In the example from the previous section, with the discount rate r ¼ 0:05 and b ¼ 0:4, the reservation wage and profit functions are w01 ðsÞ ¼ 0:490s1

a

z01 ðkÞ ¼ 0:602ka w02 ðsÞ ¼ 0:702s1 z02 ðkÞ ¼ 0:880ka

0:490s11

a

þ 0:325

0:602ka1 þ 0:484 a

0:702s11

a

þ 0:377

0:880ka1 þ 0:563

ð32Þ ð33Þ ð34Þ ð35Þ

where k1 and s1 are the optimal dividing values from the previous section. The surplus of production over the sum of reservation wage and profit can be calculated at each value of k and s in Block One and Block Two. The surplus must be positive for a worker and an employer to be willing to form a match. In this example, the surplus is positive for all matches in Blocks One and Two. The reservation wage and profit functions therefore support the condition that all matches within a block are formed.  However, the optimal dividing values s 1 and k1 cannot be equilibrium values. The reservation wage and profit functions are shown graphically in Figures 2  and 3. The reservation wages and profits at s 1 and k1 in 32–35 are shown separately in Table 2. As shown by the gaps in figures and the amounts in Table 2, w^ 01 ow^ 02 and z^01 o^z02 . A worker in Block One with s close to s 1 would then have an incentive to move to Block Two, and an employer in Block One with k close to k 1 would have an incentive to move to Block Two. The gaps cannot be eliminated by moving the dividing values downward, so that the marginal workers and jobs would move into Block Two. There would still be marginal workers and jobs, and the gap would persist.

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Michael Sattinger

Figure 2.

Reservation wage

Reservation Wage

0.7

Block Two

0.6 0.5 0.4 0.3 0.2

Block One

0.1 0.6

0.8

1

1.2

1.4

1.6

1.8

2

Worker Parameter

Figure 3.

Reservation profit

0.8

Reservation Profit

0.7 0.6

Block Two

0.5 0.4 0.3

Block One

0.2 0.1 0.6

0.8

1

1.2

1.4

1.6

1.8

2

Job Parameter

Table 2.

w^ 0i z^0i

Reservation values at division Block One

Block Two

0.325 0.484

0.377 0.563

Consider the problem of finding k1 and s1 such that w^ 01 ¼ w^ 02 and z^01 ¼ z^02 . The condition w^ 01 ¼ w^ 02 yields one condition on k1 and s1 and z^01 ¼ z^02 yields a second condition. However, the two conditions do not yield a solution because the two relations between k1 and s1 do not intersect. Figure 4 shows the two relations, one for equal reservation profits and one for equal reservation wages, in a small area to illustrate better what is going on. The two relations are parallel

179

Block Assignments

Figure 4.

Lines of equal reservation wages and profits

1.3

Profit

Worker

1.28

1.26

Wage 1.24

1.22

1.2

1.22

1.24

1.26

1.28

Job

so there is no solution in the range s0–s2 and k0–k2. The basic reason for the persistent gaps between reservation values is that marginal agents are generally better off in the block with higher agent parameter values. The results in Figures 2 and 3 and in Table 2 arise from specific functional forms (including a CobbDouglas production function) and so do not prove a general result. Whether there is a general ‘‘Theorem of Falling Dividing Lines’’ is unknown. It is possible that taxes and transfers could make it possible for a block assignment to be consistent with individual agent choice of blocks. Such a system is described in Sattinger (1995) in an assignment context. In that system, the taxes and subsidies alter agents’ reservation wages and profits so that they equal efficient levels. Then agents will chose the efficient dividing lines. In the absence of such a system, block assignments would not be sustainable, even if efficient. Another possibility for achieving voluntary division into blocks is that a value of b could be found that would yield efficient choice of blocks. In the case of a labor market with homogeneous workers and jobs, Hosios (1990) has shown that efficiency would arise if b, the parameter for labor bargaining strength, equals the elasticity of the matching function with respect to the number of unemployed. This mechanism for achieving efficiency does not directly apply to block assignments, however, because of the presence of heterogeneous workers and jobs within each block. Consider the problem of finding a value of b such that the division of workers and jobs yields an optimal division into two blocks. For the  optimal dividing values s 1 and k1 from Section 3, setting b ¼ 0:262 yields @ðp1 þ p2 Þ=@s1 ¼ 0. However, this value of b does not yield @ðp1 þ p2 Þ=@k1 ¼ 0. Instead, setting b ¼ 0:585 yields @ðp1 þ p2 Þ@k1 ¼ 0. Therefore a common value

180

Michael Sattinger

of b for the two blocks cannot be found such that decisions of workers and employers would yield an efficient block division. It is also possible to consider different values of b for the two blocks. In the case considered here, such a solution cannot be found.2

5. Conclusions This paper has undertaken analysis and computation to determine whether the behavior of workers and employers would divide an aggregate labor market into smaller, separate labor markets with narrower ranges of worker and job characteristics. Assuming agents can find appropriate blocks and choose them costlessly (i.e., they do not need to engage in search to find blocks), block assignments could generate an increase in production and agent asset values by providing for a more accurate assignment. With the assumptions about functional forms and parameters adopted here, the dividing values for workers and jobs are not consistent with individual choice of blocks in a standard search framework. Since only specific functional forms were considered, we can at most conclude that there is no robust mechanism dividing up aggregate labor markets into separate, smaller markets with ranges of worker and job characteristics. The results do not rule out the possibility that a division of the aggregate labor market could occur with different functional forms and parameter values. Institutional features that would generate dual labor markets could also divide the labor market into blocks, although there is no reason to suppose that the division would be efficient. It is further possible that some credential could be required for workers entering Block Two, again generating block assignments. A signaling equilibrium could then arise. These results are fully consistent with previous analysis of complete markets, in which workers are able to choose the specific type of job that they apply to. The difference between complete markets and block assignments is that blocks contain heterogeneous workers and jobs, rather than single types. One sloppy alternative to block assignments is an overlapping labor market (Sattinger, 2005). In an overlapping labor market, some workers do not seek some types of jobs, but the labor market does not break up into separate blocks. With overlapping labor markets, labor market conditions are not uniform but instead vary among types of workers and types of firms. Unlike block assignments, the existence of equilibrium in overlapping labor markets can be

2 The procedure is to consider combinations of b1 and b2 such that the reservation wages at s1 are the same (where b1 and b2 are the values of b in Blocks One and Two, respectively). These combinations form a curve in (b1,b2) space. A second curve is formed by the combinations of b1 and b2 that yield reservation profits at k1 that are the same in both blocks. These curves do not intersect for values of b1 and b2 between 0 and 1.

Block Assignments

181

established. A number of other phenomena arise once one departs from simple descriptions of labor markets. Acknowledgment The author is indebted to participants at the conference on ‘‘Labour market models and matched employer–employee data’’ and an anonymous referee for helpful comments. Any errors are the responsibility of the author. References Bloch, F. and H. Ryder (2000), ‘‘Two-sided search, marriages, and matchmakers’’, International Economic Review, Vol. 41(1), pp. 93–115. Burdett, K. and M. Coles (1997), ‘‘Marriage and class’’, Quarterly Journal of Economics, Vol. 112(1), pp. 141–168. Chade, H. (2001), ‘‘Two-sided search and perfect segregation with fixed search costs’’, Mathematical Social Sciences, Vol. 42(1), pp. 31–51. Hosios, A.J. (1990), ‘‘On the efficiency of matching and related models of search and unemployment’’, Review of Economic Studies, Vol. 57, pp. 279–298. Masters, A. (1999), ‘‘Wage posting in two-sided search and the minimum wage’’, International Economic Review, Vol. 40(4), pp. 809–826. Moen, E.R. (1997), ‘‘Competitive search equilibrium’’, Journal of Political Economy, Vol. 105(2), pp. 385–411. Mortensen, D. and C. Pissarides (1999), ‘‘New developments in models of search in the labor market’’, in: O. Ashenfelter and D. Card, editors, Handbook of Labor Economics, Vol. 3B, Amsterdam: Elsevier. Petrongolo, B. and C. Pissarides (2001), ‘‘Looking into the black box: a survey of the matching function’’, Journal of Economic Literature, Vol. 39(2), pp. 390–431. Pissarides, C. (2000), Equilibrium Unemployment Theory, 2nd edition, Cambridge, MA: MIT Press. Sattinger, M. (1995), ‘‘Search and the efficient assignment of workers to jobs’’, International Economic Review, Vol. 36(2), pp. 283–302. Sattinger, M. (2005), ‘‘Overlapping labour markets’’, forthcoming, Labour Economics. Shi, S. (2001), ‘‘Frictional assignment, Part I: Efficiency’’, Journal of Economic Theory, Vol. 98, pp. 232–260. Shimer, R. and L. Smith (2000), ‘‘Assortative matching and search’’, Econometrica, Vol. 68(2), pp. 43–69.

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CHAPTER 9

Social Security and Intergenerational Redistribution Joydeep Bhattacharya and Robert R. Reed Abstract Many countries around the world have large public pension programs with significant cross-cohort redistribution. This paper provides a rationale for such programs in a lifecycle framework with search and matching frictions in the labor market. In the model, public pension programs alter the age composition of the labor force by inducing the jobless elderly to retire. This improves the allocation of workers to jobs, raises firm entry and may also improve welfare. By requiring a long history of labor market attachment as a precondition to receiving benefits, these programs raise the future value of current employment for the young. This redistributes bargaining strength and income from the young to the old.

Keywords: search, labor market efficiency, unemployment, lifecycle, pensions JEL Classifications: J41, J64, E24 1. Introduction At least 150 countries around the world have public pension programs. As documented in Gruber and Wise (1999) and Mulligan and Sala-i-Martin (2004), these programs share several common striking features. First, over 70% of countries pay pension benefits in a way as to discourage work by their elderly citizens. This

Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75009-4

183

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Joydeep Bhattacharya and Robert R. Reed

is starkly evident from the fact that retirement, nearly everywhere, is a necessary condition for receiving full public pension benefits. In addition, governments use a variety of ‘‘stick and carrot’’ measures to dissuade the elderly from seeking work: high implicit taxes and earnings penalties on income earned beyond a certain age act as sticks while generous benefits act as carrots. Second, an important prerequisite for receiving public pension benefits in almost every case is a documented long history of labor market participation. Finally, public pension programs generally have pay-as-you-go features implying substantial intergenerational redistribution. In this paper, we provide a rationale for pension programs with these features. We also offer a novel channel by which cross-cohort redistribution may take place. Explanations for the existence of social security are classified by Mulligan and Sala-i-Martin (2004) as either political theories or efficiency theories. The former view social security as redistribution, ‘‘the outcome of a political struggle.’’ The elderly are the winners of a political contest in which the reward is a pension (see Boldrin and Rusticini 2001). Efficiency theories, on the other hand, identify market inefficiencies (e.g. imperfect financial markets) and explain how a pension program might be created to alleviate them. We follow the latter route and isolate a novel market inefficiency arising out of search frictions in the labor market. We go on to demonstrate that public pensions can increase aggregate welfare by partly removing such distortions. Consider a labor market characterized by search and matching frictions and in which individuals at different positions along the lifecycle compete with each other for the same jobs. In such a setting, assuming a fixed stock of available job vacancies, increased labor market participation by the old would restrict the employment prospects of the young. Public pension programs, by encouraging retirement of the elderly, may therefore improve the ability of young workers to find jobs. Moreover, such programs can also provide incentives for firms to create more vacancies. As discussed in Oi (1962), firms incur fixed costs while establishing employment relationships with workers. If the age composition of the labor force is heavily skewed toward the elderly, then firms may not find it profitable to create vacancies.1 Why? While post-match bargaining with the worker ensures a correct division of the post-match surplus created by the job, it does not compensate the firm for its pre-match sunk cost. The firm can better spread this expense if it gets matched with a young worker (one with a longexpected tenure) than an old worker (one with a short tenure). In short, firms would earn higher profits by hiring young workers rather than the old. In such an environment, it is possible that the allocation of workers to jobs would be improved by removing some old jobless workers from the labor market. We

1

This assumes that age discrimination laws limit firms’ abilities to sort across workers.

Social Security and Intergenerational Redistribution

185

argue that pension programs act in this way so as to foster job creation and raise welfare.2,3 These channels establish that social security programs can help to alleviate distortions in the labor market along the extensive margin. We also illustrate how these programs may be responsible for reducing inefficiencies along the intensive margin. To see this, notice that young jobseekers with more time remaining in the job market are in a superior bargaining position compared to their older jobless counterparts. As such, firms would have to part with a large fraction of the post-match surplus in order to attract these people to work. On its own, this effect would hamper firm entry and potentially reduce aggregate welfare. We identify an entirely novel route by which pension programs may undo this bargaining advantage of the young and foster cross-cohort redistribution. By requiring a long employment history as a prerequisite for participation, pension programs raise the future value of current employment thereby inducing the young to work for less (in exchange for future transfer payments). They also raise the option value to not working for the old and eligible. In this manner, pension programs effectively transfer income away from the young and toward the elderly and eligible. In our setup, therefore, cross-cohort redistribution can take place indirectly via firms in the form of higher (lower) wages for the old (young) in addition to the more standard pay-as-you-go transfers. The potential substitutability of one form of redistribution for another is an important, yet neglected dimension in the debate about social security reform currently in progress in all Organization for Economic Co-operation and Development (OECD) countries.

2

In related work, Shimer (2001) also studies the implications of population aging for the labor market. He shows that young workers work for less due to their lack of labor market experience, while we show this effect can be entirely induced by public pension programs. In contrast to our work, all workers in his model are infinitely lived and in each period, a new generation of workers is born. Our methodology is most closely related to Pissarides (1992) who utilizes a two-period overlapping generations model with labor market frictions to study the implications of the loss of productivity that may accompany long-term unemployment. In contrast to our framework, all jobs in his model only last for one period, and there are no costs to labor market participation. While his analysis provides a number of interesting implications for aggregate labor market outcomes, it does not address the important interactions between wages at each stage of the lifecycle, age-targeted labor market policies (such as public pension programs), and retirement decisions. 3 The paper that is closest in spirit to our paper is Sala-i-Martin (1996). In his setup, the old are assumed to be less productive than the young. Moreover, since there are spillovers in the production technology resulting from the average level of labor productivity, the old lower the productivity of the young in the economy. Social security helps induce the old to pull out of the labor force, thereby raising the average level of labor productivity in the economy and promoting economic growth. In contrast, we abstract away from possible differences between young and old workers, except for their naturally different positions along the lifecycle and their labor market experiences. In our setting, gross output from a match with either a young or a newly employed old worker is the same.

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The plan for the rest of the paper is as follows. In the next section, we outline the model environment, specify the time line of events, describe the various search-related costs, and compute payoffs to firms and workers. In Section 3, we compute wages and discuss the properties of the wage function for young workers, especially its connection to pension benefits. Section 4 defines an equilibrium in our model and describes a result on existence and uniqueness. As a benchmark for the effects of social security and induced retirement, Section 5 outlines an equilibrium in which there are no public pensions and all workers participate in the labor market. Section 6 establishes the ‘‘positive’’ aspect of our analysis by demonstrating that economies can obtain higher welfare under public pension programs that cause retirement to occur. Section 7 contains some concluding remarks. Proofs of important results are contained in the appendices. 2. The model 2.1. Environment Consider an economy consisting of an infinite discrete sequence of two-period lived overlapping generations populated by two types of agents: workers, and firms. There is no population growth. In each period, there are workers of two different ages – the young (with measure 12) and the old. At birth, all workers are jobless. Old workers may be in one of three possible states: the long-term unemployed (those who did not find jobs when young), displaced (they were employed while young, but have involuntarily lost their job; see discussion below), or employed.4 All workers are risk-neutral. There are no saving instruments. Firms produce a homogeneous consumption good each period using labor as the sole factor of production. Production is the result of pair-wise matching between one worker and a firm. Firms are infinitely lived with a total population of measure F in each period. They each have access to the same technology and seek to maximize the present discounted stream of revenues net of all costs. Workers and firms share the same discount factor b 2 ð0; 1Þ. 2.2. Time line The time line is as follows. At the start of each period, the labor market opens. At that time, jobless workers, be they old or young (the new born agents), choose whether to search for vacancies or not. If they decide to search, they incur a search cost, s, which is expressed in terms of disutility of search. As

4

Following Pissarides (1992), we refer to those who did not find jobs when young as the long-term unemployed. Since displaced workers are individuals who found job matches when young, but incurred a job separation, we can also refer to them as ‘‘separated’’ workers. Hence, we use the terms ‘‘displaced’’ and ‘‘separated’’ interchangeably.

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described in Pissarides (2000), s represents the imputed value of leisure in terms of output (utility). On the other side of the market, firms make the decision whether to pay some up-front costs (described below) and enter the labor market to look for employees. Each firm may employ at most one worker. Let U (Fv) denote the total mass of unemployed workers (unfilled vacancies) at the start of a period. A stochastic matching technology connects all job seekers with open vacancies. The technology does not discriminate on the basis of age, and therefore, any job seeker (old or young) faces the same (endogenous) probability a of getting matched with a vacancy.5 Once the labor market opens, firms and workers have at most one opportunity to meet and match. At the end of any period, the employment relationship between a worker and a firm ends involuntarily with a given probability b.6 Put differently, a given match lasts for a minimum (maximum) length of one (two) period(s). At the beginning of the period, an old worker finds himself in one of three possible employment categories: employed [attached to a match from the previous period with probability að1 bÞ], unemployed [with probability (1 a)], or displaced (working when young, but lost the job with probability ab).7 Onthe-job search is disallowed by our assumption regarding timing of labor market openings. For future reference, note that the long-term unemployed, unlike displaced workers, have no prior history of labor force attachment. This will create a distinction between them if governmental transfer payments are contingent on their employment history.8 At the end of the period, young employed workers learn their employment status for the following date (i.e., whether their

5 Our matching structure bears many similarities to Pissarides (1992). As in his framework, workers and firms may make at most one job contact each period, and the probabilities of matching are the same for each type of worker irrespective of age (i.e., we also assume a non-discriminating matching technology). 6 All job separations in the model are exogenous and outside of the worker’s influence. In this sense, b is a measure of the frequency of involuntary job separations, and therefore, parameterizes the degree of job security. See Gottschalk and Moffitt (1999) for related discussion. 7 Long term job attachment is an important feature of labor market behavior. For example, 34% of U.S. male workers aged 25 and over had worked for their current employer for 10 years or more in February 2000; for workers aged 55–64, 28% had worked for their current employer 20 years or more. In addition, Hall (1982) finds that after a job has lasted for 5 years, the probability that it will eventually last 20 years or more in all rises to close to 0.5 among workers in their early 30s. These data imply that tenure with a firm can be quite long. The low-frequency nature of our overlapping generations setup is well-suited to capture this aspect of the labor market. It bears emphasis that job turnover in our framework is entirely involuntary. 8 This is one of the benefits of our deterministic, discrete-time model. Since each worker receives only one job contact each period, it is very easy to trace an old worker’s employment status to his employment history. The linkages between eligibility for transfer payments (such as social security) and a worker’s prior labor market history are clearly important, yet often ignored in models of the labor market.

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current match survives to the next period or gets dissolved); at this time, old workers die. 2.3. The labor market As discussed in the introduction, public pension programs in many countries aim to induce retirement by the elderly so as to alleviate unemployment among the young. In this paper, we focus solely on the role played by public pension programs in encouraging the jobless elderly to withdraw from the labor market.9 In many European countries, for example, workers can collect early retirement benefits after an involuntary separation (see Gruber and Wise, (1999) and especially, Boldrin et al., 1999, among others). In France, the ‘‘contrat de solidarite´’’ recognizes the ‘‘double need to encourage 55–59 year-old workers to stop work and to bring young workers into the labor market, as rising youth unemployment was a growing concern to society as a whole.’’ A precondition to receiving unemployment benefits for people over the age of 55 is that they stop ‘‘seeking employment.’’10 We formally motivate these ideas in a setting where an individual’s position along the lifecycle affects his opportunities in the labor market. Furthermore, the participation decisions of all workers have general equilibrium implications through their impact on the number of job vacancies created by firms. En route to studying the possible desirability of policies that affect labor market participation by the elderly, we analyze a setting where a particular subset of workers chooses to retire. In particular, we consider the general equilibrium consequences of public policies that encourage displaced workers to withdraw from the labor market. In terms of deriving the endogenous labor market participation decisions of all workers (in particular, old workers), we adopt the following algorithm. We first condition on a set of strategies where all separated workers have chosen to withdraw from the labor market by accepting retirement benefits rather than incurring the costs of job search. We then study how public pensions must be designed in

9

Displacement is an important route toward retirement in many OECD countries. For example, Chan and Stevens (2002) show that displacement increases the probability of retirement in the U.S. labor market. Specifically, they emphasize that this may be due to the costs of job search and loss of firm-specific human capital. O’Leary and Wandner (2000) conclude that while less than 10% of displaced workers under the age of 55 permanently exit the labor force, more than 25% between the ages of 55 and 64 and almost half of workers over the age of 65 opt for retirement instead of searching for alternative sources of employment upon displacement. Diamond and Hausman (1984) also discuss how job loss among older workers leads to retirement. 10 In Britain, the Job Release Scheme which ran between 1977 and 1988, ‘‘specifically encouraged older workers to stand down to make way for younger ones. Once out of employment, changes to the unemployment benefit regime in 1983 removed the requirement for men over 60 to look for work, encouraging them to see themselves as retired.’’ For more details, see the OECD (1995) study on ‘‘The Labor Market and Older Workers.’’

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order to support the conjectured steady-state equilibrium. We proceed by verifying that a separated worker is better off choosing to collect pension benefits rather than searching for a job. This is the algorithm we adopt in order to endogenize labor force participation for every type of worker at each stage of the lifecycle. 2.4. Costs Firms incur sunk costs of posting vacancies, denoted by a. Once they have incurred this cost and searched for workers, all firms are equally likely to find a worker. The probability that a vacancy finds a worker is y (to be determined in equilibrium below). The probabilities of meeting a given type of worker, however, will depend on the proportion of each type in the labor market. In our conjectured equilibrium, only the young and the long-term unemployed actively search for jobs. While the total measure of unemployed workers is U, the total measures of the young and long-term unemployed are uy and uo. The probability of finding a young unemployed worker is yu~ y , where u~ y  uy =U. Similarly, the probability that a vacancy locates a long-term unemployed worker is yu~ o . The next lemma reports these population proportions for future use. Lemma 1. U¼ u~ y 

uy U

u~ o  uUo

1 þ ð1 aÞ 2 1 ¼ 1 þ ð1 aÞ ð1 aÞ ¼ 1 þ ð1 aÞ

An important point to note here is that the population proportions are all endogenous variables since they depend on a. An implication of this is that policies aimed at altering the age composition of the labor force also change these proportions, and thereby affect the probability with which firms encounter workers of other age groups. This general equilibrium effect is at the heart of our analysis. Following the insights of Oi (1962) and Hutchens (1986), we posit that there are costs which must be incurred at the beginning of an employment relationship. We refer to these as ‘‘hiring’’ costs, and denote them as h. Let the exogenously determined market value of the firm’s output be normalized to 1. Matches with new hires require the firm and the worker to incur the costs of ‘‘hiring and training’’ so that the net output from new matches is ð1 hÞ, while net output from a match with an old, retained worker is 1.11 Under this

11

Note that our framework differs from the standard search-theoretic model with ex-ante heterogeneity. Although one may view the old, retained workers in our setup as ‘‘high’’ types and the displaced and the long-term unemployed workers as ‘‘low’’ types, the probability of becoming a ‘‘high’’ type is endogenous. This is an important distinguishing feature of our model. In particular, as we demonstrate below, the chance of becoming a ‘‘high’’ type will be crucially affected by policy.

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interpretation, one may view h as a cost that is incurred each time a firm makes a new hire. Alternatively, h may proxy a productivity differential between new and old matches. In the latter sense, one may also interpret h as a parameter which reflects the importance of firm-specific human capital. Firms therefore derive higher net revenues from employing workers with longer expected tenure. The wage rate(s) for the different types of workers are determined (see below) in accordance with the protocols of Nash bargaining. As shown there, the presence of age-targeted labor market policies and the aforementioned accumulation of firm-specific human capital will cause the wages of workers (with different employment histories) to vary. 2.5. Specification of labor market policies We incorporate various aspects of real-world age-specific labor market policies, such as public pension programs and long-term old-age unemployment insurance programs, into our model. These take a particularly simple and stylized form. Old workers, currently or previously employed, are eligible for transfer payments from the government. As is common in many countries, these payments are tied to a worker’s prior attachment to the labor market. In that vein, we assume that an individual who worked when young is potentially eligible for a fixed lump sum benefit of B0. We also allow for aspects of earnings reductions, as observed in many programs, in our framework. We capture the notion of an ‘‘earnings test’’ by asserting that workers who work when old receive only a fraction d of benefits due to them.12,13 For example, suppose that an old worker who retained her job from a previous match receives a wage of weo . Then gross of pension benefits, she obtains total income in the amount weo þ dB0 . Since we conjecture that displaced workers choose to retire, in equilibrium, they earn total income B0. The longterm unemployed are not eligible for benefits since they have no documented history of labor force attachment.14

12

The ‘‘earnings test’’ that was applied in the United States until 2000 could be described as follows. In 1999, a worker age 62–65 could earn up to $9,600 without the loss of any benefits, then benefits were reduced $1 for each $2 of earnings above this amount; for workers age 65–69, the earnings test floor was $15,500 and benefits were reduced at a rate of $1 for each $3 in earnings. Although our framework is not suited to capture the specific features of various versions of the earnings test, we can consider its implications, more broadly defined, for retirement behavior and wages of older workers. 13 In our specification, the higher the value of d, the lower is the implicit tax rate on elderly work. Gruber and Wise (1999) found that while this tax is relatively low in the United States (around 20%), it is much higher in a number of European countries (as much as 80%). 14 For now, we ignore issues relating to funding of these programs. The consequent analytical tractability allows us to explicitly endogenize the labor market participation patterns on the basis of the design of public pension programs and formally prove existence of the conjectured steady-state equilibrium. In Section 6, we study the effects of pension programs under a balanced budget constraint.

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In order to study the desirability (or lack thereof) of public pension programs that induce workers of certain ages and employment histories to withdraw from the labor market, we will henceforth construct the model under the conjecture that public policies successfully induce only the separated workers to leave the labor market, and collect B0. In our analysis below, we will provide a set of sufficient conditions under which this conjectured equilibrium exists.15

2.6. Workers’ payoffs Let Jy denote the expected lifetime utility accruing to a worker who decides to search when young, J eo the expected utility of an old worker who begins the period employed and continues his employment, J uo the expected utility of an old worker who did not get matched when young and is back in the labor market seeking employment, and J so the expected utility of an old displaced worker. Then, it is easy to see that
 ð1Þ J y ¼ s þ a wy þ ð1 bÞbJ eo þ bbJ so þ ð1 aÞbmaxf0; J uo g J uo ¼

s þ awuo ;

J so ¼ Bo

J eo ¼ weo þ dBo

ð2Þ

ð3Þ

It is instructive to explore the economic interpretation of equation (1), as the explanations of the other value functions follow straightforwardly. A young worker seeking employment incurs an up-front cost s. Upon entering the labor market, he gets matched with a firm with probability a. In that case, he gets a wage wy and also the expected discounted continuation payoffs from possible employment and separation the following period. If he is unsuccessful in finding a job, he will find himself in the state of being a long-term unemployed worker. From (1), it is also clear that the value of a job to a young worker is much more than just the current wage. Because jobs are potentially durable (longlasting), a match today bestows certain continuation privileges to the worker, a fact that will play a prominent role during the wage-bargaining phase.16

15

Pissarides (1976), in an infinite-horizon model with sequential search, also studies the choice of labor market participation. He derives the optimal number of times individuals will choose to search for jobs before becoming ‘‘discouraged’’ and withdrawing from the labor market. However, he does not consider the role of the lifecycle in his analysis. His model also does not address how labor market participation is influenced by pension or labor market policies. 16 Davidson et al. (1994) demonstrate how the durability of jobs results in a social surplus when workers have finite lives in an infinite-horizon economy. However, unlike their paper, we embed this idea into an overlapping generations framework. In addition, we explicitly introduce important features of pension programs which reinforce the role of employment beyond current compensation, thereby leading to intergenerational income redistribution.

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2.7. Payoffs to firms Firms begin each period in one of the two possible states. They may currently have a vacancy, or they may be matched with an old worker from a previous employment relationship. Letting Pv (Pf) be the expected lifetime profits of a firm that has an unfilled (filled) vacancy at the beginning of the period, the following equations describe the associated expected present discounted profits of a firm in each state:   Pv ¼ a þ yu~ y ð1 h wy Þ þ ð1 bÞbPf þ bbPv 
  ð4Þ þ yu~ o 1 h wuo þ bPv þ ð1 yÞbPv  Pf ¼ 1

 weo þ bPv

ð5Þ

As indicated above, if the firm is currently matched with an old worker, it will have a vacancy next period it if incurs the cost a. Note that the firm does not face any hiring costs if the employment relationship from the previous period is retained. Also note that firms take the proportions, u~ y and u~ o , as given when deciding whether to enter the labor market. The following closed form expression for the steady-state payoff to entry will be of considerable use below: Pv ¼

a þ yu~ y ð1 h wy Þ þ yu~ y ð1 bÞbð1 weo Þ þ yu~ o ð1 h
 1 yu~ y ð1 bÞb2 yu~ y bb yu~ o b ð1 yÞb

wuo Þ

ð6Þ

2.8. Matching Unemployed workers and unfilled vacancies are brought together each period through a stochastic matching technology. The matching technology describes the total number of matches, m ¼ mM(U,Fv), that are formed at the beginning of each period, depending on the total masses of unemployed workers and unfilled vacancies. Since a represents the probability that an unemployed worker will find any vacancy in the time period and y is the probability that any unfilled vacancy will find an unemployed worker, it follows that the total number of workers who find employment (a, U) must equal the total number of firms that filled their vacancies (y
Fv): a
U ¼ y
Fv. It is important to note that a and y are determined in equilibrium, and that both workers and firms take them as given when making their decisions. Noting that m ¼ y
F v , we have aU ¼ yF v ¼ m ¼ mMðU; F v Þ

ð7Þ

the matching condition. It is standard to assume that the matching technology takes the Cobb–Douglas form: m ¼ mðUÞ1 f ðF v Þf where f 2 ½0; 1Š. Noting that yF v ¼ mðUÞ1 f ðF v Þf , it follows that y ¼ ½mðU=F v ފ1 f . An increase in either the number of unemployed workers or unfilled vacancies increases the number of matches each period, but at a decreasing rate. Ceteris paribus, more matches occur when m is higher.

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For analytical tractability, in all of the algebra we report below, we will assume f ¼ 1. Then, y ¼ m  1 obtains. In fact, it is easiest to conduct our analysis (and obtain closed form solutions to various endogenous variables) for the case where y ¼ m ¼ 1. This implies that MðU; F v Þ ¼ F v . Vacancies always find a worker, but workers may find a vacancy only with a probability a 2 ð0; 1Þ that will be determined below.17 In such an economy, the congestion problems facing unemployed workers are severe. Below, we will remark on how our results depend on the extent of congestion problems encountered by both workers and vacancies by studying numerical examples with fo1. 3. Bargaining and wage determination The friction inbuilt into the job–firm matching process creates the possibility that a firm may remain unproductive or a worker may remain unemployed in any period. Firms and workers must therefore weigh the implications of finding themselves in these states and their outside options when bargaining over their share of current and future surplus produced. Two important things deserve mention here. First, the outside options available to workers are crucially affected by policy, and second, these outside options are dependent on past employment history and on one’s position in the lifecycle. Below, we will demonstrate the powerful implications of this last observation. To foreshadow, we will establish the presence of a ‘‘skewness’’ in bargaining position toward the young, and the role played by pension programs in ‘‘undoing’’ some of the resultant inequities. 3.1. Wage functions We now turn to the determination of the wage offer functions for both young and old workers. Matches between workers and unfilled vacancies leads to a surplus that is to be divided between the worker and the firm. Nash bargaining dictates that the total match surplus be shared by the firm and the worker; principally for analytical tractability, we assume symmetric Nash bargaining. For an old worker with an unbroken employment relationship from the previous period, the gain from the match is ½weo þ dB0 Š B0 . The corresponding gain to the firm is ð1 weo þ bPv Þ bPv ¼ 1 weo .18 Then, Nash bargaining implies weo ¼

17

1 þ ð1

dÞBo 2

ð8Þ

We only study equilibria in which there are more unemployed workers than vacancies. Hence, we have both U4F v and ao1 in our analysis. 18 We assume that even when a match survives on to the second period, wages are determined by a fresh process of bargaining at the start of the second period. Also note that the outside option, due to the timing assumptions in the model, is discounted.

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Analogously, it follows that the wage to a long-term unemployed worker (one who has no history of labor force attachment) is given by wuo ¼

ð1

hÞ

ð9Þ

s

Finally, we turn to the wage determination for a young worker. The gains from trade for the firm are given by
 1 h wy þ b bPv þ ð1 bÞPf þ Pv bPv ð10Þ

while the young worker’s surplus from finding employment is given by
 wy þ b bJ so þ ð1 bÞJ eo J uo

ð11Þ

The analytical expression for the wages accruing to a young worker is described in the following lemma. Lemma 2. (a) Under the assumption of linear matching (f ¼ 1), the expression for the wage function for the young is given by   ba 2wy ¼ ð1 hÞ 1 þ bðB0 þ sÞ ð12Þ 2 (b) If ð1

hÞ4bðB0 þ sÞ, then wy 40.

Note that part (b) of Lemma 2 is a sufficient condition for young workers to earn positive wages, since it holds for any possible value of a. Ceteris paribus, higher search costs s, reduce the wages to the young by reducing the option value to waiting and searching in the future. It also follows that, ceteris paribus, a higher pension benefit B0, reduce the wages to the young, an issue to which we now turn. 3.2. Discussion of the wage function for the young Suppose for the moment that all public pension programs are absent, i.e., B0 ¼ 0. In this case, using (8)–(9), we have: wuo ¼

ð1

hÞ 2

;

weo ¼

1 2

Also, using (11), the young worker’s surplus from finding employment is given by: wy þ bð1

bÞweo

bJ uo

and, using (10), the firm’s surplus (assuming free entry) from hiring the young worker is given by: 1

h

wy þ bð1

bÞð1

weo Þ

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Equating, we get wy ¼

wy ¼

ð1

ð1

hÞ

1 þ bð1 2

hÞ

b þ J uo 4wuo 2

2

2

bÞð1

weo Þ

1 bð1 2

b bÞweo þ J uo 2

ð13Þ

ð14Þ

A young worker can expect that his job will last beyond the current period. The wage function for young workers reflects this via the fact that the value of employment this period is more than just the current wage. It is now apparent that inequities in bargaining strength over the lifecycle arise, purely because of agents’ positions on the lifecycle. Young workers, who have the option of searching for jobs when old, will have a higher threat point in negotiating over wages than old workers (who have no such outside option).19 We are now in a position to isolate a key social function played by pension programs toward reducing the aforementioned inequity. To see this, recall that young workers (by virtue of the fact that they likely have a period ahead of them) have a higher bargaining position than the old. Also a fundamental eligibility criterion for receiving pensions when old is a history of labor force attachment. Employment when young therefore raises the worker’s expected net income in the future. The threat point of a young worker (arising from their position in the lifecycle) is therefore partially reduced because the firm is aware that having a job today implies current (and future) benefits to the employee; the firm naturally extracts part of that surplus. It is in this sense that public pensions help redistribute bargaining strength from young to old workers, raising the wages for the old and eligible and reducing the wages of the young.20,21 The above discussion was based entirely on ‘‘ceteris paribus’’ arguments, since a was held constant throughout the discussion. We now turn to the determination of a along with all other endogenous variables.

19

It is important to note here that past private earnings do not affect a worker’s current bargaining strength because of our earlier assumption ruling out any form of asset accumulation or saving. 20 Black (1987) also finds that social security affects age-earnings profiles. In his model, workers would rather receive private pension payments than wages as a result of social security taxes. As workers become older, they switch from pension payments to wages since the returns from pension savings would be lower. Therefore, social security tends to generate upward-sloping age earnings profiles. In his work, the retirement date is exogenous (he does not explore the early retirement incentives in the social security system). In addition, there is no unemployment in his model. 21 In contrast to symmetric Nash bargaining, we could posit that agents’ bargaining weights vary across the lifecycle. If old individuals receive higher weights than young workers, young agents’ outside options would have less impact on wage determination. Consequently, there would be less need for intergenerational redistribution.

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4. Equilibrium 4.1. Definition and existence We focus exclusively on time-invariant equilibria. This will allow us to investigate the properties of long-run equilibria in the labor market. A steady-state equilibrium with no labor market participation by the displaced (separated) workers is formally defined below. Definition 1. A steady-state equilibrium with no labor market participation by displaced workers [an ‘‘induced retirement equilibrium’’] consists of wage functions wy, weo , and wuo [defined in (8), (9), and (12)], policy parameters, B0 and d, and a quadruple ða; y; U; F v Þ satisfying the following conditions: (i) Symmetric Nash bargaining; (ii) (Unrestricted Entry for firms): Pv ¼ 0; (iii) (Steadystate): aU ¼ yF v ¼ mMðU; F v Þ, with linear matching, y ¼ 1, and (iv) the labor market participation/non-participation constraints hold: J uo 40; J y 4bJ uo ; J eo 4Bo and the displaced worker constraint holds (see below).

4.2. Labor market participation conditions As stated in the definition of the equilibrium, we impose a pattern of labor market participation across workers of different age groups and employment histories and then state conditions under which this pattern emerges as an equilibrium. In particular, we study a steady-state equilibrium in which old workers who have experienced job loss during the course of their careers choose to accept their pension benefits and withdraw, rather than incur the costs of job search. Old individuals who have retained their jobs continue working since they have higher productivity than when they were initially employed. In contrast, the long-term unemployed with no access to pension benefits choose to look for jobs. We also provide conditions to ensure that the young actively search for jobs. We begin with a discussion of the participation conditions for old workers who have retained jobs from their youth. In order for them to continue working, we must have J eo ¼ weo þ dBo 4Bo

ð15Þ

which using (8) reduces to 14ð1 dÞB0 , a sufficient condition for which is 14B0 . The next step is to find conditions under which displaced workers choose to accept pension benefits and retire rather than incur the costs of job search. If a separated worker chooses to accept pension benefits, his expected utility is: J so ¼ Bo . However, the decision to withdraw from the labor force must yield higher expected utility. Therefore, the following condition must hold: Bo 4

s þ aws þ adBo þ ð1

aÞBo

ð16Þ

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Under the assumption that an individual displaced worker chooses to search for a job (an ‘‘individual’’ deviation), with probability a the displaced worker would be able to obtain employment and would earn total income ðws þ dB0 Þ. In this event, the wage he would earn is given by: hÞ þ ð1 dÞBo ð17Þ 2 Alternatively, if unable to find employment, the worker would still be able to collect pension benefits. Using (17) in (16), it follows that policy-induced withdrawal by displaced workers occurs if ð1 hÞ4ð1 dÞB0 and ws ¼

ao

ð1

½ð1

hÞ

2s ð1

dÞBo Š

ð18Þ

holds. This provides an upper-bound for a. In contrast, for the long-term unemployed to search for jobs, we require that: J uo ¼

s þ awuo 40

Using (9), this condition may be rewritten to provide a minimal value for a for which the long-term unemployed remain active in the labor market: a4

2s ð1

hÞ

ð19Þ

Obviously, if the search costs are too high, the long-term unemployed would be better off choosing not to search for jobs. A quick comparison of (18) and (19) reveals the following insight regarding the earnings test. Lemma 3. For an induced retirement equilibrium to exist, it is necessary but not sufficient that there be an earnings penalty, i.e., do1 must obtain. If d ¼ 1, an induced retirement equilibrium does not exist. Finally, in order for young workers to search for jobs, the expected utility of participation when young must exceed the value of waiting and looking for a job when old. This implies that:
 J y ¼ s þ a wy þ ð1 bÞbJ eo þ bbJ so þ ð1 aÞbJ uo 4bJ uo ð20Þ

The following lemma provides conditions on a such that young workers choose to actively search in the labor market.

Lemma 4. (a) Suppose that the displaced worker constraint is satisfied and ð1 hÞ4ðB0 þ sÞ. In addition, let b1  ½ð1 hÞ bðB0 þ sފ. If awy ðaÞ4s, then (20) is satisfied and young workers will choose to search for jobs, which obtains when qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 þ b21 þ 4bð1 hÞs a4aL  40 ð21Þ bð1 hÞ (b) aL 4 ð12shÞ

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To summarize, a valid induced retirement equilibrium value of a under linear matching must satisfy (18), (19), and (21); additionally, the level of benefits B0 must satisfy ð1 hÞ4max½ð1 dÞBo ; bðB0 þ sފ. Henceforth, we will maintain the assumption: (a)

ð1

(b)

3ð1

hÞ4max½ð1

dÞBo ; bðB0 þ sފ

hÞ44a

ð22Þ ð23Þ

4.3. Equilibrium entry condition Firms enter the labor market in search of employees until all profit opportunities from new jobs are driven to zero. This ‘‘free-entry condition’’ dictates that the expected present value of future profits attributable to filling the marginal vacancy must equal the cost of vacancy-posting and hiring the next worker. Utilizing the wage functions described above, along with Pv ¼ 0 [see (6)], we have   a 1      u~ o  ¼ 1 h wy þ ð1 bÞb 1 weo þ 1 h wuo ð24Þ y u~ y u~ y Then, setting y ¼ 1, candidate equilibrium values of a are derived from (24) using Lemma 1, (8), (9), and (12). One of the major benefits of using a simple matching technology like ours is that closed-form solutions to (24) can be analytically derived.22 The following proposition describes the conditions required for existence of an induced retirement equilibrium. Proposition 5. (a) The unique solution to (24) in terms of a, is given by

bs þ bð1 bÞ þ bB0 ½1 ð1 bÞð1 dފ þ 2½ð1 hÞ 2aŠ a  aðB0 ; dÞ ¼ 2 ½ð1 hÞð2 þ bÞ 4aŠ ð25Þ (b) Suppose Assumption 1 holds. Then, for a in (25) to be part of an induced retirement equilibrium with y ¼ 1, (18), (19), and (21) must hold or, more

22

As stated above, each worker experiences the same probability of finding a job vacancy. This assumes strict enforcement of age discrimination laws. Alternatively, one could consider that firms sort across different types of workers. In this manner, there would effectively be two different labor markets: a market for young workers and a separate market for old. Under equilibrium entry, firms earn the same expected net profits. However, more vacancies would be posted in the young labor market.

199

Social Security and Intergenerational Redistribution

compactly, the condition  2s a 2 aL ; min ½ð1 hÞ ð1

dÞBo Š

;1



holds.23

4.4. Partial equilibrium effects of increasing the generosity of benefits As discussed in Mulligan and Sala-i-Martin (2004), many OECD countries have munificent pension programs that induce the jobless elderly to retire and make way for the young. In this section, we establish the effects of varying the generosity of pension programs within an induced retirement equilibrium. Assuming that an equilibrium exists, we are able to analytically derive the effects of pension benefits on employment and the age-composition of the labor force. We begin by reporting the results of some comparative static exercises conducted with respect to B0. It bears emphasis here that all the upcoming results in this subsection assume away any issues relating to funding of B0 and are hence to be understood as being ‘‘partial equilibrium’’ in nature. As Section 6 will demonstrate, these insights are robust to settings in which pension benefits are funded endogenously by payroll taxes. The principal benefit of the ‘‘partial equilibrium’’ perspective is that it allows us to derive a number of interesting clean results analytically. Proposition 6. Under Assumption 1, an increase in B0 raises the probability of finding employment. In particular, we have: @a 2b½1 ð1 bÞð1 dފ 40 ¼ @B0 ½ð1 hÞð2 þ bÞ 4aŠ

ð26Þ

An increase in B0 reduces the wages of the young and at the same time raises the wages of old workers with jobs. Since the young constitute the bulk of the jobseekers, this raises the benefit from firm entry, and more firm entry makes it easier for any given worker to find a vacancy thereby raising the employment rate. Therefore, we refer to this transmission channel of pension programs as the ‘‘vacancy creation effect.’’ On the face of it, Proposition 6 is a formal statement of the type of argument governments use to defend generous pension programs ostensibly intended to free up jobs for the young. An immediate consequence of Proposition 6 is the following corollary.

23

For a generic y, it is easily checked that (25) is given by a ¼

2 4a½2ð1 ð1 hÞð2þbÞ y

h

2a y Þþ

ð1 ð1 bÞð1 dÞÞbB0 þ bs þ bð1 bފ which we use later when doing numerical computations with a non-linear matching technology.

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Joydeep Bhattacharya and Robert R. Reed

Corollary 7. An increase in B0 changes the age-composition of the labor force via   @u~ y 1 @a ¼ 40 @B0 @B0 ðu~ y Þ2 @u~ o 1 1 @a o0 ¼  ð Þ 2 2 @B @B0 ð1 aÞ 0 1 1þ ð1 aÞ As established by Proposition 6, higher pension benefits increase the probability that any worker is able to obtain employment. In particular, since there will be less workers who are unable to find jobs when young, the pool of the long-term unemployed will be smaller. Consequently, more generous pension benefits raise the probability with which firms are likely to encounter a young worker. As we describe below, from examining (8), (9), and (12), this ‘‘vacancy creation effect’’ has implications for age-earnings profiles in the economy: Proposition 8. An increase in B0 raises the wages to the old and employed and has no effect on the wages of the never-before-employed. The effect on young wages is ambiguous. The effect on wy can be seen from the expression for wy, reproduced here for convenience:   ab 2wy ¼ ð1 hÞ 1 þ bðB0 þ sÞ 2 On the one hand, a higher B0 raises a which serves to raise wy [the ‘‘vacancy creation effect’’ i.e., workers can find jobs more easily and hence their ‘‘price’’ must go up], but on the other, a higher B0 serves to reduce wy [this is the ‘‘bargaining strength redistribution effect’’ that was discussed in Section 3]. The net impact is ambiguous and depends on the relative strength of the two aforementioned effects. In particular, the vacancy creation effect somewhat compromises the income redistributive goal of social security. Under the linear matching technology, the effect of an additional vacancy on the probability of finding of a job can be substantial. This may even cause the wages of the young to rise with benefits. Numerical computations confirm that with a small degree of non-linearity (fo1) in the matching function, the ‘‘vacancy creation effect’’ (an indirect influence) is muted and dominated by the direct ‘‘bargaining strength redistribution effect.’’ Intuitively, the latter effect will dominate the vacancy creation effect as long as firms also encounter congestion problems in the labor market, i.e., there is some degree of diminishing returns to the addition of another vacancy.24

24

See Bhattacharya et al. (2004) for further discussion.

201

Social Security and Intergenerational Redistribution

Example 1. Let s ¼ 0.15, a ¼ 0.2, h ¼ 0.4, d ¼ 0.2, b ¼ 0.95, m ¼ 0.4, B0 ¼ 0.2, b ¼ 0.2, and f ¼ 0.7. For this parametric specification, a ¼ 0.61867, and for this value of a, all other conditions outlined in the definition of the induced retirement equilibrium hold. Using this configuration, it is apparent from Figure 1 that raising the level of benefits lowers the wages to the young and raises the wages to the old and eligible, thereby accomplishing cross-cohort income redistribution. This, in some sense, is a major punch line of the paper. In the model, pension programs raise the incomes of the old with jobs and tend to reduce the incomes of the young, thereby engineering an intergenerational income distribution toward the elderly. The novelty of our paper lies in the fact that we can demonstrate the presence of such intergenerational income distribution in the complete absence of any equity or political economy concerns. 5. The absence of policy In this section, as a benchmark for considering the effects of induced retirement, we briefly outline the environment in the absence of any policy intervention. Since much of the basic structure of the economy remains the same, we choose to minimize detailed discussion of the analysis. We start by revisiting the value functions describing the expected lifetime utility of workers.
 Jy ¼ s þ a wy þ ð1 bÞbJ eo þ bbJ uo þ ð1 aÞbJ uo ; J eo ¼ weo ; J uo ¼ s þ awuo

% change in wages

Recall that in the absence of policy, the old separated are indistinguishable from the old never-before-employed, and can be lumped into the single category of jobless elderly. The value functions for the firms are the same as before.

1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 -1.6 -2 -2.4 -2.8 -3.2

0.2

0.205

Figure 1.

Response of wages to benefits

0.21

0.22

0.215

0.225

0.23

0.235

0.24

B0 weo

wy

ws

0.245

0.25

0.255

0.26

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Joydeep Bhattacharya and Robert R. Reed

It is clear that the wage functions for old jobless workers and retained individuals are the same as in the previous section since they do not depend on the probability of finding a job. In contrast, the wage paid to the young is however, different. Even though the gains from trade to the firm from hiring a young worker remain the same as before, a young worker’s surplus from finding employment is now given by wy þ bbJ uo þ ð1

bÞbJ eo

bJ uo ¼ wy þ bð1

bÞs

bð1

bÞawuo þ ð1

bÞbweo

In particular, the surplus reflects that young workers will choose to search for jobs when they become old since induced retirement does not occur. Under symmetric bargaining, and free entry, it can be shown that   ð1 hÞ bð1 bÞa bð1 bÞs 1 þ ð27Þ wnp ¼ y 2 2 2 where the superscript ‘‘np’’ signifies ‘‘no policy.’’ The most crucial difference between the environment with and without policy is in the nature of the equilibrium. As we have discussed earlier, the case with policy focuses on an equilibrium in which pension benefits successfully induce the old and separated to withdraw from the labor force. The appropriate comparison is with a setting without policy intervention in which every jobless worker is in the labor force. Definition 2. A steady-state equilibrium without policy intervention and with labor market participation from all workers consists of wage functions wuo , weo , and wy [defined in (8), (9), and (27)], and a quadruple ða; y; U; F v Þ satisfying the following conditions: (i) Symmetric Nash bargaining; (ii) (Unrestricted Entry for firms): Pv ¼ 0; (iii) (Steady-State): aU ¼ yF v ¼ mMðU; F v Þ, with ynp ¼ 1, and (iv) the labor market participation constraints hold: J uo 40; J y 4bJ uo , and J eo 40. As before, firms enter until Pnv ¼ 0. Analogous to the equilibrium entry condition derived in the model with policy, it can be checked that anp ¼

4ð1 hÞ 8a þ 2ð1 bÞð1 þ sÞb ð1 bÞ½ð1 hÞb þ 2ð1 hÞ 4aŠ

ð28Þ

We conclude this section with an important result. Proposition 9. Young workers always earn higher wages in the absence of induced retirement. That is, wnp y 4wy where wy is defined in Equation (12). This is the crux of the income redistribution argument. Pension policies, by their very nature, raise the future value of employment, and thereby reduce wages to the young. In the absence of such policies, as Proposition 9 indicates, the wages of the young are relatively high. Since they form the bulk of the job seekers, ceteris paribus, a higher wage to the young adversely affects firm entry, and possibly reduces aggregate worker welfare.

Social Security and Intergenerational Redistribution

203

Furthermore, by encouraging old displaced workers to retire, social security programs can play an important role in improving the allocation of workers to jobs since they allow young workers to more easily find jobs and accumulate firm-specific human capital. In what follows below, we first aim to demonstrate our efficiency rationale for public pensions and induced retirement. In order to accomplish this objective, we explicitly introduce a government budget constraint into our framework so that pensions are funded within the economy. Specifically, we present a setting where payroll taxes imposed on both firms and workers are used to pay for pension benefits. Section 6 below establishes that public pensions through induced retirement can lead to higher welfare than when public pensions are absent. 6. Are pension programs welfare enhancing? The principal point of this paper is to argue that pension programs, through their effect on the wage structure, their inducement to pull the old displaced workers out of the labor market, and thereby encourage firms to create more job vacancies, can improve the operation of the labor market and might therefore be desirable on efficiency grounds alone (abstracting from the more standard equity and political economy motives). To that end, before discussing the overall effects of pension programs and their interactions with labor market conditions, we first seek to demonstrate that endogenously funded pension programs and publicly induced retirement can lead to higher welfare than having no pension program at all. Below we sketch a version of our model that introduces payroll taxes on workers and firms which are then used to pay the old, separated to stay away from the labor market. We compute aggregate welfare (defined below) for this economy and compare it to aggregate welfare for the economy described in Section 5. Much of the analysis set forth above will remain valid in this section. To begin with, the value functions for workers of different types are given by
 J y ¼ s þ a ð1 tÞwy þ ð1 bÞbJ eo þ bbJ so þ ð1 aÞbJ uo J uo ¼

s þ að1

tÞwuo ;


J eo ¼ ð1

tÞweo þ ð1

tÞdBo



and, under the conjectured equilibrium that displaced workers do not search, J so ¼ ð1

tÞBo

where t is the common tax rate on wage and benefit income. In addition, in the steady state we observe: 
  Pv ¼ a þ yu~ y 1 h ð1 þ tÞwy þ ð1 bÞbPf þ bbPv 
  þ yu~ o 1 h ð1 þ tÞwuo þ bPv þ ð1 yÞbPv Pf ¼ 1

ð1 þ tÞweo þ bPv

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Joydeep Bhattacharya and Robert R. Reed

It is easily verified that Pv ¼


a þ yu~ y 1



h ð1 þ tÞwy þ yu~ y ð1 bÞb 1 ð1 þ tÞweo þ yu~ o 1 h
 1 yu~ y ð1 bÞb2 yu~ y ð1 bÞbb2 yu~ o b ð1 yÞb

ð1 þ tÞwuo



Using the process of wage determination analogous to the one described in Section 3 above, it can be shown that wuo ¼

ð1

hÞ 2

weo ¼

;

1 þ ð1

dÞð1 2

tÞBo

ð29Þ


  2wy ¼ ð1 hÞ þ ð1 bÞb 1 ð1 þ tÞweo
 bbð1 tÞBo þ ð1 bÞb ð1 tÞweo þ dð1

tÞBo



að1

 tÞwuo ð30Þ

Utilizing the wage functions described above, along with Pv ¼ 0, we have the same equilibrium entry condition as in (24) given by  
a 1 ¼ 1 y u~ y

þ

h

 ð1 þ tÞwy þ ð1

  u~ o
1 u~ y

h

ð1 þ tÞwuo


bÞb 1

ð1 þ tÞweo





ð31Þ

What remains for us to describe is the government budget constraint. The payroll taxes paid by firms are given by F v yu~ o twuo þ F v yu~ y twy þ F f tweo since some job vacancies will be filled by the long-term unemployed and others by the young. In addition, some taxes will be paid by firms with retained workers from prior established employment relationships. In contrast, taxes paid out by all the workers are given by atwy að1 þ 2

  bÞt weo þ dBo ab þ auo twuo þ tBo 2 2

The expenditure by the government on workers is given by bÞ

að1 2

dBo þ

ab Bo 2

205

Social Security and Intergenerational Redistribution

We assume that the government balances its budget. It is also easy to verify that: bÞ 2 að1 bÞð2 aÞ Fv ¼ y½2ð1 bÞ þ að1 aފ að1 bÞ að1 bÞ 2 þ F¼  b 1 a 2 y 1 2 a 2 Ff ¼

að1

For completeness sake, we define an equilibrium below. Definition 3. A steady-state equilibrium with no labor market participation by displaced workers and internally funded pensions consists of wage functions wuo , weo , and wy [defined in (29), and (30)], and a quadruple (a, y, U, Fv) satisfying the following conditions: (i) Symmetric Nash bargaining; (ii) (Unrestricted entry for firms): Pv ¼ 0; (iii) (Steady-state): aU ¼ yF v ¼ mMðU; F v Þ, (iv) the labor market participation constraints hold: J uo 40; J y 4bJ uo , and J eo 4ð1 tÞBo and the discouraged worker constraint holds,25 and (v) the government’s budget is balanced. We choose a population-based average of expected lifetime utility of each group of workers as our welfare criterion. In particular, we adopt the following measure of social welfare as our welfare criterion:26 1 1 W  W ðB0 ; dÞ ¼ J y þ abð1 2 2

1 1 bÞJ eo þ abbJ so þ bð1 2 2

aÞJ uo

The task ahead is to compare aggregate welfare in the presence and absence of policy. In the presence of (internally funded) policy, the old and separated stay out of the labor market. In the absence of such policy intervention, every worker participates in the labor market but there are no pension payments or taxes. The question for us is: is aggregate welfare higher when no pension programs are present and all workers remain active in the labor force? As is readily apparent, a number of non-linearities enter the model with the introduction of the government budget constraint especially when benefits are

25

This case, the discouraged worker constraint requires

ð1 where

tÞBo 4

ws ¼ 1 26

s þ a½ð1

tÞws þ dð1

tÞBo Š þ ð1

aÞð1

hþð1 dÞð1 tÞBo 2

See Davidson et al. (1994) for a similar welfare criterion.

tÞBo

ð32Þ

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Joydeep Bhattacharya and Robert R. Reed

funded by distortionary taxes. Therefore, we use numerical computations for the general case of non-linear matching to illustrate our reasoning. Example 2. Let s ¼ 0:15, a ¼ 0:2, m ¼ 0:4, h ¼ 0:25, f ¼ 0:5, B0 ¼ 0, b ¼ 0:9, and b ¼ 0:2. Under this parametric specification, pension programs are not present, and it can be checked that all workers stay active in the labor market. The aggregate welfare in this case is 0.2095. Now consider an otherwise identical parametric specification except that B0 is allowed to go from 0.25 to 0.4 and d ¼ 0:8. By the government budget constraint, it follows that t varies from 0.108781 to 0.17043. For this specification, it can be verified that all the conditions defined in the definition of equilibrium in this section are satisfied. As illustrated in Figure 2, aggregate welfare under induced retirement is higher than when there are no public Figure 2.

Comparing across policy and no-policy regimes (Example 2) Prob. of Employment

Vacancy filling rate

0.484

0.3318 0.3316 0.3314 θ 0.3312 0.331 0.3308 0.3306 0.25

0.4835 α

0.483 0.4825 0.482 0.25

0.3

0.35

0.4

0.3

0.35

alpha

alpha-np

theta

theta-np

Young Wages 0.4 0.35 wy

uy

Youth employment rate 0.6596 0.6594 0.6592 0.659 0.6588 0.25

0.4

B0

B0

0.3 0.25

0.3

0.35

0.2 0.25

0.4

0.3

0.35

uy

0.4

B0

B0 wy

uy-np

wy-np

Welfare 0.53 0.51 0.49 0.47 0.45 0.25

W

weo

Old employed wages

0.3

0.35

0.25

0.3

0.35 B0

B0 weo

0.4

0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2

weo-np

AggW

AggW-np

0.4

Social Security and Intergenerational Redistribution

207

pension benefits and retirement does not occur. Importantly, the probability of employment (a) is higher and wages to the young are lower under policy than in the absence of pension programs. Note that policy-induced retirement occurs as long as pension benefits are sufficiently generous and the implicit tax on elderly work is sufficiently high (d ¼ 0:8o1). Once pension benefits are equal to 0.25, induced retirement occurs. In particular, we observe that publicly induced retirement generates higher welfare since the vertical intercept in Figure 2 is equal to 0.247. As mentioned in the above example, welfare in the absence of pension programs is 0.2095.

7. Conclusion Most countries have large public pension programs. Traditionally, these programs have been used to induce retirement by the elderly in order to free up jobs for the young and to redistribute income across generations. This paper provides an efficiency rationale for the intergenerational income redistribution focus of such programs in a framework which explicitly accounts for the role of the lifecycle in the labor market. It develops a model of the labor market characterized by search and matching frictions and embeds it into an overlapping generations framework. In our model, public pension programs alter the age composition of the labor force by inducing the jobless elderly to retire in exchange for pension benefits. By requiring a long history of labor market attachment in order to receive benefits, these programs raise the future value of current employment for the young. In turn, this raises the future value of current employment which alters the bargaining positions of agents and effectively redistributes income from the young to the old. In addition, depending on the design of the pension program, we show that the redistribution can take place directly via the government (explicit transfer payments) or indirectly via firms in the form of higher wages. We believe that careful general equilibrium analysis of the underlying issues can shed important light and offer some guidance to policy makers. In this regard, we have ventured to study the efficiency and desirability of publicly funded pension programs within the context of a dynamic general equilibrium model. In order to consider how age-targeted labor market policies such as social security should be designed in light of the ongoing trend toward an increasingly older population, we adopted the OG setup because it allows a natural and explicit separation of the workforce into young and old workers. The framework captures an important intergenerational conflict between the young and old since, in the model, these groups concurrently compete for the same jobs; additionally, the bargaining positions of the two during wage negotiations are different due to their different stages along the lifecycle. Moreover, the OG structure is naturally conducive to studying pension programs that tie in with the lifecycle and other

208

Joydeep Bhattacharya and Robert R. Reed

‘‘low frequency’’ aspects of the labor market, such as, long job tenure, and the accumulation of firm-specific human capital. We used the search framework in the labor market for three important reasons. First, it allows us to endogenize both the supply side (through labor market participation choices) and the demand side (via endogenous creation of vacancies) of the labor market, a clear departure from the ‘‘lump-of-labor’’ line of thought in which there is a fixed stock of job vacancies. Importantly, we see that the amount of job creation responds to the design of pension programs through their impact on age earnings profiles in the economy. Second, the retirement literature suggests that social security programs are designed to reduce labor market congestion problems for the young. The diminished prospects for job search are also a prominent factor in the labor market participation decisions of older workers. In this regard, we argue that a model with undirected search is appropriate since it allows us to demonstrate how labor market congestion contributes to potential intergenerational conflicts in the labor market. If firms were perfectly able to discriminate on the basis of age, this would imply that there are not any intergenerational congestion difficulties between workers and therefore, there would be little role for policyinduced retirement. Finally, the decentralized notion of wage bargaining used in our framework allows us to study the effects of public pension programs on wage determination at each stage of the lifecycle. This is especially important given the fact that most real-world pension benefits are generally related in some way to the number of years worked and tend to increase with lifetime earnings. In this context, an important new effect that we identify is the role of social security in redistributing agents’ outside options over the lifecycle. In our setup, younger (longer tenure) workers have the option of waiting while older (equally productive but with shorter tenure) workers do not. This inequity translates into high wages for the young, escalating labor costs (since young workers constitute the largest pool of the unemployed from which firms will have to find workers), and reduces firm entry. Positive replacement rates, raise the lifetime value of working when young and thereby reduces this inefficiency. It is true that the effect we isolate is especially strong in our two-period model but the basic qualitative insight definitely extends to a model in which different sets of workers have different expected tenures with firms. Acknowledgments We thank Casey Mulligan for stimulating our interest in this topic. We have also benefited from detailed comments by Todd Keister and Martin Schindler on an early draft of the paper. We also acknowledge useful feedback from Satyajit Chatterjee, John Duffy, Jim Heckman, Ben Heijdra, Lars Nesheim, Aarti Singh, Chris Waller, Dave Wildasin, Randy Wright, and seminar participants at the Center for European Integration Studies at the University of Bonn, University

Social Security and Intergenerational Redistribution

209

of Groningen, University of Kentucky, Southern Methodist University, University of Alberta, Tilburg, 2003 Tinbergen Week at Rotterdam, the 2003 Midwest Macro Meetings at Chicago, the Midwest Theory Meetings at Indiana University, the Federal Reserve Bank of Cleveland Central Bank Institute Summer Program on Monetary Economics (August 2002), 2003 Winter Meetings of the Econometric Society in Washington, the 2002 UT-ITAM conference, and the 2004 Canadian Economics Association Meetings in Toronto. References Bhattacharya, J., C.B. Mulligan and R.R. Reed (2004), ‘‘Labor market search and optimal retirement policy’’, Economic Inquiry, Vol. 42, pp. 560–571. Black, D. (1987), ‘‘The social security system, the provision of human capital, and the structure of compensation’’, Journal of Labor Economics, Vol. 5(2), pp. 242–254. Boldrin, M., S. Jimenez-Martin and F. Peracchi (1999), ‘‘Social security and retirement in Spain’’, in: J. Gruber and D. Wise, editors, Social security and retirement around the world, Chicago: University of Chicago Press (for NBER). Boldrin, M. and A. Rusticini (2001), ‘‘Political equilibria with social security’’, Review of Economic Dynamics, Vol. 3, pp. 41–78. Chan, S. and A.H. Stevens (2002), ‘‘How does job loss affect the timing of retirement?’’ NBER Working Paper, 8780. Davidson, C., L. Martin and S. Matusz (1994), ‘‘Jobs and chocolate: Samuelsonian surpluses in dynamic models of unemployment’’, Review of Economic Studies, Vol. 61, pp. 173–192. Diamond, P. and J.A. Hausman (1984), ‘‘The Retirement and unemployment behavior of older men’’, pp. 97–126 in: H.J. Aaron and G. Burtless, editors, Retirement and Economic Behavior, Washington, DC: The Brookings Institution. Gottschalk, P.M. and R. Moffitt (1999), ‘‘Changes in job instability and insecurity using monthly survey data’’, Journal of Labor Economics, Vol. 17(4), pp. S91–S126. Gruber, J. and D. Wise (1999), Social Security and Retirement Around the World, Chicago: University of Chicago Press (for NBER). Hall, R.E. (1982), ‘‘The importance of lifetime jobs in the US economy’’, American Economic Review, Vol. 72(4), pp. 716–724. Hutchens, R. (1986), ‘‘Delayed payment contracts and a firm’s propensity to hire older workers’’, Journal of Labor Economics, Vol. 4, pp. 439–457. Mulligan, C.B. and X. Sala-i-Martin (2004), ‘‘Political and economic forces sustaining social security,’’ Advances in Economic Analysis and Policy, Vol. 4(1), Article 5. O’ Leary, C.J. and S.A Wandner (2000), ‘‘Unemployment Compensation and Older Workers’’, W.E. Mimeo, Upjohn Institute for Employment Research. Oi, W. (1962), ‘‘Labor as a quasi-fixed factor’’, Journal of Political Economy, Vol. 70, pp. 538–555.

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Organization for Economic Co-operation and Development (OECD) (1995), The Labour Market and Older Workers, Paris: OECD. Social Policy Studies No. 17. Pissarides, C. (1976), ‘‘Job search and participation’’, Economica, Vol. 43, pp. 333–349. Pissarides, C. (1992), ‘‘Loss of skill during unemployment and the persistence of employment shocks’’, Quarterly Journal of Economics, Vol. 107, pp. 1371–1391. Pissarides, C. (2000), Equilibrium Unemployment Theory, Cambridge, MA: MIT Press. Sala-i-Martin, X. (1996), ‘‘A positive theory of social security’’, Journal of Economic Growth, Vol. 2, pp. 277–304. Shimer, R. (2001), ‘‘The impact of young workers on the aggregate labor market’’, Quarterly Journal of Economics, Vol. 116, pp. 969–1007. Appendix A: Proof of Lemma 1 In a steady state, contribution to the unemployed pool come from two sources, young workers (measure 0.5) and never-before-employed workers, of measure ð1 aÞ=2. Then, it follows that U¼

ð1

uy U uo u~ o  U u~ y 

1 1 þ ð1 ¼ 2 2 1 ¼ 1 þ ð1 aÞ ð1 aÞ ¼ 1 þ ð1 aÞ aÞ

2

aÞ

þ

Appendix B: Proof of Lemma 2 (a) Using (10), we can compute the gains from trade for the firm from hiring a young worker as: 1

h

wy þ bbPv þ ð1

bÞbPf þ dbPv

bPv

which using (5) and rearrangement yields ¼ ð1

h

wy Þ þ ð1

bÞbð1

weo Þ

Using (11), and (1)–(3), the young worker’s surplus from finding employment is given by: wy þ bbJ so þ ð1 ¼ wy þ bbBo þ ð1

bÞbJ eo

bJ uo

bÞbweo þ ð1

bÞbdBo þ bs

bawuo

Social Security and Intergenerational Redistribution

211

and further to bÞbweo þ ð1

¼ wy þ bbBo þ ð1

bÞbdBo þ bs

bað1 hÞ 2

Then, equating the gains from trade, we get 2wy ¼ ð1

hÞ þ ð1 ð1

2weo Þ bbBo bað1 hÞ bs þ 2

bÞbð1

bÞdbBo

which simplifies to ¼ ð1

hÞ

Notice that ½ð1 2wy ¼ ð1

bB0 ½ð1

bÞð1

dÞ þ b þ ð1

bÞdŠ

bs þ

bað1 hÞ 2

bÞð1 dÞ þ b þ ð1 bÞdŠ ¼ 1. Then, we have   ab bðB0 þ sÞ. hÞ 1 þ 2

(b) obvious. Appendix C: Proof of Lemma 3 (a) It is easily seen that, as long as the discouraged worker constraint holds, the young will choose to search as long as there are positive gains from entry within the period: awy ðaÞ4s which reduces to ð1

hÞa2 þ 2½ð1

hÞ

ðB0 þ sފa

4s40

Solving the above condition for a value of a in which it holds with equality provides us with a minimal value for a. Note that if ð1 hÞ4ðB0 þ sÞ, then we must use the positive root as the solution for a. Define aL as the value of a which satisfies the condition (note, this is the same sufficient condition that was required for young wages to be positive). It is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1 hÞ bðB0 þ sފ þ 2 ½ð1 hÞ bðB0 þ sފ2 þ 4bð1 hÞs aL  bð1 hÞ which is always positive as long as there are some search costs to be incurred by workers. (b) We seek to find conditions where aL 4ð12shÞ. First, recall b1  ½ð1 hÞ ðB0 þ sފ. Thus, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 þ b21 þ 4bð1 hÞs 2s 4 aL  ð1 hÞ bð1 hÞ

212

Joydeep Bhattacharya and Robert R. Reed

reduces to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 þ b21 þ 4bð1 hÞs42bs and further to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b21 þ 4bð1 hÞs4b1 þ 2bs

Next, squaring both sides yields: b21 þ 4bð1

hÞs4ðb1 þ 2bsÞ2

which upon simplifying obtains: 04

2bB0

bs

which always holds. Appendix D: Proof of Lemma 4 (a) Using (24), and setting y ¼ 1, we get         a u~ o  1 ¼ 1 h wy þ ð1 bÞb 1 weo þ u~ y u~ y

wuo

h



We begin by substituting in the steady-state population conditions from Lemma 1. Then, we use (8), (9), and (12) to get   1 þ ð1 dÞBo a½1 þ ð1 aފ ¼ ð1 h wy Þ þ bð1 bÞ 1 2   ð1 hÞ þ ð1 aÞ 1 h 2 Simplification yields 2a þ 2að1

aÞ ¼ ð1

hÞ þ bB0 þ bs

þ bð1

bÞ

bð1

bað1 hÞ 2 bÞð1 dÞBo þ ð1

aÞð1

hÞ

which further simplifies to ð1

abð1 hÞ 2 hÞ þ bs þ bð1 bÞ þ bB0 ½1

aÞ½2a

¼ ð1

ð1

hފ þ

ð1

bÞð1

dފ

2a

and finally to a¼ 2ð1

hÞ þ 2bs þ 2bð1

bÞ þ 2bB0 ½1 ð1 bÞð1 ½ð1 hÞð2 þ bÞ 4aŠ

dފ

4a

½4a

2ð1

hފ

213

Social Security and Intergenerational Redistribution

Notice that 2ð1

hÞ

4a

½4a

2ð1

hފ ¼ 4½ð1

2aŠ

hÞ

Then, it follows that bs þ bð1 bÞ þ bB0 ½1 ð1 bÞð1 dފ þ 2½ð1 a¼2 ½ð1 hÞð2 þ bÞ 4aŠ

hÞ

2aŠ

Appendix E: Proof of Proposition 5 Recall that firms enter until Pv ¼ 0, or until the revenue from entry equals the up-front cost a. Also recall that firms take u~ y and u~ o as given, and that wuo does not depend on B0. Then the revenue from entry R can be written as R ¼ u~ y ð1

h

wy Þ þ u~ y ð1

bÞbð1

weo Þ þ u~ o ð1

Using (8), (9), and (12), we get     ð1 hÞ ab bðB0 þ sÞ 1þ þ R ¼ u~ y ð1 hÞ 2 2 2   1 þ ð1 dÞBo þ u~ y ð1 bÞb 1 þ u~ o ð1 h 2 which simplifies ultimately to   ð1 hÞ ð1 hÞ ab ðB0 þ sÞ R ¼ u~ y þ 2 2 2 2 u~ y ð1 bÞb ½1 ð1 dÞBo Š þ u~ o ð1 þ 2 It follows that

h

h

wuo Þ

wuo Þ

wuo Þ

u~ y u~ y ð1 bÞbð1 dÞ @R ¼ 2 @Bo 2 u~ y ½1 ð1 bÞbð1 dފ40 ¼ 2 so, an increase in B0 raises the benefit from entry but does not raise the cost. Hence, ceteris paribus, there will be more firm entry with higher B0. The fact that more firm entry leads to a higher probability of employment follows immediately from differentiating (25) to get @a 40 @B0

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III. MICROECONOMETRIC PAPERS

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216

CHAPTER 10

The Job Ladder Audra J. Bowlus and George R. Neuman Abstract Most wage growth occurs via job changes rather than on-the-job wage growth. The Burdett–Mortensen (1998) model is an extreme version of this with no onthe-job wage growth. We estimate a version of this model using NLSY data. We simulate 10 years of data on wages, job spells, and unemployment spells and compare them to actual experience. The agreement is remarkably good in several respects, but fails to explain the amount of downward wage changes. Keywords: wage growth, equilibrium search, job ladder JEL classifications: J31, J64 1. Introduction Modern search and matching theories view labor markets as job ladders. Workers search for good matches while unemployed and while employed. Employment with a particular firm ends either when the job is destroyed or when a worker finds a better opportunity. This simple structure yields several strong predictions. Onthe-job wage growth is zero. Incomes increase as workers move from lower paying to higher paying jobs, occasionally interrupted by spells of unemployment as some jobs are destroyed and the worker has to start over again on the lower rungs of the ladder. Of course, this job ladder view of the labor market is necessarily false in some dimensions. Understanding what those dimensions are and how and why the job ladder paradigm fails is the purpose of this paper.

Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75010-0

217

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

218

Audra J. Bowlus and George R. Neuman

Textbook treatments of earnings growth over time rarely refer to the effects of the job ladder.1 Instead, most focus on the role of firm-specific human capital in generating earnings increases over time. Curiously, the contribution of on-the-job wage growth to overall wage growth has rarely been examined. Altonji and Shakotko (AS) (1987) estimate that on-the-job wage growth is 6.6% per decade. Altonji and Williams (1997) after surveying alternative estimates of wage growth state: ‘‘Our main conclusion is that the data used by both AS and Topel imply a return to ten years of tenure of about .11 ,[i.e., 11% gain over a 10 year period, or 1.1% per year] y .’’ Is this big or small? We demonstrate in Section 2 using earnings data from seven U.S. Census decades (1940–2000) that the on-the-job wage growth component is a small fraction of overall wage growth, which suggests that job mobility is the more important component in earnings growth. To pursue the contribution of the job ladder requires adopting a specific model of the job ladder. We do so using the stylized Burdett–Mortensen model and data on employment and unemployment spells from the National Longitudinal Survey of Youth (NLSY) data described in Sections 3 and 4, respectively. This analysis follows that of Bowlus et al. (2001).2 Using this structure in Section 5 we simulate earnings trajectories for a period of 10 years and then compare these results to the earnings trajectories observed over 10 years in the NLSY. The evidence we review suggests that the job ladder approach fits the observed wage distributions at the beginning and end of the period amazingly well, even assuming parameter stability over the 10-year period. A modification that allows only productivity to grow does even better. Finally, in Section 6 we study the means by which the wage growth occurs via job transitions over the 10-year period in the data as compared to the model. Here the simple story of the job ladder fails. To conduct this analysis we study the appearance of anomalies in the data and attempt to explain their cause. This is an important topic in this area because the use of measurement error models frequently results in the data being explained as error rather than structure. By anomalies we mean occurrences of workers transiting from job to job and incurring a loss in earnings. More sophisticated search models can account for this event,3 but it is of some value to understand how frequently it occurs and when and for what reason. Because the NLSY asks questions about

1

Compared to the volumes of work done on wage growth via human capital accumulation very little has been done on wage growth via job changes. A few exceptions are Abbott and Beach (1994), Keith and McWilliams (1999), Gottschalk (2001), and Yankow (2003). 2 We re-estimate the model used in Bowlus et al. (2001) because data edit checks changed the observations included somewhat. This is described in Section 3. 3 Examples of models with the potential for negative wage growth via job-to-job changes include Connolly and Gottschalk (2002), and Postel-Vinay and Robin (2002) who emphasize heterogeneous wage growth across jobs; Gorgens (2002), Dey and Flinn (2003), and Sullivan (2003) who emphasize other job attributes; and Bowlus and Vilhuber (2001) who emphasize mandatory notice of job displacement.

The Job Ladder

219

the reason for job separation we can examine what types of transitions lead to earnings reductions. Section 7 concludes.

2. Earnings growth in U.S. census data For a first look at wage growth, we examine data on labor earnings in the US from the 1940 through 2000 Censuses.4 As information on weeks worked per year and hours worked per week are only available for a subset of these years, we use annual earnings as our wage measure. We focus on white males, aged 26–65, classified by education (college graduate/non-college), so the effects of labor supply on earnings should be minimal.5 Table A1 in the Appendix shows annual income for all years, classified by education and age. One way to interpret these earnings distributions is that they are snapshots of the steadystate wage distribution in each period. Under this interpretation, changes in earnings between age groups represent wage growth that would occur over time under whatever model of labor market behavior generated the data. Table 1 shows wage growth by age group, classified by education. For each year, t, wage growth of age group j is defined as Growtht,j ¼ wt,j+1/wt,j. Thus in Table 1 for the year 2000, observe that a college educated worker aged 26–30 can expect to see his wage increase by 82.6% over the next 10 years. Over 20 years the growth is expected to be 93.0%. Note that the equivalent numbers for 1940 are 62.6% and 77.1%, which suggest that the wage-experience gradient has steepened over time. For non-college graduates wage growth has been significantly less. In 2000, a non-college worker aged 26–30 could expect a 10-year wage growth of 36.3% and a 20-year growth of 50.3%. This pattern is in accord with most analysis of the evolution of the skill premium (e.g. Juhn et al., 1993; Ingram and Neumann, forth coming). If we take the 11% on-the-job wage growth from Altonji and Williams (1997) as the consensus estimate, then on-the-job wage growth accounts for at most 13% of college wage growth and 30.3% of non-college wage growth. This must be an upper bound, because the 11% Altonji–Williams estimate is based on the worker being continuously employed with the same employer for 10 years. In Census data there is no way to check whether an employee has been with the same firm for 10 years, although we provide evidence on this below when we examine the NLSY data. In those data we find that the average non-college job lasts a bit over three years, implying that such workers will be on their fourth job at the end of a 10-year period. Thus it seems reasonable to assert that onthe-job wage growth can account for only a small share of total wage growth. Of

4

These data are provided by the Integrated Public Use Microdata Series (IPUMS) project described in Ruggles et al. (2004). 5 For 1980–2000 we obtain the same patterns whether we use weekly wages or annual earnings.

220

Audra J. Bowlus and George R. Neuman

Table 1.

Wage growth 1940–2000, by education and age

Relative wages by year and age Year 26–30

Age

31–35

36–40

41–45

46–50

51–55

56–60

61–65

1.456 1.403 1.404 1.416 1.422 1.349 1.351

1.826 1.630 1.763 1.614 1.629 1.648 1.626

1.979 1.822 1.918 1.759 1.764 1.631 1.713

1.930 2.010 1.987 1.804 1.755 1.716 1.771

1.989 2.045 1.989 1.849 1.786 1.751 1.745

1.941 1.889 1.890 1.695 1.726 1.568 1.683

1.643 1.631 1.658 1.572 1.634 1.438 1.586

(B) Non-college graduates 26–30 31–35

36–40

41–45

46–50

51–55

56–60

61–65

2000 1990 1980 1970 1960 1950 1940

1.363 1.381 1.361 1.193 1.235 1.190 1.342

1.449 1.510 1.400 1.226 1.245 1.213 1.411

1.503 1.603 1.412 1.227 1.206 1.201 1.395

1.540 1.561 1.392 1.187 1.170 1.187 1.324

1.472 1.453 1.327 1.103 1.126 1.127 1.252

1.237 1.196 1.113 0.964 1.050 1.020 1.165

(A) College graduates 2000 1.000 1990 1.000 1980 1.000 1970 1.000 1960 1.000 1950 1.000 1940 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.213 1.241 1.222 1.133 1.161 1.147 1.231

Source: Calculated from Table A1.

course other forces, such as pure productivity growth, may effect the wage distribution causing it to shift over time. We study this issue below. 3. The Burdett–Mortensen equilibrium search model If the standard firm-specific human capital model can explain only a small fraction of wage growth, it becomes interesting to see how well an equilibrium search model can perform. In the simple case of observationally equivalent workers and firms and with constant parameters, the equilibrium is completely described by the four basic parameters: l0, l1, d, p, and R, which are the arrival rate of job offers while unemployed, the arrival rate of job offers while employed, the job destruction rate, firm productivity, and the reservation wage of workers (Burdett and Mortensen, 1998).6 More complicated versions of this model allow for heterogeneity in firm productivity (e.g. Bowlus et al., 1995, 2001; Bontemps et al., 2000). In this model the steady-state earnings distribution stochastically dominates the wage offer distribution. Thus if one starts with a group of workers who are all

6

Equivalently, we could substitute b, the value of home production or unemployment benefits, for R and calculate R from knowledge of the other parameters.

The Job Ladder

221

unemployed and follows them until the steady-state is achieved their mean wage level will grow from the mean of the wage offer distribution to the mean of the earnings distribution. This transition can take many years depending on the offer arrival rates. Once steady-state is achieved individuals will still see wage growth through job changes and wage declines via unemployment, but the overall mean will not change unless a parameter change disrupts the equilibrium. Below, we estimate the parameters of the model using a sample of young workers making the transition from school to work and then predict their wage growth using the model and compare that to the actual growth found in the data. 4. Data description To conduct the analysis we use data from the 1979–1994 NLSY. The NLSY is an ideal data set to study wage growth within and between jobs because it follows the same individuals over a long period of time, starting from labor market entry, allowing researchers to construct full employment histories at the job spell level. To conduct the analysis in this paper we construct two samples. The first, called the initial spell sample, is similar to that in Bowlus et al. (2001), who used data from the NLSY to study the school-to-work transition. It collects information on first jobs following completion of education. The second, called the continuing spell sample, continues to follow individuals in the initial sample through the next 10 years collecting information on job lengths, on-the-job wage growth, job transitions and wage changes across transitions. As in Bowlus et al. (2001), we examine white males who graduated from high school but did not pursue further education.7 The initial sample contains information on the duration of unemployment from graduation until the first full-time job,8 the starting weekly wage,9 the length of the first job spell and the transition following the first spell either to unemployment or to another job.10 This is the basic information needed to identify the standard on-the-job search model. To be included in the initial sample an individual must take up full-time work11 within three years of graduating, must not be engaged in self-employment or unpaid

7

As in Bowlus et al. (2001) we exclude general educational development (GED) recipients and those who graduated from high school before 1978. In addition, in order to follow individuals for a 10-year period of time, here we exclude individuals who graduate after 1984. Given the age restriction of the NLSY, 14–22 in 1979, this latter restriction excludes very few individuals. 8 Full time refers to 35 h per week or more. 9 All reported wages are converted to weekly wages and reported in real 1982 dollars. 10 In Bowlus et al. (2001) the second job had to entail 20 h of work or more. Because we are interested in examining only full-time work in the continuing sample, here we require the second job to be full time. This leads to a lower job-to-job transition rate in our initial sample. 11 Jobs that end within two months of graduation are not used as the first job spell in order to eliminate summer and temporary jobs held while in school. Also jobs must last longer than three weeks to be considered.

222

Table 2.

Audra J. Bowlus and George R. Neuman

Sample statistics from the initial sample and predicted moments from the model (Q ¼ 4) Sample Statistics Predicted Moments

Mean unemployment duration (weeks) Mean accepted weekly wage Mean job duration (including censored spells in weeks) Job spell censoring rate Fraction of completed job spells ending in a job transition

35.56 239.14 114.01 0.07 0.38

35.27 245.92 115.80 — 0.36

work, and must have reported a valid starting wage rate.12 Since in this paper we examine wage growth over 10 years starting from graduation, for jobs that start before graduation we depart from Bowlus et al. (2001) and use the wage reported at the time of graduation as the first wage rather than the first wage ever reported for that job. We also do not trim the top of the wage distribution. Column 1 in Table 2 provides sample statistics for the initial sample. With the few exceptions already noted, the means are similar to those in Bowlus et al. (2001). The continuing sample includes all of the first jobs in the initial sample as well as all full-time jobs the respondents in the initial sample hold for the next 10 years. There are three reasons why we may not be able to follow individuals for the full 10 years: (1) attrition from the sample, (2) job spells that are censored due to incomplete information in subsequent interviews, or (3) transitions to self-employment.13 In each of these cases, we follow the individual up until his last valid observation and record how long we are able to observe them. Thus below we record annualized growth rates rather than total growth rates in order to accommodate differences in observed period lengths. When computing wage growth rates and changes, only valid wages, as determined above, are included in the computations. 5. Fitting the earnings distribution 10 years in the future We begin our analysis by re-fitting the model used by Bowlus et al. (2001) to the initial sample. The parameter estimates are shown in Table 3. They do not differ much from the equivalent specification in Bowlus et al. (2001). Workers are homogeneous with a common value of non-market time b. They search both on

12

To determine valid pay and time rate responses we cross checked them against bounds from the Current Population Survey. See Section 5.3 of Bowlus et al. (2001) for details concerning this procedure. 13 For respondents in categories 2 and 3 we often can observe a wage rate 10 years after graduation. We have compared annualized wage growth rates using our sample and a sample that contains growth rates computed on the maximum observable time period, up to 10 years, for each respondent. The average rates do not differ and thus we use our sample throughout the analysis.

223

The Job Ladder

Table 3.

Maximum likelihood estimates of the equilibrium search model

Parameter l0 l1 d P1 P2 P3 P4 r wH1 wH2 wH3 wH4 g1 g2 g3

Value

S.E.

0.0284 0.0077 0.0047 296.40 404.64 600.45 2342.97 115.97 214.58 300.95 384.77 781.99 0.524 0.807 0.927

0.0023 0.0005 0.0003 4.9216 8.1275 18.7360 148.3386 — — — — — — — —

Note: No standard errors are reported for the parameters R g3 because these estimates converge pffiffiffi faster than n and the likelihood function, although continuous, is not differentiable in these parameters.

and off the job receiving wage offers while unemployed at rate l0 and while employed at rate l1. Jobs are destroyed at rate d. Firms face constant returns to scale in production and maximize profits by choosing the wage to pay. There are Q types of firms with productivity P1oP2oyoPQ. The fraction of firms having productivity Pj or less is gj ¼ g(Pj). Mortensen (1990) shows the equilibrium wage offer distribution to be F ðwÞ ¼ fj ðwÞ8w, with fj defined by " 1 þ k1 1 fj ðwÞ ¼ k1

ð1Þ
  # 1 þ k1 1 g j 1 Pj w 1=2 , 1 þ k1 Pj wHj 1

ð2Þ

wLj ow  wHj where k0 ¼ l0/d, and k1 ¼ l1/d. The values for the wage cuts, wLj and wHj, can be solved for using the equilibrium properties and the fact that F(wHj) ¼ gj. They are given by wHj wLj wL1

¼

¼ ¼

ð1

wHj r

Bj ÞPj

Bj wHj

1

1

where Bj ¼ ½ð1 þ k1 ð1

gj ÞÞ=ð1 þ k1 ð1

gj 1 Þފ.

224

Audra J. Bowlus and George R. Neuman

Table 4.

Average real wage growth Full Sample Weekly

Annual growth rate over 10 years Annual growth rate on the job Wage growth between jobs

0.056 0.011 0.134

Hourly 0.050 0.007 0.103

Restricted Sample Weekly

Hourly

0.063 0.025 0.089

0.054 0.014 0.089

The initial sample characteristics and the predicted moments generated by the estimated model are shown in column 2 of Table 2. As expected, the predicted moments are quite close to the actual moments, although the predicted mean wage is a little high. We use the continuing sample to generate observations on the 10-year period following graduation. Table 4 shows wage growth over the full observation period as well as average wage growth within job spells and across jobs. The former is measured by taking the difference between wage observations recorded at the start and stop dates of the job spell divided by the job duration, while the latter is the difference between the starting wage of a new job and the stopping wage of an old job. Because many of the job changes are immediate this growth rate is not divided by a time measure. Thus the two do not sum together to the 10-year annual growth. In Table 4 and throughout the analysis we present rates for both the full sample and a restricted sample. The restricted sample excludes outliers and individuals who are only in the sample for a short period.14 Overall, young white males see wage growth of 5–6% per annum over the first 10 years of labor market experience. As we argued in Section 2, the majority of this growth does not stem from on-the-job wage growth. On average, on-the-job wage growth ranges from negligible to at most 2.5% per annum. In contrast, a single job change, including one via unemployment, increases wages between 9% and 13% on average. That on-the-job wage growth does not explain the lion’s share of wage growth does not imply that an equilibrium search model will. To see whether or not it does, we simulated the wage and transition process from the model estimated in Table 3 for 10 years for each observation. Because each simulation/observation is a random variable (or a collection of random variables) we conducted 500 replications of each simulation and took averages. There are many possible

14

For annual wage growth rates, the restricted sample excludes individuals with less than five years of wage growth and with annualized growth rates that are more than 100% or less than 50%. For onthe-job growth rates the restricted sample excludes job spells that are shorter than one year or have annual growth rates greater than 100% or less than 50%. For between job wage changes, the restricted sample excludes wage changes that are greater than 100% or less than 50%.

225

The Job Ladder

Table 5.

Average wage after 10 years Average annualized wage growth Average duration of unemployment spells (weeks) Average number of unemployment spells Average duration of job spells (weeks) Average number of job spells Fraction of completed job spells ending in a job transition

Comparison of actual and simulated data Continuing Sample

Simulated Data without Growth

Simulated Data with Growth

332.070 0.056

327.540 0.035

332.060 0.048

42.660

37.300

37.300

2.100

2.830

2.840

160.550

165.640

165.640

4.160

4.170

4.180

0.473

0.396

0.396

comparisons to make. Table 5 provides a summary of several measures. Here we focus on the wage after 10 years. Row 1 of Table 5 shows the mean wage after 10 years in the data (column 1) and the model (column 2) and Figure 1 shows the percentiles of the actual distribution of wages and those predicted by the equilibrium search model. The fit is amazingly good, although it is apparent that neither the lower tail nor the upper tail is captured adequately. The assumption that the parameters of the model remain unchanged for 10 years seems exceptionally strong, but with so many possible changes it is not obvious where to start. Changes in l0 or in l1 can be detected by changes in spell lengths so this might be one avenue to pursue. We take a somewhat simpler approach and consider the case where the productivity parameters Pjj, j ¼ 1 y 4 are the only parameters that change. In particular we assume that Pj;t ¼ Pj;0 ð1 þ gÞ0 , and that all wages are updated immediately with the growth in productivity. This specification ensures that individuals’ ranks in the wage offer distribution do not change with the wage growth and thus the transition probabilities remain constant. We choose g by the method of simulated moments. The moment we match is the mean of the earnings distribution. Figure 2 shows the squared-error function generated by the simulation. In each case we generated 500 replications for each of the 468 observations. The optimization was done by a grid search over g. It was not possible to determine in advance how many random draws would be needed because the number of transitions varies with the parameter g. This leads to an extra amount of variability as can be seen in the figure. In any event, the graph indicates clearly that

226

Audra J. Bowlus and George R. Neuman

Figure 1.

Actual and predicted wage distributions, g ¼ 0.0

$600 W_G ⫽ 0 W_Actual

Weekly Wage

$500 $400 $300 $200 $100 $0 10

20

30

40

50

60

70

80

90

Percentiles

Figure 2.

Error sum of squares.

12000

10000

ESS

8000

6000

4000

2000

0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 G

227

The Job Ladder

Figure 3.

Actual and predicted wages distributions, g ¼ 0.015

$600 W_G W_Actual

Weekly Wage

$500 $400 $300 $200 $100 $0 10

20

30

40

50 60 Percentiles

70

80

90

the value that will line up the sample and predicted means is an annual productivity growth of about 1.2%. Using this estimate we re-simulated the model to generate the 10 year wage distribution assuming that only productivity varied. Column 3 of Table 5 verifies the mean match, although as expected none of the other measures change as the productivity change implemented here has no impact on transition rates. A graphical comparison of the wage distributions is shown in Figure 3. Now the fit is even tighter, although the predictions are still too high in the left tail. This is likely due to the greater presence of downward wage mobility in the data than in the model, a hypothesis we investigate further in the next section. 6. Downward wage mobility In this section, we examine the path between the starting wage distribution and that 10 years later by investigating further the job transitions made by the respondents in the NLSY and the accompanying wage changes as compared to those predicted by the model. For estimation of the search model, we divided all job transitions into two categories: job-to-job transitions and job transitions via unemployment, where our definition of job-to-job transitions was any job transition made within two weeks or less. This has been a standard way to identify transitions due to job destruction, i.e. those via unemployment, and transition due to on-the-job search. The NLSY provides an alternative source of information by asking individuals for the reason why a job ended. We divided the reasons into those that indicate an involuntary transition, i.e., layoff, plant closure, fired, seasonal/temporary job, and those that indicate a voluntary

228

Audra J. Bowlus and George R. Neuman

Table 6.

Cross-tabulation of job transition categories

Transition/Reason

Involuntary

Voluntary Job Related

Voluntary Non-Job Related

Total

Via unemployment Job-to-job Total

29.98 10.01 39.99

20.45 34.15 54.60

3.38 2.03 5.41

53.81 46.19 100.00

transition, i.e., quits. We further divided voluntary transitions into those that are made for job-related reasons and those that are not. Table 6 shows the crosstabulation between these two ways of categorizing job transitions using the continuing sample. Clearly, the off-diagonal elements are not zero suggesting that, as expected, job transitions are more complicated than is suggested by the simple on-the-job search model. Overall there is almost an even split between job transitions via unemployment and job-to-job transitions, while the majority of exits are voluntary. Three-quarters of job-to-job transitions are composed of voluntary exits, while one-quarter are involuntary suggesting that some individuals who are let go are able to find re-employment almost immediately. For transitions via unemployment we find that the majority is involuntary, but the split is more even as many respondents give voluntary reasons for the exit. Interestingly, the vast majority of these voluntary reasons are job related. That is, individuals indicate that they have to quit unemployment to find a better job. Returning to wage growth, Table 7 breaks down the wage changes across jobs by types of transitions and reasons for leaving jobs. In general, wage growth is observed to be higher when individuals make immediate job transitions rather than transitions via unemployment. Likewise, voluntary transitions that are job related realize substantially more growth than those associated with being laid off or for non-job-related reasons. Interestingly, on average all transitions result in positive wage growth. Table 8 explores the wage changes further by examining the frequency of positive and negative wage changes across unemployment and job-to-job transitions and involuntary and voluntary reasons using the continuing sample. While the fraction of negative wage changes is lower for job-to-job transitions, it is still substantial as 40% of all job-to-job transitions result in lower wages and almost 30% result in a wage decline of more than 5%.15 This evidence contradicts the model prediction that job-to-job transitions always result in positive wage changes. With respect to reasons given, Table 8 indicates that 66% of job changes associated with job-related voluntary reasons result in a positive wage change, while the figure is only 40% for involuntary and non-job-related voluntary

15

Gottschalk (2001) also finds ample evidence of wage declines via job changes using data from the Survey of Income and Program Participation.

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The Job Ladder

Table 7.

Wage growth between jobs Full Sample

Type Via unemployment Job-to-job transition Involuntary transition Voluntary transition for job reasons Voluntary transition for non-job reasons

Table 8.

Restricted Sample

Weekly

Hourly

Weekly

Hourly

0.122 0.147 0.072 0.180 0.110

0.066 0.139 0.029 0.155 0.065

0.058 0.119 0.034 0.129 0.053

0.056 0.120 0.023 0.133 0.065

Frequency of positive and negative wage changes by transitions and reasons Continuing Sample

Transitions Via unemployment Positive Greater than 5% Increase Greater than 5% Decrease

0.484 0.430 0.381

Job-to-job Positive Greater than 5% Increase Greater than 5% Decrease

0.619 0.581 0.275

Reasons Involuntary Positive Greater than 5% Increase Greater than 5% Decrease

0.407 0.366 0.420

Voluntary for Job Reasons Positive Greater than 5% Increase Greater than 5% Decrease

0.662 0.607 0.289

Voluntary for Non-job Reasons Positive Greater than 5% Increase Greater than 5% Decrease

0.392 0.350 0.215

changes. Again, however, almost 30% of voluntary changes for job-related reasons result in a wage decline of more than 5%. Clearly, wages are not the only factor when workers move to better jobs. To end this section, we examine a fairly stark prediction of the on-the-job search model. Because of the up and out rule governing job-to-job transitions in the search

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Table 9.

Frequency of job-to-job transitions by wage decile Continuing Sample

Simulated Data without Growth

All job-to-job transitions 1st decile 2nd decile 3rd decile 4th decile 5th decile 6th decile 7th decile 8th decile 9th decile 10th decile

0.408 0.449 0.452 0.432 0.458 0.565 0.483 0.601 0.435 0.463

0.598 0.570 0.537 0.498 0.453 0.398 0.332 0.249 0.142

Only job-to-job transitions with positive wage change 1st decile 2nd decile 3rd decile 4th decile 5th decile 6th decile 7th decile 8th decile 9th decile 10th decile

0.321 0.337 0.276 0.312 0.216 0.314 0.290 0.323 0.193 0.129

0.598 0.570 0.537 0.498 0.453 0.398 0.332 0.249 0.142

model, the model predicts that the fraction of completed spells that end in a job transition should fall as the wage increases. Table 9 examines this prediction in the data by calculating the fraction of job-to-job transitions in each decile of the wage offer distribution. The first panel gives the results counting all observed job-to-job transitions as legitimate, while the second panel only counts those transitions that result in a positive wage change in line with the model. These figures are given in column 1. In column 2 we present the resulting predictions from the search model. As expected, the model exhibits a declining trend. This is generally not the case in the data. To illustrate the difference we present the patterns in Figure 4. For the most part the raw data exhibit a flat profile that is too low at the bottom of the wage distribution and too high at the top. The only exception is the decline at the very top of the wage distribution for the measure with only positive wage changes. This clearly highlights a failure of the model, and it is a critical failure because the driving force in equilibrium search models is the inter-firm competition in wages.16

16

Absent wage competition the model breaks down to the Diamond equilibrium with no wage dispersion.

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The Job Ladder

Figure 4. 0.7

Percentage of job-to-job transitions

Model data_all data_pos

0.6

% J-J

0.5 0.4 0.3 0.2 0.1 0.0 0.1

0.2

0.3

0.4 0.5 0.6 Wage Offer Decile

0.7

0.8

0.9

In constructing the statistics on job-to-job transitions we have assumed that individuals, stratified only by level of education, are homogeneous in labor market productivity. One possible explanation for the patterns shown in Figure 4 is that the workers actually are heterogeneous. To explore this possibility we used each individual’s Armed Forces Qualification Test (AFQT) score17 and compared job-to-job transition rates for workers with above average scores to those with below average scores. As in Table 9, we also examine the rates defining only those who had a positive increase in wages as making a job-to-job transition. Table 10 shows the results and Figure 5 plots the predicted behavior along with the empirical transition rates for workers receiving positive wage changes (columns 2, 6 and 7 of Table 10). There is some indication that workers with greater than average scores on the AFQT are less likely to quit higher paying jobs, but this is only evident for the top half of the wage distribution. Overall, there is not much difference in transition rates by AFQT score, and low-wage workers clearly do not exhibit a jobto-job transition rate that declines with the wage. 7. Conclusions In this paper, we document the relative importance of wage growth via job changes as compared to the accumulation of firm-specific human capital. After

17

We used age-corrected standardized AFQT scores.

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Audra J. Bowlus and George R. Neuman

Table 10. Decile

Model

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

0.598 0.570 0.537 0.498 0.453 0.398 0.332 0.249 0.142

Frequency of job-to-job transitions by wage decile Below Mean Afqt

Above Mean Afqt

Top 1/3 AFQT

Below Mean AFQT (Positive Change)

Above Mean AFQT (Positive Change)

0.485 0.422 0.462 0.521 0.337 0.594 0.521 0.538 0.472 0.312

0.269 0.451 0.452 0.351 0.526 0.531 0.423 0.686 0.406 0.630

0.154 0.332 0.411 0.251 0.573 0.397 0.350 0.700 0.400 0.645

0.338 0.286 0.302 0.374 0.087 0.356 0.316 0.339 0.131 0.059

0.266 0.381 0.255 0.276 0.282 0.281 0.256 0.331 0.263 0.197

Figure 5.

Percentage of job-to-job transitions

70% MODEL ABOVE MEAN+ "BELOW MEAN+"

60%

% J-J

50% 40% 30% 20% 10% 0% 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wage Offer Decile

establishing the importance of the job ladder, we test the ability of a simple general equilibrium search model with on-the-job search to match the observed 10-year growth rate. The model does surprisingly well and only needs an annual productivity growth rate of 1.2% to match the mean wage of the 10-year distribution exactly. This confirms our notion that much more attention should be paid to job transitions when studying wage growth.

The Job Ladder

233

While the model can match the growth in mean wages, it is by no means perfect. In particular, it fails to produce enough downward wage mobility resulting in a poor fit to the bottom of the 10-year wage distribution. We document the presence of substantial downward wage mobility in the data even for job-to-job transitions and job changes for job-related reasons. This is in direct contrast to the up-and-out rule governing most on-the-job search models, including the one used here. It suggests that the search model would need even greater productivity growth to match the 10-year mean if the degree of downward wage mobility seen in the data was incorporated in the model. The literature is just beginning to develop models that have as a feature job changes with negative wage changes. These can be categorized into models where the pull factor to change jobs is different from the offered wage (non-wage amenities and wage growth) and models where there is a push factor to leave the old job. With respect to the former, examples include Dey and Flinn (2003) where the pull factor is health insurance and Postel-Vinay and Robin (2002) where the pull factor is wage growth on the new job.18 Bowlus and Vilhuber (2001) model the push factor where workers who are pink slipped lower their reservation wage below their current wage due to their impending displacement. The evidence presented here confirms that both the push and non-wage pull factors are at work in explaining downward wage mobility via job-to-job transitions. We find laid off workers making job-to-job transitions and job-to-job transitions with negative wage changes that have been made for job-related reasons. In addition we also find exits to unemployment for job related reasons reviving the idea that some workers may find unemployment a more attractive state from which to search. All of this suggests that the development of more sophisticated models of labor market transitions is an important area of future research. Acknowledgments Audra Bowlus would like to thank the CBIC Program on Human Capital and Productivity at the University of Western Ontario for financial support. The authors thank Yahong Zhang for helpful research assistance. References Abbott, M. and C. Beach (1994), ‘‘Wage changes and job changes of canadian women: evidence from the 1986-1987 labor market activity survey’’, Journal of Human Resources, Vol. 29(2), pp. 429–460.

18

For other examples see footnote 2.

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Altonji, J.G. and R.A. Shakotko (1987), ‘‘Do wages rise with job seniority?’’, Review of Economic Studies, Vol. 54(3), pp. 437–459. Altonji, J.G. and N. Williams (1997), ‘‘Do Wages Rise with Job Seniority? A Reassessment’’, NBER Working Paper, No. 6010. Bontemps, C., J.-M. Robin and G.J. Van den Berg (2000), ‘‘Equilibrium search with continuous productivity dispersion: theory and non-parametric estimation’’, International Economic Review, Vol. 41(2), pp. 305–358. Bowlus, A.J., N.M. Kiefer and G.R. Neumann (1995), ‘‘Estimation of equilibrium wage distributions with heterogeneity’’, Journal of Applied Econometrics, Vol. 10, pp. S119–S132. Bowlus, A.J., N.M. Kiefer and G.R. Neumann (2001), ‘‘Equilibrium search models and the transition from school to work’’, International Economic Review, Vol. 42(2), pp. 317–343. Bowlus, A. J. and L. Vilhuber (2001), ‘‘Displaced Workers, Early Leavers and Re-employment Wages’’, CIBC Working Paper Series, No. 2001–5. Burdett, K. and D. Mortensen (1998), ‘‘Wage differentials, employer size, and unemployment’’, International Economic Review, Vol. 39(2), pp. 257–273. Connolly, H. and P. Gottschalk (2002), ‘‘Job search with heterogeneous wage growth – transition to ‘better’ and ‘worse’ jobs’’, Boston College Working Paper, No. 543. Dey, M. and C. Flinn (2003), ‘‘An equilibrium model of health insurance provision and wage determination’’ C.V. Starr Center for Applied Economics, New York University Working Paper. Gorgens, T. (2002), ‘‘Reservation wages and working hours for recently unemployed US women’’, Labor Economics, Vol. 9(1), pp. 93–123. Gottschalk, P. (2001), ‘‘Wage mobility within and between jobs’’, Mimeo, Boston College. Ingram, B. and G.R. Neumann (forthcoming), ‘‘The Returns to skill’’, Labour Economics. Juhn, C., K.M. Murphy and B. Pierce (1993), ‘‘Wage inequality and the rise in returns to skill’’, Journal of Political Economy, Vol. 101(3), pp. 410–442. Keith, K. and A. McWilliams (1999), ‘‘The returns to mobility and job search by gender’’, Industrial and Labor Relations Review, Vol. 52(3), pp. 460–477. Mortensen, D.T. (1990), ‘‘Equilibrium wage distributions: a synthesis’’, in: J. Hartog, G. Ridder and J. Theeuwes, editors, Panel Data and Labor Market Studies, Amsterdam: North-Holland. Postel-Vinay, F. and J.-M. Robin (2002), ‘‘The distribution of earnings in an equilibrium search model with state-dependent offers and counteroffers’’, International Economic Review, Vol. 43(4), pp. 989–1016. Ruggles, S., M. Sobek, T. Alexander, C.A. Fitch, R. Goeken, P.K. Hall, M. King and C. Ronnander (2004), Integrated Public Use Microdata Series: Version 3.0, Minneapolis, MN: Minnesota Population Center. http:// www.ipums.org.

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Sullivan, P. (2003), ‘‘A dynamic analysis of educational, occupational and interfirm mobility decisions’’, Mimeo, University of Virginia. Yankow, J.J. (2003), ‘‘Migration, job change and wage growth: a new perspective on the pecuniary return to geographic mobility’’, Journal of Regional Science, Vol. 43(3), pp. 483–516. Appendix Table A1.

Nominal earnings (in $) for white males, 26–65, by education

Year

Age 26–30

31–35

36–40

41–45

46–50

51–55

56–60

61–65

College graduates 2000 40,990 59,696 1990 28,798 40,396 1980 15,470 21,722 1970 8,954 12,678 1960 5,407 7,689 1950 2,989 4,032 1940 1,599 2,160

74,857 46,953 27,278 14,451 8,806 4,927 2,599

81,133 52,478 29,674 15,753 9,539 4,876 2,739

79,110 57,876 30,734 16,154 9,490 5,128 2,831

81,525 58,887 30,775 16,560 9,654 5,234 2,791

79,566 54,410 29,234 15,180 9,331 4,687 2,691

67,352 46,983 25,646 14,072 8,834 4,297 2,537

Non-college graduates 2000 27,505 33,359 1990 19,871 24,670 1980 13,314 16,270 1970 7,472 8,464 1960 4,457 5,176 1950 2,560 2,935 1940 988 1,216

37,501 27,440 18,117 8,913 5,504 3,047 1,326

39,860 30,003 18,644 9,163 5,548 3,104 1,394

41,332 31,844 18,802 9,169 5,374 3,073 1,379

42,364 31,020 18,528 8,869 5,214 3,039 1,308

40,479 28,871 17,674 8,239 5,017 2,886 1,237

34,023 23,758 14,820 7,203 4,681 2,611 1,152

Source: Calculated from data of Ruggles et al. (2004).

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CHAPTER 11

Heterogeneity in Firms’ Wages and Mobility Policies$ J.M. Abowd, F. Kramarz and S. Roux Abstract We study the simultaneous determination of worker mobility and wage rates using a model that allows for both individual and firm-level heterogeneity. The model is estimated using longitudinally linked employer–employee data from France. The structural results for mobility and wages show remarkable heterogeneity. For instance, the average structural return-to-seniority are essentially zero but, again, this masks enormous heterogeneity with positive seniority returns found in low starting-wage firms. A factor analysis of the firm-specific wage and mobility parameters estimates the strongest association as the contrast between high-turnover, low-wage, and high return-to-seniority firms with low-turnover, high-wage, and low return-to-seniority firms. This result is strongly reminiscent of Job Search models a` la Mortensen, which are structured along such lines. Keywords: mobility, wages, matched employer–employee data JEL classification: J30 1. Introduction We study the connections between firm-level compensation, promotion, and retention policies in a model with firm heterogeneity in each of these policies. We

$

The data used in this paper are confidential but the authors’ access is not exclusive. For additional information, contact Francis Kramarz ([email protected]). Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75011-2

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

237

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J. M. Abowd, F. Kramarz and S. Roux

relate a worker’s inter-firm mobility to firm-specific compensation policies. Then, in our empirical analysis we use newly developed econometric methods and fully integrated French employer–employee data to estimate some of the firm-specific parameters of the model. As others have noted, particularly for France and the United States, the results tend to show enormous individual-and firm-level heterogeneity in compensation, promotion, and retention policies. We characterize this heterogeneity by modeling its joint distribution in the populations of individuals and firms. Finally, we recover some of the structural parameters of the firm-level policies such as ‘‘returns-to-seniority’’ parameter. The labor economics literature has attempted to measure the average return to seniority in models with limited heterogeneity. Abraham and Farber (1987) were the first to demonstrate that heterogeneity in the model for employment duration induced an upward bias in the measured average return to seniority, specifically, jobs with a longer expected duration were likely to be better-paying jobs and, therefore, longer seniority would be associated with higher pay but the return to an additional year of seniority, holding constant the expected duration of the job, was much smaller than the measured average return-to-seniority, ignoring expected job duration. Brown (1989) showed that the return-to-seniority is not constant; rather, it is higher during the first years of a job and diminishes to zero at the end of the employee’s self-declared ‘‘training period.’’ In a series of articles, Altonji and co-authors (1987, 1992, 1997) applied various econometric techniques that attempted to remove the bias in the average return to seniority due to unobserved heterogeneity in individual job durations. These estimates, very much in the spirit of Abraham and Farber, also indicated that the measured average return was upward biased and that the true return was closer to zero. In contrast, Topel (1991) used a model that included the possibility of bias arising from individual job search. This bias goes in the opposite direction of the jobduration heterogeneity bias leading Topel to entertain both upward and downward biases. He concluded that the bias was downward in the uncorrected average return to seniority. More complete models of the sources of heterogeneity in the return to seniority lead to distributions of estimates that display individual, firm, and within-firm heterogeneity as in, for example, Abowd, Kramarz, and Margolis, (1999, AKM hereafter), who find substantial heterogeneity in the returns to seniority in France (all of the previously cited papers used American data) with an average return of zero for men and women. More recent work by Margolis (1996) and Dostie (2005), using French data, confirm that simultaneous modeling of individual- and firm-level heterogeneity produces estimates of the average return to seniority that are lower than the uncorrected estimates. We begin by laying out a very simple theory in Section 2 that relates individual and firm heterogeneity in wages, productivity, and mobility. This theory translates directly into estimating equations laid out in Section 3. Section 4 describes our data. Section 5 presents statistical results on compensation and mobility parameters that take account of potential mobility and heterogeneity biases that have plagued other analyses. This first set of results, which properly

Heterogeneity in Firms’ Wages and Mobility Policies

239

accounts for each of the heterogeneity biases, delivers estimates of the central tendency of compensation parameters, such as the return to seniority and the structural job duration, that may be interpreted as structural in the same sense as those given in the studies cited above. The second set of results summarizes all the above information into four factors. They show that the main factor appears to order firms very similarly to the equilibrium distribution of wages in a Job Search equilibrium model a` la Mortensen. 2. A simple theory of wages, productivity, and mobility The ‘‘theoretical’’ section presented here attempts to give a rationale to the estimating equations presented in the next section and is to be considered as a departure point for interpretations of the estimates proposed at the end of the paper. Although very simple – and because of this – the theory does not allow us to identify among the large number of competing theories that relate the wage and the mobility processes. The approach adopted in this paper is mostly descriptive and atheoretical. In this section, we try to explicit the relation between a worker’s wage, productivity, and mobility decisions. Our model is relatively straightforward. We assume that there exist firms that do not make use of technologies that have firm-specific components. These firms are homogeneous and any worker can find a job in such a firm at any period. These firms, that will thereafter be denoted as ‘‘simple’’, pay market wages. Their profits are zero.1 All other firms have heterogeneous technologies that make use of firm- and worker-specific components. These technologies are known to workers and firms who agree at all times on the value of a given worker–firm match. These firms consider market wages as exogenous and given. When entering a firm, a worker contributes to the value of the match by bringing the following productivity sequence (pij0, y , pijS), pij0 being the productivity of the worker i at the beginning of her job spell within firm j, and pijS being her productivity at the end of the job spell. Once again, we assume for simplicity that wages in ‘‘simple’’ firms equal their productivity. In these firms, productivity of individuals only depend on their innate, fixed, abilities yi and on their labor market experience Expit. Thus, basic wages or productivities can be written as pBit ¼ wBit ¼ expðyi þ bExpit Þ

ð1Þ

where b is the return to experience. The dependence of basic productivity with respect to experience may be non-linear: it has a quartic form in the empirical

1

Other specifications specifying market wages are possible. The simplest assumption, selected here, is to assume that market wages are given.

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J. M. Abowd, F. Kramarz and S. Roux

section. In simple firms, productivity and wage increase as experience increases: Expitþ1 ¼ Expit þ 1. These productivity or wage terms are not firm-specific, since all simple firms are endowed with the same technology and wage policies.2 Since productivity equals wage, there is no mobility issue within these firms. With this specification, wBit can be viewed as the reservation wage of all workers.3 Finally, notice that in these firms, the value of a match equals 0. In non-simple firms (complex, hereafter), technology is firm-specific and productivity paths may differ across firms. Overall, individual productivity may depend on worker’s seniority in the firm j.4 In addition, as in simple firms, worker’s productivity will depend on innate abilities yi, on experience as follows:

 B pNB ð2Þ ijt ¼ exp yi þ bExpit þ bj Senijt þ nijt ¼ pit exp bj Senijt þ nijt where Senijt denotes the seniority of worker i in firm j at time t. Hence, worker i entered into the firm at time t0 ¼ t Senijt . The expected value of the match at time t is the actualized sum of the difference between worker’s productivity and reservation wage as long as the job continues. ! 1 X s NB B d ðpijtþs witþs ÞRijtþs V ijt ¼ E t ð3Þ s¼0

where d is the discount rate and Rijt a variable that indicates whether worker i works in firm j at time t, in which case it is equal to 1 (and 0 otherwise). Obviously, Rijt is a decision variable. It may result from optimal decisions from both the worker and the firm. But, Rijt may also be affected by some worker or firm’s features. For instance, some workers may have a large specific propensity to move from firm to firm. In this case, this would result in a relatively high probability for Rijt to be equal to 0, with some potentially inefficient separations. Moreover, we assume that once a worker has left the firm, she cannot come back (with the same match value). Should she come back, she is endowed with a new match value, a new productivity corresponding to this new spell. Thus, if t>t0, where t0 is the job starting date, the relation Rijt ¼

2

tY t0 s¼0

Rijt0 þs

ð4Þ

Once again, we could dispense with this assumption. Papers such as Burdett and Mortensen (1998), Robin and Roux (2002), and Burdett and Coles (2003) show that ex-ante identical firms may have very different wage policies. Integration of such theories is beyond the scope of this paper. 3 We do not account for other job opportunities that may affect this value of the reservation wage as in on-the-job search theory. We assume that these aspects are contained in the individual heterogeneity parameter yi. 4 We abstract from match-specific learning, but see Woodcock (2003) for a full treatment, and from optimal sorting, but see Mortensen (2003) for a full treatment.

Heterogeneity in Firms’ Wages and Mobility Policies

241

holds. The value of the match is equal to 0 when the job spell is over. As mentioned above, when a worker reenters a firm, she is considered to be a new individual. The relation (3) corresponds to the expected realized value of the match Vijt, for the sequence of mobility decisions (Rijt, y , Rijt+s). Using (4), this value can also be written in an intertemporal way   V ijt ¼ Rijt E t pNB wBit þ dV ijtþ1 jRijt ¼ 1 ¼ Rijt V~ ijt ð5Þ ijt where V~ ijt is the expected continuation value of the match at time t, that is Vijt conditional on Rijt ¼ 1. Thus, should the choice of Rijt be efficient (not imposed here), we would have the following continuation rule:   Rijt ¼ 1 if V~ ijt ¼ E t pNB wBit þ dV ijtþ1 jRijt ¼ 1  0 ijt   ð6Þ wBit þ dV ijtþ1 jRijt ¼ 1 o0 ¼ 0 if V~ ijt ¼ E t pNB ijt

As explained above, this rule can be perturbed by worker’s or firm’s actions, potentially unrelated to the maximization of the match value. However, we believe that most of the time, as this rule exemplifies, worker’s mobility decisions are affected by shocks on her current or expected productivity. The continuationexpected value V~ ijt in Equation (5) has two components. The  NB B first component, E t pijt wBit , reflects the flow value of wit Rijt ¼ 1 ¼ pNB ijt firm-specific productivity in excess of the market value generated by the match at time t. The second component captures the expected value of the match E t ðdV ijtþ1 =Rijt ¼ 1Þ after time t. Assume, for the sake of discussion, that firmspecific productivity can be decomposed into pNB ijt ¼ f ijt þ Bijt , where the first term is the known productivity of the match and the second term an independent (over i, j, t) random shock with zero mean. Similarly, assume that the external wage rate in a firm with no firm-specific productivity is wBit ¼ git þ Zit , where the first component reflects the known market value of the worker and the second is another random shock (independent over i, t) with zero mean. Finally, assume that both fijt and git are (weakly) increasing and concave functions of time in the match s ¼ t t0, where t0 is the starting date of the job. Many theories, for example Jovanovic’s (1979) learning model of firm-specific capital, predict that productivity in the early years of a match is lower than worker’s productivity in alternative firms where productivity has no firm-specific component. To summarize these effects, consider two seniority values s and sþ with 0os osþ . Let us suppose that Eðpijt0 þs Þ ¼ f ijt0 þs is smaller than EðwBit0 þs Þ ¼ gijt0 þs for sos . Expected returns to the match are positive after s until seniority s+ when worker’s productivity stops increasing. For s>s+ expected outside productivity can, again, exceed worker’s firm-specific productivity so that at least one of the partners will terminate the match. In addition to these systematic elements, the random shocks Bijt0 þs and Zit0 þs will also affect the separation decisions. To summarize, the worker remains at firm j as long as the expected value Et (Vijt+1) is large enough

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J. M. Abowd, F. Kramarz and S. Roux

  B to exceed any current productivity disadvantage captured in pNB w ijt it . This part of the value function reflects the technology of the firm, promotion practices, human resource management, and other firm-specific factors. This reasoning applies also at the beginning of the job. Before the beginning of the job at time t0, both firm and workers who meet consider the non-optimized expected value of the match V~ ijt0 . If V~ ijt is positive the worker will be hired, which would have for consequence Rijt0 ¼ 1. If not, the worker is not hired. Until now, nothing has been said about wages in complex firms. Let us consider the wage sequence ðwijt0 ; . . . wijt0 þs ; . . . Þ. These wages reflect a breakdown of the match value between the value accruing to worker i if employed in firm j; VW and the value accruing to the firm, when j employs worker i, VF. ! 1 X s B W d ðwijtþs witþs ÞRijtþs V ijt ¼ E t ð7Þ s¼0

V Fijt ¼ E t

1  X ds pNP ijtþs s¼0



wijtþs Rijtþs

!

ð8Þ

A large number of theories explain wage formation. The most simple theories state that wages always equal productivity, in which case accumulation of specific capital leads to a direct relationship between wages and seniority. At the opposite, firms could always pay workers’ reservation wages. On-the-job equilibrium search models provide a rationale for wages being higher than reservation wages (Burdett and Mortensen, 1998) and possibly dependent on seniority (Burdett and Coles, 2003) even without accumulation of specific capital in worker’s productivity. Nash bargaining or personnel economics (Lazear, 1995) provide other rationales for wages being higher than reservation wages and possibly dependent on productivity. The point here is clearly not to choose among all these theories or even to test them. The previous representation of the mobility process is flexible enough to be consistent with all these theories. We now present a flexible description of the wage formation process consistent with as many theories as possible as well as with the separation rule described above. The optimal continuation rule discussed above tells us nothing about the optimal compensation system for the employing firm. Therefore, we use a simple flexible sharing rule that can reflect the details of many pay systems in a matching or firm-specific capital environment:5   ln wijt ¼ ln wBit þ gijt ln pNB ln wBit ð9Þ ijt With this sharing rule, the sharing parameter gijt has many potential interpretations. Using our above specification of the productivity in complex firms (2),

5

The log sharing rule specification is choosen mainly for practical empirical reasons.

Heterogeneity in Firms’ Wages and Mobility Policies

243

we get ln wijt ¼ yi þ bExpit þ gijt bj Senijt þ gijt nijt ¼ yi þ bExpit þ g~ ijt Senijt þ n~ ijt

ð10Þ

Hence, g~ ijt now corresponds to i’s specific returns to seniority in firm j at time t. n~ ijt is a firm and individual specific shock that affect wages. As it is written, the model cannot be identified. One way to add some structure is to eliminate the individual heterogeneity in the components n~ ijt and g~ ijt . Instead of being directly indexed by i, these components will depend on observable characteristics Xit of i: so g~ ijt ¼ g~ jt ðX it Þ and n~ ijt ¼ n~ jt ðX it Þ þ
ijt , where eijt is a shock on productivity or on the sharing rule. Therefore, unobserved individual heterogeneity will affect wages only through yi, transferable from firm to firm. The dependence of g~ jt and n~ jt on t potentially reflects the dependence of the sharing rule with seniority (as in personal economics theories), even in absence of pure seniority effects on workers’ productivity. We will assume that these functions are firm-specific. They are to be estimated in what follows. Even though no assumption is made about the choice of a compensation system in our modeling strategy, we will use the term ‘‘firm-specific wage policies’’ in our estimation section. These policies will reflect both the sharing rule and the technologies prevailing in each firm. To allow for a more flexible specification, the wage equation adopted in the following is: ln wijt ¼ yi þ bExpit þ g~ j ðSenijt ; X it ÞSenijt þ n~ j ðX it ; Senijt Þ þ
ijt

ð11Þ

From (10), the starting-wage equation can be written as ln wijt0 ¼ yi þ bExpit0 þ n~ ijt0

¼ yi þ bExpit0 þ n~ j ðX it0 ; 0Þ þ
ijt0

ð12Þ

Until now, nothing has been said about the relationship between mobility and wages. The link goes through productivity. A shock on productivity clearly affects the probability that a worker stays in a firm through the continuation rule (6). It also affects wages, hence inducing a potential correlation between the decision to continue in a firm and wages. Consider continuation rule (6) which states that the differential between productivity and the reservation wage must be greater than a function representing the future matched value, estimated using information available at time t. This condition can be written as ln ðpNB ijt Þ

ln ðwBit Þ  f j ðZ ijt Þ

ð13Þ

where Zijt are all characteristics, firm-level or worker-level, that may enter the continuation rule. Notice that the condition may differ from firm to firm. Using complex firms’ productivity equation (2) and reservation wages (1), this condition turns to bj Senijt þ nijt  f j ðZ ijt Þ

ð14Þ

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Thus, in this specification, a positive shock on the unobservable component of productivity increases the probability to stay in the firm. But, by using Equation (10), this component also affects wages. Obviously, there is endogenous selection. Moreover, because seniority depends on past immobility decisions, it is endogenous in the wage equation. These effects may be different from firm to firm because productivity shocks are likely to have differential effects on wages, depending on the firm-specific sharing rule and technologies in use in the firm. This simple framework provides us with a framework for thinking about a representation of firm-specific wage and retention policies. The wage policy correspond to functions g~ j and n~ j , possibly depending on individual characteristics and seniority. The retention policy is also captured by firm-specific parameters and functions ( fj), also depending on individual characteristics and seniority. The next section presents a methodology to estimate these two policies. 3. A general set of wage and mobility equations Based on the above model, we first estimate the reservation wage equation (1). Then, we estimate the firm-specific wage and retention policies. Because market wages cannot directly be observed, simple and complex firms are essentially identical at entry. Hence, the reservation wages estimates come from estimating a starting-wage equation. In addition, once again because we cannot separate simple and complex firms, the estimation of the firm-specific wage and retention policies will be restricted to large firms (i.e. those for which enough observations are available in our data sources) assuming therefore that all such firms are complex w.l.o.g. 3.1. Starting-wage equation Based on the above model, consider a wage equation in which there are two main components: 

An opportunity wage that reflects the time-varying market value of the workers – value that they keep when moving from firm to firm. This corresponds in the previous section to the wage wBit that workers get in simple firms. This opportunity wage may depend on some characteristics that may change with time, X ð1Þ it , such as the experience, as in the theoretical section, and on a fixed individual term that reflects all observed and unobserved timeinvariant heterogeneity. Thus, log wBit ¼ X ð1Þ it b þ yi

ð15Þ

b includes all person characteristics that may affect the market wage, including temporal indicators. The variables entering this equation are described in Section 4.  Firm-specific parameters which capture the firm’s compensation policy, in particular, the role of seniority Senijt, education, and other characteristics X ðit2Þ ,

Heterogeneity in Firms’ Wages and Mobility Policies

245

some possibly interacted with seniority (for instance, education and gender) in the firm’s productive activities. These components are lost when leaving firm j. Using the above notations, this leads to the following specification:   ð2Þ Senijt þ n~ j ðX ð2Þ g~ j Senijt ; X ð2Þ it it ; Senijt Þ ¼ cj þ jj X it ðSenijt Þ þ gj ðSenijt Þ ð16Þ

where gj is a non-linear function, introduced to account either for a possible non-linear dependence between productivity and seniority and/or of the sharing rule. Slightly departing from Equation (11) but using this extended specification, we obtain log wijt ¼ log wBit þ g~ j ðSenijt ; X it ÞSenijt þ n~ j ðX it ; Senijt Þ þ
ijt ð2Þ ¼ X ð1Þ it b þ yi þ cj þ jj X it ðSenijt Þ þ gj ðSenijt Þ þ
ijt

ð17Þ

Hence, for each employment spell, the wage equation (17) generates a startingwage equation at date t0, when worker i enters firm j, given by log wijt0 ¼ X itð1Þ0 b þ yi þ cj þ jj X itð2Þ ð0Þ þ gj ð0Þ þ
ijt0 ¼ X ð1Þ it0 b þ yi þ cj þ
ijt0

ð18Þ

under the assumption that the initial component of the firm-specific compensation policy can be summarized by cj, a constant irrespective of workers’ characteristics. As in Topel (1991), the returns to experience are directly estimated from this equation. Hence, X ð1Þ it0 includes experience variables (see Section 4 for a full description of the explanatory variables) as well as other individual characteristics. Because some workers have one job over the period whereas others have many more, we introduce the number of previous jobs as an additional explanatory variable.6 The starting-wage equation is estimated by full least squares based on the technique described in Abowd et al. (2003). The observations included are those ^ y^ i , and c are known for which seniority s is equal to 0. Hence, the coefficients b, j parameters for the following steps where the firm-specific compensation policy, as described by Equation (17), is examined. The identification of these parameters requires supplementary assumptions that we explicit now. y^ i is the estimation of the individual transferable productivity. It can be identified only for the individuals who had at least two different jobs in at least two different firms during the observation period. These

6

For instance, Dustmann and Meghir (2003) restrict their estimation of such an equation to displaced workers.

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J. M. Abowd, F. Kramarz and S. Roux

individuals correspond to 60% of the whole sample. There is clearly an identification problem for the others. For those ones, we consider ^ , which corresponds to the assumption that
ijt ¼ 0. y^ i ¼ log wijt0 X it0 b^ c j 0 There exists a similar problem for some firms whose number of sampled workers is very low: only 44% have more than one observation. For the firms for which ^ ¼ 0. As for the other parameters to only one observation is present, we set c j ^ there is no identification problem since the number of degrees of estimate, b, freedom is very large (around 2.5 millions), even accounting for the estimation of individual-and firm-fixed effects. Obviously, our estimated individual-fixed effects y^ i will be affected by some measurement error. Therefore, we will restrict its use as much as possible in what follows. 3.2. The firm-specific model for wages and mobility After entry into firm j, and at each date t, the worker and firm jointly decide to separate or to continue the match. In our approach, and given the available data, quits and layoffs are empirically identical. The worker’s wage is observed after entry if and only if the worker–firm pair jointly decide to continue the match. This process is very much in the spirit of work by Jovanovic (1979), Flinn (1986), Topel and Ward (1992), and Buchinsky et al. (2002) for micro-matching models or earlier work by Lillard and Willis (1978) or Mincer and Jovanovic (1981) for the whole economy. At date t for a worker with seniority s, after subtracting the effect of the market variables as measured by X it b^ þ y^ i , the mobility process can be expressed using the following equations that correspond to the firm-specific continuation rule (13) and wage Equation (11):
 Rnijt ¼ bj Senijt f j Z ijt þ nijt ð19Þ log wijt

^ X ð1Þ it b



 y^ i ¼ cj þ jj X ð2Þ it Senijt þ gj Senijt þ
ijt

where Rnitj is a latent variable expressing staying in the firm j at date t. These equations can be rewritten as R Rnijt ¼ aR j Z ijt þ nijt

log wijt

^ X ð1Þ it b

ð20Þ W y^ i ¼ aW j X ijt þ
ijt

where Z R ijt is a vector of (possibly) seniority-dependent variables that affect the continuation decision, aR j the firm-specific parameter vector describing the dependence of the separation decision on Z R ijt , and nijt the productivity shock from the theoretical model. aR describes the whole firm-specific retention policy; X W j ijt a vector of (possibly) seniority-dependent variables that affect the difference between wage and market wage; aW j the firm-specific parameter vector describing W the dependence of firm-specific wage on X R ijt . aj describes the firm-specific wage policy. For example, a worker hired at date t0, who stays for two periods then

Heterogeneity in Firms’ Wages and Mobility Policies

247

separates, has mobility and log wage equations given by R Rnijt0 þ1 ¼ aR j Z ijt0 þ1 þ nijt0 þ1 40

log wijt0 þ1

X it0 þ1 b^

W y^ i ¼ aW j X ijt0 þ1 þ
ijt0 þ1

log wijt0 þ2

X it0 þ2 b^

W y^ i ¼ aW j X ijt0 þ2 þ
ijt0 þ2

R Rnijt0 þ2 ¼ aR j Z ijt0 þ2 þ nijt0 þ2 40 R Rnijt0 þ3 ¼ aR j Z ijt0 þ3 þ nijt0 þ3 o0

ð21Þ

R The indexation of explanatory variables X W ijt and Z ijt means that their senioritydependent components increase with time as long as the job continues. In order to model the statistical relations between past wages and mobility or, similarly, future wages and mobility, we assume that the following correlation structure holds: 0 2 31 1 0 1 r1j 0 0 0 nijt0 þ1 C B 6r C B
2 0 7 B 6 1j sj r2j 0 7C B ijt0 þ1 C B 6 7C C B B nijt0 þ2 C*N B0; 6 0 r2j 1 r1j 0 7C ð22Þ B 6 7C C B B 6 7C C B 2
@ 4 0 @ ijt0 þ2 A 0 r1j sj r2j 5A nijt0 þ3 0 0 0 r2j 1

A simple rewriting of the correlation matrix based on the normality assumption is useful for estimation, since the likelihood does not involve multiple integration of the normal distribution. Actually, the true correlation matrix that would support the process described above should not be limited by the absence of the individual into the firm. In the above example, the continuation rule at date t0+3 should be affected by the wage anticipation at date t0+3. What is described here is how we deal with the censorship induced by the job separation. Since the wage at date t0+3 is not observed, it has to be taken account for this. This correlation structure assumes the absence of correlation with the starting wage. It would have been more correct to allow the residual at entry
ijt0 to be correlated with nijt0 þ1 . We did not include this supplementary information to simplify the estimation method. To account for it, we should have considered as supplementary variable in the continuation equation an estimation of the residual
ijt0 . Since this residual is assumed to be equal to 0 for more than 40% of workers, this possibility has been ruled out. The crucial point in our approach should now be clear: all parameters of the wage and mobility equation, apart from the starting-wage equation, are firmspecific. The estimation method is maximum likelihood. For instance, in the wage equation, there are return-to-seniority parameters that are similar for all workers employed in the firm and may differ from returns estimated in other firms. In addition, these firm-specific return-to-seniority are allowed to vary with the sex and education of the workers in that firm. More generally, since the estimation is done firm-by-firm, the estimated parameters can be used to characterize the hiring, promotion, and retention policies of the firm.

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The estimation of this process does not rely on the parametric assumptions on the residuals. We introduce also two types of instrumental variables or exclusion restrictions that are supposed to affect continuation probability and not wages. The first type captures individual propensity to change job: these are the duration and the number of previous jobs. The second type is firm-specific, it is the position of the worker at entry in the firm-specific distribution of ages. This variable is assumed to give some evidence on the existence of internal labor markets (Doeringer and Piore, 1976). These instrumental variables are detailed in Section 5, devoted to the presentation of the results. 4. Data description We use data from the De´clarations annuelles des donne´s sociales (DADS), a 1/ 25th sample of the French work force with information from 1976 through 1996 on the matched worker–firm side. 4.1. The DADS The ‘‘De´clarations Annuelles des Donne´es Sociales’’ are a large collection of matched employer–employee information collected by INSEE (Institut National de la Statistique et des Etudes Economiques). The data are based on a mandatory employer report of the gross earnings of each employee subject to French payroll taxes. The universe includes all employed persons. Our analysis sample covers all individuals employed in French enterprises who were born in October of even-numbered years, with civil servants excluded. Our extract runs from 1976 through 1996, with 1981, 1983, and 1990 excluded because the extracts were not built for those years. The initial data set contained 16 million observations. Each observation corresponds to a unique enterprise-individualyear combination. The observation includes an identifier that corresponds to the employee (called NNI below) and an identifier that corresponds to the enterprise (SIREN). For each observation, we have information on the number of days during the calendar year the individual worked in the establishment as well as the full-time/part-time/intermittent/at home work-status of the employee. Each observation also includes, in addition to the variables listed above, the sex; month, year, and place of birth; occupation; total net nominal earnings during the year; and annualized net nominal earnings during the year for the individual as well as the location and industry of the employing establishment. Having done various selections and imputations similar to those described in Abowd et al. (1999), the final data set that we use contains 13,770,082 observations, corresponding to 1,682,080 individuals and 515,557 firms. The estimation of the firm-specific mobility and wage process requires enough observations for each firm. Thus, we had to restrict the estimation to the

Heterogeneity in Firms’ Wages and Mobility Policies

249

firms with 200 observations at least. This leaves only 5000 firms that can be used in this purpose. These are the biggest firms and cover around one-third of all workers in the private sector. To estimate the starting-wage equation, i.e. a wage equation for all observations with zero seniority, we concentrate on all observations at entry in a new firm. This leaves us with 4,616,093 observations, which correspond to 1,535,758 individuals (some persons are only employed by their 1976 employer and never leave it) and 480,360 firms. 5. Estimation results This section is devoted to the presentation of the results. As stated in Section 3, the estimation is two-stage, a first stage estimating a starting-wage equation and the second estimating the firm-specific model for wages and mobility. The presentation of this section thus begins with the starting-wage equation, then goes to the firm-specific model. Different results are presented. Distribution of parameters for mobility and wage equations are presented, then the parameters reflection at the firm-level relationship between wages and mobility. Finally, this section presents a synthesis of all these results using principal component analysis. 5.1. Starting wages The explanatory variables in Equation (18) are experience (up to a quartic), an Ile de France region indicator, a full-time vs. part-time indicator, and year indicators. These variables are all full interacted with sex. To control for the endogenous number of starting-wage observations (some workers churn more and therefore have more entry jobs), we control for the past number of jobs in the equation (using indicator functions). In addition, we control for both a person and a firm effect using the Abowd et al. (2003) estimation technique. Notice, however, that for firms which have one observation (one worker in an entry job), we replace the firm identifier with the two-digit (NAP 100) industrial classification. Starting-wage results are presented in Appendix B. This equation captures initial heterogeneity, at entry in the firm, in the spirit of Heckman (1981). 5.2. The firm-specific wage and mobility equations For the approximately 5000 firms in the sample for which there are enough individual observations to perform a firm-by-firm estimation (at least 200 person–year observations), we estimate by maximum likelihood the set of equations similar to those described in Equation (21). Convergence occurs for 3951 firms using the automated maximization programs. We did not try to reestimate

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models for those firms in which the algorithm did not converge.7 Since we want to correct the principal component analysis correlation matrix for the estimated nature of the parameters, we also lose some firms because their parameters were too imprecisely estimated. The distributions of estimated parameters for the mobility equation and the wage equation are given in Tables 1 and 2. Most parameters are easy to interpret. Notice though that the parameters for seniority in the mobility equation and in the wage equation are estimated as splines. For the mobility equation, these parameters induce the following functional form for the probability of moving from the firm: PrðStayingjsenÞ ¼ F fa1 sen  1ð0  seno2Þ þ ½2a1 þ a2 ðsen þ½2a1 þ 3a2 þ a3 ðsen

2ފ  1ð2  seno5Þ

5ފ  1ð5  seno10Þ

þ½2a1 þ 3a2 þ 5a3 þ a4 ðsen

10ފ  1ð10  senÞg

where 1(
) denotes an indicator function and sen denotes seniority. These spline functions are set up so that the function is linear, continuous everywhere, with changing slopes at 2, 5, and 10 years of seniority. For instance, a positive coefficient for a1 means that workers are more likely to stay after 1 year than after the initial year. Then, coefficient a2 applies and measures a difference (either increasing or decreasing depending on its sign) from coefficient a1. For instance if a2 is not significantly different from zero, it means that workers with three to five years of seniority have the same continuation probability as workers with two years of seniority. Similarly, we implement spline functions of seniority for the wage equation. To assess the significance of these parameters, we also present the corresponding distributions of Student statistics for these two equations in Tables 3 and 4. The results are striking. The amount of dispersion across firms is daunting. Most models that only incorporate a minimum amount of heterogeneity are way off the mark. This heterogeneity cannot be considered as a result of noisy estimations. Should it be the case, the distribution of Students would reflect this. It appears that, for a large number of variables, the distribution of coefficients is far more dispersed than would allow a normal distribution assumption. 5.2.1. The continuation equation The continuation equation is the probability that the job continues, hence a positive coefficient corresponds to a larger propensity to stay. Starting from the

7

In previous versions of this research, we succeeded in obtaining convergence for 95% of firms after multiple attempts, but none of the subsequent results were changed by inclusion of these firms. In addition, we were less confident in some of the coefficients obtained for those firms where convergence was obtained in the supplementary searches. For these reasons, in this article, we decided to restrict attention to firms for which the maximum likelihood procedure converged immediately using a grid search for the starting values of the correlation coefficients r1j and r2j.

Table 1.

Distribution of the estimated parameters for the firm-by-firm mobility equation Mean 0.0161 0.3628 0.2607 0.0382 0.0363 0.1035 0.0083 0.0712 0.0553 0.0010 0.1796 0.3839 0.1683 0.1788 0.0074 0.1774 0.1037 0.2575 0.3246 0.4194 0.3016 0.1996 0.0522 0.1936

0.3941 0.6742 1.2618 6.3362 6.4468 0.2629 1.1331 2.7688 0.7575 1.6188 24.8821 24.6520 19.0219 4.4410 5.0466 31.1999 17.5362 1.8298 1.4654 0.6432 0.5181 0.3867 1.0663 0.4149

q1 0.8390 0.8001 2.9530 3.5082 3.2716 0.6697 0.3339 4.6965 1.3101 0.2925 2.5486 1.6256 1.5974 1.3999 0.4647 1.4458 1.6970 2.4651 3.3289 2.1209 1.7105 1.1937 0.3287 0.0030

q5 0.4055 0.2063 1.5661 1.9353 1.7774 0.2669 0.1644 2.3532 0.5314 0.1307 1.2529 0.7018 0.7267 0.7433 0.1119 0.2843 0.4159 1.2942 1.5403 1.3098 1.0220 0.7134 0.1646 0.0392

q25

q50

q75

0.1364 0.1141 0.3271 0.8422 0.8102 0.0168 0.0478 0.6320 0.2116 0.0358 0.2385 0.2156 0.1687 0.2716 0.0147 0.0190 0.0409 0.5095 0.6040 0.6852 0.5058 0.3659 0.0565 0.0670

0.0056 0.3320 0.3854 0.0323 0.0133 0.0890 0.0005 0.0146 0.0039 0.0041 0.0307 0.0619 0.0035 0.1266 0.0007 0.0498 0.0233 0.2116 0.2616 0.4609 0.3016 0.2111 0.0277 0.0930

0.0912 0.5661 0.8989 0.4819 0.4572 0.2221 0.0743 0.6348 0.2472 0.0199 0.2315 0.0601 0.0691 0.0374 0.0295 0.1318 0.0853 0.0170 0.0095 0.1124 0.0824 0.0384 0.0102 0.1972

q95

q99

0.3597 1.0100 1.9496 1.3875 1.4282 0.4912 0.2149 1.5255 0.9559 0.0545 0.4720 0.4240 0.2717 0.0377 0.1348 0.4384 0.5356 0.7692 0.8746 0.4548 0.3991 0.3132 0.0145 0.6890

0.7923 1.7191 3.1858 2.6178 2.9012 0.8511 0.4069 3.4281 2.0674 0.1587 0.8842 1.1096 0.9527 0.4514 0.4469 1.1699 1.4725 1.8120 2.2733 1.2314 1.1092 0.8599 0.1097 1.3476

Heterogeneity in Firms’ Wages and Mobility Policies

Male Part-time First period indicator Second period indicator Third period indicator Person effect (from starting-wage equation) Experience (Experience/10)**2 (Experience/100)**3 (Experience/1000)**4 Tenure (less than 2 years) Tenure (2–5 years) Tenure (5–10 years) Tenure (more than 10 years) Tenure*male Tenure*low general education Tenure*high general education Low general education High general education Entry in Q1 of the age at entry distribution Entry in Q2 of the age at entry distribution Entry in Q3 of the age at entry distribution Number of previous jobs Duration of the previous job

Std. Error

Note: Between-firms distribution of the estimated parameters for the mobility equation. The model is estimated by maximum likelihood separately firm-byfirm. For each firm in the sample, there is a set of estimated parameters used to compute the distribution. Parameters are only estimated for those firms in which there is enough within-firm variability. Number of observations (firms): 3951. Source: DADS. 251

252

Table 2.

Distribution of the estimated parameters for the firm-by-firm wage equation Mean 0.0344 0.0138 0.0151 0.0084 0.0391 0.0014 0.0056 0.0189 0.0212 0.1544 0.0005 0.0211 0.0074 0.1302 0.3139 0.0260 0.9099

0.4644 0.6353 0.7096 0.7198 1.5918 0.2508 0.2556 0.3298 0.1438 7.7711 0.3524 1.9502 0.4688 11.9309 1.2977 0.1637 0.8357

q1 0.7983 1.9025 1.9277 2.2397 0.3602 0.3470 0.2956 0.2742 0.3919 0.1321 0.9861 0.3320 1.4712 0.4659 4.8606 0.5884 2.4347

q5 0.5281 1.0553 1.6657 1.3609 0.1425 0.1297 0.0953 0.0721 0.1822 0.0434 0.4865 0.1073 0.6056 0.1535 3.6833 0.2421 2.0403

q25

q50

q75

0.2270 0.2934 0.3627 0.3812 0.0243 0.0323 0.0163 0.0078 0.0324 0.0115 0.1353 0.0206 0.1625 0.0125 0.0743 0.0747 1.4549

0.0583 0.1022 0.1234 0.1683 0.0138 0.0003 0.0032 0.0145 0.0184 0.0017 0.0148 0.0040 0.0059 0.0082 0.0048 0.0237 1.0676

0.1978 0.3829 0.3943 0.4382 0.0674 0.0318 0.0266 0.0338 0.0771 0.0081 0.1178 0.0138 0.1584 0.0391 0.0650 0.0315 0.5762

q95

q99

0.9165 0.8695 0.9038 0.9387 0.2227 0.1393 0.1219 0.1343 0.2226 0.0448 0.4481 0.1113 0.6571 0.1695 0.3359 0.1661 0.9425

1.5304 1.2988 1.2977 1.4081 0.4025 0.3395 0.3143 0.4395 0.4771 0.1280 1.0047 0.3698 1.3720 0.5464 2.6381 0.3723 1.1945

Note: Between-firms distribution of the estimated parameters for the wage equation. The model is estimated by maximum likelihood separately firm-by-firm. For each firm in the sample, there is a set of estimated parameters used to compute the distribution. Parameters are only estimated for those firms in which there is enough within-firm variability. Number of observations (firms): 3951. Source: DADS.

J. M. Abowd, F. Kramarz and S. Roux

Part-time First period indicator Second period indicator Third period indicator Tenure (less than 2 years) Tenure (2–5 years) Tenure (5–10 years) Tenure (more than 10 years) Male Tenure*male Low general education Tenure*low general education High general education Tenure*high general education Correlation between mobility and f Correlation between mobility and p Standard error of the wage shock

Std. Error

Table 3.

Distribution of the estimated student statistics for the firm-by-firm mobility equation

(Experience/10)**2 (Experience/100)**3 (Experience/1000)**4 Tenure (less than 2 years) Tenure (2–5 years) Tenure (5–10 years) Tenure (more than 10 years) Tenure*male Tenure*low general education Tenure*high general education Low general education High general education Entry in Q1 of the age at entry distribution Entry in Q2 of the age at entry distribution Entry in Q3 of the age at entry distribution Number of previous jobs Duration of the previous job

Std. Error

0.8117 4.2133 0.6768 0.7344 1.0553 1.7315 0.4688 1.0354 1.3839 1.2024 1.4053 1.0196 0.0788 3.3559 0.0035 1.7749 0.6063 1.3314 1.5891 3.0635 2.5399 2.6000 2.4686 23.0341

4.6330 7.7969 8.3624 6.4521 6.1903 3.5785 3.0357 4.0817 5.0645 5.2846 6.9969 3.9677 4.3181 3.4848 1.7796 2.3458 1.9080 2.0393 2.7773 4.4751 3.3882 4.1588 2.6466 30.8499

q1 4.5846 8.2593 29.2288 26.3768 22.0264 7.9233 9.8025 4.9620 21.0915 4.7710 12.9560 10.8282 11.9530 14.2948 5.2250 2.1896 2.8912 8.9120 11.3924 15.1124 11.6351 16.3312 10.5123 0.6994

q5 2.6651 1.4457 19.7934 14.7496 20.1132 2.5961 6.9150 2.9832 13.8944 2.9442 9.3060 7.2990 8.5697 13.0720 2.4542 0.9771 1.8786 4.4944 9.7713 14.7924 10.8439 14.7258 6.4336 1.8674

q25 0.9781 0.6292 0.5738 1.4135 1.5007 0.3158 1.3716 1.0722 1.2761 1.2377 1.2807 2.5344 1.5431 5.1823 0.9155 0.2316 0.5306 2.1132 2.5382 4.0094 3.3930 2.8080 3.9635 5.6677

q50

q75

q95

q99

0.0320 2.0488 0.9161 0.0497 0.0054 1.1439 0.0000 0.0230 0.0410 0.1353 0.2008 0.6255 0.0210 2.3665 0.0119 1.3170 0.2803 0.9830 0.9414 1.7155 1.5074 1.2474 1.9461 11.0840

1.0997 4.6859 3.1356 1.4600 1.5153 2.9645 1.0995 1.2908 1.0507 1.1844 2.2533 0.5190 1.6305 0.9545 1.0286 2.7654 1.3293 0.0397 0.0076 0.3604 0.3576 0.3372 0.7086 28.7513

5.1979 18.2803 13.2404 9.2332 6.3767 9.0843 3.3827 11.9717 2.5297 13.3020 19.3344 4.3250 8.0322 0.2945 2.6630 7.3194 3.7927 1.1336 1.3711 1.5673 0.9300 0.9689 0.7917 70.0884

21.0560 34.6720 22.4660 10.0240 7.7360 11.1400 8.7410 15.3340 4.3750 22.1950 25.4720 10.1230 10.5820 1.3840 4.8060 7.7430 6.6810 2.3000 2.3580 3.9560 1.9040 2.0480 2.9220 162.1100

Heterogeneity in Firms’ Wages and Mobility Policies

Male Part-time First period indicator Second period indicator Third period indicator Person effect (from starting-wage experience

Mean

Note: Between-firms distribution of the estimated Student statistics for the mobility equation. The model is estimated by maximum likelihood separately firm-by-firm. For each firm in the sample, there is a set of estimated parameters used to compute the distribution. Parameters are only estimated for those firms in which there is enough within-firm variability. Number of observations (firms): 3951. Source: DADS. 253

254

Table 4.

Distribution of the estimated student statistics for the firm-by-firm wage equation Mean 1.1127 1.1297 0.1831 1.5633 1.1347 0.1226 0.1058 1.5590 0.5180 0.7141 0.0995 0.4878 0.0618 0.4587 5.2210 0.2850 124.7990

17.0783 9.4462 13.8692 11.7271 4.7704 2.4116 2.6798 3.8481 2.3120 3.1378 2.5634 2.2550 2.1536 2.1630 22.5500 4.8550 207.9740

q1 51.0257 28.1657 59.8403 40.2238 6.1039 7.5739 6.1009 3.9260 4.4312 11.8777 9.1418 9.9137 4.4692 4.9640 22.7840 13.0730 957.1740

q5 34.3249 16.3387 19.6869 25.1032 4.8529 3.1071 3.8120 2.4861 2.5663 5.7531 3.6319 4.1598 3.6446 3.4729 5.9166 8.8196 561.5850

q25 8.0100 1.6538 1.6918 1.6738 0.8270 1.0027 1.0224 0.2296 0.9346 1.4278 0.9258 1.1489 1.2043 0.5072 1.1875 0.7943 142.9060

q50 0.9373 1.0256 0.9407 1.0696 0.2309 0.0019 0.1469 0.6551 0.3282 0.1832 0.1302 0.1639 0.0510 0.3880 0.0516 0.5977 48.2124

q75 4.7356 4.4142 4.4304 5.1741 1.6982 1.1426 1.1765 1.9420 1.5950 0.8077 1.1549 0.6306 1.0697 1.5273 1.4760 2.2450 15.6818

q95 24.0932 21.0527 23.3272 25.1945 13.4474 4.4182 5.0952 13.3941 4.3710 2.8793 3.0289 2.1636 3.1759 4.3626 78.7770 6.4730 73.1256

q99 74.0798 23.7489 25.2641 27.3398 21.7162 6.1592 7.6454 13.5813 9.3833 6.3390 5.8824 4.6151 8.1428 6.0300 95.1290 12.0560 153.5680

Note: Between-firms distribution of the estimated Student statistics for the wage equation. The model is estimated by maximum likelihood separately firmby-firm. For each firm in the sample, there is a set of estimated parameters used to compute the distribution. Parameters are only estimated for those firms in which there is enough within-firm variability. Number of observations (firms): 3951. Source: DADS.

J. M. Abowd, F. Kramarz and S. Roux

Part-time First period indicator Second period indicator Third period indicator Tenure (less than 2 years) Tenure (2–5 years) Tenure (5–10 years) Tenure (more than 10 years) Male Tenure*male Low general education Tenure*low general education High general education Tenure*high general education Correlation between mobility and future wage Correlation between mobility and past wage Standard error of the wage shock (in log)

Std. Error

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Student statistics in Table 3, we see that some variables almost never matter in the continuation outcome. As our model shows, this outcome results from the interaction of workers’ behavior and firms’ policies. These are equilibrium behavior and it is often difficult to disentangle the firms behavior from the workers’ decision. Examining the results, we start with the best summary of a firm retention policy: its propensity to keep or to separate workers. This policy is measured by the three period indicators (there is no constant in our model). These three indicators are very highly correlated within firms (not shown in table) and therefore reflect a long-term component of the firm mobility policy. A large negative indicator means that the firm has many separations or, equivalently, that it is a high-turnover firm. Interestingly, results in Table 3 show that more firms (around 35%) are low-turnover firms, whereas 10–20% of firms are highturnover firms. The retention policy of firms may well depend on workers’ types. For instance, we see that the worker’s sex appears to matter for less than 30% of firms. For more than 90% of firms, workers with long tenure in their previous job stay longer in their current job. Tenure in the current firm often has the opposite consequences for mobility; that is, our results show negative duration dependence for job seniority but with substantial heterogeneity. Focusing on variables that reflect potential human resource policies of the firm, we consider first the relation between the continuation probability and the individual effect from the starting-wage equation. For 30–40% of the firms there is a strong positive relation between the starting-wage person effect and the continuation probability, as indicated by the distribution of Student statistics in the row labeled ‘‘person effect’’ in Table 3. Hence, firms keep the best workers, as measured by their value on the market. An alternative measure of the heterogeneity of human resource policies comes from examining the evidence for internal labor markets (Doeringer and Piore, 1976). In this view, firms can create labor markets within their own organization. There are privileged ports of entry and the whole career takes places within the firm through moves between jobs. To assess this theory, we have created for each firm a distribution of ages at entry. In Tables 1 and 3, this firmspecific distribution of age at entry is summarized by the variables labeled ‘‘Entry in Qn of the age at entry distribution,’’ where n is the first, second, or third quartile, respectively (with n ¼ 4 as the reference group). If entry at a young age is associated with a career within the organization, we should see a negative relation between mobility and workers who enter in Q1 or Q2 of this age-atentry. For instance if a firm hires workers for some jobs on a short-term basis and other workers for core jobs on a long-term basis at a specific age (mostly young), then one expects to see that entry in the first or the second quartile of the age-at-entry distribution is associated with lower separation probabilities. Direct examination of the relevant rows of Tables 1 and 3 shows that there is not much evidence for this interpretation. There is actually a strong negative relation between entry in the first quartile of the age-at-entry distribution and the

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continuation probability for about 50% of the firms and a strong positive relation (as predicted by internal labor markets) for less than 5%. By contrast, when firms hire workers at various ages and when the worker–firm pair is concerned about the quality of the match, then one expects to see more separations for workers entering in the bottom of the age distribution. As noted above, this is precisely the case. In more than 30% of firms, workers entering in the first three quartiles of the firm-specific age-at-entry distribution move more often than workers entering in the last quartile of the age distribution (unreported results show that these three coefficients are highly positively correlated). Virtually no firm (less than 5%) keeps workers entering in the bottom of the age distribution. These results are largely inconsistent with most versions of the internal labor market theory. Notice that these variables are included in the continuation equation and excluded from the wage equation, hence they are one of our exclusion restrictions granting non-parametric identification of the model. Finally, it is interesting to note that some firms, approximately 30%, try to keep workers with technical degrees (as opposed to workers with general education, either low or high), as can be seen from the rows labeled ‘‘low general education’’ and ‘‘high general education’’ in Tables 1 and 3 since ‘‘technical degrees’’ is the omitted category. Workers with general education (the coefficient for low and for high are very strongly positively correlated) separate from those firms more often than those with technical degrees. 5.2.2. The wage equation At this stage, it is important to recall that we have jointly estimated firm-specific wage and mobility equations (19). The dependent variable in the firm-specific wage equation is the actual log wage rate less the effects that depend upon coefficients estimated from the starting-wage equation (18). This difference captures the firm-specific log wage less the opportunity wage that a worker would receive, in expectation, on the market at an employer with no firm-specific compensation component at hire. Furthermore, since we subtract y^ i from the wage, problems of unobserved person heterogeneity are controlled. Notice also that some variables that were present in the mobility equation are not included in the wage equation. These exclusions serve as identification restrictions. In particular, the position of the entrant within the firm-specific distribution of ageat-entry is excluded from the wage equation since most theories would predict that all effects of this variable work through the propensity to stay in the firm. Similarly, the seniority with the previous employer is excluded from the wage equation since the returns to experience capture this effect. Results for the firm-specific wage equation are presented in Table 2 (for the coefficient distribution) and Table 4 (for the Student statistic distribution). Many variables have more than half of the coefficients that are statistically significant at the 5% level. Only tenure in its various guises (as a spline or interacted with education and sex) displays 40% or less of the estimated coefficients that are

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significant at the same level. Even more striking is the almost completely symmetric (around zero) distribution of many wage coefficients. As before, the best summary of the compensation policy of the firm is captured by the period indicators (there is no constant in the wage equation).8 A positive coefficient corresponds to a high-wage firm and a negative coefficient to a low-wage firm. Roughly 20% of the firms are low-wage firms (at the 5% level) and 30–40% of the firms are high-wage firms (again, at the 5% level). Studying the rest of the Table 2, some results deserve further comment. That the coefficient for full-time compensation is negative may appear surprising. However, one must remember that this variable is present in the starting-wage equation. Hence, what is estimated here is the difference between part-time compensation on the market (included in the opportunity cost of time) and the firm-specific policy vis-a`-vis part-time. Hence, a positive coefficient means that the firm pays a larger differential to full-time workers than the market rate, and conversely a negative coefficient means that the firm pays its part-time workers better than the market rate. Returning to the symmetry around zero of most coefficients, it is striking to see that returns to seniority in the first two years of a job – to take a question that has attracted a lot of attention – are significantly negative for 15–20% of the firms, whereas they are positive for 20% of the firms. These negative returns are still present at higher tenure levels. Furthermore, the estimated seniority coefficients are strongly positively correlated. Hence, around 20% of French firms have negative returns to seniority, 20% have positive returns to seniority, and 60% of French firms have returns to seniority that are virtually zero (not significantly different from zero at the 5% level). This result confirms previous findings of AKM (1999) or, more recently of Dostie (2005) using a similar data set, but completely different estimation techniques. Notice also that comparing the 5th percentile with the 95th percentile for the male-specific returns to tenure we see that 20% of firms provide higher returns to tenure to women and 10–15% reward male tenure more than female tenure. Results are roughly similar for returns to tenure for our different levels of education. In general, some firms appear to favor low-education workers, other firms appear to favor technical education (the omitted category), and finally some firms focus on the higheducation group. The correlations in Table 6 show that those firms that pay loweducation workers high wages also pay their high-education workers high wages. 5.2.3. Relation between pay and mobility In Tables 2 and 4, we present the estimated coefficients and estimated Student statistics for the correlation between the mobility error term with the wage error term (future, r1 and past, r2). Here again, the heterogeneity is daunting: 20% of

8

As before, unreported computations show that these indicators are highly positively correlated.

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firms have a significant (at the 5% level) positive r1, whereas slightly less than 20% of firms have a significant negative r1. Similarly, more than 30% of firms have a significant positive r2, whereas 15% have a significant negative r2. A low r1 means that workers who face a positive shock to mobility, which decreases their probability to stay, also face a positive shock to their future wage. This is a potential reflection of firms trying to counteract workers’ decisions to accept outside offers. Alternatively, the result may mean that workers who have a tendency to move, face very good prospects in their origin firm. Conversely, a high r1 means that workers who face a negative shock to their mobility also face a positive shock to their future wage. Once again, the mobility decision is an equilibrium outcome in which workers who will not get promoted may decide to move. The question of who initiates the potential separation is virtually impossible to solve. Now, r2 captures the correlation between the past shock on wages and the continuation decision. When negative, workers move after an unexpected wage increase. Apparently, this move is induced by the workers’ decision. When positive, workers stay after an unexpected wage increase, resulting from a joint decision. As already mentioned, the latter case (joint optimization) is much more common: an unexpected wage hike is with associated workers staying in most firms. 5.2.4. Correlation analysis To gain a better understanding of potential relations among these various parameters, we performed a principal component analysis of all these estimated coefficients, i.e. those that characterize the mobility policy, those that characterize the pay policy, and those that characterize their relations. Estimates, using the covariance matrix that has been corrected for the fact that all within-firm parameters were estimated (see Appendix C), are presented in Tables 5 and 6. Table 5 shows the eigenvalues, whereas Table 6 shows the factor loadings for the first four axes. The results can be summarized as follows. The first four axes Table 5.

1 2 3 4 5 6 7 8 9 10

Factor analysis of the firm-by-firm parameters of the mobility and wage equations; eigenvalues Eigenvalue

Difference

Proportion

Cumulative

8.5768 4.7583 4.5062 4.1114 2.6482 2.0473 1.8432 1.6517 1.3416 1.0833

3.8185 0.2521 0.3948 1.4632 0.6009 0.2041 0.1915 0.3102 0.2582 0.1433

0.2382 0.1322 0.1252 0.1142 0.0736 0.0569 0.0512 0.0459 0.0373 0.0301

0.2382 0.3704 0.4956 0.6098 0.6834 0.7402 0.7914 0.8373 0.8746 0.9047

259

Heterogeneity in Firms’ Wages and Mobility Policies

Table 6.

Factor analysis of the firm-by-firm parameters of the mobility and wage equations; factor loadings Factor1

Factor2

Factor3

Factor4

Mobility equation Male Full-time First period indicator Second period indicator Third period indicator Person effect (from starting-wage equation) Experience Experienced**2 Experience**3 Experience**4 Tenure (less than 2 years) Tenure (2–5 years) Tenure (5–10 years) Tenure (more than 10 years) Tenure*low general education Tenure*high general education Low general education High general education Entered in Q1 of age at entry distribution Entered in Q2 of age at entry distribution Entered in Q3 of age at entry distribution Number of previous jobs Duration of the previous job

0.0546 0.1944 0.7950 0.7877 0.6613 0.1142 0.0215 0.0582 0.0371 0.0739 0.4400 0.3606 0.5099 0.2875 0.3427 0.1976 0.5644 0.6033 0.8451 0.7938 0.8217 0.0513 0.6476

0.3818 0.1237 0.1154 0.1230 0.1476 0.1159 0.6514 0.7565 0.7738 0.7764 0.3540 0.1688 0.2869 0.5369 0.5515 0.4691 0.3350 0.1447 0.0473 0.0238 0.1516 0.2263 0.0622

0.1798 0.0593 0.1732 0.1547 0.0601 0.2308 0.3178 0.4127 0.4240 0.4373 0.1326 0.0463 0.0822 0.0964 0.3015 0.4085 0.1004 0.0996 0.2928 0.2961 0.2051 0.1599 0.0794

0.1933 0.0606 0.0732 0.1381 0.0940 0.2744 0.1150 0.1286 0.1725 0.2043 0.2981 0.0662 0.2645 0.1514 0.1436 0.2759 0.2020 0.4848 0.2711 0.3535 0.4010 0.2406 0.3952

Wage equation Full-time First period indicator Second period indicator Third period indicator Tenure (less than 2 years) Tenure (2–5 years) Male Low general education High general education Tenure*high general education Correlation between mobility and future wage Correlation between mobility and past wage Standard error of the wage shock (in log)

0.1383 0.5328 0.6220 0.5988 0.4467 0.3198 0.1853 0.1580 0.0217 0.8412 0.5646 0.2295 0.6217

0.0747 0.1030 0.0493 0.0019 0.1651 0.2129 0.0463 0.5198 0.4907 0.6744 0.1012 0.0496 0.0305

0.3109 0.4236 0.3499 0.3360 0.0335 0.1430 0.0622 0.8486 0.8542 1.0749 0.0574 0.0356 0.0993

0.3358 0.5723 0.5999 0.5161 0.3241 0.5459 0.2192 0.4522 0.4446 0.0678 0.6084 0.5463 0.3681

capture 61% of the variance. These four dimensions are built on the following linear combinations. The first axis contrasts high-wage and low-mobility firms with those that pay low wages and are high-mobility firms. It captures 24% of the total variance. The following factor captures only 13% of the variance. These high-wage firms

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also hire relatively older workers, whereas the high-mobility firms mostly hire workers at younger ages, as measured by the firm-specific age-at-entry coefficients. This is clearly consistent with the existence of many short-term formal contracts. Among high-mobility firms, because workers increasingly move with seniority, return-to-seniority are high in the first five years. This axis is consistent with some of the theoretical predictions of on-the-job equilibrium search models. For instance, Postel-Vinay and Robin (2002) exhibit an on-the-job-equilibrium search model in which employers can respond to outside job offers received by their employees. Some predictions of this model are consistent with firm heterogeneity as structured in our first axis. Firms with low wages have a high turnover rate because workers employed in low-wage firms receive better offers more often than workers employed in high-wage firms. Because wage increases within each firm depend on the number of job offers received by its employees, low-wage firms should exhibit higher returns to seniority, exactly what is observed here. At the opposite extreme, high-wage firms do not offer returns to tenure because no firms can compete away their workers. Another explanation for these facts is also found in Burdett and Coles (2003). They present a model with no ex-ante heterogeneity in firms in which firms offer wage tenure contracts. In this model, under the assumption that workers are risk-averse, the distribution of contract offers is non-degenerate and the equilibrium is characterized by a baseline scalary scale, which corresponds to the wage/tenure profile of a firm offering the lowest starting wage. Predictions of this model are also consistent with our characterization of the first axis. Low-wage firms offer a wage/tenure profile that begins at low-starting wages. They are the ones that experiment the most important quits of workers who receive better job offers. Since the baseline salary scale is concave, the wage tenure profile of the workers who stay in these firms exhibit higher returns to seniority. These theories may also help resolve a puzzle in our estimates: the existence of negative returns to seniority in some firms. They can be understood in the presence of market failures such as search frictions. Notice that returns to tenure are estimated on top of the returns to experience that reflect potential gains due to better offers when employed. Negative return-to-seniority appear to often exist in high-wage firms, the ones for which expected gains of receiving higher offers are clearly weaker. The second factor loading axis combines experience and education. The contrast is between firms in which mobility is increasing with experience and where workers with technical education are better paid with firms in which stability is decreasing with experience and where workers with general education are better paid. The third axis contrasts low-paying firms that are high-wage firms for workers with a general education with those that are high-wage firms except for workers with a technical education. Interestingly, the fourth axis also revolves around pay choices for the different education types. It contrasts high-wage firms with relatively low wages for the technically educated and low return-to-seniority with

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low-wage firms with high wages for the technically educated and large return-toseniority. Such firms are neither low-nor high-turnover. 6. Conclusion In this paper, we have used a simple descriptive, theoretical framework to help us think about the relation between mobility and wage for an individual, both from the worker’s own perspective as well as from the employer’s perspective. This framework helped us to set up the estimating equations. The data sources were based on a very large, longitudinal employer–employee data set for France, the DADS. The system of equations was estimated firm-by-firm, very much relying upon the perspective adopted by authors such as Baker et al. (1994ab), with the distinctive feature that we capture elements of the outside labor market, at entry through an entry wage equation with both person and firm effects as well as at exit by explicitly modeling the joint mobility and wage processes, whereas these authors could not. The results are destructive of the homogeneous view of the labor market in which firms adopt very similar strategies. In fact, the amount of heterogeneity in the policies adopted by the firms is daunting. After documenting this heterogeneity, we tried to characterize what compensation and retention strategies could be in such a world. To do so, we used a simple factor analysis that was able to guide us and show that four factors gave a useful summary view of the heterogeneity. We focus here on the first factor, which appears to give a very simple and clear-cut overview of our results. The main contrast between high-wage, low-mobility firms where returns to seniority are low (even negative) and low-wage, high-mobility firms where returns to seniority are relatively high (in a country where the average returns to seniority are lower than in the United States, even compared with Altonji’s results) gives a good first-order approximation of the French landscape. Recent job search and matching models (Postel-Vinay and Robin, 2002; Burdett and Coles, 2003; Woodcock, 2003) with person or/and firm heterogeneity appear to be able to generate exactly this type of effect. Other dimensions contrast firms that favor general education with firms that favor more technical education. On the methodology side, this paper uses some newer, recently developed, techniques for analyzing the matched employer–employee data. It also contains a non-trivial number of methodological firsts. To name but a few, the firm-byfirm (maximum likelihood) estimation strategy, the correction for the estimated nature of the parameters characterizing the firm policies, the joint modeling of wages and mobility at the firm level, and the identification strategy relying on exclusion restrictions based on variables that can only be constructed using the matched worker–firm aspect of the data (for instance the age at entry within the firm-specific age distribution). We believe that the analysis presented here opens more avenues of research than it closes doors and solves problems. But, we see it as a first step in our understanding the substance as well as the methods to use when analyzing firms’

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hiring, retention, compensation, or more generally human resource management policies. New methods should also be developed that would allow us to perform an analysis of workers’ firm-to-firm movements. Acknowledgment Abowd acknowledges financial support from the National Science Foundation (grant SBER 96-18111 to the NBER and SES 99-78093 to Cornell University) and from the National Institute on Aging (R01-AG18854-01) References Abowd, J.M., R. Creecy and F. Kramarz (2003), ‘‘Computing person and firm effects using linked longitudinal employer–employee data’’, Cornell University Working Paper, revised. Abowd, J.M., F. Kramarz and D.N. Margolis (1999), ‘‘High wage workers and high wage firms’’, Econometrica, Vol. 67, pp. 251–334. Abraham, K.J. and H.S. Farber (1987), ‘‘Job duration, seniority and earnings’’, American Economic Review, Vol. 77, pp. 278–297. Altonji, J.G. and R.A. Shakotko (1987), ‘‘Do wages rise with job seniority?’’, Review of Economic Studies, Vol. LIV, pp. 437–459. Altonji, J.G. and N. Williams (1992), ‘‘The effects of labor market experience, job seniority and job mobility on wage growth’’, NBER Working Paper Series, No. 4133. Altonji, J.G. and N. Williams (1997), ‘‘Do wages rise with job seniority? A Reassessment’’, NBER Working Paper Series, No. 6010. Baker, G., M. Gibbs and B. Holmstrom (1994a), ‘‘The internal economics of the firm: evidence from personnel data’’, Quarterly Journal of Economics, Vol. CIX(November), pp. 881–919. Baker, G., M. Gibbs and B. Holmstrom (1994b), ‘‘The wage policy of a firm’’, Quarterly Journal of Economics, Vol. CIX(November), pp. 921–955. Brown, J.M. (1989), ‘‘Why do wages increase with tenure? On-the-job training and life-cycle wage growth observed within firms’’, American Economic Review, Vol. 79, pp. 971–991. Buchinsky, M., D. Fouge`re, F. Kramarz and R. Tchernis (2002), ‘‘Interfirm mobility, wages, and the returns to seniority and experience in the US’’, CREST Working Paper, 2002–29, Paris. Burdett, K. and M. Coles (2003), ‘‘Equilibrium Wage-Tenure Contracts’’, Econometrica, Vol. 71(5), pp. 1377–1404. Burdett, K. and D.T. Mortensen (1998), ‘‘Wage differentials, employer size and unemployment’’, International Economic Review, Vol. 39, pp. 257–273. Doeringer, P.B. and M.J. Piore (1976), Internal Labor Markets and Manpower Analysis, Lexington: CD Health. Dostie, B. (2005), ‘‘Job turnover and the returns to seniority’’, Journal of Business and Economic Statistics (forthcoming), Vol. 23(2), pp. 192–199.

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Dustman, C. and C. Meghir (2003), ‘‘Wages, experience and seniority’’, Unpublished paper, Institute for fiscal studies. Flinn, C.J. (1986), ‘‘Wages and job mobility of young workers’’, Journal of Political Economy, Vol. 94, pp. S88–S110. Heckman, J.J. (1981), ‘‘Heterogeneity and State Dependence’’, in: S. Rosen, editor, Studies in Labor Market, Chicago: University of Chicago Press. Jovanovic, B. (1979), ‘‘Job matching and the theory of turnover’’, Journal of Political Economy, Vol. 87, pp. 972–990. Lazear, E.P. (1995), Personnel Economics, Cambridge, MA: MIT Press. Lillard, L.A. and R.J. Willis (1978), ‘‘Dynamic aspects of earnings mobility’’, Econometrica, Vol. 46, pp. 985–1012. Margolis, D.N. (1996), ‘‘Cohort effects and the return to seniority in france’’, Annales d’Economie etdestatistique, Vol. 41/42, pp. 443–464. Mincer, J. and B. Jovanovic (1981), ‘‘Labor Mobility and Wages, Heterogeneity and State Dependence’’, in: S. Rosen, editor, Studies in Labor Market, Chicago: University of Chicago Press. Mortensen, D.T. (2003), Wage Dispersion, Cambridge, MA: MIT press. Postel-Vinay, F. and J.-M. Robin (2002), ‘‘Equilibrium wage dispersion with worker and employer heterogeneity’’, Econometrica, Vol. 70, pp. 2295–2350. Robin, J.-M. and S. Roux (2002), ‘‘An equilibrium model of the labour market with endogenous capital’’, Annales d’Economie et de Statistique, Vol. 67-68, pp. 257–297. Topel, R.H. (1991), ‘‘Specific capital, mobility, and wages: wages rise with job seniority’’, Journal of Political Economy, Vol. 99, pp. 145–175. Topel, R.H. and M.P. Ward (1992), ‘‘Job mobility and the careers of young men’’, The Quarterly Journal of Economics, Vol. 107(3), pp. 439–479. Woodcock, S.D. (2003), ‘‘Agent heterogeneity and learning: an application to labor markets’’, Chapter 1 of Ph.D. thesis, Cornell University. Appendix A: The likelihood function for the firm-specific model of wages and mobility Consider the log wage equation (17), the starting-wage equation (18), and the firm-specific wage and mobility equations (19) in the text with all definitions associated with those equations. We derive the likelihood for the firm-specific model of wages and mobility in this Appendix. After entry in firm j, and at each value of seniority s(i, t), the worker and firm mutually decide to continue or R terminate the employment relation. The latent variable Rnijt ¼ aR j Z ijt þ nijt operationalizes the condition given by the value function in equation (14). Let t0 be the starting year of the job, a wage rate is observed for s ¼ t t0 40 if and only if the employment relation continues. At date t for a worker with seniority s (after subtracting the effect of the market variables as measured by X it b^ þ y^ i ), the mobility process can be expressed by the equations (19) in the text. Consider the s ¼ 2 example from equations (21). From this structure of correlation,

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multivariate normality implies that 1 1 0 0 R r Zijt0 þ1 nijt0 þ1 s1j2
it0 þ1 j C C B B C B ZW C B
ijt0 þ1 C B ijt0 þ1 C B C C B B r1j r2 j C C Bn B ZR B ijt0 þ2 C ¼ B ijt0 þ2 s2j
ijt0 þ1 s2j
ijt0 þ2 C C C B B C B W C B
ijt0 þ2 C B Zijt0 þ2 C B A A @ @ r2j R
n ijt0 þ3 Zijt0 þ3 s2j ijt0 þ2 0 2 r21j 1 s2 0 0 B 6 j B 6 B 6 0 s2j B 6 0 B 6 B 6 2 r B 6 *N B0; 6 0 0 1 s2j2 j B 6 B 6 B 6 0 0 0 B 6 B 6 @ 4 0 0 0

r21j s2j

0

0

0

0

0

0

s2j

0

0

1

31

r22j s2j

7C 7C 7C 7C 7C 7C 7C 7C 7C 7C 7C 7C 7C 5A

where the vector Z has components with subscripts ijt denoting the individual–employer match and superscripts R or W denoting the equation number (R for the continuation equation; W for the wage equation). This last result is useful for estimation since the likelihood does not involve multiple integration of the normal distribution as shown by R Rnijt0 þ1 ¼ aR j Z ijt0 þ1 þ

log wijt0 þ1

X it0 þ1 b^

log wijt0 þ2

X it0 þ2 b^

r1j
ijt0 þ1 þ ZR ijt0 þ1 40 s2j

W y^ i ¼ aW j X ijt0 þ1 þ
ijt0 þ1 r2j r1j R R Rnijt0 þ2 ¼ aR j Z ijt0 þ2 þ 2
ijt0 þ1 þ 2
ijt0 þ2 þ Zijt0 þ2 40 sj sj W y^ i ¼ aW j X ijt0 þ2 þ
ijt0 þ2 r2j R W Rnijt0 þ3 ¼ aR j Z ijt0 þ3 þ 2
ijt0 þ2 þ Zijt0 þ3 o0 sj

or R Rnijt0 þ1 ¼ aR j Z ijt0 þ1 þ

r1j s2j

h

log wijt0 þ1

y^ i

i W R aW j X ijt0 þ1 þ Zijt0 þ1 40

W W X it0 þ1 b^ y^ i ¼ aW j X ijt0 þ1 þ Zijt0 þ1 h i r2j W R X it0 þ1 b^ y^ i aW ¼ aR j X ijt0 þ1 j Z ijt0 þ2 þ s2 log wijt0 þ1

log wijt0 þ1

Rnijt0 þ2

X it0 þ1 b^

j

Heterogeneity in Firms’ Wages and Mobility Policies

r1j h þ 2 log wijt0 þ2 sj

Rnijt0 þ3

X it0 þ2 b^

y^ i

265

i

W R aW j X ijt0 þ2 þ Zijt0 þ2 40

W W log wijt0 þ2 X it0 þ2 b^ y^ i ¼ aW j X ijt0 þ2 þ Zijt0 þ2 i r2j h W W W R ^ ^ þ Z ¼ aR log w X b y a X ijt0 þ2 it0 þ2 i ijt0 þ3 j j ijt0 þ2 þ Zijt0 þ3 o0 s2j

The contribution to the log-likelihood of this sequence of observations is
 log L Rijt0 þ1 ¼ 1; wijt0 þ1 ; Rijt0 þ2 ¼ 1; wijt0 þ2 ; Rijt0 þ3 ¼ 0 ¼  h i r1j R W W ^ ^ b y þ log F aR Z log w X a X ijt0 þ1 it0 þ1 i ijt0 þ1 j ijt0 þ1 j s2j   W log wijt0 þ1 X it0 þ1 b^ y^ i aW j X ijt0 þ1 þlog j sj h i1 0 r2j R W aR X it0 þ1 b^ y^ i aW j Z ijt0 þ2 þ s2 log wijt0 þ1 j X ijt0 þ1 j C B þlog F@ A h i r1j W X þ s2 log wijt0 þ2 X it0 þ2 b^ y^ i aW j ijt0 þ2 j   W X log wijt0 þ2 X it0 þ1 b^ y^ i aW ijt0 þ2 j þlog j sj   h i r2j R W W ^ ^ þlog 1 F aR b y Z þ log w X a X ijt0 þ2 it0 þ2 i j ijt0 þ3 j ijt0 þ2 s2 j

where Rijt0 þs ¼ 1 when Rnijt0 þs 40: More generally, the log-likelihood for person i who arrived at date t0 in firm j and stayed exactly S periods (i.e. with one entry wage and S 1 observed wages in firm j after this initial date) is
 log L Rijt0 þ1 ¼ 1; wijt0 þ1 ; ::: ; Rijt0 þS ¼ 0 ! i r1j h W W R R ¼ Rijt0 þ1 log F aj Z ijt0 þ1 þ 2 log wijt0 þ1 X it0 þ1 b^ y^ i aj X ijt0 þ1 sj ( ! W S 1 X log wijt0 þs X it0 þs b^ y^ i aW j X ijt0 þs log j þ sj s¼1 h i 19 0 r 2j R ^ y^ i aW X W > b aR log w X Z þ ijt þs it þs 2 = 0 0 j ijt0 þs j ijt0 þsþ1 sj B C þRijt0 þsþ1 log F@ h i A > r W ; þ s1j2 log wijt0 þsþ1 X it0 þsþ1 b^ y^ i aW j X ijt0 þsþ1 j
 þ 1 Rijt0 þS ! i r2j h R R W W aj Z ijt0 þS log wijt0 þS 1 X it0 þS 1 b^ y^ i aj X ijt0 þS 1 nlog F s2j

266

J. M. Abowd, F. Kramarz and S. Roux

Appendix B: Starting-wage equation estimates Table B1 presents the results for the starting-wage equation. We present only the coefficients and not the standard errors, which are not directly delivered by the Abowd et al. (2003), estimation technique. Standard errors could be obtained by subtracting the estimated person and firm effects from the wage and rerunning the regression on the observed characteristics contained in this table. Appendix C: Estimation of the corrected covariance matrix The principal component analysis requires the covariance matrix of the observed variables. The problem here is that these variables are not directly observed but estimated at the firm-level. We thus need to correct the covariance matrix for the measurement errors. Table B.1.

Entry wage equation Coefficients Male

Experience Experience**2 Experience**3 Experience**4 Year ¼ 1977 Year ¼ 1978 Year ¼ 1979 Year ¼ 1980 Year ¼ 1982 Year ¼ 1984 Year ¼ 1985 Year ¼ 1986 Year ¼ 1987 Year ¼ 1988 Year ¼ 1989 Year ¼ 1991 Year ¼ 1992 Year ¼ 1993 Year ¼ 1994 Year ¼ 1995 Year ¼ 1996 Region ¼ lle de France Full-time ¼ yes First job Second job Third job Fourth job or more

0.1024 0.5612 0.1436 0.0138 0.1545 0.1293 0.1454 0.1742 0.1987 0.1658 0.1823 0.1915 0.1967 0.2230 0.1955 0.1581 0.1715 0.2309 0.3225 0.3448 0.3691 0.0583 0.7893 0.0987 0.1464 0.1732 0.2038

Coefficients Female 0.0667 0.3347 0.0891 0.0090 0.1381 0.0976 0.1201 0.1670 0.1693 0.1476 0.1668 0.2040 0.1989 0.2296 0.2135 0.1952 0.2175 0.2543 0.3222 0.3404 0.3859 0.0656 0.7526 0.0780 0.1268 0.1619 0.2036

Note: DADS, 4,616,093 observations. The regression also includes a person and a firm effect. Estimated by a conjugate gradient algorithm. See Abowd et al. (2003).

Heterogeneity in Firms’ Wages and Mobility Policies

267

X is a set of parameters that comes from a first step equation and, therefore, is measured with error following X^ ¼ X þ n

in which ni ! N ð0; Si Þ. We know the probability distribution of n for each observation, since the first-step estimation delivered a variance-covariance matrix for each firm (set of parameters). Furthermore, 0 X^ i X^ i ¼ ðX i þ ni Þ0 ðX i þ ni Þ ¼ X 0i X i þ X 0i ni þ n0i X i þ n0i ni

By taking the average over the observations, the above implies 1 X ^0 ^ 1X 0 X iX i ¼ X i X i þ X 0i ni þ n0i X i þ n0i ni M X^ X^ ¼ N N Then, by noting that Xi and ni are uncorrelated among themselves, we see that the second and third components of the above equality tend to zero. An empirical counterpart for the last component is needed. Even though we do not know ni, we know its law. We estimate the mean of the variance of the residuals by its empirical counterpart 1X 0 1X ni ni ! Si N N Hence, an estimator of the true covariance matrix S is X 1X ^ ð1Þ Si ¼ M X^ X^ N Unfortunately, the above estimators pose practical problems because some ð1Þ estimates of X^ are too imprecise. Those X^ that are the least precise make S^ not being positive. One way to address this difficulty is by weighting the estimator presented above by the inverse of the variance of the X^ i . In practice, it is much easier to use the inverse of the trace of the variance-covariance matrix estimated at the firm-level as a weight for each observation. Therefore, we have the following estimator for the covariance matrix that will be used for the principal components analysis:  X X 1  0 1 ^ ð2Þ ^ X^ i Si ¼ X i trSi N P1 1 trSi

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268

CHAPTER 12

The Empirical Content of the Job Search Model: Labor Mobility and Wage Distributions in Europe and the U.S.$ Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin Abstract Job search models of the labor market hypothesize a very tight correspondence between the determinants of labor turnover and individual wage dynamics on one hand, and the determinants of wage dispersion on the other. This paper offers a systematic examination of whether this correspondence is present in the data by estimating a rudimentary partial equilibrium job search model on a three-year panel of individual worker data covering 10 European countries and the U.S. We find that our basic job search model fits the data surprisingly well. This also allows us to point at a number of interesting empirical regularities about wage distributions. Our results suggest that cross-sectional data on individual wages contain the basic information needed to obtain a reliable measure of the ‘‘magnitude of labor market frictions’’, as measured by a parameter of the canonical job search model. Finally, we use our results in a cross-country comparison of the intensity and nature of job-to-job turnover. We arrange countries into two different groups according to their turnover intensity. We further show that the nature of job-to-job turnover is very different between those two groups: turnover is predominantly voluntary in low-turnover countries, whereas it is to a large extent involuntary in high-turnover countries. Keywords: labor market frictions, wage distributions, wage dynamics, job mobility JEL classifications: J64, J31

$

A preliminary version of this paper was circulated under the title: ‘‘Wage Distributions and Wage Dynamics in Europe and the U.S.: Lessons from a Simple Job Search Model’’. Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75012-4

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

269

270

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

1. Introduction In their review of the job search literature, Mortensen and Pissarides (1999) present job and worker flows, together with wage dispersion, as the two main empirical phenomena making the search framework relevant for labor market analysis. Although the job search literature offers numerous and varied sets of assumptions under which to look at these phenomena, a close correspondence between the determinants of labor turnover and wage mobility on one hand, and the determinants of cross-sectional wage distributions on the other, is inherent to the basic structure of most job search models. The main objective of this paper is to closely and systematically scrutinize the empirical validity of that correspondence. The general intuition behind it is that in a typical frictional labor market the degree of competition between firms for workers is inversely related to the extent of frictions limiting the ability of workers to find new job opportunities (matching inefficiencies). As a corollary, the cross-sectional distributions of wages contain information on the dynamics of individual trajectories. The complementary observation of individual worker movements should therefore be a source of overidentification which in principle would allow specification testing. Pursuing this idea, we consider a prototypical stationary job search model which encompasses the structures of many of the labor market models subject to informational frictions restricting the employer–employee match possibilities that one can find in the search literature. We use data from a panel of 10 European countries and the U.S. to test for the overidentifying restrictions implied by the stationary search model.1 Our European data comes from the European Community Household Panel (ECHP), while our source for the U.S. is the Panel Study of Income Dynamics (PSID). This allows a number of interesting intercountry comparisons. We find that labor markets of English-speaking countries and Denmark are the most flexible, job durations being shorter on average there than in other ‘‘continentalEuropean’’ countries. There is evidence that mobility is more likely to be driven by voluntary quits where mobility is less frequent. Finally, among the countries where mobility is more often constrained (as opposed to freely chosen), Great Britain and Ireland distinguish themselves from Italy, Spain and Portugal by having substantially shorter nonemployment spell durations. We then perform a series of (formal and informal) goodness-of-fit tests. Here we find that, stylized though it may be, the basic job search model that we use is by and large successful when confronting those fit tests. More precisely, the

1

Most search models in the literature make a steady-state assumption. Remarkable exceptions are Van den Berg (1990) and Burdett and Coles (2003).

The Empirical Content of the Job Search Model

271

model is remarkably good at replicating wage distributions on one hand, and average transition rates between employment states on the other. The conclusion about its ability to replicate job and nonemployment spell duration data is somewhat more mitigated. Yet overall, given the model’s parsimony, our results lead us to advocate job search models as simple, tractable and useful tools to describe typical labor force survey data. We continue by conducting an in-depth analysis of the sources of identification of the parameters of the model. Here our main finding is that, somewhat surprisingly, the suspected overidentification lying in the joint observation of wage and worker mobility data is in fact not there. Specifically, it takes both types of data to get separate identification of all the model’s parameters. Roughly, as a simple intuition would otherwise suggest, worker turnover data alone identify parameters measuring the frequency of individual transitions between employment states, whereas wage data are needed to infer the nature (voluntary or not) of these transitions. While attempts at estimating job search model (or extensions thereof) on single-country data are many, systematic cross-country studies are very few. In fact, as far as we are aware, the only contribution explicitly aimed at comparing estimates of the job search model across several countries is Ridder and Van den Berg (2003). Our paper differs from theirs in several respects, though. First, they use nonhomogenized data from five different countries (four European plus the U.S.), whereas we use homogenized data from 10 European countries, which we supplement with ‘‘similar’’ U.S. data. Second, we use an extension of their model that allows for job-to-job transitions associated with wage cuts. Ultimately, their scope is different from ours in that their main goal is to come up with a method for measuring the extent of labor market frictions using readily available, ‘‘macro’’ data and the structure provided by the Burdett and Mortensen (1998) model, whereas we want to go into systematic testing of the structure of the job search model that we use. The paper is organized as follows: Section 2 presents the contents of our analysis sample in the form of a collection of facts about labor turnover and wage distributions. Section 3 then presents the simple partial equilibrium job search model that is to be estimated. Section 4 explains the baseline estimation method of our structural model, shows parameter estimates and compares the extent and nature of search frictions across countries. Section 5 is devoted to a meticulous analysis of the capacity of the structural model to fit various aspects of the data. Section 6 addresses identification issues. Finally, Section 7 concludes.

2. Facts about worker turnover and wages In this section, we emphasize a number of salient facts about worker turnover and wages in modern labor markets. To this end, we conduct a simple

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Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

descriptive analysis of a multi-country sample of individual worker panel data (the precise construction of which is presented in the appendix). We begin by pointing out these facts, first because they are interesting in their own right, and second because they will serve as a guide for the construction of a simple aggregate model of the labor market – which obviously has to be able to replicate those facts – in later sections.

2.1. A brief description of the sample The analysis sample consists of a cohort of male and female workers between 20 and 50 years of age from 11 countries: Belgium (BEL), Denmark (DNK), Spain (ESP), France (FRA), Great Britain (GBR), Germany (GER), Ireland (IRL), Italy (ITA), the Netherlands (NLD), Portugal (PRT) and the U.S. (USA). The European data is taken from the ECHP survey, and the U.S. data is from the PSID.2 We select workers who are found to be either not working (i.e. nonemployed) or working more than 15 h per week in paid employment and in the private sector with nonzero income from work3 at the time of their initial interview.4 We follow those individuals for up to three years or until their first change of status in the labor market which can either correspond to a job-to-job, a job-to-nonemployment or a nonemployment-to-employment transition. We thus observe the worker’s status (employed or nonemployed) at the initial observation date t ¼ 0, a (job or nonemployment) spell duration, the wage at t ¼ 0, a censoring indicator (if the individual experiences no transition before leaving the panel or before the end of the three-year observation window), a transition indicator (which can take on three values: job-to-job, job-to-nonemployment and nonemployment-to-job) and a new wage if the individual has moved to a job. Our sample contains the basic information that can be found in a typical labor force survey. A quick statistical description of that information is available from Table 1. All rows but the last show statistics on workers who are employed at t ¼ 0. We first give the number of employed workers observed, then the proportion of job spells that are censored or end with a job-to-job or a

2

See the appendix for a more detailed description of the sample. Let us mention here that the European data is ex-ante homogenized by a common questionnaire. Therefore, our European sample is as good as it gets in terms of international comparability. The American PSID data is obviously less comparable. Yet the ECHP was constructed in a similar spirit to the PSID. Again, see the appendix for a more detailed discussion of these issues. 3 We use the net hourly wage as the income variable. 4 The corresponding year is 1994 for the ECHP data, and 1993 for the PSID. Due to the start- and end-dates of the two source panels, one cannot construct two perfectly overlapping three-year subpanels. This is one of the reasons for choosing to follow workers for no longer than three years. See the appendix.

The Empirical Content of the Job Search Model

Table 1. Country

273

Descriptive statistics

BEL DNK ESP FRA GBR GER IRL ITA NLD PRT

# Employed workers 756 893 1737 2339 1602 2066 799 1923 1811 1756 % of censored spells 83.5 67.6 70.1 89.5 58.6 78.5 66.5 80.2 79.1 76.2 % of job-to-nonemployment transitions 9.8 12.3 22.5 4.0 16.5 11.2 17.0 14.1 8.8 15.2 % of job-to-job transitions 6.8 20.0 7.4 6.5 24.9 10.3 16.5 5.7 12.2 8.6 % of instantaneous job reaccessions 42.1 63.0 25.4 62.6 61.6 48.0 50.4 29.1 58.5 36.9 % of job-to-job transitions with ay yWage increase 62.8 59.8 57.4 51.3 64.4 60.4 65.2 58.7 66.4 60.9 yWage decrease 19.6 34.6 28.7 34.9 31.3 36.3 20.5 27.5 29.6 18.5 # Job entrants 246 342 779 486 403 771 288 466 707 498

job-to-nonemployment transition.5 Among job-to-job transitions, we show the share of wage increases and wage cuts. These two numbers do not add up to 100%, because the wage corresponding to the second job is missing in some cases. Finally, the last row in Table 1 gives the number of job entrants in each country, a category of workers that we define in Section 2.3. The remainder of this section is devoted to establishing a few stylized facts based on the information summarized in Table 1.

2.2. Worker turnover As row 2 in Table 1 shows,6 many – most, in fact – of the workers we initially observe in employment are still in their initial job spell by the end of the threeyear period. Two countries distinguish themselves particularly: France as a very ‘‘static’’ labor market and Great Britain as a very ‘‘dynamic’’ one. The observed proportions of job spells ending with a job-to-job (respective job-to-nonemployment) transition within the three-year observation window is an indicator of the intensity of job-to-job (respective job-to-nonemployment) reallocation. The fourth row of Table 1 shows that one can divide our set of countries into a clearly ‘‘high job-to-job turnover’’ category which comprises Denmark and the U.K., a clearly ‘‘low-turnover’’ group with Belgium, France, Italy, Portugal and Spain, and finally a ‘‘middle-range’’ group, with Germany, the Netherlands, the U.S. and Ireland – the latter two countries being closest to the ‘‘high-turnover’’ category. While it is not exactly obvious where the dividing

5

See the appendix for a precise statistical definition of what we mean by a job-to-job vs. a job-tononemployment transition. Roughly, a job-to-job transition is defined as a job change for which the individual declares no intervening unemployment spell between the ending date of the first job and the starting date of the second one. 6 Figures 6 and 7 are graphical representations of the country ordering induced by rows 2–7 of Table 1. Looking at these figures makes the classification of countries into ‘‘low-’’, ‘‘middle-’’ and ‘‘high’’turnover groups clearer.

274

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

line between the high- and low-turnover groups should be drawn, the striking fact is that there is a three- to four-fold increase in our job-to-job turnover indicator from one end of the spectrum to the other. In other words, the intensity of job-to-job worker turnover varies widely across countries. One can take a similar look at job-to-nonemployment transitions (Table 1, row 3). Interestingly, there does not seem to be a strong correlation across countries between these job loss rates and the job-to-job turnover indicators. In fact, contrary to the job-to-job turnover rate, the job loss rate as it is computed in Table 1 exhibits little cross-country variation: it lies roughly between 9% and 15% in all countries, save for France (where it is noticeably low at 4%) and Spain (where it is noticeably high at 23%). These three indicators are average turnover indicators in the sense that they average worker mobility over a discrete period of time (three years). In order to get a sense of instantaneous turnover, we count the number of transition between two consecutive jobs with an observed duration of one month or less and for which the interviewee reports that the second job was not preceded by a period of nonemployment.7 The ratio of this count to the number of job spells ending before the end of the observation period provides an estimate of the probability that a job spell be immediately followed by another job. Row 5 of Table 1 shows the results. Anglo-Saxon countries contrast with Latin countries where nonemployment is clearly more frequent as a destination. However, France seems to play a very solitary game as it turns up in the group of countries where nonemployment is least likely as a destination. Mobility is a rare event in France, but when it occurs it is very likely to be a job-to-job movement. Another interesting feature of the process governing worker turnover appears on Figure 1, which plots the nonparametric Kaplan–Meier estimates of the job spell hazard rates, together with a smoothed version of this estimator obtained by locally weighted regression. Given the scarcity of uncensored job spells, those estimates are somewhat imprecise. Yet, the impression that they give is that of a small amount of negative duration dependence in most countries: at this level of aggregation, it seems that workers with longer job tenure are a bit less likely to have their jobs terminated at any given point in time. Figure 2 brings up a final observation about worker turnover. It plots nonparametric (Kaplan–Meier) estimates of the job re-accession rates after a job separation. For the construction of this graph, we take all job separations that occur in our sample and simply ‘‘count’’ the number of job re-accessions at all durations (at a monthly frequency). Similar patterns of duration dependence are observed in all countries: job re-accession rates are high at very short durations (zero and one month), then abruptly drop at two months to remain roughly

7

In the U.S., the only information that we have is through a monthly calendar of activities. We therefore retain job-to-job transitions with no intervening nonemployment period (which, given the structure of the calendar of activities, can hide nonemployement spells of less than three weeks).

275

The Empirical Content of the Job Search Model

Figure 1.

Job hazard rates

BEL

DNK

ESP

FRA

GBR

GER

IRL

ITA

NLD

PRT

USA

.03 .02 .01 0

.03 .02 .01 0 0

10

20

30

40

.03 .02 .01 0 0

10

20

30

40

0

10

20

30

40

0

10

20

30

lowess KM

40

Durations (months)

Figure 2.

Job reaccession hazard rates

BEL

DNK

ESP

FRA

GBR

GER

IRL

ITA

NLD

PRT

USA

.6 .4 .2 0

.6 .4 .2 0 0

.6 .4 .2 0 0

10

20

30

40

0

10

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30

40

0

10

20

30

Non employment durations (months)

40

10

20

30

40

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Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

constant at all longer durations. Surely, many of the quick job re-accessions at very short durations correspond to voluntary job changes (where by ‘‘voluntary’’ we mean that it is the result of an unconstrained choice of the worker). Yet some of them are likely to reflect involuntary reallocation – essentially job losses followed by the immediate finding of a replacement job. 2.3. Wages Rows 6 and 7 in Table 1 report the proportions among job-to-job transitions that are associated with a wage increase (respectively a wage cut). The most obvious striking fact here is that a very substantial share (between 25% and 40%, with substantial variation across countries) of job-to-job transitions are associated with wage cuts. One can think of many reasons why a job change can be associated with a wage cut.8 We shall go into a more detailed theoretical discussion of this issue in the following sections, yet those numbers suggest that not all job-to-job transitions are a ‘‘positive’’ event from the workers’ viewpoint. This reinforces our earlier conjecture that some of the observed quick job re-accessions – which in many cases will be recorded as job-to-job transitions – are in fact involuntary job changes. Another well-documented fact about wages is that more senior or more experienced workers tend to earn higher wages than their junior/less-experienced counterparts. This broad kind of phenomenon can be illustrated using the information that we have in our sample by comparing the distribution of wages in the whole population of employed workers to the distribution of wages among ‘‘job entrants’’ (the distribution of entry wages, for brevity). We define job entrants as workers who were just hired after a period of nonemployment. In practice, there are two types of workers that we consider to be job entrants. First, initially nonemployed workers can be followed until they first get a job – those are job entrants by definition. Second, in order to increase the per-country number of observations on which to base our nonparametric estimate of the distribution of entry wages, we also consider workers that are employed at the time of their first interview but who report that their job only started a short while ago – six months in practice – and that they were nonemployed prior to holding this job. By shedding those latter workers into the category of job entrants, we obtain a reasonable number of observed wage draws from the distribution of entry wages in each country. This number is reported in the bottom row of Table 1. Figure 3a is a by-country plot of the two distributions (cdfs) of wages among job entrants (dashed line) and in the initial cross section of all employed workers (solid line). The striking fact appearing in Figure 3a is that the distribution of wages

8

Those wage cuts are often substantial. The percentage of job-to-job transitions with a wage cut of more than 10% varies from country to country between 10% and 20% (with a high 29% in France), and the share of such transitions with a wage cut exceeding 20% varies between 7% and 20%.

277

The Empirical Content of the Job Search Model

Figure 3.

(a) Wage cumulative distributions; (b) Wage densities

BEL

DNK

ESP

FRA

1 .5 0 100 200 300 400

500

40

60

GBR

80 100 120

0

2000

GER

0

50

100

IRL

150

200

ITA

1 .5 0 0

5

10

15

0

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20

NLD

30

40

0

5

PRT

10

15

5

10

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20

USA

1 .5 employed workers job entrants

0 0

10

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0

500

1500 0

50

Wage (local currency)

(a)

0

0

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200

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600

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0 0

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0 10

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0 0

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0

0 0

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0

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0

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0

DNK

0

BEL

(b)

1000

1000

0

50

employed workers job entrants

Wage (local currency)

in the population of workers as a whole systematically first-order stochastically dominates the distribution of entry wages. This is a particular materialization of the broad idea of positive returns to seniority. A second observation about Figure 3a is that the extent to which the cross-sectional wage distribution dominates the

278

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

distribution of entry wages – as measured by the horizontal distance between the two cdfs at various quantiles – varies across countries. Figure 3b displays nonparametric kernel density estimates. The densities of the wage distributions among job entrants and among all employees are both positively skewed (long tail in the positive direction). The density for all employees is located to the right of the density for job entrants, as expected given the relative positions of the cdfs. More interestingly, the distribution of wage offers is systematically less dispersed than the distribution of wages among all employees and is more positively skewed. 2.4. Summary A successful formal description of worker turnover and individual wage dynamics should thus be able to account for the following broad facts: (1) Workers transit from job to job or in and out of employment (2) Most job-to-job transitions are associated with a wage increase, yet a sizeable fraction of those transitions are still associated with a wage cut. (3) Job separation hazards exhibit (slightly) negative duration dependence. (4) Job re-accession hazards after a job separation are high during the first few weeks, then drop to a lower value and remain approximately constant at longer durations. (5) Wages are dispersed. Moreover, the distribution of wages in a cross section of employed workers first-order stochastically dominates the distribution of entry wages and is less positively skewed. In the next section, we present a candidate model that accounts for all the above phenomena in a very simple qualitative fashion. We then systematically investigate its ability to quantitatively fit the data. Our candidate model is formally inspired by the theory of job search. 3. A simple model of worker turnover 3.1. The environment The labor market under study has a unit-mass continuum of homogeneous, infinitely lived workers, a fraction u of which are unemployed.9 Time is continuous. Unemployed workers sample job offers sequentially at some

9

In the theory, we only have one nonemployment state, i.e. we do not distinguish between unemployed and out of the labor force. Consistently with the vocabulary used in the previous section, the empirical counterpart of the theoretical state of unemployment is nonemployment (which includes nonparticipation).

The Empirical Content of the Job Search Model

279

exogenous Poisson rate l0>0. We authorize on-the-job search, so that job offers also accrue to employed workers at a rate l1>0. Each job is characterized by a constant flow wage w, that the hiring firm is committed to pay until the job is terminated. Upon receiving a job offer, a worker draws the associated wage w from a continuous sampling distribution with cumulative function F and density f. Given this environment, workers optimally follow a reservation wage policy. Therefore, an employed worker whose current wage is w and who receives an offer associated with a wage w0 is willing to take the offer if and only if w0 >w, which has probability F ðwÞ ¼ 1 F ðwÞ. We further assume that the unemployment income flow is low enough for all job offers to be accepted by the unemployed.10 The unemployment outflow rate thus equals l0. In addition to receiving outside job offers – which they can either accept or turn down – at rate l1, employed workers face two types of shocks. First, the conventional job destruction shock: at rate d>0, employed workers are hit by a negative productivity shock that makes their job unproductive and forces them back into unemployment. Second, we introduce a ‘‘reallocation shock’’: at rate l2Z0, employed workers receive a job offer with an associated wage drawn from the sampling distribution F, which they cannot reject (i.e. for which the only alternative is to become unemployed, which by assumption is never preferable). When hit by a reallocation shock, an employed worker is thus forced to leave his/her current job for another job, with a wage drawn at random from F. This reallocation shock is formally equivalent to a layoff immediately followed by a job offer. As a matter of structural interpretation, the latter can result from an employer-provided outplacement program, or from the worker’s job search activity during the notice period. In terms of data description, its purpose is to make the model consistent with the observed positive share of job-to-job movers that experience a wage cut while changing jobs (see Table 1 in Section 2) and to the particular nonstationarity pattern previously documented for unemployed workers’ re-employment rates (Figure 2). Note that this reallocation shock is absent from conventional job search models.11

10

This would naturally happen in a homogeneous worker equilibrium search model, where a firm offering a wage strictly below the common reservation wage of unemployed workers would never attract any worker. 11 In fact, our simple setup encompasses the Burdett and Mortensen (1998) job search model as a special case where there are no reallocation shocks (i.e. l2 ¼ 0). We should also mention that such reallocation shocks were considered before in Ridder and Van den Berg (1993, 1997). An observationally similar concept of ‘‘immediate re-employment probability’’ is also explored theoretically and empirically using U.K. data in a recent contribution by Coles and Petrongolo (2003).

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280

3.2. Individual labor market transitions Summing up, unemployed workers exit unemployment at a constant rate of l0, while employed workers can experience three types of mobility: (1) A mobility from employment into unemployment following a layoff (a ‘‘d shock’’). (2) A voluntary mobility from a job into another job following an outside job offer (a ‘‘l1-shock’’). The rate at which this happens to an employed worker at a job paying a wage w is l1 F ðwÞ. Clearly, workers only move up the wage scale when they receive outside job offers. (3) An involuntary mobility from a job into another job following a reallocation shock (a ‘‘l2-shock’’). Reallocation shocks cause job-to-job movements with either a gain or a loss of wage. What the quadruple of parameters (d, l0, l1, l2) thus essentially governs is the frequency and nature of individual labor market transitions. We shall henceforth refer to (d, l0, l1, l2) as the transition parameters. The set of assumptions listed above immediately implies the following:  Hazard rates. The hazard rate for unemployment termination equals l0, and

the hazard rate for the termination of a job with associated wage w equals d þ l2 þ l1 F ðwÞ.  Types of transition. Consider a worker initially employed at a job with wage wi. Conditional on wi and on job termination, J the probability that this worker becomes unemployed equals d ; d þ l2 þ l1 F ðwi Þ J

the probability that she/he becomes employed at a job paying a wage wf>wi is ðl2 þ l1 Þf ðwf Þ ; d þ l2 þ l1 F ðwi Þ

J

the probability that she/he becomes employed at a job paying a wage wfowi is l2 f ðwf Þ , d þ l2 þ l1 F ðwi Þ

since this can only happen following a l2-shock.

The Empirical Content of the Job Search Model

281

3.3. Stationary worker flows and stocks From now on, we assume that the labor market is in a steady state.12 The steadystate assumption implies a series of flow-balance equations, from which various stocks and distributions of interest for the empirical analysis can be derived. Starting with the balance of unemployment in- and outflows, we get: l0 u ¼ dð1

uÞ

3

u¼

d . d þ l0

ð1Þ

The LHS in (1) is the unemployment outflow, which equals the measure of unemployed workers times the offer arrival rate l0 (recalling that the acceptance rate of offers by unemployed workers equals 1). The RHS in (1) is the unemployment inflow, given by the layoff rate d times the measure of employed workers, 1 u. We now consider the distribution of wages in a cross-section of employed workers. Let G denote its cdf and g its density. The stock of employed workers earning w or less is thus (1 u)G(w). Workers leave this stock either because they are laid off (which happens at rate d), or because they receive an outside offer of a job with associated wage greater than w (which happens at rate l1 F ðwÞ), or finally because they are hit by a reallocation shock, but are lucky enough to draw a wage greater than w (this last event occurs at rate l2 F ðwÞ). On the other hand, workers enter the stock (1 u)G(w) either because they were unemployed and got an offer with a wage draw below w (the measure of such entrants is l0uF(w)) or because they were employed and earning a wage greater than w, were hit by a reallocation shock and drew a replacement job associated with a wage below w (the measure of such entrants is l2(1 u)[1 G(w)]F(w)). Constancy of the stock (1 u)G(w) thus implies:
 d þ l1 F ðwÞ þ l2 F ðwÞ ð1 uÞG ðwÞ ¼ l0 uF ðwÞ þ l2 ð1 uÞ½1 G ðwފF ðwÞ, ð2Þ

which, together with (1), implies the following relationship between F and G: GðwÞ ¼

F ðwÞ 1 þ kF ðwÞ

gðwÞ ¼


3 F ðwÞ ¼

1þk 2 f ðwÞ 3 1 þ kF ðwÞ

ð1 þ kÞGðwÞ , 1 þ kGðwÞ

f ðwÞ ¼

1þk gðwÞ, ½1 þ kGðwފ2

ð3Þ ð4Þ

where k ¼ l1/(d+l2). Obviously, G and F have equal support. Looking at (3), one sees that the combination of parameters k ¼ l1/(d+l2) seems to play a particular role. This ratio has a simple interpretation: it is the

12

Conventional though it may be, this is an obviously strong assumption. One of our empirical goals in this paper is to assess its validity.

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282

average number of job offers that a worker receives between two ‘‘adverse’’ shocks, an adverse shock being either a layoff (d) or a reallocation shock (l2). In other words, an adverse shock is defined as an event that forces the worker to move (either to unemployment or to a different job) whether she/he likes it or not. Now going back to (3), a straightforward manipulation shows that k

F ðwÞ GðwÞ . GðwÞF ðwÞ

ð5Þ

Hence, k is also a measure of the extent to which G first-order stochastically dominates the sampling distribution F. It can thus be seen as a summary measure of the competitive forces that put upward pressure on the workers’ wages.13 If k tends to zero, then G becomes confounded with F, meaning that employed workers never get higher wages than what firms are willing to offer to them. Conversely, as k becomes large, then the distribution G becomes more and more concentrated at high wages. In the limit where k tends to infinity, employed workers tend to move immediately to the highest-paying job or firm in the market: the labor market becomes Walrasian. We will pay particular attention to this ratio k when we get to the discussion of our estimation results. We shall therefore keep in mind this last interpretation of k and use it as our ‘‘summary index of labor market frictions’’.14 3.4. Discussion The model outlined in this section is qualitatively consistent with the set of facts presented in Section 2. First, by assumption, workers experience transitions from job to job or in and out of employment at discrete, random intervals. Second, some job-to-job transitions – those caused by a l2-shock and followed by an ‘‘unlucky’’ wage draw – are associated with a wage cut. Third, job spell hazard rates are declining with tenure, as workers with longer tenure tend to be those holding better-paying jobs and are therefore less likely to receive attractive outside offers.15 Fourth, nonemployment hazard rates are constant (equal to l0) after an initial peak at very short durations corresponding to job-to-job transitions – caused by either a l1- or a l2-shock. Finally, the distribution of entry wages, which exactly corresponds to the sampling distribution F in the model, is first-order stochastically dominated by the cross-sectional wage distribution G, as Equations (3) and (5) show.

13 A similar insight is developed by Christensen et al. (2005). The two differences between (5) and their Equation (9) are their consideration of an endogenous search effort and our modeling of reallocation shocks. 14 The empirical job search literature generally uses l1/d as an index of labor market frictions (see, e.g. Ridder and Van den Berg, 2003). Our index k ¼ l1/(d+l2) simply generalizes this approach. 15 Formally, they have a smaller value of the l1 F ðwÞ term in their job hazard rate. See Section 5.2 for a more formal analysis of negative duration dependence.

The Empirical Content of the Job Search Model

283

4. Structural estimation The aim of this section is to estimate the parameter vector y ¼ (d, l0, l1, l2) of the structural model described in Section 3. We use the estimation technique of Bontemps et al. (2000), who treat the distribution of wages among employees, G, as a nuisance parameter which can be nonparametrically estimated beforehand. The distribution of wage offers, F, will then be deduced from G using the steadystate restriction (3). 4.1. Estimation procedure We estimate the model using all the structural restrictions implied by the steadystate hypothesis. This includes in particular the relationships between sampling and cross-sectional wage distributions implied by (3). We shall refer to the resulting estimator of the model parameters as the structural or constrained estimator, and denote it by yc ¼ ðdc ; lc0 ; lc1 ; lc2 Þ. For any given country in our sample, the data is a set of N workers who are initially either employed or nonemployed, and whom we follow over time until the end of their first observed (job or nonemployment) spell. In order to clearly exhibit the sources of identification in this constrained estimation, we now spell out the individual likelihood contributions. Here, a typical observation for a worker i ¼ 1, y , N is a vector xi ¼ ðe0i ; w0i ; t0i ; cs0i ; e1i ; w1i Þ,

ð6Þ

where     



e0i is the worker’s initial state (e0i ¼ 1 if employed at t ¼ 0, and ¼ 0 otherwise), t0i the worker’s observed spell duration (t0i ¼ T if spell is right-censored), w0i the worker’s initial wage (available only if e0i ¼ 1), cs0i a censoring indicator of the worker’s spell (cs0i ¼ 1 if spell is rightcensored), e1i indicates worker i’s employment state in his second observed spell (obviously, this is only available if cs0i ¼ 0, i.e. the first observed spell is uncensored),16 and w1i the worker’s wage observed after his/her first transition (which can be either job-to-job or nonemployment-to-job, depending on the initial state e0i).

16 Note that when e0i ¼ 1 and cs0i ¼ 0 (i.e. the individual’s first spell is an uncensored job spell), then e1i is an indicator of job-to-job or job-to-nonemployment transitions, e1i ¼ 1 meaning job-to-job transition.

284

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

Conditional on initial state e0i and wage w0i, the contribution of worker i to the sample likelihood is given by: h ie0i cs0i  ‘ðxi je0i ; w0i ; y; F Þ ¼ e ½dþl2 þl1 F ðw0i ފt0i h
ie0i ð1 cs0i Þ    d þ l2 þ l1 F ðw0i Þ e ½dþl2 þl1 F ðw0i ފt0i  e0i ð1 cs0i Þð1 e1i Þ d  d þ l2 þ l1 F ðw0i Þ  e0i ð1 cs0i Þe1i l2 þ l1 1fw1i  w0i g  f ðw1i Þ d þ l2 þ l1 F ðw0i Þ
l t ð1 e0i Þcs0i
ð1 e0i Þð1 cs0i Þ  e 0 0i  l0 e l0 t0i f ðw1i Þ ,

ð7Þ

where 1{
} designates the logical indicator function. The first line of (7) is the likelihood of the worker’s job spell duration t0i conditional on their wage w0i. Note that the possible right-censoring of the spell is accounted for (cs0i ¼ 1). The second line is the probability of the destination state given that a transition occurs: Conditional on not being censored the job spell can end with a job-to-nonemployment transition (e1i ¼ 0) or a job-to-job transition (e1i ¼ 1). In the event of a job-to-job transition, a second wage w1i is observed which conveys information about the cause of the job-to-job transition: if w1iow0i, then the transition was involuntary – i.e. caused by a l2-shock – for sure; in the opposite case (w1i>w0i), the cause of the transition cannot be inferred and is either a l1- or a l2-shock. Finally, the third line of (7) concerns initially nonemployed workers: it contains the joint likelihood of the (possibly censored, cs0i ¼ 1) unemployment spell duration t0i and the accepted wage w1i when a transition is observed. The individual likelihood contribution (7) involves the vector of transition parameters as well as the distributions G and F. However, these two distributions are interrelated through the structural relationships (3) and (4). In other words, we really need to observe only one of those two distributions in order to compute (7). In practice, we estimate G nonparametrically by the empirical cdf of wages in the population of initially employed workers: N 1 X ^ ½e0i 1fw0i  wgŠ, GðwÞ ¼ N G i¼1

ð8Þ

where N G ¼ SN i¼1 e0i is the number of individuals employed at t ¼ 0. Then, we replace F and f in (7) by the following expressions: ^ ^  ð1 þ kÞGðwÞ , F ðwjk; GÞ ^ 1 þ kGðwÞ

ð9Þ

The Empirical Content of the Job Search Model

1þk ^ f ðwjk; GÞ  h i2 gðwÞ, ^ iÞ 1 þ kGðw

285

ð10Þ

for k ¼ l1/(d+l2).17 We then obtain our baseline set of parameter estimates yc ¼ ðdc ; lc0 ; lc1 ; lc2 Þ by maximizing the sample log-likelihood function Lc ðyÞ ¼ ^ SN i¼1 ln lðxi je0i ; w0i ; y; F ð
jk; GÞÞ separately for each country. 4.2. Results The results are gathered in the first four rows of Table 2, which contain, in addition to yc (durations being measured in months) the ‘‘summary index of search frictions’’ kc ¼ lc1 =ðdc þ lc2 Þ. All parameters are precisely estimated and one sees that the estimated values of all parameters vary substantially across countries, thus suggesting that labor market frictions differ in both intensity and nature from one country to another. For a better understanding of what these numbers mean, we construct the following functions of the parameters. Transition rate parameters determine both spell durations and the relative probabilities of transiting toward such or such particular labor market state. We thus compute average job duration as in the formula: Z dGðwÞ d þ l2 þ l1 =2 JobDur ¼ , ð11Þ ¼ d þ l2 þ l1 F ðwÞ ðd þ l2 Þðd þ l2 þ l1 Þ

dþl2 þl1 GðwÞ 1 where Equation (3) was used to substitute ðdþl for dþl þl  . The 2 Þðdþl2 þl1 Þ 2 1 F ðwÞ different transition probabilities at the end of a job (i.e. conditional on job termination) are: Z d dGðwÞ Prfnonemploymentjtransitiong ¼ ¼ d
JobDur, ð12Þ d þ l2 þ l1 F ðwÞ Z l2 dGðwÞ ¼ l2
JobDur, ð13Þ Prfreallocation shockjtransitiong ¼ d þ l2 þ l1 F ðwÞ

and

Prfvoluntary mobilityjtransitiong ¼

Z

¼1

l1 F ðwÞ dGðwÞ d þ l2 þ l1 F ðwÞ ðd þ l2 Þ
JobDur ¼

l1 =2 .ð14Þ d þ l2 þ l1

17 The implicit assumption made in the sequel is that we can measure G(
) without error. Otherwise stated, the standard errors on our various estimators shown below do not account for the presence of ^ ^ a nuisance variable Gð
Þ. Also note that the density gðwÞ only appears in the expression of f(w|k,G) in ^ a multiplicatively separable way. Since gðwÞ is independent of the parameters, we can thus ignore it in our likelihood maximization.

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286

Table 2. Country c

d

lc1 lc2 kc lc0

Constrained model estimates ðper annumÞ

BEL

DNK

ESP

FRA

GBR

GER

IRL

ITA

NLD

PRT

USA

0.0353 (0.0040) 0.0522 (0.0104) 0.0080 (0.0023) 1.2046 (0.2682) 0.3367 (0.0253)

0.0504 (0.0049) 0.0520 (0.0114) 0.0577 (0.0060) 0.4814 (0.1187) 0.6483 (0.0442)

0.0878 (0.0044) 0.0512 (0.0067) 0.0136 (0.0021) 0.5053 (0.0704) 0.5971 (0.0201)

0.0129 (0.0013) 0.0476 (0.0050) 0.0105 (0.0012) 2.0300 (0.2158) 0.5614 (0.0227)

0.0803 (0.0049) 0.0764 (0.0110) 0.0822 (0.0058) 0.4698 (0.0754) 0.7195 (0.0330)

0.0394 (0.0026) 0.0660 (0.0069) 0.0189 (0.0019) 1.1326 (0.1265) 0.7705 (0.0303)

0.0716 (0.0060) 0.0840 (0.0131) 0.0318 (0.0050) 0.8119 (0.1412) 0.4455 (0.0260)

0.0526 (0.0032) 0.0259 (0.0045) 0.0110 (0.0019) 0.4073 (0.0765) 0.4140 (0.0152)

0.0324 (0.0025) 0.0666 (0.0073) 0.0220 (0.0022) 1.2259 (0.1435) 0.4552 (0.0231)

0.0548 (0.0034) 0.0205 (0.0049) 0.0194 (0.0029) 0.2768 (0.0721) 0.6373 (0.0293)

0.0547 (0.0035) 0.1028 (0.0103) 0.0320 (0.0032) 1.1853 (0.1308) 1.7143 (0.0885)

Figure 4 plots the probability of a voluntary mobility (given job termination) as a function of average job durations. First, average job durations vary a lot across the different countries: less than 10 years for the U.K., Denmark, Ireland, the U.S. and Spain, around 10–15 years for Portugal, Italy, Germany and the Netherlands, around 15–20 for Belgium and way more for France where we find that average job duration is somewhere between 25 and 30 years.18 Second, one notices that relative to involuntary mobility (reallocation shocks and layoffs), voluntary mobility is a rather rare event: the probability of voluntary mobility given that a transition occurs varies from a low value of 10–15% (Denmark, Italy, Portugal, Spain and the U.K.) to a high 33% for France, with an intermediate value of 25% for a third group of countries (Belgium, Germany, Ireland, the Netherlands and the U.S.). Nevertheless, the general impression is that of a negative correlation between the extent of job turnover (short job durations) and the relative chances that job mobility be voluntary.19 Third, we observe a very significant and negative correlation between the probability of a nonemployment shock and the probability of a reallocation shock (see Figure 5). Denmark and Britain, in particular, stand out both as

18

A natural question to ask at this point is whether all countries are at the same stage of the business cycle. For a rough assessment, we looked at Hodrick–Prescott-filtered OECD quaterly series of GDP growth for all countries in our sample, bar Ireland and Portugal for which the data are not available prior to 1996. The broad message delivered by those filtered series is that all continental European countries are in the early ascending phase of the business cycle – i.e. within 12–18 months of the last trough – at the beginning of the observation period, while the following peak occurs 12–24 months after the end of the observation period. The U.S. and the U.K. seem to be ahead of continental European countries by a few quarters, with a trough at the beginning of 1991 and a peak in the first quarter of 1997. Remember, however, that our observation window for the U.S. starts a year earlier than the one we have for Europe, so that the only potentially problematic country (as far as differences in business cycle phases go) is the U.K., which is still in an upward phase over the observation period, yet a few quarters ahead of the others. 19 Removing France from the regression reduces the R2 by a large amount, but does not change the slope.

287

The Empirical Content of the Job Search Model

Figure 4.

Probability of voluntary job turnover vs. job duration

.35 FRA

voluntary mobility

.3 NLD GER

USA

BEL

.25 IRL

.2 GBR

.15

ESP DNK ITA PRT

.1 5

10

15

20

25

30

job duration (years)

Figure 5.

The anatomy of job destruction shocks

.5 DNK GBR reallocation shock

.4

.3

FRA

NLD USA GER

IRL

PRT

.2 ITA

BEL

ESP

.1 .4

.5 .6 nonemployment shock

.7

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Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

intense-turnover countries and as countries with high shares of involuntary jobto-job transitions. The group of countries exhibiting a very low rate of voluntary turnover is thus heterogeneous as involuntary mobility is dominated by instantaneous job-to-job reallocation in Denmark and the U.K., whereas it predominantly reflects entry into longer unemployment spells in Italy, Portugal and Spain. 5. Fit In this section, we investigate how the structural model fits the data. 5.1. Transitions across employment states We start this study by looking more specifically at worker flows. There, we distinguish worker turnover averaged over the observation period (three years) from instantaneous turnover. 5.1.1. The intensity of worker turnover Considering a sample of initially employed workers that we follow over an observation period of length T, the predicted share of workers who leave their job during the observation period is given by: Z  C ¼ ð1 e ½dþl2 þl1 F ðwފT Þ dGðwÞ. ð15Þ C is the model-predicted share of completed or uncensored spells, i.e. one minus the share appearing in row 2 of Table 1. The predicted share of workers whose first transition is job-to-job (voluntary or not) and occurs before the end of the observation window is: Z   l2 þ l1 F ðwÞ  J¼ 1 e ½dþl2 þl1 F ðwފT dGðwÞ. ð16Þ d þ l2 þ l1 F ðwÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Prfjob-to-jobjtransition; wg Prftransition occurs before Tjwg This is an indicator of the intensity of job-to-job turnover over T periods. J is designed to be the theoretical counterpart of the data shown in row 4 of Table 1. Lastly, one can define a similar indicator of the job destruction rate corresponding to the theoretical prediction of the numbers contained in row 3 of Table 1, i.e. the share of initially employed workers whose first transition occurs before the end of the three-year observation window and is from job to nonemployment. Letting D denote this indicator, one has: Z d  D¼ ð1 e ½dþl2 þl1 F ðwފT Þ dGðwÞ. ð17Þ  d þ l2 þ l1 F ðwÞ Note in passing that the sum D+J ¼ C.

289

The Empirical Content of the Job Search Model

By substituting (3) in the last series of definitions (15–17), it turns out that one can get a closed-form expression of J, D, and C as functions of the transition parameters (d, l1, l2) alone:20  ðdþl2 þl1 ÞT ðd þ l2 þ l1 Þðd þ l2 Þ 1 e x E 1 ðxÞ C¼T , ð18Þ l1 x ðdþl2 ÞT dT 2 ðd þ l2 þ l1 Þðd þ l2 Þ D¼ l1 2



1

e x

x

e x þ E 1 ðxÞ x

ðdþl2 þl1 ÞT ðdþl2 ÞT

,

ð19Þ

and of course J ¼ C D. As a first test of fit, one can plot these indicators of worker turnover – constructed for each country in the sample using our estimates ðdc ; lc1 ; lc2 Þ – against their empirical counterparts (the Figures in rows 2–4 of Table 1). This is done in Figures 6 and 7. One sees that the model is very good at capturing the intensity of worker flows. In particular, the classification of countries as either ‘‘high-’’, ‘‘intermediate-’’ or ‘‘low-turnover’’ that one could establish from Table 1 is the same as the classification that one would obtain using the predicted indicator J. 5.1.2. Instantaneous transition probabilities If one looks at the formula for J, for example (Equation (16)), one clearly sees that J is the average product of the probability of completing a job spell before the end of the period and of the relative probability of quitting the job voluntarily given that mobility accrues (instantaneous transitions). To analyze the ability of the model to match observed transitions, as they were reported in the fifth row of Table 1, we now compute the unconditional probability (say, j) that a job that has just been terminated be immediately followed by another employment spell: Z l2 þ l1 F ðwÞ l2 d l1 =2 j¼ þ dGðwÞ ¼ . ð20Þ  d þ l d þ l2 þ l1 d þ l d þ l2 þ l1 F ðwÞ 2 2 Figure 8 plots the actual vs. the predicted values. Once again, the fit is remarkably good.

20

For example, C¼w ¼

Z

w

w

ð1

e

½dþl2 þl1 F ðwފT

ðd þ l2 Þðd þ l2 þ l1 ÞT l1

Z

Þ dGðwÞ ðdþl2 þl1 ÞT

ðdþl2 ÞT

ð1

e xÞ

dx x2

after changing w into x ¼ ½d þ l2 þ l1 F ðwފT. This is an exponential R þ1 integral that can be easily computed using tabulated exponential integral functions, E 1 ðxÞ ¼ x e t =t dt.

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

290

Figure 6.

Average job-to-job turnover (J) GBR

.25

Predicted

.2

DNK

IRL

.15

USA GER

.1

ESP BEL FRA ITA

.05 .05

NLD

PRT

.1

.15

.2

.25

Observed

Figure 7.

Average job-to-non-employment turnover (D)

.25 ESP

.2

Predicted

GBR IRL PRT

.15 DNK

ITA

USA

.1

BEL NLD

GER

.05 FRA

.05

.1

.15

.2

.25

Observed

Following a job termination, a worker instantaneously transits to another job with unconditional probability j derived in Equation (20), and enters an unemployment spell of positive duration, from which she/he exits at a constant rate l0, with probability 1 j. It is interesting at this point to look at job

291

The Empirical Content of the Job Search Model

Figure 8.

Instantaneous job reaccession (j)

.6 NLD

FRA DNK GBR

USA GER

Predicted

.5

IRL BEL

.4 PRT

.3

ESP

ITA

.2 .2

.3

.4

.5

.6

Observed

re-accession hazards which were displayed in Figure 2. The predicted job re-accession hazards is equal to j at duration zero, and to l0 at all positive durations. Figure 9 superimposes this predicted hazard (using the constrained estimates yc) and the corresponding Kaplan–Meier estimates from Figure 2. The excellent fit of instantaneous exit probabilities (exit at duration zero) simply confirms what we already showed in Figure 8. At longer durations, one sees that the horizontal lines at lc0 correctly capture average unemployment exit rates. Moreover, a casual look at the various panels of Figure 9 suggests that a constant unemployment hazard rate at unemployment spell durations longer than one month is not too bad an approximation – at least in our analysis sample. 5.2. Job durations We now turn to an analysis of the model’s ability to fit duration data. Let l^ e ðtÞ denote the Kaplan–Meier estimates of the hazard function for job durations that can be smoothed using a local weighted regression. We want to compare l^ e with the corresponding model prediction. Given a set of values of the transition parameters (d, l1, l2), and conditional on a wage w0 at t ¼ 0, we know  that the probability for a job spell to last more than t is e ½dþl2 þl1 F ðw0 ފt . Therefore, the unconditional survival function of employment spells reads: Z  S e ðt; d; l1 ; l2 Þ ¼ e ½dþl2 þl1 F ðw0 ފt dGðw0 Þ. ð21Þ

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

292

Figure 9.

Job reaccession hazard rates

BEL

DNK

ESP

FRA

GBR

GER

IRL

ITA

NLD

PRT

USA

0.6 0.4 0.2 0

0.6 0.4 0.2 0 0

10

20

30

40

0.6 0.4 0.2 0 0

10

20

30

40

0

10

20

30

40

0

10

20

30

40

KM predicted hazard

Non employment durations (months)

Using (3), Se(t) can be expressed as a function of t and the transition parameters:  ðdþl2 þl1 Þt ðd þ l2 þ l1 Þðd þ l2 Þ e x þ E 1 ðxÞ , ð22Þ Se ðt; d; l1 ; l2 Þ ¼ t l1 x ðdþl2 Þt and one deduces hazard rates as: le ðt; d; l1 ; l2 Þ ¼

d ln Se ðt; d; l1 ; l2 Þ . dt

Substituting ðdc ; kc1 ; kc2 Þ in (21), we get our predicted hazard function le ð
; dc ; kc1 ; kc2 Þ, which we can plot as a function of job spell duration together with its empirical counterpart l^ e in order to assess the fit of our model to job duration data. This is done in Figure 10, which simply adds a plot of le ð
; dc ; kc1 ; kc2 Þ on top of the empirical plot already shown in Figure 1. Durations (on the x-axes) are in months. Looking at Figure 10 from a distance, it appears that the various job spell hazard rates are correctly reproduced by our model (with the notable exception of Britain and Spain). Yet, a closer look at those figures reveals that in some cases there seems to be more negative duration dependence in the data than the model can predict. In other words, in a number of countries the Kaplan–Meier estimate of the hazard function, l^ e ðtÞ, looks steeper than the predicted le ðt; dc ; kc1 ; kc2 Þ. The only source of duration dependence in our structural model is the fact that workers get paid different wages, and that lower-paid

293

The Empirical Content of the Job Search Model

Figure 10.

Job hazard rates

BEL

DNK

ESP

FRA

GBR

GER

IRL

ITA

.03 .02 .01 0

.03 .02 .01 0 0 NLD

PRT

10

20

30

40

USA

.03 .02 .01 0 0

10

20

30

40

0

10

20

30

40

0

10

20

30

40

lowess predicted hazard KM

Durations (months)

workers tend to accept outside offers more often; this source of duration dependence turns out to be quantitatively weak. The more important (even though still moderate) amount of duration dependence that we observe in the data can result either from transition rates that are truly nonstationary, or from the fact that more individual heterogeneity is involved in the job search process than just heterogeneity in wages. A promising lead in terms of modeling was opened by Christensen et al. (2005) – even though these authors are not focusing on the issue of duration dependence. It consists of letting heterogeneity in wages spill over to another determinant of worker mobility, namely the arrival rate of offers l1, simply by introducing an endogenous, worker-chosen search intensity. The idea is that if search is costly, then high-wage workers are less induced to search than low-wage workers, given that the returns to search – which are essentially measured by the probability of drawing a higher wage from F – are lower for the former than for the latter. Clearly, this would reinforce the mechanism through which our simpler model already predicts negative duration dependence. 5.3. The sampling distributions of wage offers We then question the quality of the fit of the predicted wage offer distribution ^ to the observed one, F^ . Figure 11a is a by-country plot of the F ð
jkc ; GÞ empirical sampling distribution F^ , which we already constructed in Section 2 (Figure 3a) from the sample of wages among job entrants (solid line), together

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

294

Figure 11.

(a) Wage cumulative distributions; (b) Wage densities

BEL

DNK

ESP

FRA

1 .5 0 100 200 300 400 500

40

60

GBR

80 100 120

0

2000

GER

0

50

100

IRL

150

200

ITA

1 .5 0 0

5

10

15

0

10

20

NLD

30

40

0

5

PRT

10

15

5

10

15

20

USA

1 .5 F predicted 0 0

10

20

30

40

0

500

1500 0

G F

50

Wage (local currency)

(a)

0

0

ESP

200

400

600

0

50

100

150

5

10

15

0

10

20

30

40

20

30

40

0

500

0

50

100 150 200 ITA

5

10

15

0

5

10

15

USA

0 10

0

PRT

0

1000

0

NLD

0

500 IRL

0

0 0

0

GER

0

GBR

0

0

FRA

0

DNK

0

BEL

(b)

1000

1000

0

Wage (local currency)

50

F' predicted G' F'

20

The Empirical Content of the Job Search Model

295

^ The empirical cdf of wages, G, ^ was with the predicted distribution F ð
jk ; GÞ. also put on the graphs (dash-dot line) for comparison. For our results to be more convincing, we also plot the corresponding wage densities in Figure 11b. While f^ and g^ are kernel density estimates of wage densities among job entrants ^ is the structural prediction. and employed workers, respectively, f ð
jkc ; GÞ A glance at Figures 11a and b immediately confirms that the model-predicted wage offer distribution is very close to the empirical one in almost every country. That being said, there are some discrepancies – most obviously in Spain, once again – which can be conveniently described using the terminology introduced by Christensen et al. (2005). These authors consider the horizontal difference between the earnings distributions G and the wage sampling distribution F, which they call the ‘‘employment effect’’ or ‘‘employment premium’’. Comparing the actual and predicted employment effects, one sees that there is a slight general tendency of the model to underpredict this employment effect at high quantiles of the sampling distribution, a result which goes in the same general direction as the findings of Christensen et al. (2005) on Danish data. We see that Spain is the country where the problem of underprediction of the employment effect at high quantiles is worst, followed by France, Ireland and Italy. (In this latter case a close look at the graphs reveals that the predicted distribution tends to dominate the observed one to a small extent throughout the entire support.) ^ is a In spite of those discrepancies, we are tempted to conclude that F ð
jkc ; GÞ ^ good overall predictor of F , which again suggests that the structural relationship (3) is consistent with the data. Moreover, this conclusion takes substantial additional force from the fact that it seems to apply to densities also (see Figure 11b). In other words, our simple search model under the steady-state assumption seems to correctly describe the connection between the wage distribution among job entrants and the wage distribution in the whole population of employed workers. The ‘‘visual’’ impression given by Figures 11a and b can be made clearer by formally testing the equality of the observed and predicted offer distributions. ^ and F^ is The results of a w2 test21 are reported in Table 3. Equality of F ð
jkc ; GÞ strongly rejected in Spain and Italy. It is marginally rejected in France (at the 5% level) and marginally accepted in Ireland. It is accepted everywhere else. Thus, the conclusions that could be drawn from a casual look at Figures 11a and b are by and large confirmed by a formal (w2) test. Overall, at our (high) level of aggregation, it thus seems that just one parameter (k) goes a long way into capturing the observed difference between the distribution of wages among tenured workers (G) and the distribution of wages among job entrants (F). While a number of different theories can certainly account for the stochastic ordering of the two distributions, we want to advocate the job search model as a simple framework in which it can be interpreted. c

21

For the w2 test we use 15 evenly spaced bins in each separate country.

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

296

Table 3. Country

BEL

DNK

^ and F^ ð
Þ w2 test of equality of F ð
jkc ; GÞ ESP

FRA

GBR GER

IRL

ITA

NLD

PRT

USA

Test statistic 15.10 13.53 85.87 23.43 9.35 9.73 22.13 78.33 14.24 20.63 18.19 p-Value 0.301 0.408 0.000 0.037 0.746 0.716 0.053 0.000 0.357 0.081 0.151

5.4. Wage mobility We conclude this fit analysis with a quick look at the model’s performance at predicting wage mobility. In order to reproduce the numbers in rows 6 and 7 of Table 1, one constructs the share of upward job-to-job turnover, J+, as:22 Z w  1 ðl2 þ l1 ÞF ðwÞ   þ 1 e ½dþl2 þl1 F ðwފT dGðwÞ J ¼
J w d þ l2 þ l1 F ðwÞ  1 l2 þ l1 l2 D J ¼ J l1 d

and the associated share of downward job-to-job turnover as J ¼ 1 J+. J+ and J are plotted against their observed values in Figure 12. It appears that the model captures the cross-country differences in J+ and J slightly less accurately than it does for other ‘‘average’’ worker mobility indicators (see the analysis of J, D and C in Section 5.1). Yet those differences are arguably small, and both the empirical and the predicted versions of the J+ and J indicators are affected by sampling/estimation errors that probably render the cross-country differences appearing in Figure 12 nonsignificant.23 6. Identification and specification analysis Having shown that the model fits the data well we must now decipher whether there is any other reason than prior intuition to believe in that particular theory of facts. For example, the model was used to predict the relative shares of voluntary and involuntary mobility. This distinction is not entirely transparent in the data and our prediction thus strongly rests on how we estimate parameters

22

Recall that the J-indicator, which gives a measure of the extent of job-to-job turnover, was defined in Equation (16). 23 At this point we made no attempt to construct confidence ellipses around the points in Figure 12. Also note that one important source of error in the ‘‘empirical’’ version of J+ and J , which is computed from the numbers in rows 6 and 7 of Table 1 is that, as we mentioned before, those numbers do not add up to 100% because of missing wage data. The values on the x-axis of Figure 12 thus assume that the share of wage raises in the set of transitions for which the accepted wage is missing is the same as the share of wage raise in the set of transitions for which it is effectively observed.

297

The Empirical Content of the Job Search Model

Figure 12.

Shares of wage increases and wage cuts BEL

.8

ESP FRA GER ITA USA NLD DNK

Predicted

.6

GBR

.4

IRL PRT

GBR

DNK

PRT IRL

NLD USA ITA

GER

FRA

ESP

.2

BEL

.2

.4

.6

.8

Observed

l1 and l2. Transition rates d, l1 and l2 are determinants of job and employment durations and of transition probabilities across employment states; and they are also determinants of the equilibrium relationship between the wage offer distribution and the distribution of wages among employees. Which data component is exploited by the estimation procedure to yield these estimates is still unclear. Moreover, at least in principle, we could come up with different parameter values according to the specific source of identification which is being used. This immediately raises the subsidiary question of the consistency of these different identification sources. If we confirm that the model parameters are indeed overidentified, then we do have the possibility – and the duty – to test the model hypotheses. The aim of this section is first to test whether these overidentifying restrictions really exist, that is whether the transition rate parameters d, l1 and l2 are in effect identified from the transition and duration data separately from the wage data. Second, once we have detected overidentification restrictions, we want to test whether they are satisfied by the data. 6.1. Inference from transition data In Section 5.1, we showed that the model predicted well the three moments: J, the frequency of job-to-job transitions in the three-year observation period; D, the frequency of job-to-nonemployment transitions; and j, the empirical probability of exiting nonemployment in the first month of nonemployment. We thus have three apparently independent moments and three transition

298

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

parameters to identify – d, l1 and l2. We observe that the observed values of empirical analogs of J, D and j exhibit substantial cross-country variation (see Figures 6–8) and our model seems to do a good job of capturing this variation. One is thus entitled to hope that the knowledge of moments J, D and j suffices to completely identify the transition rates. The surprise is that it is not the case: our attempt at estimating the three transition rates by fitting those three moments failed. More precisely, fitting J, D and j only allows identification of d on one hand, and some compound of l1 and l2 (which captures job-to-job turnover) on the other. While those moment-based estimates are consistent with the constrained estimates ðdc ; lc1 ; lc2 Þ,24 this indirect inference method does not yield a separate identification of l1 and l2. One may even set l1 ¼ 0 for all countries and still fit J, D and j using d and l2 alone as well as with the complete set of parameters ðdc ; lc1 ; lc2 Þ. The duration component in J and D that is absent from j hence does not seem to be enough to identify all parameters. We investigate this point further in the next paragraph by looking at duration dependence. 6.2. Inference from both transition and duration data Intuition suggests that a definite source of identification of l1 lies in duration dependence: in fact, the only source of (negative) duration dependence in the model is the fact that workers earning higher wages tend to accept outside offers – i.e. to respond to l1- shocks – less often. The aim of this section is to pursue this idea by taking a closer look at job duration distributions. The fully structural likelihood function (7) mixes information on wages together with information on job durations and transitions. The presence of wages in the likelihood function is a consequence of the maintained assumption from the theoretical model that the workers’ mobility decisions are made based on wage comparisons. This is implicitly assuming that the worker’s lifetime value of holding a job is adequately measured by the wage paid in that job. While this assumption is obviously attractive from an empirical viewpoint (at least because wages are directly observed in our data), it comes at a cost in terms of additional theoretical restrictions. Specifically, equality of job values to wages flows from the following combination of assumptions: (1) the labor market is in a steady state; (2) wages are constant over time within a job spell; and (3) the wage is the only argument of the workers’ instantaneous utility function (or at least the only argument thereof that varies across jobs). There are obviously a number of fair criticisms to this approach. While assumption 1 (steady state) has little theoretical content, assumptions 2 and 3 are

24

We do not report the results here. They are available upon request. By ‘‘consistent’’ we mean that the moment-based estimates of d are close to the constrained estimates dc, and that if one fixes l1 at its constrained value lc1 , then l2 becomes identified in the indirect inference method, which then delivers an estimate close to lc2 .

299

The Empirical Content of the Job Search Model

much more restrictive. Assumption 2 is an ad hoc restriction on the set of contracts that can be offered by employers,25 while assumption 3 is a restriction on worker preferences.26 Taking the sole wage into account is therefore at best an approximation, and possibly completely misleading. In this paragraph, we want to take this argument seriously and confirm whether l1 (and the transition rates in general) can be identified without appealing to wage data, i.e. from transition and duration data alone. Formally, this amounts to treating wages as unobserved heterogeneity parameters, something that was first proposed in the context of a job search model by Ridder and Van den Berg (2003). The first thing to do is thus to integrate wages out of the likelihood function. As shown by Ridder and Van den Berg (2003), the model’s structure makes this integration rather easy, yielding a closed-form solution for the integrated likelihood functions that can be easily maximized. Formally, let us consider all the N G ¼ SN i¼1 e0i individuals employed at t ¼ 0. Taking up the notation xi from (6), we now write worker i’s likelihood conditional on e0i ¼ 1 but not on w0i as:  ‘ðxi je0i ¼ 1; d; l1 ; l2 Þ ¼ gðw0i Þe ½dþl2 þl1 F ðw0i ފt0i

dð1

cs0i Þð1 e1i Þ

½ðl2 þ l1 1fw1i  w0i gÞf ðw1i ފð1

cs0i Þe1i

.

ð23Þ

Combining (23) with (3) and (4), one realizes that the wages w0i and w1i only appear through F and f: ð1 þ kÞf ðw0i Þ ½dþl2 þl1 F ðw0i ފt0i ‘ðxi je0i ¼ 1; w0i ; d; l1 ; l2 Þ ¼
2 e 1 þ kF ðw0i Þ dð1

cs0i Þð1 e1i Þ

½ ðl2 þ l1 1fw1i  w0i gÞf ðw1i ފð1

cs0i Þe1i

.

ð24Þ

Now, we are only interested in worker i’s likelihood contribution unconditional on wages as we want to know if duration dependence together with transitions is enough to identify (d; l1 ; l2 ), that is ‘ðxi je0i ¼ 1; d; l1 ; l2 Þ ¼ ‘ðxi je0i ¼ 1; w0i ; y; F Þ dw0i dw1i .

25

ð25Þ

Offering a flat wage profile is not generally optimal for the employer. This argument was taken seriously in a number of recent papers, where the consequences of allowing more sophisticated wage contracts are explored. Stevens (2004) and Burdett and Coles (2003) look at nonconstant wage-tenure profiles. Postel-Vinay and Robin (2002a,b,2004) introduce the possibility of ex-post wage bargaining. 26 Against which one can argue that workers value jobs based on a set of job characteristics, which the wage surely enters together with a number of other arguments (working time, working conditions, distance from home, ‘‘amenities’’y). This idea is pursued in (inter alia) Hwang et al. (1998).

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

300

Table 4. Country u

d

lu1 lu2

Unconditional model estimates ðper annumÞ

BEL

DNK

ESP

FRA

GBR

GER

IRL

ITA

NLD

PRT

USA

0.0366 (0.0042) 0.0008 (31.734) 0.0248 (15.721)

0.0492 (0.0047) 0.0376 (1.0858) 0.0632 (0.4306)

0.0920 (0.0045) 0.0783 (0.2810) 0.0000 (0.0820)

0.0144 (0.0015) 0.1706 (0.3305) 0.0000 (0.0096)

0.0776 (0.0047) 0.0190 (2.8611) 0.1082 (1.3283)

0.0429 (0.0028) 0.1594 (0.2390) 0.0000 (0.0238)

0.0686 (0.0058) 0.0900 (0.5071) 0.0320 (0.1429)

0.0532 (0.0032) 0.0569 (0.2543) 0.0000 (0.0673)

0.0328 (0.0026) 0.0958 (0.3197) 0.0152 (0.0558)

0.0533 (0.003) 0.0045 (3.468) 0.0279 (1.667)

0.0540 (0.0036) 0.2483 (0.1521) 0.0075 (0.0130)

First integrating w.r.t. w1i and then using the change of variables x ¼ F(w0i) in the second integral, one finally obtains: ‘ðxi je0i ¼ 1; d; l1 ; l2 Þ ¼ dð1 cs0i Þð1 e1i Þ Z 1 ð1 þ kÞ e ½dþl2 þl1 ð1  xފ2 0 ½1 þ kð1 ½l2 þ l1 ð1 ¼d

ð1 cs0i Þð1

xފt0i

xފð1 cs0i Þe1i dx (  ðdþl2 þl1 Þt0i )ð1 e x e1i Þ E 1 ðxÞ  t0i x ðdþl2 Þt0i

ðd þ l2 þ l1 Þðd þ l2 Þ l1 ( ðdþl2 þl1 Þt0i )ð1 e x  dt0i þ ð1 dt0i ÞE 1 ðxÞ x ðdþl2 Þt0i

cs0i Þð1 e1i Þþcs0i



cs0i Þe1i

.

ð26Þ

Maximization of (26) yields the ‘‘unconditional’’ estimates ðdu ; lu1 ; lu2 Þ gathered in Table 4. Once more, this is a disappointment. While the unconditional estimates du of the job loss rate are precise enough and close to the constrained estimates dc, the job-to-job transition rate estimates ðlu1 ; lu2 Þ are affected by standard errors so large that any formal test of the joint equality of ðdu ; lu1 ; lu2 Þ and ðdc ; lc1 ; lc2 Þ would be uninformative (since probably no formal test would reject this joint equality). Under those circumstances, the only reasonable conclusion is that the point estimates ðlu1 ; lu2 Þ are meaningless, thus corroborating the ‘‘no identification’’ result of the preceding paragraph. 6.3. Inference from cross-sectional wage data From the above paragraph and the previous section one has to conclude that, somewhat surprisingly, duration dependence in job spell hazards and job transitions (supplemented by the steady state assumption summarized in (3)) do not contain the information needed to identify our three transition parameters (d, l1, l2) altogether. Moreover, this negative result does not seem to be entirely attributable to a lack of data as (unreported) experiments on simulated data

301

The Empirical Content of the Job Search Model

Table 5. Country kF

BEL

DNK

Estimate from sampling distribution

ESP

FRA

GBR

GER

IRL

ITA

NLD

PRT

USA

0.6586 0.4899 1.0225 2.6795 0.5671 1.3003 0.9385 0.5730 1.3539 0.1828 1.2874 (0.1907) (0.1507) (0.1386) (0.2917) (0.1388) (0.1435) (0.2162) (0.1290) (0.1507) (0.0972) (0.1926) Generalized Wald test of H0: kF ¼ kc

Test statistic 2.7525 0.0020 11.0660 3.2029 0.3797 0.7678 0.2404 1.2194 0.3781 0.6042 0.1922 p-Value 0.0971 0.9646 0.0009 0.0735 0.5378 0.3809 0.6239 0.2695 0.5386 0.4370 0.6611

showed that l1 and l2 are indeed very poorly identified if one cannot distinguish gains from losses in job values upon observing a job-to-job mobility. The obvious implication of this result is that wage data are really needed for the separate identification of l1and l2. In other words, the suspected source of overidentification lying in the ‘‘correspondence’’ between determinants of job durations and transitions on one hand, and wage distributions on the other that we pointed out in the introduction is not really there. In order to investigate to what extent the steady state constraint (9) linking F to G constitutes an important source of identification of the remaining parameter, we compare and contrast the structural predictions of F and f, using Equations (9) and (10), to the nonparametric estimates, F^ and f^. We address the identification question by testing whether kc is equal (close) to the ^ and f ð
jk; GÞ. ^ value of k that provides the best fit of F^ and f^ by F ð
jk; GÞ The value of k that provides the best fit of the empirical distribution of wages among job entrants is kF that maximizes the log likelihood LF ðkÞ ¼

NF X

^ ln f ðwi jk; GÞ,

i¼1

where NF is the number of workers in the subsample of job entrants. The estimates kF are displayed in row 1 of Table 5. Rows 2 and 3 in that same table contain the test statistic and p-value of a generalized Wald test of equality between kc and kF. The point estimates kc and kF are close in most countries. Indeed, the test p-value goes under 10% in only three countries: Belgium, France and Spain. For the first two, equality between kc and kFthe is still accepted at the 5% level. Only Spain comes up with a significant difference between the two estimates.27 For now, it seems fair to say that our structural estimator kc is indeed close to the

27

This is not unexpected given the differences between the empirical and predicted wage ^ in Spain (cf. Figures 11a and b). distributions, F^ and F ð
jkc ; GÞ,

302

Gre´gory Jolivet, Fabien Postel-Vinay and Jean-Marc Robin

value of k best fitting the empirical distribution of wages among job entrants. We thus conclude that the steady state constraint (3) is a strong source of identification of the index of search friction k, and consequently of l1 or l2. 6.4. Inference from wage mobility The harvest of the preceding paragraphs is rather meager in terms of overidentifying restrictions. The only nontrivial theoretical prediction, the steady state relationship between F and G, seems to be an important source of identification of the model. We have shown that the model provided a rather good description of employment transitions and of the various cross-sectional wage distributions but so far we have not been able to exhibit an indisputable test of specification. The last remaining data component that we have not yet considered is wage mobility. In this final paragraph, we show that employment transition data can be used together with wage transition data to identify the structural parameters independently of the cross-sectional distribution of wages among employed workers. The procedure that we propose here yields an estimate of the index of search frictions that can be compared with the corresponding estimate that we obtained from cross-sectional wage data. This will generate the specification test we are looking for. We again consider all the NG individuals employed at t ¼ 0, and the individual likelihood contribution (23), which we repeat here for convenience:  ‘ðxi je0i ¼ 1; w0i ; d; l1 ; l2 ; F Þ ¼ e ½dþl2 þl1 F ðw0i ފt0i

dð1

cs0i Þð1 e1i Þ

½ðl2 þ l1  1fw1i  w0i gÞf ðw1i ފð1

cs0i Þe1i

.

Maximization of Lm ðd; l1 ; l2 Þ ¼

NG X i¼1

ln ‘ðxi je0i ¼ 1; w0i ; d; l1 ; l2 ; F^ Þ,

ð27Þ

m c c c yields a new triple of estimates ðdm ; lm 1 ; l2 Þ, which differs from ðd ; l1 ; l2 Þ only by the fact that likelihood (27) does not impose to F to be related to G by the steady-state restriction (3). Effectively, this final estimation does not get rid of wages, but unlike in the structural estimation – by maximization of (7) – we do not use (3) in order to substitute F in the likelihood function. Rather, we compute the value of F at w0i using our nonparametric estimate F^ of the sampling distribution (see Section 2 and Section 5.3). Apart from job durations – which we saw in Section 5.2 are not enough to separately identify the job-to-job transition rates l1 and l2 – the extra bit of information used in (27) is about wage mobility, i.e. about whether a job-to-job transition is accompanied by a wage increase or a wage cut. Even though it also pertains to wages, this last bit of information is a source of identification which is completely separate from the cross-sectional

303

The Empirical Content of the Job Search Model

relationship between the sampling distribution of wage offers F and the distribution of wages in a cross section of employed workers, G, that was used in the previous Section (5.3). We first compare these estimates with the ones we obtained from the constrained estimation. The first three rows of Table 6 contain the point m estimates and standard errors of dm, lm 1 and l2 . Those can be compared with the benchmark values obtained from the constrained estimation. The following two rows in that table show the test statistics and p-values of a Wald test of joint m c c c equality of ðdm ; lm 1 ; l2 Þ and ðd ; l1 ; l2 Þ. m c c c Once again it appears that the point estimates ðdm ; lm 1 ; l2 Þ and ðd ; l1 ; l2 Þ are close almost everywhere. In fact, equality between the two sets of parameters is frankly rejected at the 5% level in one country only: the U.K. (while acceptance is borderline in Germany, where the p-value is just over 5% at 5.9%). The failure of the Wald test in the U.K. seems to be due mainly to a substantially higher point estimate of l1 on wage mobility data alone than in the constrained m estimation. Overall, it thus seems fair to conclude that ðdm ; lm 1 ; l2 Þ and F c c c m m m m ðd ; l1 ; l2 Þ, and consequently also k and k ¼ l1 =ðd þ l2 Þ, are consistent. An alternative way of verifying the steady-state assumption is to test for the m m equality of kF and km ¼ lm 1 =ðd þ l2 Þ. It may be a more direct test of the steadyF state assumption as k only depends on the restriction placed on the link between F and G by the structural model and the steady-state assumption, i.e. Equation (3), whereas km depends on wage and mobility data but does not force F to be tied to G by Equation (3). Table 6 presents the results of such a formal Wald test. The equality of kF and km is rejected at the 5% level in three countries: Spain, France and Germany. Rejection in Spain and France was not Table 6. Country d

m

lm 1 lm 2

BEL

DNK

0.0366 0.0492 (0.0042) (0.0047) 0.0399 0.0775 (0.0086) (0.0133) 0.0095 0.0493 (0.0029) (0.0061)

Estimates from wage mobility ðper annumÞ ESP 0.0920 (0.0045) 0.0406 (0.0068) 0.0162 (0.0025)

FRA

GBR

0.0146 0.0776 (0.0015) (0.0046) 0.0362 0.1255 (0.0057) (0.0125) 0.0136 0.0656 (0.0018) (0.0057)

GER

IRL

ITA

NLD

PRT

USA

0.0429 0.0686 0.0532 0.0328 0.0533 0.0552 (0.0028) (0.0058) (0.0032) (0.0026) (0.0032) (0.0036) 0.0445 0.1134 0.0235 0.0654 0.0341 0.1010 (0.0065) (0.0150) (0.0047) (0.0076) (0.0054) (0.0108) 0.0236 0.0260 0.0118 0.0225 0.0144 0.0326 (0.0026) (0.0047) (0.0020) (0.0027) (0.0025) (0.0036)

m c c c Genearlized Wald test of H0: ðdm ; lm 1 ; l2 Þ ¼ ðd ; l1 ; l2 Þ

Test statistic 0.9375 2.3337 2.0029 p-Value 0.8164 0.5061 0.5718 km 0.8644 0.7868 0.3754 (0.2278) (0.167) (0.0684)

4.9123 9.8344 0.1783 0.0200 1.2828 0.8766 (0.2512) (0.1114)

7.4547 2.6322 0.1784 0.0448 3.8082 0.0395 0.0587 0.4519 0.9810 0.9975 0.2829 0.9979 0.668 1.1987 0.3613 1.1846 0.504 1.1498 (0.1135) (0.1997) (0.0803) (0.1753) (0.0938) (0.1542)

Generalized Wald test of H0: km ¼ kF Test statistic 0.4795 1.7415 17.5 13.2 p-Value 0.923 0.628 0.0006 0.004

3.0226 11.9366 0.7820 1.9389 0.5363 5.6584 0.3108 0.388 0.008 0.854 0.585 0.911 0.130 0.958

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unexpected, given the somewhat mitigated success of our model at passing the ‘‘eye-ball’’ goodness-of-fit tests for these two countries (see in particular Figures 11a and b and Section 5.3). The negative result for Germany is more disappointing, as the model seemed to do a very good job of fitting the German data (but the consistency test between wage mobility and the constrained model, km ¼ kc, was only marginally accepted). Yet in spite of these few discrepancies, we are again tempted to conclude that the job search model’s success at passing such a demanding test in 8 out of 11 countries is quite remarkable, particularly given the model’s parsimony. 7. Conclusion In this paper, we have evaluated in detail the capacity of stationary job search models to successfully describe individual employment and wage trajectories and we have precisely analyzed how the different aspects of these data allowed identification of the model parameters and whether there existed overidentifying restrictions upon which specification tests could be based. We found that our prototypical steady-state search model fits with well the individual employment transition and wage data. In particular, the relationship between the distribution of wage offers and the cross-sectional distribution of employees’ wages that is predicted by the steady-state assumption is remarkably well accepted by the data. Moreover, despite the many institutional differences between the different countries that we consider, the specification tests that we are able to construct pass rather well. We are tempted to interpret this evidence in the following way. In a more complete model, the transition parameters should respond to aggregate productivity shocks. Yet we found evidence of stationarity in the wage distributions, suggesting that transitions between steady states are quick (here one should bear in mind that the typical business cycle lasts less than 10 years). This is good news as working out the equilibrium outside the steady state with nondegenerate wage offer distributions is still out of analytical reach. This paper therefore brings some empirical support to standard macroeconomic policy exercises analyzing labor market reforms by comparing pre- and post-reform steady states. Acknowledgment Repeated conversations with Guy Laroque about job search were very useful for the preparation of this paper. We also wish to thank Luc Behaghel, Gerard van den Berg, He´le`ne Turon, conference participants from the 3rd CEPR/DAEUP meeting in Bristol (April 2004), seminar participants at CREST, Johns Hopkins, Northwestern, Penn State, Paris I and Toulouse for stimulating questions. The usual disclaimer applies.

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References Bontemps, C., J.-M. Robin and G.J. Van den Berg (2000), ‘‘Equilibrium search with continuous productivity dispersion: theory and non-parametric estimation’’, International Economic Review, Vol. 41(2), pp. 305–358. Burdett, K. and M.G. Coles (2003), ‘‘Wage/tenure contracts and equilibrium search’’, Econometrica, Vol. 71(5), pp. 1377–1404. Burdett, K. and D.T. Mortensen (1998), ‘‘Wage differentials, employer size and unemployment’’, International Economic Review, Vol. 39, pp. 257–273. Christensen, B.J., R. Lentz, D.T. Mortensen, G.R. Neumann and A. Werwatz (2005), ‘‘On-the-job search and the wage distribution’’, Journal of Labor Economics, Vol. 23, pp. 31–58. Coles, M. & B. Petrongolo (2003), ‘‘A test between unemployment theories using matching data, Manuscript’’, University of Essex and L.S.E. Hwang, H., D.T. Mortensen and W.R. Reed (1998), ‘‘Hedonic wages and labor market search’’, Journal of Labor Economics, Vol. 16, pp. 815–847. Mortensen, D.T. and C.A. Pissarides (1999), ‘‘New developments in models of search in the labor market’’, pp. 2567–2624 in: O. Ashenfelter and D. Card, editors, Handbook of Labor Economics, Vol 3B, Amsterdam: Elsevier. Postel-Vinay, F. and J.-M. Robin (2002a), ‘‘The distribution of earnings in an equilibrium search model with state-dependent offers and counter-offers’’, International Economic Review, Vol. 43(4), pp. 1–26. Postel-Vinay, F. and J.-M. Robin (2002b), ‘‘Equilibrium wage dispersion with worker and employer heterogeneity’’, Econometrica, Vol. 70(6), pp. 2295–2350. Postel-Vinay, F. and J.-M. Robin (2004), ‘‘To match or not to match? Optimal wage policy with endogenous worker search intensity’’, Review of Economic Dynamics, Vol. 7, pp. 297–331. Ridder, G. and G.J. Van den Berg (1993), ‘‘On the estimation of equilibrium search models from panel data’’, in: J.C. Van Ours, G. Pfann and G. Ridder, editors, Panel Data and Labor Market Dynamics, Amsterdam: NorthHolland. Ridder, G. and G.J. Van den Berg (1997), ‘‘Empirical equilibrium search models’’, in: D.M. Kreps and K.F. Wallis, editors, Advances in Economics and Econometrics, Cambridge: Cambridge University Press. Ridder, G. and G.J. Van den Berg (2003), ‘‘Measuring labor market frictions: a cross-country comparison’’, Journal of the European Economic Association, Vol. 1(1), pp. 224–244. Stevens, M. (2004), ‘‘Wage-tenure contracts in a frictional labor market: firms’ strategies for recruitment and retention’’, Review of Economic Studies, Vol. 71, pp. 535–551. Van den Berg, G.J. (1990), ‘‘Nonstationarity in job search theory’’, Review of Economic Studies, Vol. 57, pp. 255–277.

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Appendix: Data U.S. data We use the PSID for the analysis of the U.S. labor market. The PSID is a longitudinal data set in which individual members of an initial sample of 4800 families are interviewed once a year since the starting year 1968. Individuals are followed over the years and, as young adults from the original sample form their own families, the sample expands (through births, marriages, etc.) The survey contains abundant information on individual characteristics, incomes and labor market statuses. Conveniently for our purposes, individuals are asked retrospectively every year about their monthly ‘‘calendar of activities’’ for the year just elapsed. Individual labor market statuses are thus recorded at a monthly frequency. A number of important changes to the PSID occurred in 1997 as the sample became too large. First, the number of families was reduced from 8500 to 6100 and families of post-1968 immigrants were introduced into the sample. Second, and more problematically, data collection became biennial although the calendar of activities kept covering a retrospective period of 12 months only. It thus becomes impossible to follow individuals at a monthly frequency after 1997. Given those problems, we are able to build a three-year panel of workers running from 1993 to 1996 (the latest exploitable year). Thanks to the calendar of activities, we observe individual labor market states (employed or nonemployed) on a monthly basis. Merging this information with wages and working hours that are observed at each yearly interview, we can, to a certain extent, associate each employment spell with an hourly wage. We choose to restrict our analysis to a three-year sample for three reasons. First, we want to maximize the overlap between our U.S. and European data, which only start in 1994. Second, many LFS data sets are short in their panel dimension (typically, they are three-year rotating panels), and we want to show that the model can be estimated with reasonable precision on such short panels. Third, the model assumes that the labor market is at a steady state, an assumption that would be harder to defend over a long period of time. European data For European countries, we use the ECHP. The ECHP is an eight-wave panel of ex-ante homogenized (common questionnaire) individual data covering 15 EU countries from 1994 to 2001. By construction, the ECHP is similar in spirit to the PSID: households are interviewed each year and every individual present in the initial sample is followed over the seven waves. It is designed as a standard household socioeconomic survey, with a rich set of variables. Each observation includes in particular basic information about individual characteristics (age, sexy) as well as, when the individual is employed, a

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description of their job at the time of the interview that includes the wage, the date when the job has started or if it was preceded by unemployment. What proves to be useful in this data is a group of variables about individuals’ previous jobs (which is also available for the currently unemployed). Combining ending dates of previous jobs and starting dates of current jobs, we are able to construct job spell durations and to label labor market transitions as either jobto-job or job-to-unemployment without resorting to retrospective calendars of activities (which are likely to be more vulnerable to memory biases, and are unavailable or poorly reported in many countries). We follow a cohort of workers between 1994 and 1997. We choose this particular three-year subperiod, because it maximizes the overlap with the American sample (which runs from 1993 to 1996). Due to the structure of both the PSID (which changes dramatically in 1997) and the ECHP (which only starts in 1994), this is the best we can do. Considering this three-year observation window forces us to restrict the original 15-country panel to a 10-country sample, in particular because the initial years are missing for Austria, Finland and Sweden (which only joined the ECHP in its second or third year). We also had to do without Greece due the poor quality of a number of crucial variables. Finally, we should mention that Germany, Luxembourg and the U.K. have left the ECHP in 1997. Fortunately, the missing original ECHP data for Germany and the U.K. have been replaced by ex-post harmonized data from the German SOcio Economic Panel (GSOEP) and the British Household Panel Survey (BHPS).28 In the end, we are left with the following 10 countries: Denmark, the Netherlands, Belgium, France, Ireland, Italy, Spain, Portugal, Germany (SOEP) and the U.K. (BHPS).

The definition of job-to-job transitions Since we focus on worker mobility, we have to be careful about our empirical definition of job-to-job transitions. As explained in the main text (Section 2), Figure 2 plots the re-employment hazard rates of workers who are observed leaving the job they had in 1994 (1993 for the U.S.). Looking at Figure 2, one sees that in almost every country, the job re-accession hazard rate is high in the first two months, then drops abruptly at three months to finally stay roughly constant at all longer durations. We think that many of those nonemployment spells that are observed to last one month or less29 can be transitions between two jobs, the start- and end-dates of which do not coincide.

28

Missing ECHP data from Luxembourg were only replaced by Panel Socio-Economique du Luxembourg (PSELL) data from 1995 onwards, so we also had to drop Luxembourg. 29 Due to the structure of the ECHP, observed durations of one month or less correspond to actual durations of up to two months.

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We thus define a job-to-job transitions as follows. In Europe, we consider any transition between two jobs with an observed duration of one month or less and for which the interviewee reports that the second job was not preceded by a period of nonemployment as a job-to-job transition. In the U.S., the only information that we have is through a monthly calendar of activities. We therefore retain as a job-to-job transition any job change with no intervening nonemployment period (which, given the structure of the calendar of activities, can hide nonemployment spells of less than three weeks).

CHAPTER 13

Job Changes and Wage Growth over the Careers of Private Sector Workers in Denmark$ Paul Bingley and Niels Westergaard-Nielsen Abstract We describe and model careers for Denmark’s population of men and women who are continuously full-time employed. Individual pay and mobility between firms is assumed to be a function of firm and industry pay and retention policies. These policies are characterized by firm and industry specific returns to education, experience and tenure. Workers solve the dynamic programming problem of mobility between firms by accounting for the expected future consequences of current decisions until retirement. This highly complex problem is considered as a problem of choosing between different wage trajectories believed to represent different wage offers. We use information on all private sector white collar full-time employed men and women aged 20–54 in 1980. They are followed until 2000, and matched throughout to their employers and industries. The data allow us to study all wage changes for a very large group of workers and whether these differences happened between different work places or within the same work place. Substantive findings are: the average person has 12 different employers between ages 20 and 60, experiences a doubling of real earnings, three quarters of which is achieved before the age of 40 in the first 8 jobs. One third of wage growth is between jobs. In estimations of a model for moving or staying, we find that a high expected wage level in the current firm means lower probability of moving. The higher the expected

$

This study is funded by a grant from the Danish Social Science Research Council through Centre for Corporate Performance and grant 24-02-0064. The paper has benefited from comments at presentations at IZA, Bonn, and at Universta Catolica, Milan, EALE Conference in Lisbon and Sandbjerg. Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75013-6

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

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wage growth in the current firm, the less likely is a move. The higher expected wage or wage growth in competing firms, the more likely is a move. Keywords: wage mobility, job mobility, returns to tenure JEL classifications: J31, J64 1. Motivation Recent studies have reported high rates of worker turnover in a number of countries OECD (1997) and European Commission (1998). This has a microeconomic as well as a macroeconomic dimension. The micro dimension is the mechanism that determines the mobility decision of each worker, within a given framework of labour protection. Previous studies for Denmark have shown very high turnover rates for most labour force participants but we have not been able to explain the large level of turnover, (Albæk and Sørensen, 1998l; Bingley et al., 1999, Bingley and Westergaard-Nielsen, 2003). The second dimension is how these mechanisms are affected by different sets of labour protection laws in various countries. Denmark seems to be an outlier in continental Europe, with relatively weak labour protection. OECD (1997) demonstrates that differences in worker turnover between countries are related to differences in regulations of layoff possibilities. However, it is probably only a small part of the total labour turnover which is due to regular layoffs, whereby an employer takes the initiative and consequentlylabour protection rules may apply. More often, an employee receives a wage offer from another firm and decides to move to the better paying firm. The most important explanation for worker mobility is therefore the search for new and better jobs. But what does employment protection have to do with mobility decisions, if employer-initiated layoffs are relatively uncommon? One likely explanation is that workers may loose their protection shield when moving between employers, because it takes several months before new hires are subject to protection. Consequently, workers will be less tempted to move, even if they may get a pay rise by moving. Though it is fairly straight forward to argue that liberal labour market regulations will be positively correlated to mobility because it makes it easier for employers to layoff their workers, researchers have not yet been able to come up with a plausible explanation of why employees apparently engage in so many job changes as is the case in Denmark. If we look at job turnover from the employers’ perspective, it is hard to accept that Danish employers should be less interested in retaining workers than their European competitors who operate under a worker protection shield. In this paper, we focus on individual worker gain from turnover. We present an empirical model that explains a part of worker between-firm mobility within

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a country with its regulations and subsidies. The explanation, in this setting, is via quits and the search for new and better paid jobs.

2. The literature Several theories have predictions regarding turnover. For example, Parsons (1977) applies a search model. The factors of importance in the job search model are the distribution of wage offers, the cost of search, the explicit or implicit job duration, workers state of knowledge of market conditions, the contracting terms and the flexibility of work hours on the current job. According to the search model, the worker searches and chooses the job where the expected total wage income is highest. For the employed searcher, this means that he/she will move whenever the expected lifetime wage in a new job is higher than the expected life time wage in the current job. Search is also a central element in a model of wage growth and mobility by Borjas (1981). The earnings profile is influenced by a search premium and experience. He assumes that the earnings profile is discontinuous across jobs because job mobility results in a wage premium if the job switch has been voluntary. The argument is that there is a gain in connection to job mobility representing the return to the investment in search. Furthermore, the path of investment in own training over time will also tend to have discontinuities across jobs because different jobs provide different learning opportunities and hence the fraction of time that can be invested is likely to vary among jobs. His main findings using data from the US ‘‘National Longitudinal Survey of Mature Men’’ are that labour turnover may lead to a significantly higher wage in the new job than in the old job. But by the time men reach their middle age, the short run advantages of labour mobility are less important, leaving immobile men with significantly higher wage rates. Furthermore, individuals who have experienced a substantial amount of job mobility over their working lives tend to have smaller rates of growth of earnings within each job. Now, turning to the employer side, the mechanisms described by the search model suggest that firms will use the wage rate as a policy instrument that can regulate how many workers leave the firm. Thus, increasing the wage rate of a particular firm will accordingly mean that fewer receive wage offers motivating them to move. We might expect that higher pay actually reduces worker turnover. However, even if this is observed as an average effect, it is more likely that the firm uses this wage policy selectively in order to retain key workers. This is exactly what happens in the case where a firm decides to match outside wage offers. Each time a worker comes to his superior and tells her that he has received a wage offer above the current wage, the supervisor has to decide if she wants to match it or not. The guideline will be the value of the productivity of this worker compared to the potentially increased wage. Thus, the equilibrium wage that will come out of this process will depend on the productivity in the

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incumbent and the competing firms. Of course, the firm will only use the wage instrument to retain workers if it expects this to be profitable. For firms there is another issue at stake, since they face the cost of recruiting and training new workers. At the incumbent firm, the costs are recruitment and training of the replacement for the lost worker. Furthermore, human capital related to experience in that particular firm, firm-specific human capital, is destroyed. At the raiding firm the costs are those of recruitment and training. The costs of hiring are usually substantial, averaging 1/3 of an annual income according to consultancy firms and Bingley and Westergaard-Nielsen (2003). Of course it varies with the type of job, the person, etc. The gain for firms from raiding is the future value of marginal productivity minus the future wage. One of the important elements may be that the person hired may carry valuable human capital with him or her. Allowing this mechanism to work contributes to keeping a society on the efficient production frontier, because it allocates workers to jobs where they are most cost effective. However, this offer matching process may not be as smooth as described above. General concerns about the overall wage distribution at the firm and the reluctance of the local Trade Union to accept more unequal wages may constrain the willingness of firms to give higher wages even to workers who might be profitable, in expectation, to retain. This is the subject of Hall and Lazear (1984) where it is shown that worker turnover may become sub-optimal due to the fact that firms cannot pay each individual worker a wage so that they can retain them because of trade unions (wage compression). The claim is that there are too many inefficient layoffs when conditions are poor in the firm but even worse in the market, and that there are too many (inefficient) quits when conditions in the market are good but even better in the firm. The consequence is that some degree of wage compression results in excessive layoffs in bad times and excessive quits in good times. However, it is not clear whether we can meaningfully distinguish between layoffs and quits. The classification depends on whether the demand for a wage revision is initiated by the worker or the firm. In the model of McLaughlin (1991), separations are censored wage revisions, where the worker knows his outside wage offer but does not know his productivity with his incumbent employer. Wages are governed by a counter-offer-matching contract. If productivity in the incumbent firm exceeds the outside offer, then new pay is equal to the value of productivity, if not the worker quits, due to a censored wage increase. If the firm observes that worker productivity falls, then it initiates a wage cut. If the new wage is below the value of productivity and the worker rejects it, there is what McLaughlin calls a censored wage cut. This has an empirically testable hypothesis, which McLaughlin tests on data about self-reported reasons for separations. First, it is found that the effect of wages prior to separations is zero in a separation probit, but the quit rate conditional on separation is falling in the same wage. Second, higher alternative wages are predicted to increase both separations and the fraction of separations

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labelled as quits. He finds that high predicted wages in the current job reduce separations and that lower wages prior to the separation increase the quit probability compared to the layoff probability. These are in accordance with the predictions of the model. At the same time, he finds that wages prior to separation are negatively correlated with the separation probability, contrary to the prediction of his model. He reasons that this is because the model neglects match-specific capital. His main conclusion is that the distinction between quits and layoffs is not economically meaningful since a simple story about wages and expected wages actually seems to account for the difference. The same view was supported in Becker et al. (1977), the conclusion being that turnover (in a different market) is always efficient or joint wealth maximizing and the voluntary/involuntary distinction is inappropriate. The firm and worker dissolve their match if and only if their total value as separated exceeds the combined value of the match. In this world, potentially inefficient separations result in wage adjustments or side payments to preserve optimal assignments. The focus of our paper is on wage growth within job and job-to-job. The importance of job-to-non-employment transitions is illustrated in Royalty (1998), and between-firm wage growth ignoring growth within the firm is Christensen et al. (2005). The latter is particularly relevant because it also uses Danish data. The descriptive work in Topel and Ward (1992) is closest to our contribution here. Their main empirical finding was that job-specific wage is a key determinant of mobility. Our contribution is to extend this empirical insight to include alternative job-specific wages. It is the measurement of this alternative wage, made possible because of the exhaustive nature of our longitudinal employer–employee data, which is the novelty of our paper. 3. The descriptive model In our descriptive model, motivated by the previous discussion, we do not attempt to distinguish between quits and layoffs. The model is formulated within the option value tradition. The idea in the model is that people maximize their expected future earnings so that they will change job if they are offered a job with a higher expected future wage. We will allow for different wage trajectories at different employers by estimating a wage function for each employer. The problem is how to model the wage offers, which are only rarely observed. Instead, we usually observe the wages accepted. In the search literature, this problem has been addressed by assuming that the accepted wage offers follow a censored wage distribution where we only observe the accepted wage offers (see Jensen and Westergaard-Nielsen, 1987). Each wage trajectory can be characterized by the wage level for the first year of employment and a wage growth element for each year of tenure, as illustrated in Figure 1. The alternative wage that a worker would receive if (s)he changed employer is not observed for workers who stay employed at the same firm. Yet, it is this

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Figure 1.

The choice between wage trajectories

wage

C B A

Time

counterfactual which may be an important determinant of the firm-to-firm mobility decision of a worker. In particular, the present discounted value of the income stream following from staying one more year with the current employer, compared with that from leaving. The wage ‘‘premium’’ in the current job comprises, among other things, contributions from occupation, industry and firm during the relevant time period. Controlling for determinants of wage with the current employer along these dimensions is often assumed to characterize the alternative wage in labour market models (see, for example, contributions in the rent sharing literature due to Blanchflower and Oswald, 1996 and Estevao and Tevlin, 2003). Recent work by Abowd et al. (2004) and Beffy et al. (2004) models worker mobility and current wages as a simultaneous process. Firm heterogeneity in terms of wages and mobility is taken seriously in these papers, which extends the work of Abowd et al. (1999) on identifying firm wage heterogeneity. The relevant contribution of these papers is modelling the mobility as a function of current wage, where both are simultaneously determined. Still, as is the case in all previous firm-to-firm mobility literature, the wage ‘‘premium’’ of the current employment is calculated regardless of the set of alternative employers. These models are silent with regard to the identity of possible destination firms, which are offering alternative wages. Wage ‘‘premium’’ are relative to the population as a whole. The achievement of the above-sited simultaneous heterogeneity papers is important, and modelling choice of destination even at the 1-digit industry level together with wages and general heterogeneity is beyond the scope of current computational capacity. The contribution of our paper is to characterize alternative employers and alternative wages in an informative and novel way. We allow for firm heterogeneity in wages and mobility, but we do not model this as a simultaneous process.

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0

In our model, potential alternative employers (F ) for each firm (F) are assumed to comprise next destination employers of all previously separated workers from the firm (F). These employers are weighted according to their relative frequency as a destination for workers leaving each firm (F). This characterization of alternative employers, in turn, provides weights for the firm-specific effect of alternative wages. Thus, the wage ’’premium’’ of each worker–firm match is a function of current wage over and above the alternative wage, so characterized. These current and alternative wage terms are included as separate determinants of the firm-to-firm mobility function, in order to distinguish their contribution. In contrast to Abowd et al. (2004) and Beffy et al. (2004), we do not allow for individual heterogeneity in wages. One criticism of our approach is that our ‘‘firm effects’’ are biased by the composition of the workforce because we do not simultaneously account for firm and worker heterogeneity (this is the contribution of Abowd et al. (1999)). However, in lieu of a simultaneous wage and mobility model, we argue that these ‘‘exogenous mobility’’ estimates of firm and worker effects could also be biased, and as such would not present an unambiguous improvement in our context. These person effects in wages are not of interest in their own right for our model, since they would, by definition, be differenced out in a comparison of current and alternative wages. However, we do allow for idiosyncratic person effects in wages to have a distinct effect on the firm mobility decision. A positive wage residual for the current person-firm-year suggests that the worker has a productivity level (which is recognized by the employer, and corresponds to a higher wage level) different than would be predicted by the researcher on the basis of observed individual characteristics and unobserved firm heterogeneity (firm fixed effects). So the worker is observed to be more productive by the employer than (s)he is predicted to be by the researcher based on observable (to the researcher) characteristics. The empirical question that our mobility model answers is as follows: Is this productivity, which is unobserved to us, but observed to the current employer, associated with greater mobility, i.e. do other (potentially alternative) employers recognize this too? Our prior is no (negative sign on the estimated coefficient) simply because what the potential alternative employer can observe is likely to be somewhere in between what we the researcher can observe (not very much) and what the current employer can observe (somewhat more). The test of the model is whether people actually tend to move according to our priors about the relative value of present wage level and growth compared to outside wage offers and potential wage growth. 4. Data The data consist of the population of residents in Denmark who have been employed in the private sector at some time during the period 1980–2000. This originates from the Centre for Corporate Performance extract of Statistics Denmark’s Integrated Database for Labour Market Research (IDA) The raw

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data give a number of insights into the mobility decision which are worthy of mention. The link between firms and individuals is established on a specific day in November. Consequently, we have excluded some non-November employments from our analysis. Furthermore, we have excluded persons characterized as unemployed at the November census point. The great virtue of this data set is that we can track all workers moving from firm-to-firm at these annual census points. In the following descriptive analysis, we have used this feature to describe mobility and its consequences for individual wages. Although we are using a window of 21 years, we have full information on the elapsed tenure since the start of the mandatory supplementary pension (ATP) registers in 1964. Descriptive statistics are presented in Table 1. Lrwage is log of hourly wage in January 2004 DKK. This is calculated by Statistics Denmark as labour earnings associated with a work attachment divided by hours worked in the attachment. Observations of wages in the top 0.1% in each year are dropped from the sample. Figures 2 and 3 show the elapsed firm tenure represented by the mean and by the 25th and the 75th percentile for different ages. The mean curve shows that a 34-year-old has on average held 5 jobs since (s)he started a career. However, a person on the 25th percentile is just about to start his/her 2nd job, while the person on the 75th percentile is about moving from his/her 7th to the 8th job in as many different firms. Tenure increases with age. The counterpart of this is the probability of changing firms. Figure 4 shows that the probability of changing firms declines over the lifecycle. It starts high in the 20s and drops down to 15 and 12% for 38 year old men and women, respectively. The mobility probability for women is Table 1.

Descriptive statistics for observations entering the wage function estimation Males

lrwage educ Exper tenu age no of observation per person no of person-years number of firms average firm size average number of observation of same firm

Females

Mean

Std.

Mean

Std.

5.184 13.064 19.069 6.145 39.132 7.586 6536.683 All firms 231.383 11.669 5.025

0.391 2.642 10.509 5.778 10.050 5.437

4.910 12.512 17.337 5.660 36.848 7.737 7119.813

0.333 2.420 10.353 4.988 9.782 5.315

Job Changes and Wage Growth over the Careers of Private Sector Workers in Denmark

Figure 2.

317

Distribution of elapsed firm tenure and age for women Tenure distribution for women

18 16 14 12 10 8 6 4 2 0 20

23

26

29

32

35

38

41

mean

Figure 3.

44

p25

47

50

53

56

59

p75

Distribution of elapsed firm tenure and age for men Tenure distribution for men

20 18 16 14 12 10 8 6 4 2 0 20

23

26

29

32

35 mean

38

41

44 p25

47

50

53

56

59

p75

almost 5% lower at the beginning of the labour market career and it is almost the same from the age of 41. The empirical mobility pattern can also be described as the mean number of jobs held so far for a specific age group as in Figure 5. Figure 6 shows that the mean average earnings gain from moving to a different firm is about twice the earnings growth obtained by staying with the same firm for a young person. This gain should be understood as the immediate

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Figure 4.

The probability of changing firm for men and women as a function of age Probability moving

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 Females

Figure 5.

Men

Mean number of jobs held until a given age Number of jobs held so far

7

count

6 5 4 3 2 20

25

30

35

40 age

45

50

55

earnings gain from one year to the next, but it says nothing about the wage growth in the periods following a move. Earnings changes are highly related to age. For men and women up to a certain age there are, on average, substantial wage gain from moving between employers. Furthermore, women stop gaining from job changes already from age 30, but there is no direct loss compared to stayers. For men the gains continue until 40, but thereafter they face a relative loss in wage compared to stayers. After 47 this turns into an absolute loss in wage. The latter result could be due to the build-up of firm-specific human capital with age, or a layoff. Finally, we have in Figure 7a described the distributions of these gains. These show that mobility is not always related to higher wages. The curves for the low gainers at the 25th percentile demonstrate that there are a number of people who take wage cuts by changing employers. There are several potential explanations

Job Changes and Wage Growth over the Careers of Private Sector Workers in Denmark

Figure 6.

319

Mean immediate earning gain from mobility between firms, by age and gender Immediate Earnings change for men 0.3

0.25 0.2 0.15 0.1 0.05 0 -0.05

20 23

26

29

35

32

38

41

Stay

44

50

47

53

56

Move

Earning gain for women 0.3 0.25 0.2 0.15 0.1 0.05 0

20

23

26

29

32

35

38 Stay

41

44

47

50

53

56

Move

for this: the person was laid off (unobserved to the researcher), or she takes a current wage cut in order to obtain a higher wage later due to specific training as indicated in Figure 1. In Figure 7b, we show the full dispersion of wage gains for a 35-year-old employee. The Graph is censored with respect to the top and bottom 1%, where outliers probably dominate. One could easily get the impression that most of the wage gain is actually obtained in connection with job shifts. This is of course not correct because after all people do not change job every year and because they also get raises by staying with the same firm. Figure 8 summarizes the sources of the cumulative wage gain for different age groups. This is calculated as the expected wage increase over time, if the person follows the mobility patterns seen in the data and

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Figure 7. (a) Distribution of wage gains by moving depending on age; (b) Distribution of wage gains for movers among 35 years olds. (10% sample) Wage gain percentiles for men 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 20

23

26

29

32

35

38

mean

41

44

p25

47

50

53

56

p75

Mean wage gain for women 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 20

23

26

29

32

35

38

41

44

47

50

53

56

-0.1 mean

(a)

p25

p75

Wage dispersion for 35 years of age 0.035 0.03 0.025 0.02 0.015 0.01 0.005 (b)

0 -0.48 -0.40 -0.32 -0.24 -0.16 -0.08 0.00 0.08 0.16 0.24 0.32 0.40 0.48

Job Changes and Wage Growth over the Careers of Private Sector Workers in Denmark

Figure 8.

321

Accumulated wages related to job mobility and on-the-job wage growth Cumulative wage growth 160 140

percentage

120 100 80 60 40 20 0 20

23

26

29

32

35

within

38

41 age

44

between

47

50

53

56

total

stays as long as the average person etc. This answers the following question: How large a proportion of the total wage growth for an average person, of a given age, is related to job changes and how much is related to on-the-job wage growth? Figure 8 shows that high wage growth in years with mobility is still outweighed by slow, but steady growth in wages over the years when the person stays with the same employer. It also illustrates that the gain from moving becomes less and the wage growth related to staying becomes greater. This pattern could be explained by the accumulation of firm-specific human capital. Figure 6 shows that the older the person is (which is correlated with high tenure), the less wage is gained by mobility. Above 47 years of age, there is even a short run loss for men. This can also be explained by the firm-specific human capital hypothesis, because the initial loss of firm-specific human capital at the time of a job shift becomes larger and larger with age. This initial loss does not preclude subsequent wage growth in the new job. These findings are in accordance with the results in George J. Borjas (1981). We find, as he does, that labour turnover may lead to a significantly higher wage in the new job, and that this short run effect is reduced with age. Our result – that there is an increasing return to staying at the current firm compared to moving – is also in accordance with his results. 5. Estimation The first step in the estimation process is to obtain estimates of the wage functions. We have estimated two sets of wage functions. First, we have estimated a

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Table 2.

The wage functions Wage level

lrwage male meduc meduc2 mexper mexper2 mtenu mtenu2 feduc feduc2 fexper fexper2 ftenu ftenu2 _cons Number of firms Number of individual-years R2(adj)

Wage growth

Coefficient

Std.

0.2320 0.1028 0.0059 0.0323 0.0006 0.0096 0.0000 0.1499 0.0077 0.0192 0.0004 0.0153 -0.0003 5.2951

0.0084 0.0009 0.0000 0.0001 0.0000 0.0002 0.0000 0.0010 0.0000 0.0001 0.0000 0.0002 0.0000 0.0061

232,230 16,104,630 0.4304

Coefficient 0.0108 0.0008 0.0000 0.0032 0.0000 0.0023 0.0001 0.0034 0.0002 0.0027 0.0000 0.0004 0.0000 0.0407

Std. 0.0059 0.0007 0.0000 0.0001 0.0000 0.0001 0.0000 0.0007 0.0000 0.0001 0.0000 0.0001 0.0000 0.0043

159,800 11,512,370 0.0375

wage function that explains the level of wages using all 16 million person-year observations. The wage function is specified with tenure, education and experience, these squared and with full gender interactions. Finally, we have estimated a firm fixed effects model. In this way, we have obtained a set of coefficients a to the X-vector of person characteristics. The firm effects are the b’s. There are altogether 232,000 different firm effects. All a coefficients are significant. The coefficients are conventional, with a lower intercept term for men but ‘‘compensated’’ by a higher coefficient to education, experience, but not tenure. The second function is the wage growth function, where wage growth is regressed on the same set of variables. This is only estimated with persons who stayed in their firm in year t and t – 1, which comprises 11.5 million observations. The wage growth equation has a set of coefficients g to the personspecific vector X and a set of firm effects d. There are 159,800 different firm effects(see Table 2). In the next stage, we have used these estimates to look at several hypotheses which are in turn related to different wage concepts. Hypothesis 1. A large wage residual gives a lower separation probability. Thus, we define the individual wage element, wageresidual: wij ¼ X ij a^ þ bj þ wageresidual i; j
 wageresidual i; j ¼ wij X ij a^ þ b j

ð5:1Þ

323

Job Changes and Wage Growth over the Careers of Private Sector Workers in Denmark

where wij is the currently observed wage. Thus, wageresiduali,j is a match-specific element related to person i while employed in firm j. A high wageresidual means that the person gets a relatively high wage due to some unobserved matchspecific characteristics. This may influence mobility in two ways: one will lower the probability of moving because he may not be able to get a similar wage at alternative employers. Hypothesis 2. A firm that pays a higher wage than other firms will have a lower separation probability. Here, we calculate the predicted wage in the current job which consists of the predicted wage for individual i plus the firm-specific element for firm j: firmwageij ¼ X ij a^ þ bj

ð5:2Þ

Thus, the hypothesis is that a high firmwage in the current (incumbent) job makes mobility less likely. Hypothesis 3. If competing firms are paying more, separation probability is higher. Here, we estimate the alternative wage for person i, who comes from firm j: alterwageij ¼ X ij ðtenure ¼ 0Þ^a þ b j

ð5:3Þ

where bj is the weighted average of the firm effects in the firms j0 ,where employees in this particular firm have moved to in the past. Thus, alterwage expresses the wage level that a person with characteristics Xij and with tenure ¼ 0 can obtain in the firms where his former colleagues and their predecessors actually moved in previous years. Hypothesis 4. Employees on fast tracks have lower separation probability. The reason for this hypothesis is that a fast track (defined as rapid wage growth in the current firm) makes workers less likely to move because they have been selected for a speedy career. However, the converse could be true if wage compression within-firm dominates. In the latter and other cases, it is possible that high wage growth may be used as a signal for raiding employers. Here, we estimate changes in wageresidual, calculated over the period [t – 1, t], called dwageresidual. Hypothesis 5. Fast wage growing incumbent firms have less separations. Firms with a high rate of wage growth are more attractive to stay in, because they offer better prospects. Here we calculate the change over [t – 1, t] in the current firm wage dfirmwage. Accordingly, we will expect a negative sign. Hypothesis 6. Fast wage growth in alternative employers will increase separations. Here we calculate the change in alternative wages from t – 1 to t, dalterwage. Finally, we have estimated two different specifications of a mobility function. Both estimations are made on a 10% sample of persons (see Table 3. Table 4.).

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Table 3.

Paul Bingley and Niels Westergaard-Nielsen

Individual worker mobility probit (without firm random effects) – model estimates Males

move

Coefficient

wageresidual firmwage alterwage dwageresidual dfirmwage dalterwage educ educ2 exper exper2 tenu tenu2 _cons log likelihood

Table 4.

0.2202 0.7730 1.3557 0.1782 0.0368 0.3719 0.0204 0.0018 0.0351 0.0004 0.0660 0.0013 3.3377 1413778

Females Std.

Coefficient

0.0034 0.0076 0.0117 0.0057 0.0241 0.0278 0.0024 0.0001 0.0005 0.0000 0.0005 0.0000 0.0452

0.1391 0.6983 1.0886 0.2489 0.2948 0.5032 0.0081 0.0013 0.0171 0.0001 0.0967 0.0027 2.3246 1520832

Males Std.

0.0036 0.0077 0.0113 0.0049 0.0237 0.0279 0.0027 0.0001 0.0004 0.0000 0.0006 0.0000 0.0466

dF/dx

Females Std.

0.0444 0.1558 0.2733 0.0359 0.0074 0.0750 0.0041 0.0004 0.0071 0.0001 0.0133 0.0003

0.0007 0.0015 0.0024 0.0012 0.0049 0.0056 0.0005 0.0000 0.0001 0.0000 0.0001 0.0000

1413778

dF/dx 0.0275 0.1381 0.2152 0.0492 0.0583 0.0995 0.0016 0.0003 0.0034 0.0000 0.0191 0.0005

Std. 0.0007 0.0015 0.0022 0.0010 0.0047 0.0055 0.0005 0.0000 0.0001 0.0000 0.0001 0.0000

1520832

Individual worker mobility probit with firm random effects—model estimates Males

wageresidual firmwage alterwage dwageresidual dfirmwage dalterwage educ educ2 exper exper2 tenu tenu2 _cons log likelihood N Number of groups

Females

Males

Coefficient

Std.

Coefficeint

Std.

0.0800 0.2941 0.3147 0.3023 0.5119 0.1977 0.0311 0.0009 0.0202 0.0001 0.0684 0.0018 0.0890 944277.67 3,796,464 65,505

0.0045 0.0146 0.0224 0.0073 0.0322 0.0430 0.0040 0.0002 0.0009 0.0000 0.0008 0.0000 0.1005

0.1216 0.3931 0.0705 0.2679 0.3650 0.2241 0.1102 0.0052 0.0091 0.0001 0.0588 0.0016 1.7696 945850 4,155,230 73,483

0.0050 0.0137 0.0214 0.0067 0.0301 0.0402 0.0045 0.0002 0.0006 0.0000 0.0009 0.0000 0.0960

dy/dx 0.0206 0.0758 0.0811 0.0779 0.1320 0.0510 0.0080 0.0002 0.0052 0.0000 0.0176 0.0005 944278 3,796,464 65,505

Females Std.

0.0012 0.0037 0.0058 0.0019 0.0083 0.0111 0.0010 0.0001 0.0002 0.0000 0.0002 0.0000

dy/dx 0.0333 0.1076 0.0193 0.0733 0.0999 0.0613 0.0302 0.0014 0.0025 0.0000 0.0161 0.0004

Std. 0.0014 0.0037 0.0059 0.0018 0.0083 0.0110 0.0012 0.0001 0.0002 0.0000 0.0002 0.0000

945850 4,155,230 73,483

6. Discussion The first type of estimation is probit ignoring firm-specific heterogeneity (Table 3) whereas the second is a firm-wise random effect probit (Table 4). The latter

Job Changes and Wage Growth over the Careers of Private Sector Workers in Denmark

325

method takes account of the fact that we observe several workers in the same firm. Thus, we are in this method controlling for unobserved firm effects. Examples could be a pleasant working environment, subsidized leisure facilities, poor leadership, decent canteen, etc. But it could also be the use of different HRM methods, see Batt (2002). The main difference between the two methods is that the wage growth at the incumbent firm, dfirmwage becomes negatively signed in the firm-wise random effect model according to priors, because we are now controlling for firm effects. This suggests that non-pecuniary elements may play an important role. Almost all of the variables are highly significant in both types of estimation and for both genders. High wage level firmwage and growth dfirmwage help to retain workers. Coefficients on match-specific effects wageresidual and dwageresidual both have negative signs, indicating that the ‘‘worker–firm unobserved component’’ explanation dominates the wage constraint hypothesis. Similarly, it is found that people on ‘‘fast track’’ are actually less likely to move than other employees. The wage level and growth of the wage in competing firms has a positive sign in all specifications confirming our Hypotheses 3 and 6. Finally, the individual control variables show that higher education induces more mobility, experience and tenure both diminish mobility. Comparing these coefficients it is reassuring that the two most important factors are the incumbent firm effect, firmwage and the wage level at the competing firm, alterwage and corresponding growth variables. For women, these variables have almost numerically the same impact on the decision to move, but it is interesting that wage growth in competing jobs has a somewhat larger impact than the wage growth of incumbent firm. For men, the immediate wage difference in levels is most important and wage growth in competing firm has less impact than wage growth in incumbent firm. Comparing the marginal effects for men and women, it can be seen that females are generally less responsive to wage differentials. This probably reflects that other issues play a bigger role for women in their decision to stay or move. Thus, this adds to the factors that may result in lower wages for women compared to men. Subsequently, we have run regressions on different subsets of data. In one case, we limited the sample to surviving firms since leaving a closing firm constitutes a different problem because there is no possibility staying on. Limiting the sample to surviving firms actually sharpens our results as we get more significant estimates with the expected signs. A similar situation arises, if the person is fired. See Kletzer (1998) for a survey of US work based on Displaced Worker Surveys. The problem here is, however, that we do not observe a firing. A simple but unjustified method would be to exclude from the estimation persons with a spell of unemployment between two jobs. The result here is also more significant, but without important changes in coefficients. Finally, we have run the probability functions for people belonging to different areas of wage bargaining. Here we are using Unemployment Insurance fund membership as a proxy for Trade Union and the bargaining procedure. Results are indicated with signs in Table 5.

326

Table 5.

Signs and significance in trade union probits (random firm effects)

wage residual

Women KAD (workers) HK (office clerks) Non-insured Other funds ns ¼ non-significant at the 5% level.

ns

+

+

alter wage

dwage residual

+ + + + + + + + + + + +

ns

dalter wage

ns ns + ns

+ + + ns + + + + ns ns + +

161,250 408,517 306,923 66,680 51,907 370,014 267,266 128,039 130,050 111,058 916,117 681,022

37,930 82,627 65,675 18,680 12,279 67,091 41,143 24,794 25,506 19,458 208,548 147,104 750,835

+ + + +

42,310 1,288,548 582,165 2,139,100

11,485 235,418 173,345 378,966

ns ns ns

+ +

+

dfirm wage

+ ns

+

+

Observation

Groups Paul Bingley and Niels Westergaard-Nielsen

Men SID (workmen) HK (office clerks) FTF (salaried employees) Kristelige (yellow) Metal workers Managers Engineers Professionals Sales agents Technicians Non-insured Other fund

firm wage

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327

Most of the signs are in accordance with our general results presented so far. The main reason for deviations seems to be differences in number of observations and size of groups. 7. Conclusions We have shown that high job mobility in Denmark may be (partly) explained by a model, where workers seek the best pay schemes and they do this by jumping between wage trajectories of different firms. This explanation is compatible with a search explanation, though our hypothesis is that the main mechanism is by competing firms making offers through raiding. We have proposed a model for mobility that for each period compares the wage obtained at the current firm with an assumed outside offer. Our task has been to test if the relative wage differential between the current job and the competing job is a determinant of the mobility decision. We have operationalized the wage differential into different elements, and these are defined according to the wage level of the current firm and the competing firm. Furthermore, we have defined concepts of wage growth in the current and the competing firm. The main result is that people tend to move if they can get a higher wage level in other firms. Differences in wage growth are also important. Men are more likely to move than women due to a competing firm offering a slightly higher wage level than the incumbent firm. The reverse is true with regard to wage growth prospects inside and outside the current firm, whereby women are more responsive than men. Our model shows that wage differentials play a key role in explaining the high level of labour turnover in Denmark. This does not exclude the possibility that other factors such as work environment, unobserved career opportunities and fringe benefits may play a role. Indeed, our intermediate results show that these firm related factors do indeed play a role. Only when controlling for these unobserved factors do we obtain sensible coefficients to all of the wage elements. The model also demonstrates that the wage policy of firms may be important in retaining key personnel. Acknowledgment The authors would like to thank an anonymous referee for excellent comments and suggestions. References Abowd, J., F. Kramarz and D. Margolis (1999), ‘‘High wage workers and high wage firms’’, Econometrica, Vol. 67(2), pp. 251–333.

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Abowd, J., F. Kramarz and C. Roux (2004), Wages, Mobility and Firm Performance: An Analysis Using Matched Employer– Employee Data from France, CREST, Paris: CREST. Albæk, K. and B. Sørensen (1998), ‘‘Worker flows and job flows in Danish Manufacturing, 1980–1991’’, Economic Journal, Vol. 108, pp. 1750–1771. Batt, R. (2002), ‘‘Managing customer services: human resource practices, quit rates, and sales growth’’, Academy of Management Journal, Vol. 45(3), pp. 587–597. Becker, G.S., E.M. Landes and R.T. Michael (1977), ‘‘An economic analysis of marital instability’’, The Journal of Political Economy, Vol. 85(6), pp. 1141–1188. Beffy, B., D. Fougere, T. Kamoinka and F. Kramarz (2004), ‘‘The returns to seniority in France (and why they are lower than in the United States)’’, CREST. Bingley, P., T. Eriksson, A. Werwatz and N. Westergaard-Nielsen (1999), Beyond Manucentrism – Some Fresh Facts About Job and Worker Flows, Aarhus: Center for Labour Market and Social Research. Bingley, P. and N. Westergaard-Nielsen (2003), ‘‘Returns to Tenure, Firm Specific Human Capital and Worker Heterogeneity’’ International Journal of Manpower, Vol. 24(7), pp. 774–788. Blanchflower, D. and A. Oswald (1996), ‘‘Wages, profit, and rent-sharing’’, Quarterly Journal of Economics, Vol. 111(1), pp. 227–252. Borjas, G.J. (1981), ‘‘Job mobility and earnings over the life cycle’’, Industrial and Labor Relations Review, Vol. 34(3), pp. 365–376. Christensen, B.J., R. Lentz, D. Mortensen, G. Neumann and A. Werwatz (2005), ‘‘On-the-job search and the wage distribution’’, Journal of Labor Economics, Vol. 23(1), pp. 31–58. European Commission (1998), Wages and Unemployment, Bruxelles: European Commission, Directorate V/A.1. Estevao, M. and S. Tevlin (2003), ‘‘Do firms really share their success with their workers? The response of wages to product market conditions’’, Economica, Vol. 71(280), pp. 597–617. Hall, R.E. and E.P. Lazear (1984), ‘‘The excess sensitivity of layoffs and quits to demand’’, Journal of Labor Economics, Vol. 2(2, Essays in Honor of Melvin W. Reder), pp. 233–257. Jensen, P. and N.C. Westergaard-Nielsen (1987), ‘‘A search model applied to the transition from education to work’’, The Review of Economic Studies, Vol. 54(3), pp. 461–472. Kletzer, L.G. (1998), ‘‘Job displacement’’, Journal of Economic Perspectives, Vol. 12(1), pp. 115–136. McLaughlin, K.J. (1991), ‘‘A theory of quits and layoffs with efficient turnover’’, Journal of Political Economy, Vol. 99(1), pp. 1–29. OECD (1997), Employment Outlook, Paris: Oecd.

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Parsons, D.O. (1977), ‘‘Models of Labor Market Turnover: A Theoretical and Empirical Survey’’, in: R. Ehrenberg, editor, Research in Labor Economics, Greenwich, CT: JAI Press. Royalty, A.B. (1998), ‘‘Job-to-job and job-to-nonemployment turnover by gender and education level’’, Journal of Labor Economics, Vol. 16(2), pp. 392–443. Topel, R.H. and M.P. Ward (1992), ‘‘Job mobility and the careers of young men’’, Quarterly Journal of Economics, Vol. 107, pp. 439–479.

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CHAPTER 14

Identification and Inference in Dynamic Programming Models Bent J. Christensen and Nicholas M. Kiefer Abstract We treat identification and inference in dynamic programming models. Many insights relevant for general dynamic programming models are motivated by earlier work on the econometrics of the search model. Issues considered include interval identification, the curses of determinacy and degeneracy, the precise role of Bellman’s equation in identification, and the dependence of parameter estimates on distributional assumptions in the random utility case.

Keywords: search, marketing, advertising campaign, likelihood, interval identification, curse of determinacy, curse of degeneracy, random utility, functional equation, optimality principle JEL Classifications: C13, C51, C61, J31, J64, M31 1. Introduction Econometrics done as a productive enterprise deals with the interaction between economic theory and statistical analysis. Theory provides an organizing framework for looking at the world and in particular for assembling and interpreting economic data. Statistical methods provide the means of extracting interesting economic information from data. Without economic theory or statistics all that

Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75014-8

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

331

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Bent J. Christensen and Nicholas M. Kiefer

is left is an overwhelming flow of disorganized information. Thus, both theory and statistics provide methods of data reduction. The goal is to reduce the mass of disorganized information to a manageable size while retaining as much of the information relevant to the question being considered as possible. In economic theory, much of the reduction is done by reliance on models of optimizing agents. Another level of reduction can be achieved by considering equilibrium. Thus, many ‘‘explanations’’ of behavior can be ruled out and need not be analyzed explicitly if it can be shown that economic agents in the same setting can do better for themselves. In statistical analysis, the reduction is through sufficiency – if the mass of data can be decomposed so that only a portion of it contains all the relevant information, the inference problem can be reduced to analysis only of the relevant data. Stochastic models are important in settings in which agents make choices under uncertainty, or, from a completely different point of view, in models which do not try to capture all features of agents’ behavior. Stochastic models are also important for models involving measurement error or approximations. These models provide a strong link between theoretical modeling and estimation. Essentially, a stochastic model delivers a probability distribution for observables. This distribution can serve as a natural basis for inference. In static models, the assumption of optimization, in particular of expected utility maximization, has essentially become universal. Methods for studying expected utility maximization in dynamic models are more difficult, but conceptually identical, and modeling and inference methods for these models are developing rapidly. The main work horses here are the fundamental dynamic programming model and likelihood analysis. The classic dynamic programming reference is Bellman (1957) who coined the term. This was followed by Bellman and Dreyfus (1962). Blackwell (1962, 1965) and Maitra (1968) provide the foundations for the modern approach to dynamic programming. Ross (1983) gives an accessible and brief introduction. Computational complexity increases rapidly with the dimension of the state space, leading Bellman (1957) to introduce the term ‘curse of dimensionality’. Treatment of identification and inference is a relatively new area. Key contributions are made by Rust (1987). A recent discussion of the curse of dimensionality with approaches to breaking the curse is Rust (1997). An early application of dynamic programming in economics is the sequential job search model, which is due to Mortensen (1970) and McCall (1970). See also Lippman and McCall (1976a, 1976b) and Mortensen (1986). Mortensen’s work has extended the range of applications of the search model from unemployment to labor turnover, research and development, personal relationships, and labor reallocation. His insight, that friction is equivalent to the random arrival of trading partners, generates immediately a stochastic model that lends itself to structural likelihood analysis. This is a key instance of fruitful data reduction through the marriage of statistical analysis and theory of optimizing agents.

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Kiefer and Neumann (1979) and Christensen and Kiefer (1991) are contributions in this direction. Mortensen (1990) and Burdett and Mortensen (1998) initiated the development of equilibrium dynamic models designed to account for wage dispersion and the time series behaviour of job and worker flows. This leads to the second level of data reduction, obtained by imposing equilibrium conditions on the statistical framework for optimizing agents. Empirical analysis and inference issues are studied by Kiefer and Neumann (1993), van den Berg and Ridder (1993, 1998), Christensen and Kiefer (1997), Bunzel et al. (2001), Christensen et al. (2005) and many others. In this paper, we treat identification and inference issues in dynamic programming models. To see the issues simply and clearly, we present them in simple discrete state/control settings. A main example running through the sections is a simple marketing model. The study of optimal advertising policy in dynamic models where current investment in advertising affects future demand was pioneered by Nerlove and Arrow (1962). An empirical investigation in the dynamic duopoly case is presented by Chintagunta and Vilcassim (1992). We draw attention to several issues that apply in wide generality in dynamic programming models but have received little or no notice previously, such as identification of parameters only up to intervals, rapid information accumulation, the additional curses of determinacy and degeneracy stemming from the fact that the model predicts a deterministic relation between the state and control, the subtle differences between measurement error, imperfect control and random utility approaches, and the dependence of the parameter estimates on the assumed shock distribution in the random utility case. Many insights derive from earlier work on the econometrics of the search model, and we review some of the relations between the motivating studies of the search model and the implications for general dynamic programming models. That relations exist is natural, of course, since components in one model frequently have equivalents in other models, e.g. unemployment in the search model corresponds to inventory in the marketing model, etc. A fuller account of all these and further general economic modeling and inference issues is provided by Christensen and Kiefer (2005). The rest of the paper is laid out as follows. In Section 2, we introduce the fundamental dynamic programming model and our marketing example. Section 3 considers discrete states and controls and provides illustrative computations. Section 4 is a preview of important identification issues. Likelihood functions are introduced in Section 5, and identification and inference treated in detail. Measurement error is introduced in Section 6. Section 7 considers the case of imperfect control. Section 8 introduces random utility models, and the case of continuously distributed shocks is treated in Section 9. The connections between the results for general dynamic programming models and the motivating work on the econometrics of the job search model are reviewed in Section 10, and Section 11 concludes.

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2. Dynamic programming: the marketing example The basic components of a dynamic optimization model are the objective function, the state variables, the control variables with any associated constraints, and the transition distribution giving the evolution of states as a stochastic function of the sequence of states and controls. These components are illustrated here in a simple marketing example. Consider a firm deciding whether to invest in a marketing campaign. Suppose that there are two states of demand, high and low. The marketing decision is to run the campaign or not. Profit in any period is given by demand less costs including marketing costs. Since there are only four configurations of states (demand) and controls (the marketing decision) the profit function is given by four numbers, u(x,c),(x,c)A{0,1}2. Low demand is indicated by x ¼ 0 and marketing by c ¼ 1 While simple, our discrete model relates naturally to continuous state/control dynamic models of investment in advertising campaigns, e.g. Chintagunta and Jain (1992). A plausible profit function might be the one in Table 1. Here, the low-demand state with no marketing generates the same profit as the high-demand state with marketing. The objective function of the firm is to maximize the expected present discounted value of profits ESTt¼0 bt uðxt ; ct Þ, where T is a ‘‘horizon’’ which may be infinite and bA[0,1) is a discount factor. Dynamics are incorporated in the model by letting the probability distribution of demand next period depend on the marketing decision this period. If the advertising campaign is effective, then the probability that xt+1 ¼ 1 is greater when ct ¼ 1 than when ct ¼ 0. A simple case has the distribution of xt+1 depending only on ct and not on the current state. For example, consider the particularly simple case p(xt+1 ¼ 1|ct ¼ 1) ¼ 1 and p(xt+1 ¼ 1|ct ¼ 0) ¼ 0. Thus the state of demand in period t+1 is determined exactly by the marketing effort in period t. The logic of dynamic programming can be illustrated by considering the 2-period problem. In the last period, clearly there is no benefit from marketing and c ¼ 0 is optimal no matter what the level of demand. Now consider the first period. Here, there is a tradeoff between current period profits (maximized by c ¼ 0) and future profits (maximized by c ¼ 1). If the current state of demand is low (x ¼ 0) then the current cost of marketing is 3 and the current period value of the gain next period as a result of marketing is 4b. Thus for b>3/4 it is optimal to invest in Table 1.

Profit function u(x,c)

x¼0 x¼1

c¼0

c¼1

7 11

4 7

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marketing in the first period when the state of demand is low. Suppose the state of demand in the first period is high. Then the marketing effort costs 4 in current profit, and gains only 4b in current value of future profits. Since bo1 this effort is not worthwhile. We have found the optimal policy. The logic of beginning in the last period in finite horizon models and working backwards in time is known as ‘‘backward recursion’’ and is generally the way these problems are solved. A more general and plausible specification would have the distribution of next period’s demand depending on both the marketing decision and the current state of demand. As a matter of interpretation, this allows for marketing to have lasting effects (though tapering off over time). This possibility is pursued further below. 3. Discrete states and controls The discrete state/control dynamic programming model has many applications, can be treated with fairly elementary methods, and allows illustration of most of the important issues of identification and estimation that appear in more general settings. The simple case allows focus on issues of substance rather than details. The latter are important, but can be better handled once the substantive issues are identified. Note that the discrete case is in many ways quite general. For practical purposes machine calculations are discrete, as are data, and indeed applications of continuous models often (but not always) require explicit discretization. In the finite horizon case it is conventional to index value functions by the number of periods left, rather than distance from the current period ð0Þ. Thus, in a T-period optimization, the value function at the outset is VT

1 ðxÞ

¼ max E T p

1;

p

T 1 t St¼0 b uðxt ; ct Þ,

ð1Þ

where p ¼ (pT 1,pT 1,y,p0) is a sequence of policy functions. A policy function at period t maps the current state xt and all previous states into the current policy ct. Thus, we are maximizing over a sequence of functions. The expectation operator is a little more subtle – it is a conditional expectation conditioning on the current value of the state variable (hence the T 1 subscript) and on the policy p. Since the transition distribution typically depends on controls, the expectation clearly depends on the policy. The value function in the final period is V0(x). With u(x,c) the immediate reward from using control value c with state x, and C the set of admissible controls, the final period value function is clearly V0(x) ¼ maxcAC u(x,c). The value function with one period left V1(x) is just the maximized value of the current reward and the discounted expected value of future rewards. But the future reward is V0(x), so the function V1(x) is given by
 V 1 ðxÞ ¼ max uðx; cÞ þ bE 1 V 0 ðx0 Þ . ð2Þ c2C

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Here, x is the next period state – a random variable whose distribution is determined by c and x, current controls and state. Iterating, we obtain the whole sequence of value functions V0, y ,VT 1 with
 V t ðxÞ ¼ max uðx; cÞ þ bE t V t 1 ðx0 Þ . ð3Þ c2C

This recursion is known as the optimality principle, or Bellman’s equation. The calculations above are simple and can be done straightforwardly using, for example, a spreadsheet program. We illustrate using the marketing model of Section 2. Here, the state variable is a state of demand, xA{0,1}, and the control variable is an advertising decision cA{0,1}. The profit function is given in Table 1. A heuristic backward recursion argument together with a trivial transition distribution was used to obtain the solution for the 2-period problem in Section 2. Here, we make the problem a little more interesting and realistic by specifying the transition distribution in Table 2. Thus, when demand is low it is unlikely that demand will increase without the marketing effort; when demand is high and marketing efforts are 0 the next period’s demand states are equiprobable; in either case marketing improves the probability of next period’s demand being high to 0.85. Discounting the future at b ¼ 0:75 we calculate the value functions and optimal policies by backward recursion for the 1 through 10-period problems and report results in Table 3. In the 1-, 2-, and 3- period problems it is never optimal to run the marketing campaign. In the 4-period problem, it is optimal to run the marketing program in the initial period if the state of demand is low. This makes sense; it is optimal to try to shock the system into the fairly persistent high-demand state, then abandon the marketing investment. In the longer problems, it is optimal to run the marketing program if demand is low and the remaining horizon is greater than 3 periods. It is never optimal to run the marketing program during the high-demand periods. Define the operator T by
 Tf ¼ max ðc; xÞ þ bEf ðx0 Þ

ð4Þ

c2C

in the infinite horizon stationary case and note that T is not subscripted with t. Then the optimality equation can be written V ¼ TV and V is the unique bounded solution to this functional equation. The value function V solving the optimality equation can be computed as the limit of T-period finite-horizon Table 2.

Transition probabilities p(xt+1 ¼ 1|xt,ct)

xt ¼ 0 xt ¼ 1

c¼0

c¼1

0.1 0.5

0.85 0.85

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Table 3.

337

Value functions and optimal policies

t

Vt(0)

Vt(1)

ct(0)

ct(1)

0 1 2 3 4 5 6 7 8 9

7.00000 12.55000 16.80250 20.14638 22.64185 24.51672 25.92201 26.97621 27.76680 28.35976

11.00000 17.75000 22.36250 25.68687 28.18746 30.06099 31.46664 32.52074 33.31135 33.90430

0 0 0 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0

value functions as T-N. Let V0(x) ¼ maxcACu(x,c), the value in the 1-period problem. Note that V 1 ¼ maxfuðx; cÞ þ bEV 0 g ¼ TV 0 is the value function for the 2-period problem, where T is the operator defined in (4). Then
 V T ðxÞ ¼ max uðx; cÞ þ bEV T 1 ðxÞ c2C

¼ TV T

1

¼ TT

1

V0

ð5Þ

has the interpretation of the value of the T+1 period problem with final period state-dependent reward V0(x). We have V T ðxÞ

V ðxÞ ! 0 as T ! 1,

ð6Þ

since Tð
Þ satisfies Blackwell’s (1965) conditions of monotonicity and discounting and hence is a contraction (see also Stokey and Lucas, 1989). To illustrate we consider Table 3 giving the value function in the marketing model for the T ¼ 1, y ,10-period problems and add results in Table 4 for the 45 through 50-period problem. Here, the value function iterations are identical to 6 digits – the improvements in the calculations are only relative changes of order 10 7. Note, however, that the policy function appears to have converged much more rapidly. This is frequently the case in discrete state/control models. It is much easier to determine a map from one finite set to another than it is to determine the exact value of a real vector. Notice, on a computer the value can only be determined to a certain level of precision.

4. Identification: a preview A central identification issue can be easily illustrated in the context of the marketing model of Sections 2 and 3. The issue will be pursued in detail in Section 5. Parameters are split into two groups: those that relate to the transition distribution at the optimal policy, and those composed of utility, discount factor, and

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Table 4. t 45 46 47 48 49

Value function iterations

V(0)

V(1)

c(0)

c(1)

30.13856 30.13857 30.13858 30.13859 30.13860

35.68311 35.68313 35.68314 35.68314 35.68315

1 1 1 1 1

0 0 0 0 0

transition probabilities for nonobserved transitions. The basic result is that a discrete state/discrete control model can only determine parameters within certain ranges. More information is required in order to determine some of the economically interesting parameters. The information is sometimes introduced in specification, and we will consider this possibility, too. Consider the data sequence fxt ; ct gTt¼0 generated by the optimal policy in the marketing model. The data can be summarized in two tables reflecting the within-period information on the control rule and the intertemporal information on the transitions (see Table 5). Here, n(x ¼ j) is the number of time periods t the state variable xt is observed in state j, and nx(jk) is the number of observed transitions from state j to k. The transition distribution can clearly be estimated at the optimal policy. Note that the components of the transition distribution at other values of the policy contribute to determination of the optimal policy. Information on these transition probabilities is available only through observation of the optimal policy and any restrictions implied by the functional equation, which depends on alternative transition probabilities and characterizes the optimal policy. This relationship is subtle and we will set it aside for purposes of this example by simply assuming that the transition distribution is known. Returns u(x,c) may also be observed, in which case the components of the return function corresponding to the optimal policy can be estimated. The other components of the return function, corresponding to state/control configurations never observed, can be identified only through restrictions implied by the optimality equation. Again, the functional Equation (4) (or (3), in the finite horizon case) depends on alternative rewards and characterizes the optimal policy. We will set this aside as well by assuming that the reward function is known. The only remaining unknown parameter is the discount factor b. Suppose we observe the entire infinite sequence fxt ; ct g1 t¼0 so that any parameter that can be estimated consistently is known. What can be said about b? Not much. We have seen that the policy c(0) ¼ 1;c(1) ¼ 0 is optimal for b ¼ 0:75. It is obviously optimal for any larger value of b. What about smaller values? It turns out that this policy is optimal for b  0:7143 (approximately) and the policy of c(0) ¼ c(1) ¼ 0 is optimal for smaller values of the discount factor. Thus, the most we can expect the data to tell us is whether b  0:7143 or not. Of course, b is

Identification and Inference in Dynamic Programming Models

Table 5.

339

Data configuration

Policy information

x¼0 x¼1

c¼0 0 n(x ¼ 1)

c¼1 n(x ¼ 0) 0

Transition information

xt ¼ 0 xt ¼ 1

xt+1 ¼ 0 nx(00) nx(10)

xt+1 ¼ 1 nx(01) nx(11)

a continuous parameter with parameter space [0,1), but all we will be able to tell from the data is which of the sets {[0,0.7143),[0.7143,1)} the parameter b is in. This feature of finite state/finite control dynamic programming models, that continuous parameters are identified only up to ranges in the parameter space, is ubiquitous and it is not specific to our example. The situation is not completely hopeless. Of course, continuous parameters corresponding to transition probabilities at the observed policy are typically identified. Similarly, if rewards are observed, rewards corresponding to state/control combinations given by the optimal policy are typically identified. It is the parameters which must be identified through the restriction imposed by the functional equation that are typically underidentified. Note that the above analysis applies to observations generated by a single decision maker over time, or to panel data on many decision makers following the optimal policy. Panel data do not help. Observations on the finite horizon control policy are more informative. Here, the policy is not stationary and information on how the horizon changes the policy is informative. However, identification is not possible – instead we increase the number of intervals to which we can assign b. With b ¼ 0:75, we saw that the policy c(0) ¼ 1 and c(1) ¼ 0 is optimal if the horizon is 4 periods or longer. At b ¼ 0:72 that policy is optimal if the horizon is 5 periods or longer. At b ¼ 0:715, optimality requires 7 periods or longer, at 0.7145 it requires 8 periods or longer, etc. Thus, if we observe panel data on decision makers and we know their horizons we can isolate the horizon at which the policy shifts. In this case it is possible to use the optimality equation to narrow the range of possible b consistent with the policy. But it remains impossible to identify b more closely than an interval. 5. Likelihood functions We begin with the likelihood function in the simplest dynamic programming case: a single binary state and a single binary control variable in a stationary

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infinite horizon problem. This case, while simple, illustrates properties which are general. In the case of accurate observations, the control is a deterministic function of the state, and hence the control policy can be taken as known after a small sample is realized (specifically, after both possible values of the state are realized). Of course, in many applications this setup would be unrealistic, in part because the model is just a model and is not purporting to be an exact description of the world, but it is a useful starting point for considering identification issues. In a sense, this is a situation of maximal information. If a parameter is not identified in this setting, it is hard to argue that additional data information somehow appears when the setting is generalized. After treating the simple case, we show that the notation and techniques extend to the general discrete model. Then, we consider extensions allowing measurement error, imperfect control and random utility. Identification issues are treated in detail. Parameters are split into two groups: those that relate to the transition distribution at the optimal policy, and those composed of utility, discount factor, and transition probabilities for non-observed transitions. Continuous parameters apart from the transition distribution parameters are typically unidentified. They are only restricted to lie in certain ranges in the parameter space, even asymptotically. The results from our marketing model in Sections 2–4 are general. We focus on the infinite horizon problem t V ðxÞ ¼ max ES1 t¼0 b uðxt ; ct Þ,

ð7Þ

p

where the expectation is over a Markov transition distribution p(xt|xt 1,ct 1) and hence the optimal policy (c(x),c(x),y) is stationary, or equivalently
 V ðxÞ ¼ max uðx; cÞ þ bEV ðx0 Þ . ð8Þ c2C

The observables are the state sequence fxr gTr¼0 and the control sequence fcr gTr¼0 , with xr and cr in {0,1}. We assume that the reward sequence is not observed (this will be treated later). The state transition probabilities are state and control dependent. Thus, we have Table 6. In Table 6, pab(c) is the probability of a transition from a to b when the period t control is c. Given the adding up constraints pa0(c)+pa1(c) ¼ 1, there are four transition probabilities in the model. The policy function is a pair (c(0),c(1))A{0,1}  {0,1} ¼ {00,01,10,11}. There are four possible policy Table 6.

Transition probabilities

ct ¼ 0 xt ¼ 0 xt ¼ 1

p00(0) p10(0)

ct ¼ 1 p01(0) p11(0)

p00(1) p10(1)

p01(1) p11(1)

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341

functions, of which only two are interesting since in the other cases the policy does not depend on the states. Given the period 0 values of the state and control, the period 1 distribution of the observables conditional on parameter vector y is p(x1,c1|x0,c0,y). Now, this is a singular distribution, in that ct is a deterministic function of xt for any t and in particular for t ¼ 0 and 1. This important point has several implications. First, it suffices to condition on one of x or c; we choose to condition on x. Second, the distribution of x and c given parameters is completely described by the distribution of x alone. Nevertheless, the information given by c in addition to x in the likelihood function is enormous. In fact, the deterministic relation between c and x is learned with certainty as soon as the different values of x are seen in the data. Thus, parameters that enter only through the deterministic relation between c and x are learned, if they are identified, rapidly. We proceed by conditioning. Let x,x0 and c,c0 refer to current and 1-period ahead values of the state and control, respectively. For the present, we suppress dependence on the parameter vector y. Then pðx0 ; c0 jx; cÞ ¼ pðx0 jx; cÞpðc0 jx0 ; x; cÞ ¼ pðx0 jxÞpðc0 jx0 ; xÞ ¼ pðx0 jxÞpðc0 jx0 Þ,

ð9Þ

since there is no point in conditioning on c as well as x, and since c0 is a deterministic function of x0 and hence x is irrelevant for c0 given x0 . The second factor is just I(c0 ¼ c(x0 ,y)) where I is the indicator function and y has been reinserted here for emphasis: given y, the policy function can be calculated and this factor of the distribution easily evaluated. The likelihood is Lðy; xr ; cr ; r ¼ 1; . . . ; TÞ ¼ PTr¼1 pðxr jxr 1 ; yÞpðcr jxr ; yÞ

¼ PTr¼1 pðxr jxr 1 ; yÞPTr¼1 pðcr jxr ; yÞ.

ð10Þ

Since the second factor is zero for values of y inconsistent with the data and the first term is positive, only values consistent with the data have positive likelihood. As a practical matter, it is useful to check early on whether there exists any parameter value consistent with the data. In many cases, using this simple specification, there will not be, and hence the model is rejected and must be modified. However, this is a good place to begin the study of identification. Suppose we have a sample of size T, indexed by subscripts r,s,tA{0, y, T 1 T 1}. Consider the state sequence fxt gt¼0 . There are 2T such sequences. Arrange these sequences in lexicographic order and index these by i,j,kA{0, y, 2T 1}. Then the ith sequence is the binary expansion of i. This is a convenient way of thinking about the problem. Our random variable is now i, the position of the realized sequence. Table 7 illustrates for the case T ¼ 3. Elements of the sequences are indexed by r,s,t. Let x(i,s) be the sth digit in the binary expansion of i. Let k(i,r) ¼ x(i,r)||x(i,r+1) be the rth pair of digits in the binary expansion of i.

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Table 7.

State transitions

i

Sequence

0 1 2 3 4 5 6 7

000 001 010 011 100 101 110 111

The parameters entering p(x0 |x) are the transition probabilities corresponding to the optimal policy. We write these pab,a,bA{0,1} without an argument to select the appropriate pieces from Table 6. Thus, there are two probabilities to be estimated. Of course, there could be restrictions relating these probabilities. Further, knowledge of the optimal policy could restrict the values of these probabilities (for example if these were the only parameters and the optimal policy was known). However, the restricted estimators can usually be written as functions of the unrestricted, and the unrestricted maximum likelihood estimators (MLEs) are easy to obtain. We will set aside consideration of these issues for the moment. The likelihood function corresponding to the first factor in (10) is T 1 pðijx0 Þ ¼ Pr¼1 pkði;rÞ .

ð11Þ

T 1 Introducing the notation Nkði; abÞ ¼ Sr¼1 Iðkði; rÞ ¼ abÞ, the number of ab pairs in the ith sequence, we have

pðijx0 Þ ¼ Pa;b pNkði;abÞ ; a; b 2 f0; 1g. ab

ð12Þ

Taking logarithms and writing pa1 ¼ 1 pa0 yields lðp00 ; p10 ji; x0 Þ ¼ Nkði; 00Þlnp00 þ Nkði; 01Þlnð1 p00 Þ þ Nkði; 10Þlnp10 þ Nkði; 11Þlnð1 p10 Þ,

ð13Þ

and the maximum likelihood estimators p^ a0 ¼ Nkði; a0Þ=ðNkði; a0Þ þ Nkði; a1ÞÞ; a 2 f0; 1g.

ð14Þ

The step of taking logarithms is justified only where the probability is positive, that is for values of pa0 consistent with the control rule. This important point is illustrated below. Consider the marketing model of Sections 2 through 4. This is a 2-state, 2-control problem. In Sections 2 and 3 the focus was on solving the dynamic program; here we consider estimation. First, suppose the only unknown parameter is the transition probability p00. Suppose we have a sample of length T, and suppose in this sample each value of x is observed at least once. Then the

Identification and Inference in Dynamic Programming Models

Table 8.

343

Marketing model Profit

x¼0 x¼1

c¼0

c¼1

7 11

4 7

p(xt+1 ¼ 1|xt,ct)

xt ¼ 0 xt ¼ 1

ct ¼ 0

ct ¼ 1

0.1 0.5

0.85 0.85

control rule c(x) ¼ (c(0),c(1)) is known. In evaluating the likelihood, only values of the parameter p00 consistent with the observed state-control sequence and the hypothesis of optimization are considered (the likelihood is zero for other values of the parameter). How does knowledge of the control rule constrain the value of p00? With the utility function and transition distributions given in Table 8 and with discount factor 0.75 we found that the optimal policy in the infinite horizon problem is c(x) ¼ (c(0),c(1)) ¼ (1,0). Here, the value of p00, the 0–0 transition probability corresponding to c ¼ 1, is 1 0.85 ¼ 0.15. Clearly, there are other values of this transition probability for which c ¼ (1,0) is also optimal. For example, any smaller value would leave the optimal policy unchanged – the purpose of the marketing campaign is to shock the system into the high demand state, so if this becomes easier, it must still be optimal. In other words, the value associated with running the campaign is increased for higher 0–1 transition probability (and hence lower 0–0, our parameter). Hence if it is optimal to run the campaign for p00 ¼ 0.15, it must also be optimal for any smaller value. In fact, c ¼ (1,0) is optimal for p00r0.20 ¼ r. Thus, once the control rule is known to be c ¼ (1,0) (i.e., after each value of the state variable has been seen), the information contained in the likelihood factor corresponding to Pt pðct jxt Þ is exactly that p00A[0,r]. This information is in one sense extremely precise. It is accumulated quickly (this is not a T-N result) and it completely rules out a portion of the natural parameter space [0,1]. It is in another sense extremely imprecise. The exact value of the parameter p00 cannot be estimated on the basis of the information in the control rule – only bounded into an interval. There is no more information in the relevant likelihood factor, even as T-N. The situation is identical to that discussed in Section 4 relative to estimation of the discount factor b. The difference here is that there is additional information on p00 available through the first factor in the likelihood function, corresponding to transition information. We turn now to this factor, Lðp00 ji; x0 Þ ¼ Nkði; 00Þlnp00 þ Nkði; 01Þlnð1

p00 Þ,

ð15Þ

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Bent J. Christensen and Nicholas M. Kiefer

which can be maximized subject to the constraint p00rr. The Lagrangian is Lðp00 ji; x0 Þ ¼ Nkði; 00Þlnp00 þ Nkði; 01Þlnð1

p00 Þ þ lðr

p00 Þ,

ð16Þ

where l is a Lagrange multiplier. The Kuhn–Tucker conditions require Nkði; 00Þ=p00

Nkði; 01Þ=ð1 r p00  0 lðr

p00 Þ

l¼0

ð17Þ

p00 Þ ¼ 0

and the solution possibilities are clearly either l^ ¼ 0 and p^ cmle ¼ p^ ml or p^ cmle ¼ r and l^ ¼ Nkði; 00Þ=r Nkði; 01Þ=ð1 rÞ. A significant nonzero value for l^ indicates that the transition information is inconsistent with the control information, and that the model is therefore likely misspecified. This point is pursued later. For the present, assume that the unconstrained MLE satisfies the constraint. In Section 4, we saw informally that the discount factor was not identified by knowledge of the control rule even when all other parameters including the transition probabilities were known. The parameter was bounded to an interval. In fact, the transition information is informative on the discount factor b, in that the boundary of the interval containing feasible estimates of b depends on the transition probabilities. We consider the case y ¼ (p00,b); unknown discount factor and transition probability. Here the natural parameter space is Y ¼ ½0; 1Š  ½0; 1Þ. As soon as the control rule is observed, this can be narrowed. Specifically, in our example with the observed control rule c ¼ (1,0), the parameter space consistent with the observed control rule is ACY, illustrated in Figure 1. This is all of the parameter information contained in the control rule. Turning now to the transition information on the two parameters, the loglikelihood is Lðp00 ; bji; x0 Þ ¼ Nkði; 00Þlnp00 þ Nkði; 01Þlnð1

p00 Þ,

ð18Þ

to be maximized subject to the constraint that yAA. Note that this portion of the loglikelihood does not depend on the discount factor b at all. Consequently, the Figure 1. β 1

Parameter space 0.45

A

0.625

Θ 0 p00 0

1

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345

constraint is once again in the form p00rr and can be imposed as above. Once again, the estimator will satisfy the constraint asymptotically if the model is wellspecified, and if the unconstrained estimator does not satisfy the constraint, the significance of the estimate of the Lagrange multiplier can be used as the basis of a specification test. Turning to the unconstrained estimator p^ 00 , we note that the transition information is informative on the discount factor b in that the interval in which b can lie and still be consistent with the known control rule depends on the value of p^ 00 . Can a general result be obtained in the simple case with the control rule known? Suppose there are K state variables, the ith taking values in the discrete set Hi with cardinality|Hi| ¼ Hi and C control variables, the jth taking values in the discrete set Jj with cardinality|Jj| ¼ Jj. The problem can be rewritten as a single discrete state/discrete control model with the single state variable x taking H ¼ Pi H i values from the set H and the single control taking J ¼ Pj J j values from the set J. The control rule is a map c:H-J, a point in L ¼ JH, a finite set. Now write the control rule c as a function of the parameter y, c(x;y), where yAYDRk. Regarded as a function of y we have c:Y-L. The identification question is whether we can map backwards from knowledge of the control rule to the parameter y, i.e. whether the map c is invertible. The general answer is no. Brouwer’s Theorem on Invariance of Domain states that there is no homeomorphism between spaces of different dimensions. Thus, if the parameter space is even just an interval in R1, there is no way to identify the unknown (in this case scalar) parameter from knowledge of the control rule (a point in a finite set). As we have seen, the parameter values can be bounded, but the parameter cannot be estimated without further information. Further information comes from the transition distribution. There are H origin states and H destination states, hence H2 transition probabilities less H from the adding-up constraint for a given value of the control rule. Since transitions are only seen under the optimal control, these are the only transition probabilities that can be estimated using transition data. Of course, the other transition probabilities enter the problem in determining the control rule. These can be estimated only from the control rule, and hence can at best be bounded into an interval. The notation developed above generalizes easily. There are HT possible sequences of states of length T. Arrange these sequences in lexicographic order and index these by i,j,kA{0, y, HT 1}. Then the ith sequence is the H-ary expansion of i. Let x(i,s) be the sth digit in the H-ary expansion of i. Let k(i,r) ¼ x(i,r)||x(i,r+1) be the rth pair of digits in the H-ary T 1 expansion of i. Finally, let Nkði; abÞ ¼ Sr¼1 Iðkði; rÞ ¼ abÞ, the number of ab pairs in the ith sequence (here a,b take on H distinct values). The log likelihood factors according to the origin state, so we have for example for origin state 0 lðp0a ; a ¼ 0; . . . ; H

H 1 1ji; x0 Þ ¼ Sa¼0 Nkði; 0aÞlnp0a ,

ð19Þ

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Bent J. Christensen and Nicholas M. Kiefer

1 defined with the constraint SH a¼0 p0a ¼ 1. The model is clearly in the exponential family. The MLE’s are H 1 Nkði; 0aÞ, p^ 0a ¼ Nkði; 0aÞ=Sa¼0

ð20Þ

if these values are consistent with the observed control rule. Thus, H(H 1) transition probabilities can be estimated from the transition data. Note however that a constraint here (e.g. some transitions are impossible, others are necessary, etc.) does not imply that there are degrees of freedom available for estimating utility function parameters. At most, these estimates can be used to refine the bounds on parameters imposed by knowledge of the control rule. Although the model predicts that a given state should always be associated with the same control, and therefore the control rule is learned rapidly (as soon as each possible state has been observed once), the data will rarely satisfy such a strong requirement. This is the ‘‘curse of determinacy.’’ The model, although stochastic, predicts a deterministic relationship between the state and control. This is one of the major difficulties in applying dynamic programming models empirically. There are a number of approaches to modify the model to be consistent with data not satisfying this deterministic constraint. The approaches are not equally successful. 6. Measurement error One natural approach to breaking the curse of determinacy is to allow for measurement error. The idea here is that the model is an accurate description of behavior, but that we are not measuring exactly the quantities entering the optimization problem. We first develop the appropriate notion of measurement for discrete models, working first with the binary situation as above. Then we consider in turn measurement error in the state and in the control. Finally, we consider the case of measurement error in both state and control. We find sensible specifications which do break the curse. A simple specification for measurement error in a binary model is to allow a constant misclassification probability (crossover probability) e. Letting x be the true state and x the observed, the model for the measurement process is illustrated in Figure 2. Suppose x is a realization of a sequence from a Markov chain with transition probabilities p00, p01 ¼ 1 p00, p11, and p10 ¼ 1 p11. Thus, trivially, P(x2 ¼ 0|x1 ¼ 0,x0 ¼ a) ¼ p00 ¼ P(x2 ¼ 0|x1 ¼ 0). To check, note that P(x1 ¼ 0|x0 ¼ a) ¼ pa0 and P(x2 ¼ 0,x1 ¼ 0|x0 ¼ a) ¼ pa0p00 and dividing gives the result. Suppose x is the sequence with measurement error, so that xt ¼ xt with probability 1 e and xt ¼ 1 xt with probability e. Let us calculate Pðx2 ¼ 0jx1 ¼ 0; x0 ¼ aÞ. We begin by calculating Pðx1 ¼ 0jx0 ¼ aÞ. Note first that all 2  2 ¼ 4 sequences of length 2 for the true state variables are consistent with observing the sequence a0. Thus, Pðx1 ¼ 0jx0 ¼ aÞ ¼ pa0 ð1


Þ2 þ pa1 ð1


Þ
þ pc0
ð1


Þ þ pc1
2

ð21Þ

Identification and Inference in Dynamic Programming Models

Figure 2. True x

347

Measurement error probability

Observed x*

(1-ε) 0

0 ε ε

1

1

(1-ε)

where the index c ¼ 1 a. Next, calculate Pðx2 ¼ 0; x1 ¼ 0jx0 ¼ aÞ. All 23 sequences are consistent with the observed pattern a00. The probability is thus the sum of 8 terms. The term corresponding for example to the true sequence a01 is pa0p01(1 e)2e, the probability that a process beginning in a is observed to be in a; (1 e), times the probability of moving from a to 0, pa0 multiplied by the probability of being observed correctly in the second period, (1 e), times the probability of moving from 0 to 1 and being incorrectly observed in the final period (p01e). Upon adding these terms and dividing we see that Pðx2 ¼ 0jx1 ¼ 0; x0 ¼ aÞ ¼ Pðx2 ¼ 0; x1 ¼ 0jx0 ¼ aÞ=Pðx1 ¼ 0jx0 ¼ aÞ aPðx2 ¼ 0jx1 ¼ 0Þ.

ð22Þ

Thus, the x process is not Markovian. This suggests a simple diagnostic before calculating estimates for a complex dynamic programming model. Namely, examine the state sequence to see whether it looks Markovian. If not, either reformulate the state variable specification (sometimes the model can be made Markovian by appropriate choice of the state variables) or consider the possibility of measurement error in the state variable. To formulate the likelihood for the observed sequence x, we first calculate the probability of seeing x conditional on the actual underlying sequence x, then essentially marginalize with respect to x. This strategy is attractive since it is easy to calculate the probability of observing the x sequences – realizations from Markov processes. We have 

pðx jiÞ ¼ PTt¼1
jxt

xði;tÞj

ð1


Þ1

jxt xði;tÞj

,

ð23Þ

the probability of observing the sequence x when the ith x sequence was actually realized (recall our convention on ordering the sequences). The probability of realizing the ith x sequence of length T is T 1 pðiÞ ¼ Pr¼1 pkði;rÞ

ð24Þ

where k(i,r) is the rth pair of digits in the binary representation of i. Hence, the marginal probability of observing the measured sequence x is T



2 1 T Pt¼1
jxt pðx Þ ¼ Si¼0

xði;tÞj

ð1


Þ1

jxt xði;tÞj

T 1 Pr¼1 pkði;rÞ ,

ð25Þ

348

Bent J. Christensen and Nicholas M. Kiefer

and in the absence of additional information this would serve as the likelihood function for the unknown parameters y ¼ (p00,p11,e). There is however additional information in both the known control rule (given parameters) and the observed control sequence. Knowledge of the control rule as we have seen restricts the range of possible parameter estimators. Observation of the control sequence, however, is equivalent to observation of the true state sequence, given the parameters. The data series consists of the sequence c and the sequence x , with joint probability distribution pðc; x Þ ¼ ¼

X x

pðc; x; x Þ

x

pðcjx; x Þpðx jxÞpðxÞ,

X

ð26Þ

conditioning on a value of the true sequence x and then marginalizing. This formulation is useful since p(c|x) ¼ p(c|x,x) is degenerate at c ¼ c(x), with c(x) from the control rule (note that this function does depend on parameters). We treat here for simplicity the 2  2 case with c(x) invertible, noting that the results apply immediately to K  K models, and treat the noninvertible case below (this is the case with more state than control variables). Thus, the probability p(c|x) is zero except for the ith sequence x, where the ith sequence satisfies ct ¼ c(x(i,t)). Hence 

pðc; x Þ ¼ PTt¼1
jxt

xði;tÞj

ð1


Þ1

jxt xði;tÞj

T 1 Pr¼1 pkði;rÞ ,

ð27Þ

^ abÞ=Sab NkðiðyÞ; ^ abÞ and yielding the MLEs y^ ¼ ðp^ 00 ; p^ 11 ;
^Þ with p^ ab ¼ NkðiðyÞ; ^ tÞj with iðyÞ ^ the index of the x sequence satisfying
^ ¼ T 1 St jxt xðiðyÞ; ^ where the presence of parameters in this condition is now explicit c ¼ cðx; yÞ, for emphasis. This is important: when the likelihood is evaluated at a different parameter value y, it may require different i, as well. The likelihood is thus only piecewise continuous in parameters, so some care must be ^ taken in ensuring that the estimators satisfy the constraint that x ¼ c 1 ðc; yÞ where y^ is the estimator, and that the estimators in fact correspond to a ^ typically global likelihood maximum. In fact, the control rule and thus iðyÞ do not depend on the unknown parameter e, which affects only observation of the data and does not enter the agent’s optimization problem. Thus, we have an explicit solution for the maximizing value of e given the other parameters. This can be substituted back into the loglikelihood
lðp00 ; p11 ;
Þ ¼ STt¼1 jxt

xði; tÞjlnð^
Þ þ ð1

T 1 lnðpkði;rÞ Þ, þ Sr¼1

jxt

xði; tÞjÞlnð1

b
Þ



ð28Þ

Identification and Inference in Dynamic Programming Models

349

to yield the profile loglikelihood function
lðp00 ; p11 jc; x Þ ¼ STt¼1 jxt ð1

þ

jxt T 1 Sr¼1

1

xði; tÞj lnðT xði; tÞjÞ lnð1 lnðpkði;rÞ Þ

¼ Tð^
lnð^
Þ þ ð1

Ss jxs

T


^Þ lnð1

1

xði; sÞjÞþ

Ss jxns

xði; sÞjÞ



T 1
^ÞÞ þ Sr¼1 lnðpkði;rÞ Þ,

ð29Þ

in which it must be emphasized that both
^ and i depend on the parameters p 00, p11. Note that the curse of determinacy has been broken, in that the observed c,x pairs need not satisfy c ¼ c(x) for every observation. Thus, the data table does not have to be in the form of Table 5 above. The extent to which this form is not satisfied is used to estimate the crossover probability e. Note also that e does not enter the optimization problem and is not restricted by knowledge of the optimal control policy. The control rule is not learned quickly and with certainty as in the completely observed case. Turning now to the case of measurement error in observation of the control, we allow misclassification with probability ec. With the state observed without error the argument is completely analogous to the case of observed controls and states with measurement error. That is, observation of the states gives the controls deterministically as a function of parameters. We repeat details briefly. The conditional distribution of the observed controls, given the state sequence is the ith, is pðcn jiÞ ¼ Ps
jcðxði;sÞÞ c

cns j


c Þ 1

jcðxði;sÞÞ cns j


c Þ 1

jcðxði;sÞ cs j

ð1

,

ð30Þ

hence the joint distribution is pðc ; iÞ ¼ Ps
jcðxði;sÞ c

cs j

ð1

T 1 Pr¼1 pkði;rÞ ,

ð31Þ

^ abÞ=Sab NkðiðyÞ; ^ abÞ and yielding the MLEs y^ ¼ ðp^ 00 ; p^ 11 ;
^c Þ with p^ ab ¼ NkðiðyÞ; 1  ^
^c ¼ T Ss jcðxðiðyÞ; sÞ cs j. Note that the function c(x) depends on unknown parameters through the optimization problem. The constraints mentioned above in the discussion of measurement error in states must be satisfied at the MLEs. Once again, the profile loglikelihood is easily obtained. It is
lðp00 ; p11 jc ; iÞ ¼ Ss jcðxði; sÞÞ cs j lnðT 1 Ss jcðxði; sÞÞ cs jÞ  þ ð1 jcðxði; sÞÞ cs jÞ lnð1 T 1 Ss jcðxði; sÞÞ cs jÞ þ Sr lnðpkði;rÞ Þ

¼ Tð^
c lnð^
c Þ þ ð1


^c Þ lnð1


^c ÞÞ þ Sr lnðpkði;rÞ Þ.

ð32Þ

Again, the dependence of
^ c and i on parameters is emphasized. The curse of determinacy is broken, in that the observed state/control sequence does not have to be degenerate. That is, the same state value can be

350

Bent J. Christensen and Nicholas M. Kiefer

associated with different observed controls without implying a breakdown of the statistical model. Further, it is not the case that the control rule is learned immediately after each state value has been realized – instead, information is accumulated over time. Both specifications, measurement error in states and measurement error in controls, break the curse of determinacy. The curse of degeneracy is still present, in that p(c|x)A{0,1}, imperfect observations of c or x simply makes it a little more difficult to learn which. Thus, for data in which the summary statistics do not have the structure of Table 5, i.e. for data in which the same state is associated at different time periods with different controls, one of these measurement error models might be appropriate. But they have different implications for observables, in that the observed sequence x is Markov, while x is not. Thus, as a practical matter, introducing measurement error in controls might be appropriate if the observed state sequence appears Markovian, measurement error in states if not. Finally, we turn to a specification with measurement error in both states and controls. Assume that the measurement error in the states and controls are independent, conditionally on the underlying realization of the process. Then pðx ; c jiÞ ¼ pðx jiÞpðc jiÞ n

¼ ðPTt¼1
jxt ¼

xði;tÞj


Þ1

ð1

 ðPs
cjcðxði;sÞÞ cs j ð1 T 1 jxt xði;tÞj Pt¼1
ð1

jxnt xði;tÞj


c Þ

Þ

1 jcðxði;sÞÞ cs j

Þ

 ct j
Þ1 jxt xði;tÞj
jcðxði;tÞÞ ð1 c


c Þ 1

jcðxði;tÞÞ ct j

. ð33Þ

T 1 The marginal probability pðiÞ ¼ Pr¼1 pkði;rÞ ; multiplying and marginalizing gives T



2 1 T pðx ; c Þ ¼ Si¼0 Pt¼1
jxt

xði;tÞj

ð1


Þ1

jxt xði;tÞj jcðxði;tÞ ct Þj
c ð1


c Þ1

jcðxði;tÞ ct j

pkði;tÞ ,

ð34Þ

leading to a likelihood function, which is substantially more complicated in that it has 2T terms and no simple closed forms for the estimators. Essentially, when either the state or control is observed without error, then the other is ‘‘known’’ in the sense that it is a deterministic function of parameters. This is not the case when both are measured with error. Nevertheless the likelihood function is not continuous in parameters. This is general, since the controls make up a discrete set. 7. Imperfect control Here we study a second approach to breaking the curse of determinacy. That is, we model a decision maker with imperfect control over the action he takes. Thus, the agent may know that c ¼ 0 is optimal for x ¼ 0, but may only be able to achieve c ¼ 0 with high probability, not with certainty. This kind of imperfect control has been used e.g. by Chow (1981).

Identification and Inference in Dynamic Programming Models

351

The extension fits easily into our simple framework. The c variables, which the agent would like to control exactly but cannot, are cA{0,1}. Define the variables aAA ¼ {0,1} as the variables the agent actually can control; if a ¼ 0 is chosen then c ¼ 0 with probability p0; if a ¼ 1 then c ¼ 0 with probability p1. Specify without loss of generality that p0>p1, so that a ¼ 0 is the natural choice if the agent would prefer c ¼ 0, etc. By choosing a, the agent chooses a probability distribution over the controls. Let pa be the probability that c ¼ 0 when action a is chosen. We can now apply the dynamic programming framework. Define the new utility function u ðx; aÞ ¼ E a uðx; cÞ ¼ pa uðx; 0Þ þ ð1

pa Þuðx; 1Þ,

ð35Þ

and the new transition distribution pðx0 jx; aÞ ¼ pðx0 jx; c ¼ 0Þpa þ pðx0 jx; c ¼ 1Þð1

pa Þ.

ð36Þ

Then we can simply do dynamic programming using a as the control instead of c. The value function satisfies Bellman’s equation
 V i ðxÞ ¼ max u ðx; aÞ þ bE i V i 1 ðx0 Þ ð37Þ a2A

in the finite horizon case, and
 V ðxÞ ¼ max u ðx; aÞ þ bEV ðx0 Þ

ð38Þ

a2A

in the infinite horizon case. In our simple marketing model from the previous sections, the new utility and transition functions using p0 ¼ 0.8 and p1 ¼ 0.2 are given in Table 9 (compare Tables 1 and 2). The first 10-value function iterations are given in Table 10 (compare Table 3). Here, we see that it is optimal to try to run the advertising policy in the low-demand period if the horizon is 3 or more periods (the exact control case required 4 or more). The policy has converged, though the value functions have not, and this is indeed the optimal policy in the infinite horizon problem. The value function converges to (approximately) V(0) ¼ 29.11526 and V(1) ¼ 34.21777. Table 9.

Imperfect control Profit

x¼0 x¼1

a¼0

a¼1

6.4 10.2

4.6 7.8

p(xt+1 ¼ 1|xt,ct)

x¼0 x¼1

a¼0

a¼1

0.178 0.570

0.700 0.780

352

Bent J. Christensen and Nicholas M. Kiefer

Table 10.

Value functions and optimal policies

t

Vt(0)

Vt(1)

at(0)

at(1)

0 1 2 3 4 5 6 7 8 9

6.40000 11.70730 15.96201 19.25980 21.72274 23.57096 24.95703 25.99659 26.77626 27.36101

10.20000 16.62450 21.08258 24.36055 26.82542 28.67345 30.05954 31.09909 31.87876 32.46351

0 0 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0

Comparing the analysis of the information on b in learning the optimal policy, we see here that this policy, (a(0),a(1)) ¼ (1,0), is optimal for b40:6694; in the perfect control case we found that (c(0),c(1)) ¼ (1,0) was optimal for b40:7143. In contrast to the case with pure measurement error, introducing imperfect control changes the solution to the optimization problem. The lesson here is that observation error and imperfect control are quite different specifications. The observables remain the {x,c}t sequence, for t ¼ 0, y ,T. Now, the state sequence xt is observed without error, so we can concentrate on a single sequence (without marginalizing with respect to all possible sequences as in the case of measurement error in both states and controls). The sequence ct is also observed without error, but it is no longer the control. The controls are the unobserved at. However, the observed ct can be regarded as noisy observations on the actual controls at. Thus, an approach very similar to the approach in the case of measurement error can be used in developing this factor of the likelihood. Finally, note that there is more reduced form of information on the transition distribution in the case of imperfect control. That is, in the case of perfect control, even with measurement error, the only transitions observed are those corresponding to x,c pairs (the conditioning variables in the transition distribution) which are optimal. The other transition probabilities enter the likelihood only through their effect on the optimal policy. Without perfect control, transitions corresponding to all x,c pairs are observed without error and can be used to estimate the transition probabilities directly. Let us develop the likelihood: pðx0 ; c0 jx; cÞ ¼ pðc0 jx; x0 ; cÞpðx0 jx; cÞ ¼ pðc0 jx0 Þpðx0 jx; cÞ,

ð39Þ

since the relation between the state and the control does not depend on lagged values, and neither does the measurement error. The first factor can be simplified to pðc0 jx0 Þ ¼ Sa0 pðc0 ja0 ; x0 Þpða0 jx0 Þ,

ð40Þ

Identification and Inference in Dynamic Programming Models

353

where the sum is over all possible {a}t sequences. Note however that the term p(a0 |x0 ) is degenerate, in that for given parameters, there is only one a sequence corresponding to the realized states. Thus, all but one of these terms are 0 and hence pðc0 jx0 Þ ¼ pðc0 ja0 ; x0 Þ ¼ pðc0 ja0 Þ

ð41Þ

for that value of a0 consistent with x0 (and parameters). Let i be the index of the observed x sequence. Then pðcjiÞ ¼ Ps2faðxði;sÞÞ¼0g p10

cs

ð1

p0 Þcs Ps2faðxði;sÞÞ¼1g pc1s ð1

p1 Þ1

cs

,

ð42Þ

and in the case p0 ¼ 1 p1, as in our example, there is further simplification to pðcjiÞ ¼ Ps p0ð1

jaðxði;sÞÞ cs jÞ

ð1

p0 Þjaðxði;sÞÞ

cs j

,

ð43Þ

very like the likelihood in the measurement error case, but with one important difference. That is, here the function a(x(i,s)) depends on all parameters, including p0. We can find an illustrative expression for the MLE for p0, namely ^ sÞ cs j, but here this is only one of many equations p^ 0 ¼ 1 T 1 Ss jaðxðiðyÞÞ; that must be solved simultaneously, since p0 enters the function a, unlike the parameter e in the measurement error case, where a similar expression can be used to obtain the profile likelihood. Summing up so far, we have developed the likelihood function in a compact notation for the discrete state – discrete control setup. The curses of degeneracy (a property of the distribution of the control given the state) and of determinacy (a requirement of the data configuration) are easily illustrated here. The control rule becomes known after a few observations (i.e., there is no sampling error – as soon as all of the states have been realized, the control rule is known). In general, knowledge of the control rule is not sufficient to identify underlying real parameters. Two approaches to breaking the curses were examined. The first was measurement error. Here there are two possibilities, measurement error in states or in controls (or, of course, both). Measurement error in states implies that the state-to-state transitions are not Markovian. Thus, this specification might be useful when the transitions are not Markovian. Here, the data do not have to satisfy the unlikely restrictions imposed by the perfect observation case (the curse of determinacy). However, the curse of degeneracy is not broken, essentially because the perfect observation of the controls identifies the states, given the parameters. Measurement error in controls is essentially the same. Observing the states without error identifies the controls, given parameters. Here, however, the state-to-state transitions remain Markovian. Combining both types of measurement error leads to a more complicated likelihood, as it is no longer possible to recover the true states and controls given the parameters. The curse of determinacy is broken in all cases, although the curse of degeneracy remains. Real parameters are typically not identified, although their ranges may

354

Bent J. Christensen and Nicholas M. Kiefer

be restricted. Imperfect control is an alternative approach. The results are somewhat different from the measurement error case, in that the optimal policies may differ from those in the perfect control setting, although the implications are the same in that the curse of determinacy is broken but that of degeneracy is not. Neither measurement error nor imperfect control is particularly appealing from an economic modelling point of view, and neither actually solves the problem we wish to solve. Economists have been led almost invariably to a random utility specification, which does solve the curse of degeneracy and allows identification of real parameters, but which does so by introducing ‘‘information’’ in the form of highly specific assumptions. That is, the random utility specification allows estimation of parameters that are not identified if utility is deterministic in a model that is otherwise the same. This is the topic of Section 8. 8. Random utility models A useful and popular approach to breaking the curse of degeneracy is to introduce a random utility specification. Here, the period utility is subject to a control-specific shock. The agent sees this shock before the control choice must be made, but it is not seen in the data. Thus, the choice of the agent may depend on the realization of the shock, and hence the observed optimal control may correspond to different observed states depending on the value of the unobserved shock. Of course, the random utility shock cannot simply be ‘‘tacked on’’ to a dynamic programming problem; it changes the problem, the value, and the optimal policy function. The approach of specifying the random utility shock as an unobserved state variable was introduced in empirical dynamic programming by Rust (1987), following McFadden (1973, 1981) on static discrete choice. Suppose now that the utility is subject to a random shock, so that  cÞ þ
ðcÞ, u ðx; c;
Þ ¼ uðx;

ð44Þ

where e(c) is a random shock. The idea here is that, at time t when the state xt ¼ k has been realized, a vector of random variables e is added to the kth row in the utility table, then the choice of control c is made. Thus c ¼ c(x,e) is a deterministic function, but given only x, c is a random variable whose distribution depends on x. If e has a nonzero mean, the mean can simply be absorbed into the utility specification, so it is a harmless normalization to set Ee ¼ 0. Consider the 2  2 model and suppose the utility shock is (e,0) where e is a scalar random variable taking the value a with probability 12 and a with probability 12. Consider for simplicity the case x ¼ 0 alone, so we can focus attention on the distribution p(c|x ¼ 0). Let p1 ¼ p(c ¼ 1|x ¼ 0); of course, p1 is a function of the parameter y characterizing preferences, etc. We have pðcjx ¼ 0Þ ¼ pc1 ð1

p1 Þ1 c ,

ð45Þ

Identification and Inference in Dynamic Programming Models

355

and with n independent observations Pt pðct jxt ¼ 0Þ ¼ pSc 1 ð1

p1 Þ n

Sc

,

ð46Þ

and Sc/n is a sufficient statistic for p1. Here, n is the number of observations with x ¼ 0. The natural parameter space for the reduced-form parameter p1 is [0,1]. But what values of p1 are consistent with the dynamic programming model with random utility shocks? In the model, the probability that c ¼ 1 is chosen is given by Prðarg maxfuð0; cÞ þ bE 0c V ðx0 ;
0 Þ þ
ðcÞg ¼ 1Þ. c

ð47Þ

This probability can be written Prð
ohðyÞÞ,

ð48Þ

where h ¼ u(0,1)+bE01V u(0,0) bE00V and the generic parameter y has been introduced as an argument in h for emphasis. Now, we have specified a binary distribution for e, so this probability can take values in {0,1/2,1}; the precise value depending on h(y). Specifically, c ¼ 0 could be optimal for both values of e(0); and c ¼ 1 could be optimal for e(0) ¼ a and c ¼ 0 for e(0) ¼ a, and finally c ¼ 1 could be optimal for both values of e(0). Thus, the likelihood function is flat for generic continuous parameters y, except where the value of y is such that the probability shifts between 2 of its 3 possible values. The most that can be obtained is that possible values of y are restricted to intervals, corresponding to values so that the implied choice probability is as close as possible to the sample fraction. Note that this does represent some improvement over the model without shocks – if h is monotonic in the parameter y, then without shocks, so p(c|x)A{0,1}, y is restricted to one of two intervals (recall estimating b in our marketing model), now with p(c|x)A{0,1/2,1}, y is restricted to one of 3 intervals. The curse of degeneracy has been broken in that p(c|x) is not necessarily in {0,1}. However, only one new point has been added to that set of possibilities, so it might be better to think of the curse as weakened, not broken. The main point, however, is that merely adding a random utility shock with a completely known distribution appears to add information about a structural parameter, but does not serve to identify fully otherwise unidentified parameters. Let us generalize the shock distribution slightly and introduce a new parameter. Suppose the distribution of e is a with probability (1 p) and a with probability p. For p not equal to 12 and a not equal to 0, this distribution has a nonzero mean. That does not really present a problem, as the mean can be absorbed into u(0,0), as we have seen. For a fixed value of p, we have that Pr(eoh(y,p))A{0,p,1}, so again y can at best be bounded in an interval. By varying p, however, it may be possible to match the sample fraction (it may not be possible, if the probability is zero or one for all y but this is not an interesting

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case). Here, the curse of degeneracy is unambiguously broken, in the sense that the sample fraction can be matched by choice of the parameter p. However, this amounts to no more than a reduced-form approach; no identifying information on y is available and it is at best restricted to an interval. 9. A continuously distributed utility shock Since adding a utility shock with a known 2-point distribution adds 1 point to the model-consistent parameter space for p1 (and hence possibly restricts the range of the generic parameter y), it is natural to ask whether a known distribution with support on an interval might tighten things up even more. In fact, the assumption of a continuously distributed utility shock is much more common in applications, for reasons which will become clear. Let us begin the analysis by supposing that, instead of a 2-point distribution on { a,a}, e has a continuous distribution with support [ a,a]. Note that we are not making innocuous assumptions by changing the shock distribution – changes here, even with the mean held constant, will affect both the value of the problem and the optimal policy function. To start with, assume f(e) ¼ 1/(2a), the uniform distribution. Then pðc ¼ 1jx ¼ 0Þ ¼ prð
ohðyÞÞ ¼ hðyÞ=ð2aÞ þ 1=2.

ð49Þ

Recall that h(y) ¼ u(0,1)+bE01V u(0,0) bE00V. This probability now does depend on y, continuously if h is continuous in y, so y can be estimated by ^ as long as setting h(y)/(2a)+1/2 equal to the sample fraction and solving for y, this is a feasible value (again, it could be that h(y) is such that|h|>a for all values of y, and therefore the probability is always zero or one; not an interesting case). By introducing randomness into the utility specification, we have managed to achieve identification of a parameter not identified without randomness. How can ‘‘introducing noise’’ serve to identify a preference parameter? What is in fact happening is that the assumption that the shock is continuously distributed with a known distribution is crucial. Our specification inserts ‘‘information’’ into the model – information on preferences. This is not necessarily inappropriate, but it is important to realize that y is being identified completely on the basis of the assumed shock distribution. Indeed, this point can be emphasized with a little further analysis. Let F be the distribution function for the utility shock. For simplicity, suppose F is a member of a 1-parameter family of distributions indexed by g. Then Z hðy;gÞ pðc ¼ 1jx ¼ 0Þ ¼ prð
ohðy; gÞÞ ¼ dF a

¼ Hðy; gÞ.

ð50Þ

In practice, g is assumed known (the distribution of the utility shock is fully ^ specified), and H(y,g) ¼ t (the sample fraction) is solved for yðgÞ. Here, the

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dependence on the assumed distribution is indicated by the explicit dependence on g. Assume F is continuously differentiable in g and h in y and g. This assumption rules out sudden shifts of the probability to zero or one (e.g. when h passes the value a); but rather than get involved in details we note that we are really concerned only with properties in the neighborhood of the solution to H(y,g) ¼ t. Alternatively, we can simply set a ¼ N so the distribution function has full support on the real line (indeed this is the most common practice). Using the implicit function theorem, ^ d y=dg ¼

H g =H y ,

ð51Þ

with Hg ¼ f(h(y,g))hg+Fg(h(y,g)) and Hy ¼ f(h(y,g))hy where f is the density function dF(x)/dx. Writing this out gives ^ d y=dg ¼

hg =hy

F g =fhy ,

ð52Þ

the first term giving the tradeoff between g and y holding h (the utility difference between using the two controls) constant and the second giving the effect of g on h through the change in the probability. This analysis indicates that the solution for the parameter y is a function of the assumed distribution, here indexed by g. Further, this derivative is typically nonzero, so the assumption matters. To push this analysis a little further, we concentrate on the parameter b, the discount factor. Then, hðb; gÞ ¼ ðuð0; 1Þ

uð0; 0ÞÞ þ bðE 01 V

E 00 V Þ,

ð53Þ

hence hb ¼ ðE 01 V E 00 V Þ þ bðE 01 V b E 00 V b Þ and hg ¼ bdðE 01 V E 00 V Þ=dg. Let us evaluate these expressions at b ¼ 0, in order to get a local tradeoff between the assumed distribution and the estimated discount factor at b ¼ 0. Here hbjb¼0 ¼ E 01 ðmaxfuðx; cÞ þ
ðcÞgÞ

E 00 ðmaxfuðx; cÞ þ
ðcÞgÞ,

ð54Þ

a function of g, and hgjb¼0 ¼ 0,

ð55Þ

so there is no direct effect of the assumed distribution g on h, the utility difference, when b ¼ 0. Hence, all of the effect is through the change in the probability, and this is given by ^ d b=dg jb¼0 ¼

ðF g =f Þ=½ðE 01 ðmaxfuðx; cÞ þ
ðcÞgÞ

E 00 ðmaxfuðx; cÞ þ
ðcÞgފ;

ð56Þ

all terms in this expression depend on g, so we have illustrated now in a simple case the correspondence between the specification of the utility shock distribution and the estimated discount factor. Note that this formula gives the effect on the estimate of b of a change in the distributional assumption, holding the data constant. The numerator is the change in the choice probability; the denominator is the density multiplied by the expected utility difference. While this

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expression clearly depends crucially on F (this is the point, after all), a little more may be said. In particular, if the expected utility difference between the two controls is smaller, then the role of the assumption on F is more important, in that this derivative is larger. It is sometimes thought that by ‘‘freeing up’’ the random utility distribution F through addition of unknown parameters, one can mitigate the effects of directly and completely specifying an unknown distribution. Let us consider this. First, in the stylized case studied above, with only one choice probability to estimate (i.e. the probability associated with one value of the discrete state variable and a binary control), this proposition is easily rejected. Our analysis shows there is a 1–1 relationship between values of the parameter g of the now unknown distribution and the preference parameter y. Clearly, these are not both identified. Specifically, there is a curve in the (g,y) space corresponding to a given value of h, and thus a set of parameter values which serve to match the fitted value h to the sample fraction. Identification requires that the functional form of the utility shock be completely specified.

10. Continuous state and optimal stopping: the search model In the previous sections, we have drawn attention to several issues in identification and inference in dynamic programming models, such as identification of utility parameters (including the discount factor b) only up to intervals, rapid (finite time) accumulation of information on these, the curses of determinacy and degeneracy, the role of the functional equation in identifying parameters off the optimal path, the subtle differences between measurement error, imperfect control and random utility approaches, and the functional dependence of the parameter estimates on the assumed shock distribution in the random utility case. These issues or close relatives apply generally (and in various disguises) in dynamic programming models and have been illustrated in the discrete state/control case. Many of the insights originate from earlier motivating work on the econometrics of the job search model, although this model differs slightly from the setup of the previous sections. It is an optimal stopping model, so it does have a discrete (binary) control, namely, whether or not to stop, but it has a continuous state variable (the wage). For statistical purposes, information is in many cases only accumulated until the process is stopped, so for asymptotics, a panel is considered. In this section, we briefly review some of the links between the material in the previous sections (general dynamic programming and marketing) and the motivating work on the statistical properties of the search model. The sequential job search model is due to Mortensen (1970) and McCall (1970). The search model was an early application of stochastic dynamic programming techniques to economic theory. It was among the first models to be estimated with econometric techniques exploiting the dynamic programming

Identification and Inference in Dynamic Programming Models

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structure. In the simplest setup, a worker is assumed to be unemployed and searching for employment. Search consists of sampling, once each period, a wage offer w from a known distribution of offers. Once accepted, a job is held forever. Once declined, an offer is no longer available. The state variable is the outstanding wage offer in the current period. The control is the decision to accept or reject the outstanding offer. Offers are assumed independently and identically distributed (iid), so the distribution of next period’s state does not depend on the current state. The state distribution does depend on the control, since offers are no longer received once a job has been accepted. The worker chooses a strategy which maximizes the expected present discounted value of his income stream ESTt bt w where in the simplest models T ¼ N. This model has been extended and refined and widely applied. The basic logic of optimal stopping can be illustrated in the infinite horizon model. The ‘‘value’’ for an unemployed worker is heuristically the maximized value of ESTt bt w, where the maximum is taken over all possible strategies the worker might follow in his effort to maximize the present discounted value. This value Vu is a constant not depending either on the current (declined, since we are assuming the worker is unemployed) wage offer or on the particular period. The value does not depend on the current offer since we assume offers are iid; it does not depend on the period since we have an infinite horizon problem and the future looks the same from any point. Now suppose the worker gets the next offer w. The value of accepting the offer w is simply STt bt w ¼ w=ð1 bÞ. If this value is greater than the value of continued search, namely Vu, the worker should accept the offer; if not he should decline and continue the search. Thus we have found the optimal strategy. The worker should decline wage offers until he receives one greater than r ¼ (1 b)Vu, then he should accept employment and stop searching. In particular, r is the reservation wage. Note that the logic of backward recursion does not apply here. It would if we considered finite horizon search, and then V u would depend on the time left before the horizon. Let b be the current utility of unemployment benefits net of search costs. Then Vu ¼ b+bEV, and with the state variable x ¼ w we have the value function   w ; b þ bEV , ð57Þ V ðwÞ ¼ max 1 b the maximum of the value of stopping and accepting the offer w and the value of continuing search. Assume offers arrive at Poisson rate l0 during unemployment, so that the probability of receiving an offer any given period is p ¼ 1 e l0 ; and write f for the density of the offer distribution. Then   Z 1 Z 1 p EV ¼ wf dw þ 1 p f dw ðb þ bEV Þ ¼ TðEV Þ, 1 b ð1 bÞðbþbEV Þ ð1 bÞðbþEV Þ ð58Þ

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with the derivative  T0 ðEV Þ ¼ b 1

p

2 ½0; bŠ,

Z

1

r

 f dw

ð59Þ

and since bo1 the operator T is a contraction and may be iterated to solve for EV and hence the optimal reservation wage strategy given by r ¼ (1 b) (b+bEV). Note the simplification: We are iterating on the scalar EV rather than the function V as in Section 3. Christensen and Kiefer (1991) study the present model from a likelihood perspective. Consider panel data of the form fd i ; wi gN i¼1 where wi is the accepted wage of the ith initially unemployed worker and di the unemployment duration. Rapid information accumulation similar to that from the observed control rule in the marketing model in Section 5 occurs in the search model, too, although accumulation does not stop at a finite sample size. Similarly, preference parameters may be restricted to intervals, as in the marketing model in Section 4. For example, if b is the unknown parameter to be estimated in the job search model, the requirement wiZr, i ¼ 1, y ,N imposes an interval restriction on b. In the search case, the interval keeps shrinking as N-N, and the estimator b^ ¼ r 1 ðwm Þ

ð60Þ

where r ¼ r(b) is inverted with respect to the parameter and wm ¼ mini{wi} is the minimal order statistic converges rapidly (at rate N, as opposed to the usual 1 N 2 ) to the true value of the discount factor. More generally, for k parameters y, including also b and parameters of f, Christensen and Kiefer (1991) show that 1

N 2 ðy^

y0 Þ ! nk ð0; B0 Þ,

rank B0 ¼ k

1,

ð61Þ ð62Þ

a reduced rank limiting normal distribution, with asymptotic variance–covariance matrix B0 and one parameter being superconsistent. Thus, the rapid information accumulation here takes the form of N-asymptotics in one direction of the parameter space. The control rule is learnt in finite time in Section 5 (discrete state/control) and at rate N in the search model. Superconsistency here is a result of the fact that with complete observability, data must satisfy wmZr, i.e. controls and states must line up such that no wage (state) is accepted (the control decision) unless the parameter dependent inequality is satisfied. This is a case of the curse of determinacy of Section 5. The strict requirement on data may be softened up by introducing measurement error, as in Section 6. This is done by Christensen and Kiefer (1994b) in an application to the 1986 Survey of Income and Program Participation (SIPP) 1 data, and it is shown that regular (rank B0 ¼ k) N 2 -asymptotics result. More general theory on information accumulation at different rates on subparameters

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may be found in Christensen and Kiefer (1994a, 2000), who introduced and studied the concept of a local cut (wm is a local cut in the search model). Mortensen (1990) and Burdett and Mortensen (1998) introduced the equilibrium version of the search model, showing that in the simplest case it gives rise to an endogenous wage offer distribution with c.d.f.   ! 1 þ l1 =d q w 1=2 1 F ðwÞ ¼ ; w 2 ½r; hŠ, ð63Þ l1 =d q r where l1 is the offer arrival rate during employment (the case of on-the-job search), d an exogeneous lay-off rate, q firm productivity and h an upper bound on the wage distribution. Christensen and Kiefer (1997) determine the minimum panel data structure sufficient for identifying all structural parameters, in particular, fd i ; wi ; j i gN i¼1 ; accepted wages along with unemployment and employment durations, and show that both the minimum and maximum wages are local cuts, 1 the reduced rank of the N 2 -asymptotic normal distribution now being k 2. Again, regularity may be restored by allowing for measurement error, and this is done in an application to Danish data by Bunzel et al. (2001). Christensen et al. (2005) apply a related model with on-the-job search to a Danish panel of matched employer–employee data and use movers and stayers in a firm between two consecutive periods to estimate l1,d, and a cost of search parameter. The results imply that on-the-job search explains the employment effect, i.e. the extent of the stochastic dominance of the cross-section wage distribution of employed workers relative to the wage offer distribution. 11. Conclusion Both statistics and economic theory provide ways to isolate to relevant portions of economic problems and data. Stochastic models are important for inference purposes, and the stochastic dynamic programming model is important when moving to the dynamic case. The sequential job search model of Mortensen (1970) and McCall (1970) is an important early application of dynamic programming in economics. By representing frictions as the random arrival of trading partners, the model is naturally stochastic and leads directly to a likelihood function. This is a key instance of useful data reduction through the productive combination of statistical analysis and theory of optimizing agents. Later theoretical developments starting with Mortensen (1990) allow imposing equilibrium constraints in the statistical analysis of optimizing agents. In this paper, we have drawn attention to several issues in identification and inference in dynamic programming models which have received little or no prior notice, such as identification only up to intervals, the precise role of the optimality equation in identification, identification off the optimal path, rapid information accumulation, the curses of determinacy and degeneracy, and the dependence of parameter estimates on distributional assumptions in the random

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utility case. Along with the search model, a simple marketing model of investment in advertising campaigns affecting future demand has been used for illustration. Our discussion shows how earlier work on the econometrics of the search model has led to insights that apply to general dynamic programming models. Acknowledgments We are grateful to the editor, referee and participants in the Conference on Labour Market Models and Matched Employer-Employee Data, Sandbjerg, 2004, for useful comments and suggestions, and to the Danish Social Science Research Council for research funding. References Bellman, R. (1957), Dynamic Programming, Princeton: Princeton University Press. Bellman, R. and S. Dreyfus (1962), Applied Dynamic Programming, Princeton: Princeton University Press. van den Berg, G.J. and G. Ridder (1993), ‘‘Estimating equilibrium search models from wage data’’, pp. 43–55 in: H. Bunzel, P. Jensen and N.C. Westerga˚rd-Nielsen, editors, Panel Data and Labour Market Dynamics, New York: North-Holland. van den Berg, G.J. and G. Ridder (1998), ‘‘An empirical equilibrium search model of the labor market’’, Econometrica, Vol. 66(5), pp. 1183–1221. Blackwell, D. (1962), ‘‘Discrete dynamic programming’’, Annals of Mathematical Statistics, Vol. 33, pp. 719–726. Blackwell, D. (1965), ‘‘Discounted dynamic programming’’, Annals of Mathematical Statistics, Vol. 36, pp. 226–235. Bunzel, H., B.J. Christensen, P. Jensen, N.M. Kiefer, L. Korsholm, L. Muus, G.R. Neumann and M. Rosholm (2001), ‘‘Specification and estimation of equilibrium search models’’, Review of Economic Dynamics, Vol. 4, pp. 90–126. Burdett, K. and D.T. Mortensen (1998), ‘‘Wage differentials, employer size, and unemployment’’, International Economic Review, Vol. 39, pp. 257–274. Chintagunta, P.K. and D. Jain (1992), ‘‘A dynamic model of channel member strategies for marketing expenditures’’, Marketing Science, Vol. 11, pp. 168–188. Chintagunta, P.K. and N.J. Vilcassim (1992), ‘‘An empirical investigation of advertising strategies in a dynamic duopoly’’, Management Science, Vol. 38, pp. 1230–1244. Chow, G.C. (1981), Econometric Analysis by Control Methods, New York: Wiley. Christensen, B.J. and N.M. Kiefer (1991), ‘‘The exact likelihood function for an empirical job search model’’, Econometric Theory, Vol. 7, pp. 464–486. Christensen, B.J. and N.M. Kiefer (1994a), ‘‘Local cuts and separate inference’’, Scandinavian Journal of Statistics, Vol. 21, pp. 389–407.

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Christensen, B.J. and N.M. Kiefer (1994b), ‘‘Measurement error in the prototypaljob-search model’’, Journal of Labor Economics, Vol. 12, pp. 618–639. Christensen, B.J. and N.M. Kiefer (1997), ‘‘Inference in non-linear panels with partially missing observations: the case of the equilibrium search model’’, Journal of Econometrics, Vol. 79, pp. 201–219. Christensen, B.J. and N.M. Kiefer (2000), ‘‘Panel data, local cuts, and orthogeodesic models’’, Bernoulli, Vol. 6, pp. 667–678. Christensen, B.J. and N.M. Kiefer, (2005), Economic Modelling and Inference, Princeton: Princeton University Press, forthcoming. Christensen, B.J., R. Lentz, D.T. Mortensen, G.R. Neumann and A. Werwatz (2005), ‘‘On-the-job search and the wage distribution’’, Journal of Labor Economics, Vol. 23, pp. 31–58. Kiefer, N.M. and G.R. Neumann (1979), ‘‘An empirical job search model with a test of the constant reservation wage hypothesis’’, Journal of Political Economy, Vol. 87, pp. 89–107. Kiefer, N.M. and G.R. Neumann (1993), ‘‘Wage dispersion with homogeneity: the empirical equilibrium search model’’, pp. 57–74 in: H. Bunzel, P. Jensen and N.C. Westerga˚rd-Nielsen, editors, Panel Data and Labour Market Dynamics, New York: North-Holland. Lippman, S.A. and J.J. McCall (1976a), ‘‘The economics of job search: a survey: part I’’, Economic Inquiry, Vol. 14, pp. 155–189. Lippman, S.A. and J.J. McCall (1976b), ‘‘The economics of job search: a survey: part II’’, Economic Inquiry, Vol. 14, pp. 347–368. Maitra, A. (1968), ‘‘Discounted dynamic programming in compact metric spaces’’, Sankhya Ser. A, Vol. 40, pp. 211–216. McCall, J.J. (1970), ‘‘Economics of information and job search’’, Quarterly Journal of Economics, Vol. 84, pp. 113–126. McFadden, D. (1973), ‘‘Conditional logit analyses of qualitative choice behavior’’, pp. 105–142 in: P. Zarembka, editor, Frontiers in Econometrics, New York: Academic Press. McFadden, D. (1981), ‘‘Econometric models of probablistic choice’’, in: C.F. Manski and D. McFadden, editors, Structural Analysis of Discrete Data with Econometric Applications, Cambridge: MIT Press. Mortensen, D.T. (1970), ‘‘Job search, the duration of unemployment, and the Phillips curve’’, The American Economic Review, Vol. 60(5), pp. 847–862. Mortensen, D.T. (1986), ‘‘Job search and labor market analysis’’, pp. 849–919 in: O. Ashenfelter and R. Layard, editors, Handbook of Labor Economics, Amsterdam: North-Holland. Mortensen, D.T. (1990), ‘‘Equilibrium wage distributions: a synthesis’’, pp. 279–296 in: J. Hartog, G. Ridder and J. Theeuwes, editors, Panel Data and Labor Market Studies, New York: North-Holland. Nerlove, M. and K.J. Arrow (1962), ‘‘Optimal advertising policy under dynamic conditions’’, Economica, Vol. 29, pp. 129–142.

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Ross, S.M. (1983), Introduction to Stochastic Dynamic Programming, New York: Academic Press. Rust, J. (1987), ‘‘Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher’’, Econometrica, Vol. 55, pp. 999–1033. Rust, J. (1997), ‘‘Using randomization to break the curse of dimensionality’’, Econometrica, Vol. 65(3), pp. 487–516. Stokey, N.L. and R.E., Lucas, Jr. (with E.C. Prescott,) (1989), Recursive Methods in Economic Dynamics, Cambridge: Harvard University Press.

CHAPTER 15

On Estimation of a Two-Sided Matching Model Linda Y. Wong1 In honor of Dale Mortensen’s 65th birthday.

Abstract This paper provides an assessment of the implementation of a canonical two-sided matching model using marriage data from the National Longitudinal Survey of Youth (1979–1998, NLSY). Results indicate that agents tend to be patient in the marriage market, and their attractiveness is more sensitive to permanent income (education) than transitory income (wage). The average marriage premium is about 1.3–1.6 percent. Results also show that using non-random samples of spouses distorts preference parameters, that matching outcomes are sensitive to singlehood utility specifications, and that segregation in sorting could limit heterogeneity in agents’ behavior. The proposed algorithm – solving for agents’ marriage sets based on analytic equations and feeding the solution into the likelihood estimation – does not demonstrate sufficient flexibility to match uncensored singlehood duration data.

Keywords: two-sided matching, structural estimation, panel data, heterogeneity JEL classifications: D83, J12, J64 1. Introduction Two-sided matching, such as that between workers and firms, buyers and sellers, investors and entrepreneurs, two business firms, and men and women, has been

1

I have benefited from suggestions of Gerard van den Berg, George Neumann, Randy Wright, a referee, and seminar participants at the conference on Labour Market Models and Matched Employer–Employee Data, Sandbjerg Manor, Denmark, 2004. All errors are my own. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75015-X

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

365

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the focus of many recent studies. Contributions to the literature on search and matching by Diamond (1982), Mortensen (1982), and Pissarides (1990) highlight market frictions and provide an excellent framework for analyzing two-sided match formation. Typically agents are assumed to be homogeneous, so the sorting aspect of matching is absent. In the last decade, a branch of literature has focused on matching problems due to heterogeneity of agents: assortative matching. Following Gale and Shapley (1962) and Becker (1973), who considered frictionless markets, this literature has centered on stylized matching markets (e.g. Burdett and Coles, 1997). Demographic data on marriage are widely available. However, studies based on survey data often use samples from male and female respondents separately in estimation. The problem with this approach is that the spouses of respondents are not randomly sampled, therefore inferences drawn on the other sex (spouses of respondents) could be misleading. The purpose of this paper is to examine an empirical method suited to the framework established in Burdett and Coles (1997), using survey data on both male and female respondents.2 Defining a method that estimates such a model would be useful not only because the model could be used to analyze various policy measures and market and behavioral fundamentals affecting outcomes, but also because it could be applied to other bilateral markets such as financial markets and product markets. The task here is not to deduce a realistic behavioral model, but to determine whether estimating a behavioral-matching model is empirically possible, and if so, how sensitive the model will be to fundamental parameters and assumptions. So, the method outlined in Section 4 shall best be thought of as the basis for further work. The presence of a reservation strategy is one defining characteristic of search and matching models. My method takes the Nash equilibrium reservation solution of who matches with whom into the likelihood function. That is, I estimate a ‘restricted’ model and impose restrictions on reservation types. Given data on duration and the characteristics of marriage matches, the log-likelihood function is maximized without imposing many restrictions on the symmetry between men and women. Marriage history data from the National Longitudinal Survey of Youth (NLSY) 1979–98 is utilized and presented in Section 3. The empirical method is guided by a canonical two-sided search and matching model (Nash equilibrium) described in Section 2. Section 4 outlines the method and Section 5 presents estimation results. Section 6 concludes.

2

This paper thus differs from Wong (2003) who considers symmetry between the two-sides of the market and employs only male respondent data that include both censored and uncensored singlehood duration elements.

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2. The model Time is continuous, and there are two groups of infinite-lived risk-neutral agents, men and women, who discount future income at rate r40. At each point in time, all agents are in one of two states: single or married. Only single agents search for marriage partners. There is an atomless continuum of agents, each indexed by his/her exogenously given type: x (male) or y (female). Types are publicly observable. The  range of types is positive and boundedly finite. Let x and y be bounded by ½x; xŠ  respectively. Let the density of type x single agents be fx(x) and the and ½y; yŠ, cumulative distribution function be Fx(x). Because agents’ marriage strategies, as well as the number of agents who get married or separate, affect the pool of single agents in the market, the distribution of single agents is endogenous. Let the meeting rate for a type x single male be lm, governed by a Poisson process. Upon meeting, agents decide whether or not to match. A match is only consummated if both accept it. To maintain a steady-state proportion of single agents and to focus on who matches with whom, the separation process is assumed to take the simplest form. A match is assumed to dissolve with an exogenous constant flow probability (Poisson rate) d40. When a match is terminated, agents flow back to the single pool. While an agent is single, instantaneous utility is U(x). When an agent is married, the two agents split the match output Q(x,y) equally, i.e., the flow match utility is non-transferable. A single agent chooses a range of acceptable types of potential partners with the objective of maximizing his expected discounted value in the future utility stream. Consider a type x single agent. The value of singlehood is his instantaneous utility U(x) and the expected benefit of marriage obtains if a partner of type Y is accepted following an optimal policy, given that a partner has arrived: rV ðxÞ ¼ UðxÞ þ lm E max½W ðx; Y Þ

V ðxÞ; 0Š,

ð1Þ

where W(x,Y) is the expected discounted value of marriage with a random partner of type Y. The ex post value of marriage consists of the flow payoff and the value of remaining single due to an exponential random separation: rW ðx; yÞ ¼

Qðx; yÞ þ d½V ðxÞ 2

W ðx; yފ.

ð2Þ

2 ðx;yÞ By the envelope theorem, W(
) is differentiable and W y ðx; yÞ ¼ Q2ðrþdÞ 40. Strategies. A steady state pure strategy for a type x agent is to choose a set of agents Ax with whom x is willing to match, Ax ¼ fyjW ðx; yÞ4 ¼ V ðxÞg. The lowest type with which x is willing to match is determined by a reservation policy defined by: W ðx; Rx Þ  V ðxÞ, where Rx represents the reservation type of type x. While x can determine with whom he desires to match, he also needs to be desired by potential partners. Define x’s matching set to be Mx  Ax \ fy x 2 Ay g, which contains the set of women types to whom he proposes and who are willing to

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match with him. A match of a pair of agents (x, y) is mutually acceptable if each party is willing to match with the other, x 2 Ay and y 2 Ax , or y 2 Mx .3 Combining Equations (1) and (2) and the reservation policy, and using the envelope theorem and integration by parts, the reservation-partner-type for type x agents is the solution to the following equation: Z Qðx; Rx Þ lm Q2 ðx; Rx Þ ¼ UðxÞ þ ½F y ðM x Þ F y ðzފ dz, ð3Þ 2 2ðr þ dÞ z2Mx where Mi represents the highest type within matching set Mi . The solution is unique because the left-hand side of Equation (3) is increasing in Rx and the right-hand side is decreasing. Because the situation is symmetric between men and women, the reservation type for women satisfies Z QðRy ; yÞ lw Q1 ðRy ; yÞ ¼ UðyÞ þ ½F x ðM y Þ F x ðzފ dz. ð4Þ 2 2ðr þ dÞ z2My Equilibrium. Definition: A Nash equilibrium is a tuple (Rx, Ry, Mx ) such that for each type of agents,  a pair of reservation types (Rx, Ry) solves Equations (3) and (4),  the matching set Mx is optimal, i.e. y 2 Mx if and only if W ðx; yÞ4 ¼ V ðxÞ.

In other words, in a steady-state equilibrium (i) every single agent maximizes the expected net benefit flow attributable to the choice of partner, taking all other strategies as given, and (ii) there is mutual acceptance that specifies who matches with whom. The equilibrium solution is a two-dimensional graph. Burdett and Coles (1997) prove the existence of an equilibrium in a closely related model. 3. Data The data for this analysis are from the NLSY79 youth cohort.4 The samples are core nationally representative random samples. Interviews were conducted yearly from 1979 through 1994; since then data have been recorded biannually. The 1979–1998 cross-sectional and supplemental samples consist of 11,774 respondents between the ages of 14 and 22 in 1979. My focus is match formation at first marriage. I keep track of single respondents until the first marriage spell, even though longer marriage histories are available. Interval-censored data are present. For singlehood spells, for example, some respondents were single, became non-responsive, and then married. First marriage spells were poorly recorded, about 30 percent of marriages were interval-censored. This could be problematic because people could fail to report

3 4

x 2 My implies y 2 Mx . Details of sample selection and description can be found in Wong (2005).

On Estimation of a Two-Sided Matching Model

369

divorces, and subsequent reports of being married could mean another marriage(s). Because treating interval-censored data is not the primary focus of this paper, I exclude such data for singlehood and marriage spells.5 Search duration can only be partially observed in data, because elapsed singlehood duration is unknown. I generate singlehood duration data by taking the difference between age at first marriage (or the censored time c, whichever comes first, i.e. minftn0i ; cg) and spousal search starting time, tsi, assumed to be a parameter to be estimated. The duration of marriage, t1, is defined as the number of years a couple stays married before or until the censored time, whichever comes first. The level of education is taken as the highest educational attainment over the 20-year period, because individuals may have foresight over the level of schooling they are to complete. Wages are taken as the average over the 20-year period. Wage data are weekly wages in constant (1987) dollars. Spouses’ education levels, wages, and ages were poorly recorded, and the bulk of observations contain missing data. Zero wages or education levels are deleted. The unavailability of adequate information on spouses rendered about 80 percent of marriage observations unusable! The final sample size contains male and female respondents who eventually married, a total 872 observations. The sample contains more female than male respondents (539 vs. 333). The tiny sample size and imbalanced respondents by sex are due to (1) the fact that most spouses have missing data, particularly spouses of male respondents, and (2) there are more non-response data for married males than for married females. A caveat of using uncensored data is that of unity in sex ratio, which differs from the typical census data of 1.05.6 Summary statistics are provided in Tables 1 and 2. Table 1 contains descriptive statistics on age at first marriage and marriage durations for male and female respondents. Age at first marriage is roughly similar between the two groups of respondents, with men’s age being slightly higher. Marriage duration contains both censored and uncensored samples. Empirical survival rates (Kaplan–Meier estimates) decline with the duration of singlehood. I use Kaplan–Meier survivor functions to perform log-rank, Wilcoxson, and Flemming–Harrington tests for the equality of survivor functions between males and females. All three tests strongly reject equality of survivor functions for all age levels, implying that restricting sexual similarity in parameters is not supported by data. The data also indicate that the marriage hazard declines with duration, which cannot be predicted by the simplest finite-horizon search models. Explaining such a pattern requires heterogeneity within each age group or some

5

See Wong and Yu (2005) for a non-parametric approach to interval-censored marriage data. The aggregate number of the unattached males (29,172,023 weighted) divided by the aggregate number of the unattached females (27,790,497 weighted) in the age group [14,42] is 1.05 in the 1990 Census.

6

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Table 1.

Age at first marriage and marriage durations

Number of respondents Min. age at first marriage Mean age at first marriage Fraction censored Mean duration

Table 2.

M

F

333 16 22.90 (4.21) 0.4805 8.65 (5.99)

539 15 22.13 (4.47) 0.3859 8.21 (5.55)

Sample mean characteristics

Respondents

M

F

Wage Education Spouses Wage Education

322.47 (140.81) 12.15 (2.26)

197.71 (104.06) 12.50 (2.12)

163.13 (139.79) 10.66 (3.90)

317.56 (211.56) 11.23 (3.66)

structural duration dependence such as modeling match quality. Table 2 contains sample characteristics that tell typical stories of male and female wage and education differentials by marital status. 4. Estimation strategy I first spell out specifications of the model and then describe the likelihood function and solution method. Sensitivity tests on certain model specifications will be presented in Section 5. 4.1. Specification 4.1.1. Heterogeneity I am interested in quantifying how responsive types are to wage and level of education, so I allow these measured characteristics to affect agents’ types. Note that using (average) wage as an observed characteristic is not necessarily inconsistent with the model specification of non-transferable utility, when wages are considered to be returns to ability. For each sex, I consider agents’ cardinal quality, z, as a function of the logarithm of education e and the logarithm of wage w: z ¼ expðbw þ ð1

bÞeÞ,

ð5Þ

On Estimation of a Two-Sided Matching Model

371

where b A [0, 1] is a parameter interpreted as the elasticity of attractiveness with respect to wage. Note that there is no intercept in (5) for the identification reason similar to qualitative response models. Agents are assumed to have identical valuations for endowments of potential partners of the opposite sex. The ranking of attractiveness is obtained by discretizing z into deciles. The purpose of discretization is to simplify the classification error problem that accounts for the dependence between the true and the observed type (see below).7 An agent’s cardinal type x is generated as the median value within the xth decile. 4.1.2. Model elements The model can only be solved if further structure is imposed. Basically, utility functions are specified as: UðxÞ ¼ x. Qðx; yÞ=2 ¼ xy=2. That is, agents’ valuation of singlehood preferences is simply their own type. Note that because I use the reservation equation to solve for agents’ reservation type, singlehood utility cannot be identified and so the assumption of U(x) ¼ x is arbitrary. Marriage is considered as a production unit as theorized in Becker (1973). The marriage utility embraces two kinds of gains: one is increasing returns to scale, and the other is complementarity between partners’ types. With the above specification on utility, Equations (3) and (4) are independent on agents’ own types: Z lm Rx ¼ 2 þ ½F y ðM x Þ F y ðzފ dz, ð6Þ r þ d z2Mx Ry ¼ 2 þ

lw rþd

Z

½F x ðM y Þ

F x ðzފ dz,

ð7Þ

z2My

which amounts to Burdett and Coles’ (1997) partial equilibrium with a shift. The equilibrium stratification is a sequence of marriage classes. Besides utility functions, the singlehood type distribution is specified as its empirical counterpart: F x ðxÞ ¼ F^ x ðxÞ, F y ðyÞ ¼ F^ y ðyÞ.

7

One can discretize z into 100 or as many categories as one wants. The tradeoff is to solve for a substantially large simplex in a linear algebra problem when classification error is considered. For 10 types, the simplex problem is much more tractable. Wong (1998) simulated markets with 5, 10, and 15 types, and found 10 types yield the best results.

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4.2. The likelihood function and solution method In its simplest form, the model has no observed explanatory variables, and dependent variables are aspects of individual marriage market histories. The dependent variables are: the duration of singlehood and marriage, the observed education and wage of the respondent and spouse (if married), and left- and right-censoring indicators. The model implies a particular distribution for all dependent variables. The likelihood of the sample is obtained by multiplication of the individual contributions: the singlehood duration, the probability that a type x matched with a type y, and the marriage duration. The parameters are (lm, lw, tsm, tsw, d, bm, bw, e), where tsm and tsw are spousal search starting time estimated by minimum order statistics, and e represents a vector of classification error probabilities illustrated below. Consider the case of a type x male who is single at first interview (females’ contributions are analogous to the present description). The distribution of singlehood duration t0m has an exponential distribution with parameter lm Sy2Mx f y ðyÞ, which represents the hazard rate of marriage. Conditioned on being type x, the individual’s likelihood contribution of singlehood duration until and including the time of exit into marriage is: " # X X lm f y ðyÞ exp lm f y ðyÞt0m ; ð8Þ y2Mx

y2Mx

where t0m 40. Events occurring after exit from singlehood are independent of the events up to exit. Therefore, their probability is independent of the likelihood of being single. The event immediately following type x’s single duration is the realization of a match with a type y spouse. It is given by the joint probability of acceptance Prðx; yjy 2 Mx ; x 2 My Þ, which I assume to equal the product of each individual’s conditional probability, Prðxjx 2 My Þ Prðyjy 2 Mx Þ. Let 1ðy 2 Mx Þ equals one if a type y woman is contained in the marriage set of a type x man. The probability that the type x man matches with a type y woman is the fraction of type y women out of all types of women acceptable to the type x man, Prðyjy 2 Mx Þ ¼

f y ðyÞ1ðy 2 Mx Þ . J Sx¼1 f y ðyÞ1ðy 2 Mx Þ

ð9Þ

The probability for type y women is defined likewise Prðxjx 2 My Þ ¼

f x ðxÞ1ðx 2 J Sy¼1 fxðxÞ1ðx

My Þ . 2 My Þ

ð10Þ

Conditional on whom to match with, the marriage duration t1 has an exponential distribution with parameter d. Let d 1 ¼ 1 if t1 is right-censored, and

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On Estimation of a Two-Sided Matching Model

d 1 ¼ 0 otherwise. So, the marriage duration contribution is dð1

d1Þ

expð dt1 Þ,

ð11Þ

where t1 40. The total contribution to the likelihood function for a type x respondent who is single at the time of the first interview equals the product of (8), (9), (10) and (11) " # X L0 ¼ lm exp lm f y ðyÞt0m f y ðyÞ1ðy 2 Mx Þ y2Mx

f x ðxÞ1ðx 2 My Þ

 PJ

y¼1 f x ðxÞ1ðx 2 My Þ

dð1

d1Þ

expð dt1 Þ.

ð12Þ

To cope with the possibility of misclassifying agents’ type that is based on their observed traits only, I adopt the classification error model proposed in Wong (2003). The error structure is flexible and it accounts for the dependence nature of the true type and the error, in oppose to typical measurement error model for wages assuming the error being stochastically independent of the true wage. Basically, if x and y are true types for the observed types k and l, respectively and the classification errors are v1 and v2, their relation is defined as k ¼ x þ v1 and l ¼ y þ v2 . Because the supports of v1 and v2 are known, only their distributions need to be estimated. I consider the problem of classification errors in terms of the distance between true and observed type, i.e. If dðk; xÞ ¼ jk xj ¼ jv1 j is the distance between the true type k and the observed type x, then q(d) denotes the classification error probability with distance/error equals d. Two assumptions are made: (1) there is symmetry, qðdÞ ¼ qðd 0 Þ for any jk xj ¼ 0 0 k x , and (2) the error distribution is independent of sex, otherwise it cannot be identified. The solution to the classification error problem requires solving a simplex problem. For 10 types of agents, the result contains five parameters. The classification error probability can be expressed as e ¼ o e0, e1, e2, e3, e4, e5, e4, e3, e2, e1 >, where e0 ¼ 1 2(e1+e2+e3+e4) e5. Not any e’s are admissible. The parameters have to be chosen such that the following two conditions must be satisfied: (a) e1, e2, e3, e4, e5 > 0, and (b) 2e1+2e2+2e3+2e4+e5 o ¼ 1. With classification errors, the likelihood function (12) becomes L1 ¼

jX k 1j

jX l 1j

v1 ¼jk 10j v2 ¼jl 10j

 f ðl  dð1

v2 Þ1ðl d1Þ

lm exp

h

v2 2 Mk

lm Sl h

v1 Þ

v2 2Mk

Sk

v1

v1 2Ml

1 expð dt1 Þeðv2 Þeðv1 Þ: , 2

where k ¼ ., 1, y ,10, and l ¼ ., 1, 2, y ,10.

f ðl

v2

f ðk

v2 Þt0m

i

v1 Þ1ðk

v1 2 Ml

v2 Þ

i

ð13Þ

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Linda Y. Wong

The model does not permit an analytical solution, but results can be numerically solved in a straightforward manner. Given a set of parameters and model specifications, I construct the likelihood function as follows: (1) Generate 10 types of agents. (2) Estimate d from the marriage duration data. (3) Solve for the matching sets Mx by invoking the matching algorithm outlined in Wong (2003), using (6) and (7). (4) Feed the model solution to the likelihood function (12) and maximize the logarithm of it. The log-likelihood function is maximized using the simulated annealing method. 5. Results Estimates and characteristics of the equilibrium are analyzed in this section. I start by choosing a small value of the interest rate and estimate the full sample. Then, I compare the results with a large interest rate to examine how mate selection responds to the relative patience of individuals. After fixing an interest rate, I estimate the model using male and female subsamples to examine whether this creates distortion in parameter estimates. Characterization of the equilibrium is described next, after the baseline model is determined. Two regimes are then considered, varying the assumptions on utility. 5.1. A low interest rate Due to lack of knowledge about the interest rate in the marriage market, r is arbitrarily assumed to be 0.06. Results using the full sample are shown in Table 3. Column 1 indicates that the relative elasticity of attractiveness with respect to wage is modest for both men and women. This reflects that agents’ attractiveness is more sensitive to their permanent incomes (education) than temporary incomes (wage). 5.2. A high interest rate To evaluate how mate selection responds to the relative patience of individuals, I estimate a model setting a high interest rate r ¼ 0:9 (column 2, Table 3).8 An impatient marriage market indicates higher arrival rates of partners, about fivefold more than when r ¼ 0:06. The preference parameter values swing to the opposite end: as people become more impatient, attraction becomes more sensitive to wage than education. In the modern days as women have wage incomes,

8

I also estimate the model using r ¼ 0:4, with qualitative results similar to r ¼ 0:9.

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On Estimation of a Two-Sided Matching Model

Table 3.

Maximum likelihood estimates using the full sample Full Sample

bm bw lm lw d e1 e2 e3 e4 e5 log L

r ¼ 0:06

r ¼ 0:9

0.1955 0.1582 0.2108 0.2166 0.0690 0.1403 0.1193 0.1285 0.0067 0.0959 6170.9484

0.8357 0.9789 1.1807 1.2001 0.0690 0.1123 0.0929 0.0988 0.1006 0.0892 6302.4539

Male-Subsample

Female-Subsample

r ¼ 0:06 (N ¼ 333)

r ¼ 0:06 (N ¼ 539)

0.0006 0.7694 0.2741 0.1871 0.0601 0.1767 0.1167 0.0899 0.0722 0.0613 2310.9816

0.0844 0.2717 0.1195 0.2292 0.0748 0.1755 0.1169 0.1008 0.0813 0.0456 3861.8086

their wage-traits become remarkably valuable as their marriageable qualities when people are impatient. However, a likelihood ratio test shows that ‘people being desperate’ (a high interest rate) is not accepted. The rest of the analysis is based on r ¼ 0:06. 5.3. Subsamples Next, the model is estimated using the male and the female subsample respectively. One problem associated with using subsamples is that spouses’ data are not randomly sampled, this may distort the preference parameter estimates. The results shown in Table 3 (columns 3 and 4) indicate that subsamples grossly underestimate the preference parameters for males but overestimate those for females. The arrival rate of partners for the spouses are also off. These results raise an alarm over using subsamples. For the rest of the results, I utilize the full sample. 5.4. Characteristics of the baseline model The baseline model uses r ¼ 0:06 and the full sample. The mean and variance of type distribution for men are 8.19 and 0.10, respectively, and those for women are 8.01 and 0.06, respectively. So, men have higher mean and variance than women in type distribution. How are agents stratified in equilibrium? There are two classes, with type 1 matching with each other and the rest form the other class. Given this, I calculate the marriage premium and the hazard rates. The marriage premium is defined to be the ratio of the value of marriage E(W(x,Y)) to the value of singlehood V(x) conditional on agents’ types. As type 1 agents marry partners of the same type, the marriage premium is 1. Agents of type 2 and above have identical and positive marriage premium. Table 4 reports that on average men gain slightly less from marriage than women, which is 1.3

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Linda Y. Wong

Table 4.

Model predictions Baseline Model

The marriage premium The average marriage hazard Predicted singlehood duration Actual singlehood duration Predicted marriage duration Actual marriage duration

Regime 1

M

F

M

F

1.0137 0.1729 5.7837 6.9

1.0164 0.1776 5.6306 7.13 8.3768 8.3784

1.0122 0.1705 5.8651 6.9

1.0168 0.1325 7.5472 7.13 8.3768 8.3784

and 1.6 percent, respectively. There is not much sexual difference in marriage premium because of the limited sexual difference in the parameter estimates. Women have a higher marriage hazard than men and the predicted singlehood duration is 1.2–1.5 years lower than the actual duration. An explanation for underestimation in singlehood duration is that only married samples are used. The estimates are likely to be higher if singlehood censoring is allowed. Another explanation is that the solution method may be too rigid to lend itself a close fit to the data. Given the tight structure of the elements in the reservation equations, it is difficult to reconcile both the acceptance probability and the probability contribution of singlehood duration. In particular, the solution method requires an analytic solution from the reservation equation as an input to the likelihood function. This problem may seem evident as we examine the marriage duration outcome: the average marriage duration from the data is 8.378 years, and the predicted value is 8.377, right on the line. The reason for the close fit is that the separation rate is estimated prior to solving the analytic model. But if we do the same for the marriage hazard rate, the present reservation equation is unlikely to be satisfied. Thus, altering the reservation equation by introducing extra parameters may provide a sound alternative. 5.5. Regime 1 In this regime, we consider outcome variables to be dependent on cardinal types. Consider UðxÞ ¼ 0:3x and Qðx; yÞ/2 ¼ y. That is, agents enjoy their own companies proportionally when single, and marriage is now similar to an exchange unit instead of a production unit so that agents obtain their partner’s types as match utility. This gives Z lm Rx ¼ 0:3x þ ½F y ðM x Þ F y ðzފ dz. ð14Þ r þ d z2Mx Ry ¼ 0:3y þ

lw rþd

Z

½F x ðM y Þ z2My

F x ðzފ dz.

ð15Þ

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On Estimation of a Two-Sided Matching Model

The only thing changed from (6) and (7) is the first terms on the right-hand side of (14) and (15), where each equation now varies with the agent’s type. This regime has strong implications on the equilibrium stratification. First and foremost, the equilibrium is no longer marked by ‘Marriage Classes’ as shown in Burdett and Coles (1997). In that study, agents within each class have the same acceptable sets of partners. Here, agents with higher types derive more utility from remaining single, and so they are prone to be more selective, i.e., when different types of agents share the same maximum-attainable-partner-type, higher-type agents tend to have higher reservation values. This specification can give rise to ‘overlapping’ matching sets. When marriage sets overlap, agents are able to match across sets. Besides generating an overlapping stratification pattern, Regime 1 can also give rise to asymmetry in the model as long as the cardinal values of types for men and women sufficiently differ. Figure 1 shows that the stratification is asymmetric between men and women, with men being more selective. This is mainly driven by the higher arrival rate of men, making their value of singlehood higher than that of women. The marriage sets also overlap. As this regime utilizes the flow utility of singlehood as 0.3  x. An important implication of such an assumption is that some lower types of potential partners (types 1 and 2 women) will be rejected. Column 1 in Table 5 shows the parameter estimates. From it, I calculate (see Table 4 for results) the average marriage premium and the hazard rates. Figure 2 Figure 1.

Equlibrium stratification – Regime 1

10 9

Type, Women

8 7 6 5 4 3 2 1

1

2

3

4

5

6 Type, Men

7

8

9

10

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Linda Y. Wong

shows the marriage premiums by types and sex. In general, middle types have higher marriage premiums than others, and women (if married) have a higher marriage premium than men. The average marriage premiums are similar to those in the baseline case. However, women have much lower hazard rates than men because of the lower estimated arrival rate of partners. Figure 3 shows the marriage hazard is concave in agents’ types. The predicted singlehood durations are improved from the baseline model, particularly for women (Table 4). Nonetheless, matching outcomes generated under this regime is a taboo because the reservation solution is not driven by pure ordinal rankings but admit cardinal values, even though this regime indicates that it may be desirable to relax segregation in the sorting outcomes. Table 5.

Maximum likelihood estimates using alternative specifications Regime 1

Regime 2

0.2190 0.2511 0.2508 0.1948 0.0690 0.1729 0.0100 0.2102 0.0589 0.0886 6322.7037

0.8884 0.9972 0.1709 0.1720 0.0690 0.0410 0.2035 0.0469 0.0230 0.2538 6422.5610

bm bw lm lw d e1 e2 e3 e4 e5 log L

Figure 2. 1.05 1.04 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95

1

2

The marriage premium – Regime 1

3

4

5

6 Type

men

7

women

8

9

10

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On Estimation of a Two-Sided Matching Model

Figure 3.

The marriage hazard – Regime 1

0.25 0.2 0.15 0.1 0.05 0

1

2

3

4

5

6 Type

men

7

8

9

10

women

5.6. Regime 2 I also re-estimate the model setting the utility of singlehood to be null, as in Burdett and Coles (1997). Results (Table 5) indicate that parameter estimates become drastically different from the baseline model as well as Regime 1. Equilibrium matching is purely random, partly because people derive no utility from being alone; as marriage is the only way to contribute positive utility, people seek a large acceptance probability by being less selective. Random matching is also driven by the low arrival rates of partners. 6. Conclusion This paper takes preliminary steps toward understanding the implementation of two-sided matching models. Results caution against using non-random samples (of spouses) in two-sided matching estimation. Estimates indicate that agents seem to be patient in the marriage market, and agents’ types are more sensitive to education than wage. The average marriage premium, the mean of the value of marriage that exceeds the value of singlehood, is about 1.3–1.6 percent. The results point out a few notable issues in estimating the partial matching equilibrium: (1) The model is sensitive to singlehood utility specifications, which may be more desirable to be treated as a parameter. (2) The segregation stratification outcome is too restrictive and could lead to limited variations in agents’ marriage behavior. When estimating the restricted model, i.e., computing reservation types, the estimation method does not demonstrate sufficient flexibility to match the data. Even though the method fits marriage duration data extremely well, the

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proposed algorithm – solving for agents’ acceptable range of partner-types based on analytic equations and feeding the solution into the likelihood estimation – does not fit singlehood duration very well, and therefore may not be the best way to tackle the problem. Including censored singlehood duration data, and/or augmenting the model to general equilibrium or to include endogenous search intensity, may be some fruitful avenues to pursue in future work. References Becker, G. (1973), ‘‘A theory of marriage: Part I’’, Journal of Political Economy, Vol. 81, pp. 813–846. Burdett, K. and M. Coles (1997), ‘‘Marriage and class’’, Quarterly Journal of Economics, Vol. 112, pp. 141–168. Diamond, P. (1982), ‘‘A model of price adjustment’’, Journal of Economic Theory, Vol. 3, pp. 156–168. Gale, D. and L. Shapley (1962), ‘‘College admission and the stability of marriage’’, American Mathematical Monthly, Vol. 69, pp. 9–15. Mortensen, D. (1982), ‘‘Property rights and efficiency in mating, racing, and related games’’, AER, Vol. 72, pp. 968–979. Pissarides, C. (1990), Equilibrium Unemployment Theory, Cambridge, MA: MIT Press. Wong, L.Y. (1998), ‘‘Three essays on matching and mating’’, Dissertation, University of Iowa. Wong, L.Y. (2003), ‘‘Structural estimation of marriage models’’, Journal of Labor Economics, Vol. 21, pp. 699–728. Wong, L.Y. (2005), ‘‘Assortative matching: estimation and an application to marriage formation’’, Working paper, Binghamton University. Wong, L.Y. and Q. Yu (2005), ‘‘A bivariate interval censorship model for partnership formation’’, Manuscript, Binghamton University.

CHAPTER 16

A Structural Nonstationary Model of Job Search: Stigmatization of the Unemployed by Job Offers or Wage Offers? Ste´fan Lollivier and Laurence Rioux Abstract We develop a structural nonstationary model of job search in the fashion of van den Berg (1990). Nonstationarity comes from duration-dependence in benefits, in the arrival rate of job offers, and in wage offers. The model is then estimated using the French sample of the European Community Household Panel (ECHP) (1994–2000). This data set provides the variables required to identify the model (reservation wages, job offers arrival rate, accepted wages, and rejected wages) and allows reconstructing the ‘‘true’’ monthly sequence of benefits for each unemployed worker. We find that duration-dependence in job offers is quite limited: the arrival rate of job offers is exactly the same after 2 years of unemployment than at the beginning of the spell. Duration-dependence in wage offers is slightly more pronounced: wages are decreasing during the first 2 years of unemployment. Nevertheless, the most important fall is observed at the beginning of the spell. We also find that those formerly employed in temporary jobs are more sensitive to duration than the other unemployed. Then we simulate the effects, on the expected duration of unemployment, of four reforms of the unemployment compensation system: (A) an increase in the amount of UI benefits, keeping unchanged the profile of benefits over the unemployment spell; (B) the replacement of the declining time sequence of insurance benefits by a constant sequence; (C) the reform B combined with the imposition of punitive sanctions; (D) a 3-month increase in the maximum duration of UI entitlement.

Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75016-1

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

381

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Ste´fan Lollivier and Laurence Rioux

Keywords: unemployment duration, insurance, reservation wages, job search JEL classifications: J64, J65 1. Introduction Various reduced-form empirical studies show a clearly decreasing pattern of the exit rate from unemployment with duration (Narendranathan et al., 1985), even when unobserved heterogeneity is controlled for (Meyer, 1990). These results support the hypothesis of nonstationarity in job search. This nonstationarity may originate from three sources. First, in most countries the level of benefits falls when the unemployed exhaust their entitlement to insurance and become assistance recipients. In some countries,1 the time sequence of insurance benefits itself is declining. Second, the arrival rate of job offers may fall with the length of the unemployment spell if employers interpret longer spells as a bad signal or if workers’ human capital depreciates over time. Finally, for the same two reasons, the distribution of wage offers can be shifted to the left during the unemployment spell. Reduced-form estimations of job search models do not allow to distinguish the respective effects of the three sources of nonstationarity cited above. Moreover, they do not allow to evaluate different reforms of the unemployment compensation system (as a change in the time sequencing of benefits or in the duration of entitlement). If we want to do it, we need to estimate a structural model of job search in a nonstationary environment. And yet, not many structural nonstationary models have been estimated until now. As far as we know, no article has identified the respective effect of the three possible sources of nonstationarity. The reason for this is simple: the data sets used in these studies do not allow to identify all these effects. Nevertheless, there exists a theoretical model. Van den Berg (1990) proposes a very general theoretical framework to model the three sources of durationdependence already mentioned. He first shows that the optimal strategy of an unemployed worker is still a reservation strategy. Next, he derives a differential equation that describes the evolution of the reservation wage over the unemployment spell. However, in the empirical implementation of the model, only one source of nonstationarity is considered: the decrease in the level of benefits that occurs when the unemployed exhaust their entitlement to insurance and become assistance recipients. The model is not identified if other causes of nonstationarity are examined. The arrival rate of job offers, the amount of insurance benefits, and the distribution of wage offers are thus supposed to be constant over the unemployment spell. Cases and Lollivier (1993) are confronted with the same problem. They estimate a structural dynamic model of job search

1

As in France from July 1992 to July 2001.

A structural Nonstationary Model of Job Search

383

where both the amount of benefits and the arrival rate of job offers are allowed to be duration-dependent. However, they have to suppose the distribution of wage offers constant for the model to be identified. In an interesting paper, Garcia-Perez (2003) extends the theoretical model developed by van den Berg to take into account that jobs do not last forever. He advocates that considering this fact may considerably change the durationdependence in reservation wages. The extended job search model is then estimated using Spanish data. This confirms the stronger duration-dependence in reservation wages when employment is not an absorbing state. However, one source of nonstationarity, the decrease in benefits over the unemployment spell, is not taken into account. The whole path of unemployment benefits, which would be required for identification, is not available in the data set he uses. The aim of this paper is twofold. Our first aim is to estimate a structural nonstationary model of job search in the fashion of van den Berg (1990) using the first seven waves of the French sample of the ECHP survey (1994–2000). Compared with the papers mentioned above, this data set has the advantage of allowing to identify the respective impact of the declining sequence of benefits, the fall in the arrival rate of job offers, and the shifting distribution of wage offers. In particular, beyond the declining sequence of benefits, we are able to assess whether the decrease in reservation wages over the spell of unemployment comes more from the fall in job offers or from the shift in the distribution of wage offers. Our second aim is to evaluate the impact of different reforms of the unemployment compensation system. The ECHP survey used in the study interviews in October of each year between 1994 and 2000, the same group of persons representative of the French population on their income and their job or job search. The characteristic of the French data set is to provide enough information to identify the model. More precisely, at each date of interview, the unemployed have to report their reservation wage, the number of job offers received during the previous month, and the corresponding wage (accepted or rejected). Those formerly unemployed are asked to report their post-unemployment wage. In addition, the respondents complete a monthly labor market history from January 1993 to December 2000. Using this information and the French rules on unemployment benefits, we are able to reconstruct the ‘‘true’’ monthly sequence of benefits for each unemployed worker. The three sources of nonstationarity are then identified. Furthermore, the model can be estimated using reported reservation wages and without using them. Comparison of these two estimations allows to evaluate the quality of this variable in the ECHP survey. Once the structural parameters of the model are estimated, alternative reforms of the unemployment compensation system can be simulated and their effects on the expected duration of unemployment and on the path of reservation wages can be evaluated. The different reforms are: (A) a 14% increase in the amount of unemployment insurance (UI) benefits, keeping unchanged the profile of benefits over the unemployment spell; (B) the replacement of the declining

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Ste´fan Lollivier and Laurence Rioux

time sequence of insurance benefits by a constant sequence; (C) the reform B combined with the imposition of punitive sanctions if two job offers are refused;2 and (D) a 3-month increase in the maximum duration of UI entitlement. In reform B, the constant monthly benefit is supposed to be equal to the ‘‘fullrate’’ benefit in the reference situation. Ex ante (i.e., keeping unchanged job search behaviors), this reform increases the financing cost for the UI agency by 14%. Therefore, reforms A and B have ex ante the same cost and only differ by the profile of benefits over the spell of unemployment. Comparison of these two reforms will thus allow for assessing the effect of time sequencing of benefits on the expected duration of unemployment. With reform C, our aim is to evaluate the impact of sanctions, such as benefit cuts, on the exit rate from unemployment. The main results are the following. Duration-dependence in job offers appears quite limited: the arrival rate of job offers is the same after 2 years of unemployment than at the beginning of the spell. Nevertheless, those formerly employed in temporary jobs are more sensitive to duration than the other unemployed. Duration-dependence is slightly more pronounced in wage offers than in job offers. Wages are decreasing during the first 2 years of unemployment. However, the most important fall is observed at the beginning of the spell. Afterwards, duration has a quite limited negative impact. Once again, those formerly employed in temporary jobs are far more sensitive to duration than the other unemployed. More generous insurance benefits have a negative, but quite limited, effect on the exit rate from unemployment, resulting in a small increase in unemployment duration. More precisely, the expected duration of unemployment goes from 14.01 months to 14.35 months (i.e. +2.42%) when the level of UI benefit is raised by 14%. Nevertheless, former high-wage workers are far more sensitive than the others. Replacing a declining time sequence by a flat profile (reform B) lengthens the spell of unemployment, which is in accordance with the theoretical literature. Duration is raised by 1.39 months (+9.92%) with reform B versus 0.34 month (+2.42%) with reform A, for the same ex ante cost. Furthermore, a flat profile of benefits has a dramatic impact on the subset of former high-wage workers, whose unemployment duration is raised by 3.92 months (+24.48%). Compared with reform B, the imposition of sanctions seems to shorten substantially the expected duration of unemployment, which goes from 15.4 months to 13.15 months (i.e. 14.61%). Once again, the effect is stronger on the subset of former high-wage workers, whose unemployment duration is decreased by 6.15 months ( 30.85%). Finally, a 3-month increase in the maximum duration of UI entitlement has a quite limited impact on unemployment duration. The paper is organized as follows. Section 2 presents the data set used and the method implemented to reconstruct the ‘‘true’’ monthly sequence of insurance

2

Reforms B and C can be seen as the soft and hard versions of the reform of the unemployment compensation system that has been implemented in France in July 2001.

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benefits for each unemployed worker. The empirical implementation of a structural nonstationary model of job search is proposed in Section 3. Section 4 presents the results of the model estimation. In Section 5, different reforms of the unemployment compensation scheme are simulated and their effects on the path of reservation wages and the expected unemployment duration are assessed. 2. The data The data used are drawn from the 1994 to 2000 waves of the French sample of the ECHP.3 This sample of households is aimed at studying the dynamics of employment and income. Two types of information are provided. At each date of interview (in October or November of each year), the 17 years old or more respondents are asked to provide information on their personal characteristics, labor market status, and income. If the individual is unemployed, she has to report the number of job offers received during the previous month (0, 1, or more than 1) and the corresponding wage.4 She is also interviewed on her reservation wage, defined as ‘‘the lowest monthly net salary accepted to take a job, divided by the desired number of hours to be worked.’’ If the individual found a job since the last interview, she has to report the accepted post-unemployment wage. In addition, all individuals are asked every year to retrospectively state their monthly labor market status from January of the past year. In that way, we obtain the monthly status from January 1993 to December 2000 of 15,711 individuals: employed in a permanent full-time job, permanent part-time job, temporary full-time job, temporary part-time job, unemployed, or inactive. About 3349 individuals (21.3%) have experienced at least one unemployment spell over the period. Since some labor market histories are incomplete, we selected only 2988 individuals from this last sample, which gives a total number of 5975 unemployment spells. Identifying the respective effects of the different causes of nonstationarity requires information on the monthly path of unemployment benefits. However, in the ECHP, the amounts of UI and unemployment assistance (UA) benefits are only annually recorded. Moreover, this information is of dubious quality: often missing5 and, when not missing, most likely subject to significant measurement error. As a result, we decided not to use this information and to proceed in a completely different way. We reconstructed, for each unemployment spell, the ‘‘true’’ monthly sequence of benefits by applying the French

3

We do not use the 2001 wave because a major reform of UI rules was decided in January 2001 and implemented in July of the same year. 4 Rejected wage offers will be important for the identification of the model. It is worth noting that this information is not available in the European version of the ECHP survey. 5 Both the proportions of UI and UA recipients are much lower than they should be.

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legislation on unemployment compensation. Fortunately, the legislation remained roughly the same over the period 1994–2000 covered by the data set.6 Two compensation systems coexist: insurance and assistance. The UI rules are the following (see Table 1). Depending on two criteria, her age and length of contribution to UI, a person who enters into unemployment is assigned to one of the eight benefit categories and receives insurance benefits for a limited duration. Eligibility for insurance benefits requires four months of contribution during the eight months that preceded the unemployment spell (category 1). The entitlement is then limited to four months. The most numerous category is the 5th category: 14 months of contribution during the 24 months that preceded the unemployment spell are required to enter this category; the maximum duration of UI entitlement is then of 30 months. The benefit paid at the beginning of the unemployment spell (named ‘‘full-rate benefit’’) is the sum of a fixed part and of a part related to the previous before tax labor earnings (40.4%). If this amount is lower than 57.4% of the previous labor earnings, then this last value applies. Lastly, the full-rate benefit cannot be lower than a minimum value, or higher than 75% of the previous labor earnings. The time sequencing of UI benefits is declining: the benefit is first paid ‘‘at full rate’’ and then reduced by a fixed percentage every 4 months until December 1996 and every 6 months from January 1997. However, a minimum allowance is guaranteed. Finally, the unemployed who fail to meet the eligibility criteria or have exhausted their entitlement can still receive an assistance benefit, provided they are over 25 years old and the resources of their household, whatever their origin, are under a threshold that depends on family composition. This benefit is not related to previous earnings and is available for an unlimited period of time. As a result, to reconstruct the ‘‘true’’ sequence of monthly unemployment benefits, we only need to know the employment history during the 24 months preceding the entry into unemployment,7 the age, the past wage, the family composition, and the resources of the household. The first variable can be easily computed using the activity history data set and the others are available in the data set. In principle, this method should be applied only to the unemployment spells beginning after December 1994. However, in practice, a very large proportion of the unemployment spells beginning in 1994 can also be reconstructed. Therefore, we restricted the sample to the spells that begin after

6

Only a minor change occurred in January 1997. Before this date, the amount of insurance benefit was reduced by a fixed percentage every four months; after this date, this was reduced every six months. 7 For the unemployed aged 55 or more, the employment history over the 37 months preceding the spell of unemployment is needed to reconstruct the ‘‘true’’ monthly profile of benefits. When this information is not available, they are treated as the unemployed over 50, with a maximum entitlement of 45 months. Note that they represent only 2.5% of the unemployment spells (Table 2).

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Table 1. Category

The unemployment insurance rules in France

Contribution Duration

Age

Duration at Full Rate (Months)

Stage Duration (Months)

% of Decrease

Compensation Duration (Months)

Indifferent

0

4

25

4

Indifferent

4

4

15

7

o50 years Z50 years

4 7

4 4

17 15

15 21

o25 years 25–50 years Z50 years 50–55 years Z55 years

7 9

4 4

17 17

30 30

15 20

4 4

15 15

45 45

27

4

8

60

Indifferent

4

0

0

4

Indifferent

4

6

15

7

o50 years Z50 years

4 7

6 6

17 15

15 21

o50 years Z50 years

9 15

6 6

17 15

30 45

50–55 years Z55 years

20

6

15

45

27

6

8

60

(A) July 1993–December 1996 1

2

3 4

5’ 5 6 7 8

4 months during the last 8 months 6 months during the last 12 months 8 months during the last 12 months 14 months during the last 24 months 27 months during the last 36 months

(B) January 1997–December 2000 1

2

3 4

5 6

7 8

4 months during the last 8 months 6 months during the last 12 months 8 months during the last 12 months 14 months during the last 24 months 27 months during the last 36 months

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Ste´fan Lollivier and Laurence Rioux

Table 2.

Descriptive statistics %

Gender Men Women

49.03 50.97

Educational level No diploma Elementary school Vocational diploma Junior high school Technical school High school graduate College

16.34 5.9 19.13 18.54 10.79 6.94 22.35

Age r25 years 26–35 years 36–45 years 46–55 years Z55 years

37.56 29.92 18.3 11.69 2.52

First year of the unemployment spell 1994 1995 1996 1997 1998 1999 2000

17.69 19.96 18.16 15.53 11.81 9.51 7.35

N

4438

Source: French sample of the ECHP, Insee, 1994–2000. Sample: Unemployment spells beginning after December 1993.

January 1994, which gives a total number of 4438 unemployment spells. Among these, 820 (18%) are right censored. The data set is thus an inflow sample and not a stock sample, which saves us having to deal with the problem of left censoring. Some summary statistics are presented in Table 2. The unemployed are young: 38% are less than 25 years old and 30% are between 26 and 35 years old. About 40% have at least completed high-school education. Figure 1 shows that the instantaneous exit rate from unemployment diminishes continuously with the duration of the spell.8 This goes from about 0.13 for a duration of 1–3

8

More precisely, this result appears for duration shorter than 40 months. For duration longer than 40 months, the population becomes too small for this result to be confirmed.

A structural Nonstationary Model of Job Search

Figure 1.

389

Instantaneous exit rate from unemployment

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

Source: French sample of the ECHP, Insee, 1994−2000. Sample: Unemployment spells beginning after December 1993.

months to around 0.02 for a duration of 40 months. The decrease is more pronounced during the first 12 months than after. Figure 2 depicts the monthly distributions of accepted wages, of reported reservation wages, and of rejected wage offers. The average (median) postunemployment monthly wage is equal to 7496 Frs9 (6685 Frs) and a large proportion of accepted wages is close to the minimum wage. The average (median) monthly reservation wage is equal to 7201 Frs (6171 Frs). The distribution of reservation wages has the same shape than the distribution of accepted wages, but is slightly shifted to the left. Reservation wages are thus slightly lower than accepted wages. Lastly, 3350 unemployed reported the number of job offers they received the previous month. Among them, 8% have received at least one offer and rejected it. The mean (median) of the 164 rejected wage offers is equal to 6823 Frs (6205 Frs). Logically, the distribution of rejected wage offers is shifted to the left of the distribution of reservation wages. 3. The empirical implementation of a structural nonstationary job search model 3.1. The model The labor market is described by a structural nonstationary model of job search in the fashion of van den Berg (1990). The workers can be in two states: employed or unemployed. Let t denote the elapsed duration of the unemployment spell.10 At date t, an unemployed agent receives a duration-dependent

9

1 Franc is equal to about 0.15 Euro. For ease of exposition, let us assume that calendar time and unemployment duration coincide.

10

Ste´fan Lollivier and Laurence Rioux

390

Figure 2.

Density functions of monthly accepted wages, reservation wages, and rejected wages Accepted wages Reservation wages Rejected wages

200

1650

3100

4550

6000

7450

8900 10350 11800 13250 14700

Note: Wages measured in French Francs. Source: French sample of the ECHP, Insee, 1994−2000. Sample: Unemployment spells beginning after December 1993.

unemployment benefit b(t). Job offers arrive according to a Poisson process with parameter l(t). The wage associated to a job offer is a random drawing from a cdf Ft(wt). When a job offer arrives, the agent has to decide whether to accept or to reject it. If the job offer is accepted, the agent keeps it forever: on-the-job search and the risk of job loss are thus excluded.11 Otherwise, she stays

11

The assumption that the worker remains employed forever at the same wage is unsatisfactory, but usual in this type of structural nonstationary models (Wolpin, 1987; van den Berg, 1990; Frijters and van der Klaauw, 2005). To our knowledge, there are only two exceptions in the literature. Joutard and Ruggiero (2000) estimate a structural nonstationary model where an employed worker can lose his job and go back to unemployment. However, they have to make strong hypotheses on the behavior of job search: the horizon is limited to three spells of unemployment; the environment becomes stationary as soon as the second spell begins. These hypotheses complicate importantly the model. A second exception is a paper by Garcia-Perez (2003) that extends the theoretical model developed by van den Berg to take into account that jobs do not last forever. The difficulty with this kind of model is that the present value of employment must integrate all the future possibilities of new unemployment spells. In Garcia-Perez, UI entitlement is not explicitly modelized and does not depend on the past employment history. This simplifies the computation of the present value of employment. In our paper, at the opposite, we take into account that the whole monthly sequence of benefits depends on the past employment history. If jobs do not last forever, the present value of employment becomes extremely complicated to compute. For this reason, we prefer to assume, as usual, that employment is an absorbing state.

A structural Nonstationary Model of Job Search

391

unemployed and continues to search for a job. The instantaneous utility function is supposed to be linear: thus the utility of an employed agent is wt and that of an unemployed is b(t). The agent discounts future utility at the subjective rate r. Nonstationarity thus originates from the duration-dependence in benefits, in the arrival rate of job offers, and in wage offers. The benefit is durationdependent for two reasons. First, because unemployment insurance pays a declining compensation. Second, because two compensation systems coexist: insurance, which pays relatively high benefits at the beginning of the unemployment spell, and assistance, which gives a relatively low compensation to those who are no more eligible for UI. The arrival rate of job offers may fall with the length of the unemployment spell if employers interpret longer spells as a bad signal or if workers’ human capital depreciates during spells of unemployment. For the same two reasons, the distribution of wage offers can be shifted to the left. In the rest of the paper, workers’ observable characteristics are supposed to be constant over the duration of unemployment. Stigmatization will thus be the only possible source of duration-dependence in both job offers and wage offers. More precisely, compared to a short-term unemployed agent, a long-term one can suffer from two things: she can receive fewer job offers; the wage associated to a job offer can be lower. We will test whether the first type or the second type of stigmatization occurs actually, and, if both occur, which one dominates the other. Let U(t) denote the expected flow of income for an agent who is unemployed for t units of time. Under the hypothesis that at most one job offer arrives in the small interval [t,t+h], the Bellman’s equation for U(t) verifies (van den Berg, 1990):
 E wt max wrt ; U ðt þ hÞ hbðtÞ ð1 lðtÞhÞUðt þ hÞ UðtÞ ¼ ð1Þ þ þ lðtÞh 1 þ rh 1 þ rh 1 þ rh At date t, the policy that maximizes the expected return from unemployment is to accept any job offer if the associated wage exceeds rU(t). Let R(t) ¼ rU(t) denote the reservation wage. Rewriting Equation (1), we obtain the value of the reservation wage at date t: ¼ hrbðtÞ þ lðtÞh

ð1 þ rhÞRðtÞ Rðt þ hÞ R1 Rðt þ hÞð1 RðtþhÞ wt dF t ðwt Þ

F t ðRðt þ hÞÞÞ

ð2Þ

We assume that there exists some date T such that the environment remains stationary after T. The variables b(t), l(t), and Ft(wt) are thus supposed to be constant on [T, N] and to take the respective values b, l, and F(w). Before T, the environment is nonstationary and the sequence of reservation wages is described by Equation (2). After T, the optimal strategy is stationary. If R denotes the constant reservation wage, then R is the solution of Z 1   rR ¼ rb þ l wdF ðwÞ Rð1 F ðRÞÞ ð3Þ R

Ste´fan Lollivier and Laurence Rioux

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 it Since the left-hand side is increasing and the right-hand side decreasing in R, follows that this equation has a unique finite solution. It is worth noting that this solution can be computed only numerically and not analytically. Moreover, since the agents differ from one another in the amount of benefit, in the arrival rate of job offers, and in the distribution of wage offers, the computation of R has to be made for each individual of the sample. In the data set, the unit of time is the month. We thus make the hypothesis that all the variables are constant over this unit time interval and we estimate a discrete time dynamic programing model. Using backward induction, the sequence of optimal reservation wages in discrete time can then be rewritten as ð1 þ rÞRt ¼ Rtþ1 þ rbt þ lt for

R1

Rtþ1

t ¼ 1; . . . ; T

wt dF t ðwt Þ 1

with

Rtþ1 ð1

RT ¼ R

F t ðRt 1 ÞÞ

ð4Þ

The probability of getting a job after an unemployment spell of length t is yt ¼ lt ð1

F t ðRt ÞÞ

Hence, the integrated hazard at date t is given by: ht ¼

t X

ð5Þ

yt

t¼1

3.2. Parameterization The arrival rate of job offers is lt ¼ l0t expðZcÞ

ð6Þ

log wt ¼ Xb þ dt þ su u

ð7Þ

where Z is a vector of observable characteristics and l0t the baseline rate. Z is constant over the duration of unemployment, which means that the agents do not suffer from any loss of human capital. This leads us to interpret any change in the arrival rate of job offers over the spell of unemployment as ‘‘stigmatization’’ of the long-term unemployed by employers. The distribution of wage offers is supposed to be log-normal (Wolpin, 1987) and to have the following form: where X is a vector of observable individual characteristics, dt a parameter that captures the effects of duration-dependence, and u an error term that is normally distributed with mean 0 and variance 1. Thus we restrict the change in the distribution of wage offers to a shift in the mean of the distribution, holding variance constant. Duration-dependence in both l0t and dt is captured by a piecewise linear function defined by four thresholds (t1 ¼ 7, t2 ¼ 13, t3 ¼ 25, t4 ¼ 36) and four

A structural Nonstationary Model of Job Search

393

different inclinations, and constant after t4 ¼ 36. Note that the parameters l0t will be estimated under the exponential link function to insure their positivity. Combining Equations (5)–(7), the hazard rate can be rewritten as    logðRt Þ Xb dt yt ¼ l0t expðZcÞ 1 F su

ð8Þ

We assume that the observed wage is measured with a multiplicative error term that is independent of the ‘‘true’’ wage offer received by the worker (Wolpin, 1987; Eckstein and Wolpin, 1995).12 Denoting the observed wage by w~ t and the ‘‘true’’ wage by wt, the two are supposed to be related by logðw~ t Þ ¼ logðwt Þ þ sv v;

with

v ! Nð0; 1Þ

The measurement error, v, is distributed N(0, 1) and independent of u. With the above assumptions, the observed wage offer distribution can be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi logðw~ t Þ ¼ Xb þ dt þ s2u þ s2v s; with s ! Nð0; 1Þ. Individuals who are unemployed at the date of the interview are asked for their lowest acceptable net wage in a job at that date. These observed reservation wages ðR~ t Þ may differ from the true ones (Rt) because of a measurement error (van den Berg, 1990). This error term is supposed to be normally distributed (with mean 0 and variance 1) and independent of duration. Thus, logðR~ t Þ ¼ logðRt Þ þ sw w;

with

w ! Nð0; 1Þ

The error terms u, v, and w are supposed to be independent. Finally, we take into account that macroeconomic events, like economic policy changes or business cycle effects, may occur. In this aim, we include a dummy variable for each calendar year. We thus consider two distinct calendars (the duration of the spell of unemployment and the usual calendar). 3.3. Likelihood of the sample Three types of information are available on a spell of unemployment: its duration, the monthly amount of benefit, and the post-unemployment wage. Moreover, at each date of interview, the unemployed report their reservation wage, the number of job offers they received during the previous month, and the wage corresponding to each rejected offer. Both types of information are used to write the likelihood.

12

If observed wages are measured without error, the maximum likelihood estimator for the reservation wage is the minimum accepted wage. The existence of very low wages in the data set has thus a very large impact on the estimated reservation wages (Flinn and Heckman, 1982). In order to deal with this problem, we assume that observed wages are measured with a multiplicative error term.

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3.3.1. Information on the spell of unemployment There are three different cases: the spell of unemployment can be right-censored; it can be completed but the accepted wage is unobserved; it can be completed and the accepted wage is observed. The likelihood contribution of a censored spell at date t is simply given by the survivor function l ¼ expð ht Þ If the wage offer is unobserved, the likelihood contribution of a completed spell at date t is equal to the density of completed unemployment duration l ¼ yt expð ht Þ Finally, if the offered wage is observed, the likelihood contribution of a completed spell at date t is obtained by multiplying the conditional density function of accepted wages and the density function of unemployment duration. The accepted wage is drawn from the wage offer distribution truncated at the reservation wage. We also need to take into account that the observed wage is measured with error and thus different from the offered wage. The probability that the observed wage is w~ t given that the offered wage exceeds the reservation wage is then given by Pðw~ t jwt  Rt Þ ¼

Pðw~ t ; wt  Rt Þ Pðw~ t ÞPðwt  Rt jw~ t Þ ¼ Pðwt  Rt Þ Pðwt  Rt Þ

where   1 logðw~ t Þ Xb dt f Pðw~ t Þ ¼ ss w~ t ss   logðRt Þ Xb dLt pðwt  Rt Þ ¼ 1 F su       u 0 1 c !N ; where Since u and s are normally distributed, 0 c 1 s  c2 ¼ s2u s2s represents the share of observed wage variation which is not explained by the measurement error. The density function associated to the conditional distribution of u, given s thus verifies: 1 pffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi exp 2p 1 c



ðu csÞ2 2ð1 c2 Þ



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A structural Nonstationary Model of Job Search

This leads to

Pðwt  Rt jw~ t Þ ¼ 1

2



6 logðRt Þ F4

Xb

dt

su

c2

logðw~ t Þ

Xb

dt

su



3

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 1 c2

3.3.2. Information provided by the annual interview Since the arrival rate of job offers follows a Poisson process with parameter lt and since we assume that at most one job offer arrives per month, the probability of receiving a job offer between the months t 1 and t is l ¼ lt

1

while the probability of receiving no offer is l¼1

lt

1

If a job offer has been received, the unemployed are asked to report the offered wage. This wage is lower than the reservation wage, since the offer has been rejected. The offered wage is thus drawn from the truncated wage offer distribution. Let w^ t 1 denote the observed rejected wage at t 1. Some computations lead to the likelihood contribution of a rejected wage offer at t 1: Pðw^ t 1 jwt 1 oRt 1 Þ 

f c

logðw^ t 1 Þ

Xb su

dt

1



c w^ t 1 su

¼

2  6 logðRt 1 Þ F4 F



Xb

dt

logðw^ t 1 Þ c2

1

su

logðRt 1 Þ

Xb su

dt

1



Xb su

dt

1



3

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 1 c2

Finally, if the reported reservation wage at the date of interview is R~ t , then the likelihood contribution is given by

PðR~ t Þ ¼

  1 logðR~ t Þ logðRt Þ f sw R~ t sw

3.3.3. Likelihood of the sample If the observations are assumed independent, the final likelihood of an unemployment spell is obtained by multiplying the different likelihood contributions. For instance, take the case of a completed unemployment spell of length t with an observed accepted wage w~ t . Furthermore, assume that a reservation wage R~ t 3 is reported at date t 3 , and a job offer with an associated

Ste´fan Lollivier and Laurence Rioux

396

wage w^ t L

4

is rejected at date t 4 . Then the likelihood can be written as

  1 logðR~ t 3 Þ logðRt 3 Þ f ~ s Rt 3 sw w 0 31 2     ~ t Þ Xb dt logðw~ t Þ Xb dt c B 1 7C 6 logðRt Þ Xb dt 2 logðw qffiffiffiffiffiffiffiffiffiffiffiffiffiffi5A f c c @1 F4 w~ t su su su su 1 c2   logðRt Þ Xb dt 1 F su 0 2 31     ^ t 4 Þ Xb dt 4 logðw^ t 4 Þ Xb dt 4 c B 6 logðRt 4 Þ Xb dt 4 1 7C 2 logðw qffiffiffiffiffiffiffiffiffiffiffiffiffiffi5A f c c @F4 w^ t 4 su su su su 1 c2   logðRt 4 Þ Xb dt 4 F su

¼ yt expð

ht Þð1

lt 4 Þ

3.3.4. Identification The identification is very much in line with Flinn and Heckman (1982). The main difference is that we observe rejected wage offers. This means that the wage offer distribution below the reservation wage can be identified. Two other points are worth to be noted. First, both reported postunemployment wages and reservation wages are supposed to be measured with error. Yet, the data do not allow to separately identify su, sw, and c2 (the share of observed wage variation which is not explained by the measurement error). In the literature, this problem is usually solved by supposing only one type of measurement error, either in observed reservation wages (van den Berg, 1990) or in observed accepted wages (Wolpin, 1987; Eckstein and Wolpin, 1995; or Garcia-Perez, 2003). It seems difficult to believe that post-unemployment wages can be observed without error. It seems even more difficult to believe that reservation wages can be correctly reported by the unemployed, while a measurement error is supposed to be in observed accepted wages. This leads us to proceed in a different way: estimate the model conditionally on c2. We will thus make different hypotheses on c2 (25%, 50%, 75%, and 90%) and estimate the other structural parameters for each of these identifying assumptions. The large set of values for c2 is justified by the very different results found in the literature. For instance, Eckstein and Wolpin (1995) show that a very small fraction of the variance of accepted wages is due to measurement error (c2>86%). The opposite result is found by Garcia-Perez (2003) (c2 ¼ 8%). It is also worth to note that reported reservation wages are needed only to identify sw. The model can thus be estimated under two alternative specifications: one using reported reservation wages data and the other not using them. Comparison of these two estimations allows to test the quality of reported reservation wages. The structural parameters of the model are estimated by maximizing the loglikelihood of the sample with respect to b, c, dt, l0t, r, su, and sw, under the

397

A structural Nonstationary Model of Job Search 2

restriction imposed by Equation (4) and the hypothesis on c . We proceed in the following way. First, the duration of unemployment T after which all exogenous variables are supposed to be constant is taken equal to 36 months.13 Equation (3) is then numerically solved for the stationary value R at the point T. R ¼ RðTÞ serves as an initial condition for the differential Equation (4). Using backward induction, the whole sequence of optimal reservation wages R(t), t ¼ 1, y ,T is then obtained. The likelihood contribution is next deduced as a complicated function of the unknown parameters. This procedure has to be repeated for each unemployed individual, since the agents differ from one another in the time sequence of benefit and in observable characteristics. 4. Results The results for different values of c2 (the share of observed wage variation which is not explained by the measurement error) are presented in Table 3. Comparison of columns 1–4 reveals that the choice of c2 has a limited impact on the parameters estimates, apart from the discount rate.14 We will, thus, only discuss the case where c2 ¼ 50%. In this case, the estimated monthly discount rate is 0.0079, which corresponds to a quite reasonable rate of 9.5% a year. We first examine the duration-dependence in job offers. The former employed in temporary jobs are likely to differ in their job search behavior from the other unemployed. In particular, they are likely to be more sensitive to unemployment duration. This explains why they are distinguished from the rest of the population in the estimation. Their probability of receiving a job offer remains stable during the first 6 months and decreases during the following 6 months. Thus, after 1 year of unemployment, the arrival rate of job offers is 30% lower than initially and does not change during the second year of unemployment. A decrease is observed during the third year of unemployment. Nevertheless, this last result has to be considered with caution, given the small number of persons with more than 2 years of unemployment in the sample.

13

This hypothesis seems quite reasonable for insurance benefits since the maximum duration of UI entitlement is equal to 30 months for the unemployed less than 50-years old (Tables 1A and 1B). After 30 months of unemployment, these latter are eligible to unemployment assistance (RMI or ASS). Unlike insurance benefits, assistance benefits are constant over the spell of unemployment and are paid as long as the person is unemployed. Only the unemployed over 50 may still be entitled to UI after 36 months of unemployment, provided their cumulated employment duration during the 24 months preceding their entry into unemployment is longer than 14 months. Few people combine all these requirements in the sample. 14 The discount rate is strongly sensitive to the hypothesis made on c2. The monthly estimation goes from 0.0033 for c2 ¼ 25% to 0.0160 for c2 ¼ 90%.

Ste´fan Lollivier and Laurence Rioux

398

Table 3.

Wage offers equation Constant Age Age2 Women Born out of France Living in Ile-deFrance Educational level No diploma or elementary school Vocational diploma Technical school graduate High school graduate College graduate Local labor market Unemployment rate Long-term unemployment rate

Estimated parameters of the structural model c2 ¼ 25%

c2 ¼ 50%

c2 ¼ 75%

c2 ¼ 90%

8.72 (204.9) 0.0092 (5.8) 0.00007 ( 1.4)

8.69 (170.6) 0.0095 (5.2) 0.00003 ( 0.6)

8.61 (143.1) 0.0088 (4.1) 0.00002 (0.3)

8.53 (121.0) 0.0075 (3.1) 0.00008 (1.0)

0.13 0.02 0.16

( 13.2) (1.0) (11.8)

0.10 0.04 0.16

( 9.3) (2.0) (10.2)

0.07 0.06 0.16

( 5.9) (2.7) (8.6)

0.05 0.08 0.16

( 3.5) (3.2) (7.6)

0.05

( 3.3)

0.03

( 2.0)

0.02

( 1.0)

0.00

( 0.2)

0.04 0.01

( 2.9) (0.7)

0.05 0.01

( 2.9) (0.4)

0.06 0.00

( 2.9) (0.0)

0.07 0.01

( 2.9) ( 0.3)

0.04 0.17

(1.8) (11.6)

0.04 0.16

(1.7) (9.7)

0.03 0.14

(1.2) (7.7)

0.02 0.13

(0.8) (6.3)

0.0039 0.0015

( 1.9) (1.1)

0.0019 0.0017

( 0.8) (1.1)

0.0002 0.0018

(0.1) (1.0)

0.00222 0.00140

(0.7) (0.7)

0.17

( 7.1)

0.15

( 5.3)

0.13

( 3.8)

0.12

( 2.8)

0.01

( 0.6)

0.01

(0.2)

0.02

(0.4)

0.02

(0.4)

Past labor market state Permanent part-time job Temporary full-time job Temporary part-time job Military service Occasionnal activities Inactivity

0.03

(1.1)

0.10

(2.9)

0.16

(3.6)

0.21

(3.8)

0.20 0.04 0.08

( 7.5) ( 1.9) ( 3.2)

0.21 0.03 0.02

( 7.6) ( 1.3) ( 0.7)

0.22 0.02 0.03

( 7.4) ( 0.8) (0.7)

0.24 0.02 0.08

( 7.4) ( 0.5) (1.7)

Temporal dummy 1994 1996 1997 1998 1999 2000

0.11 0.00 0.14 0.12 0.13 0.242

( 6.7) (0.3) (5.4) (4.6) (7.0) (12.9)

0.13 0.04 0.12 0.06 0.05 0.138

( 5.6) ( 1.8) (4.9) (2.3) (2.0) (6.0)

0.12 0.03 0.10 0.04 0.02 0.09

( 4.4) ( 1.4) (4.1) (1.6) (1.0) (3.4)

0.11 0.03 0.10 0.02 0.01 0.05

( 3.8) ( 1.1) (3.6) (0.8) (0.5) (1.9)

( 3.3) ( 0.5) ( 2.7) (5.4)

0.035 0.002 0.014 0.036

( 3.4) ( 0.2) ( 2.4) (5.7)

Duration (except for the previous employed in a temporary job) 1–6 months 0.016 ( 4.3) 0.020 ( 3.4) 0.027 7–12 months 0.006 ( 1.9) 0.006 ( 1.1) 0.004 13–24 months 0.007 ( 3.6) 0.010 ( 3.0) 0.012 25–36 months 0.005 (1.8) 0.017 (4.3) 0.028

399

A structural Nonstationary Model of Job Search Table 3 (continued ) c2 ¼ 25%

c2 ¼ 50%

Duration (for the previous employed in a temporary 1–6 months 0.029 ( 7.6) 0.040 7–12 months 0.015 ( 4.2) 0.017 13–24 months 0.005 ( 1.9) 0.005 25–36 months 0.002 (0.6) 0.011 Standard error su Job offers equation Constant Age Age2 Women Born out of France Living in Ile-deFrance Educational level No diploma or elementary school Vocational diploma Technical school graduate High school graduate College graduate Local labor market Unemployment rate Long-term unemployment rate

0.13

(55.9)

0.21

job) ( 7.4) ( 3.2) ( 1.3) (2.4) (54.1)

c2 ¼ 75%

c2 ¼ 90%

0.052 0.017 0.004 0.020

( 7.6) ( 2.4) ( 0.8) (3.2)

0.066 0.015 0.002 0.026

( 8.0) ( 1.8) ( 0.4) (3.4)

0.31

(52.5)

0.42

(50.7)

1.19 ( 5.0) 0.0005 ( 0.1) 0.00057 ( 2.2)

1.15 ( 4.9) 0.0018 ( 0.2) 0.00063 ( 2.4)

1.11 ( 4.9) 0.0014 ( 0.2) 0.00067 ( 2.7)

1.13 ( 5.1) 0.0009 (0.1) 0.00073 ( 2.9)

0.23 0.253 0.00

( 4.5) ( 2.8) (0.0)

0.25 0.251 0.02

( 4.8) ( 2.8) ( 0.2)

0.25 0.24 0.03

( 5.0) ( 2.8) ( 0.4)

0.25 0.25 0.03

( 5.2) ( 2.9) ( 0.4)

0.13

( 1.6)

0.14

( 1.7)

0.13

( 1.6)

0.13

( 1.7)

0.05 0.08

(0.7) (0.8)

0.07 0.10

(0.9) (1.1)

0.09 0.12

(1.2) (1.3)

0.10 0.12

(1.3) (1.3)

0.05 0.13

(0.5) (1.7)

0.05 0.15

(0.5) (1.9)

0.06 0.16

(0.6) (2.1)

0.06 0.17

(0.6) (2.3)

0.03 0.00

( 2.5) (0.4)

0.03 0.00

( 2.4) (0.0)

0.03 0.00

( 2.4) (-0.1)

0.03 0.00

( 2.5) (0.1)

0.19

( 1.4)

0.20

( 1.5)

0.20

( 1.5)

0.20

( 1.5)

0.44

(3.3)

0.49

(3.7)

0.52

(4.1)

0.53

(4.3)

Past labor market state Permanent part-time job Temporary full-time job Temporary part-time job Military service Occasionnal activities Inactivity

0.11

( 0.7)

0.08

( 0.6)

0.05

( 0.3)

0.03

( 0.2)

0.47 0.07 0.44

(3.7) (0.6) ( 2.8)

0.51 0.08 0.47

(4.0) (0.6) ( 3.2)

0.51 0.08 0.47

(4.2) (0.7) ( 3.3)

0.50 0.08 0.46

(4.3) (0.7) ( 3.3)

Temporal dummy 1994 1996 1997 1998 1999 2000

0.09 0.16 0.17 0.00 0.15 0.12

(1.0) (1.8) ( 1.4) (0.0) (1.6) (1.1)

0.10 0.16 0.13 0.04 0.17 0.27

(1.2) (1.9) ( 1.3) (0.4) (1.8) (2.5)

0.05 0.09 0.09 0.04 0.12 0.27

(0.6) (1.1) ( 1.0) (0.4) (1.3) (2.7)

0.02 0.05 0.06 0.06 0.10 0.25

(0.3) (0.7) ( 0.8) (0.7) (1.2) (2.7)

Ste´fan Lollivier and Laurence Rioux

400 Table 3 (continued )

c2 ¼ 25%

c2 ¼ 50%

c2 ¼ 75%

c2 ¼ 90%

Duration (except for the previous employed in a temporary job) 1–6 months 0.07 (2.5) 0.06 (2.3) 0.07 7–12 months 0.06 ( 2.4) 0.06 ( 2.5) 0.07 13–24 months 0.01 (0.6) 0.01 (0.7) 0.01 25–36 months 0.06 ( 2.6) 0.07 ( 3.4) 0.07

(2.5) ( 2.8) (0.7) ( 3.7)

0.07 0.07 0.01 0.07

(2.8) ( 3.1) (0.6) ( 3.7)

Duration (for the previous employed in a temporary 1–6 months 0.02 (1.0) 0.02 7–12 months 0.05 ( 1.7) 0.05 13–24 months 0.01 ( 0.6) 0.01 25–36 months 0.03 ( 1.2) 0.04 sw

0.25

Discount rate r

0.0033

(67.4) (8.1)

job) (0.9) ( 2.0) ( 0.6) ( 1.4)

0.02 0.06 0.01 0.04

(1.1) ( 2.4) ( 0.6) ( 1.6)

0.03 0.07 0.02 0.04

(1.5) ( 2.7) ( 0.8) ( 1.8)

0.24

(68.5)

0.24

(68.5)

0.25

(67.9)

0.0079

(11.8)

0.0126

(15.7)

0.0160

(18.2)

Note: t-ratio are in parentheses. Estimation by maximum likelihood. Reference: a man, born in France, junior high-school graduate, unemployed in 1995, previously employed in a permanent fulltime job. Source: French sample of the ECHP, Insee, 1994–2000. Sample: Unemployement spells beginning after December 1993.

Let us now examine the rest of the population. For them, durationdependence in job offers appears quite limited. More precisely, the initial increase in the arrival rate of job offers observed during the first 6 months is exactly offset by a movement in the opposite sense during the following 6 months. Afterwards, their probability of receiving an offer remains stable. As a consequence, the arrival rate of job offers is exactly the same after 2 years of unemployment than at the beginning of the spell. Not surprisingly, men and high-educated workers are offered more jobs than women and less-skilled workers. The effect of education is nevertheless lower than expected: college graduates are the only ones to receive significantly more job offers than junior high-school graduates (the reference situation). The past labor market state is also an important variable. In particular, the unemployed who were previously employed in a temporary full-time job receive much more job offers (+49%) than those who were employed in a permanent full-time job. The local unemployment rate affects negatively the arrival rate of job offers. Lastly, the dummy variables included to capture macroeconomic events become significant and positive at the end of the period, because of the more favorable economic situation at this time. Let us now examine the duration-dependence in wage offers. Wages are decreasing during the first 2 years of unemployment. However, the most important fall is observed at the beginning of the spell (1–6 months). Afterwards, duration has a quite limited negative impact. After 2 years of unemployment, the

A structural Nonstationary Model of Job Search

401

impact becomes even positive. This result, which has to be considered with caution, is due to unobserved heterogeneity. The former employed in temporary jobs can again be distinguished from the other unemployed: the decrease in wage offers is more pronounced for them. Besides duration-dependence, the equation of wage offers presents a fairly familiar picture. Not surprisingly, men and inhabitants of Ile-de-France sample wage offers from a distribution that statistically dominates that for women and inhabitants outside of Ile-de-France, respectively. For instance, all other things being equal, the wages offered to women are 10% lower than that offered to men. Living in Ile-de-France gives a gain of 16%, which is close to what is usually found in wage equations. Some other variables have a smaller impact than in usual wage equations. Among them, age (only 0.95% per year) and education: college graduates are the only ones to receive significantly higher wage offers. The model is also estimated without using reported reservation wages. Table 4 presents the estimation results for c2 ¼ 50%. Comparison of columns 1 and 2 reveals that the estimated parameters are close under the two alternative specifications. In particular, the duration-dependence in the arrival rate of job offers is not at all affected by the choice of using or not using reported reservation wages. Concerning the duration-dependence in wage offers, the only significant change occurs for the third year of unemployment.15 The discount rate is the only other variable strongly affected. This leads us to conclude that reported reservation wages are rather consistent with the reservation wages implied by the theoretical model. Nevertheless, the measurement error in observed reservation wages is high (sw ¼ 0.24). To test the validity of the parameters estimates, the model is simulated on the whole sample for c2 ¼ 50%. Figure 3 graphically depicts the observed and simulated exit rates from unemployment and the simulated arrival rate of job offers as functions of duration. The simulated arrival rate of job offers is clearly increasing during the first 6 months of unemployment, decreasing over the following 6 months, and constant during the second year. This means that the unemployed receive one job offer per period of 3.95 months during the first 6 months and one per period of 4.61 months during the following 6 months. For a duration between 12 and 24 months, the arrival rate goes to one offer per period of 5.54 months and diminishes to one per period of 7.94 months for longer durations. The simulated exit rate from unemployment decreases with duration during the first year of unemployment (from an initial value equal to 0.12). Afterwards, this remains constant, around 0.05. The simulated hazard rate is quite close to

15

The coefficient is significant when using the reported reservation wages and nonsignificant under the alternative specification. But remember that the number of persons with more than 2 years of unemployment in the sample is small.

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Table 4.

Estimations using and not using reported reservation wages (c2 ¼ 50%) Not using Reported Reservation Wages

Wage offers equation Constant Age Age2

Using Reported Reservation Wages

8.65 0.0125 0.00016

(129.4) (4.4) ( 1.7)

8.69 0.0095 0.00003

(170.6) (5.2) ( 0.6)

0.11 0.05 0.16

( 6.7) (1.7) (7.2)

0.10 0.04 0.16

( 9.3) (2.0) (10.2)

Educational level No diploma or elementary school Vocational diploma Technical school graduate High school graduate College graduate

0.05 0.07 0.01 0.04 0.13

( ( ( (

0.03 0.05 0.01 0.04 0.16

( 2.0) ( 2.9) (0.4) (1.7) (9.7)

Local labor market Unemployment rate Long-term unemployment rate

0.0083 0.0031

( 2.4) (1.4)

0.0019 0.0017

( 0.8) (1.1)

Past labor market state Permanent part-time job Temporary full-time job Temporary part-time job Military service Occasionnal activities Inactivity

0.07 0.02 0.05 0.20 0.04 0.09

( 1.5) (0.7) (1.1) ( 5.6) ( 1.0) ( 1.9)

0.15 0.01 0.10 0.21 0.03 0.02

( 5.3) (0.2) (2.9) ( 7.6) ( 1.3) ( 0.7)

Temporal dummy 1994 1996 1997 1998 1999 2000

0.13 0.02 0.15 0.17 0.20 0.41

( 4.9) ( 0.8) (4.6) (5.3) (7.1) (13.4)

0.13 0.04 0.12 0.06 0.05 0.138

( 5.6) ( 1.8) (4.9) (2.3) (2.0) (6.0)

Duration (except for the previous employed in a temporary job) 1–6 months 0.026 ( 4.4) 7–12 months 0.009 ( 1.8) 13–24 months 0.012 ( 3.9) 25–36 months 0.003 (0.6)

0.020 0.006 0.010 0.017

( 3.4) ( 1.1) ( 3.0) (4.3)

Duration (for the previous employed in a temporary job) 1–6 months 0.044 ( 7.3) 7–12 months 0.025 ( 4.3) 13–24 months 0.008 ( 1.8) 25–36 months 0.002 ( 0.4) 0.22 (51.0) Standard error su

0.040 0.017 0.005 0.011 0.21

( 7.4) ( 3.2) ( 1.3) (2.4) (54.1)

Job offers equation Constant Age

1.15 0.0018

( 4.9) ( 0.2)

Women Born out of France Living in Ile-de-France

1.18 0.0010

2.0) 2.9) 0.3) 1.0) (5.5)

( 5.0) ( 0.1)

403

A structural Nonstationary Model of Job Search Table 4 (continued ) Not using Reported Reservation Wages Age2 Women Born out of France Living in Ile-de-France

0.00054 0.23 0.23 0.02

( ( ( (

Educational level No diploma or elementary school Vocational diploma Technical school graduate High school graduate College graduate

0.13 0.05 0.07 0.03 0.11

Local labor market Unemployment rate Long-term unemployment rate

0.00063 0.25 0.251 0.02

( ( ( (

( 1.7) (0.6) (0.8) (0.2) (1.4)

0.14 0.07 0.10 0.05 0.15

( 1.7) (0.9) (1.1) (0.5) (1.9)

0.03 0.00

( 2.6) (0.2)

0.03 0.00

( 2.4) (0.0)

Past labor market state Permanent part-time job Temporary full-time job Temporary part-time job Military service Occasionnal activities Inactivity

0.19 0.46 0.09 0.47 0.10 0.39

( 1.4) (3.5) ( 0.6) (3.8) (0.8) ( 2.5)

0.20 0.49 0.08 0.51 0.08 0.47

( 1.5) (3.7) ( 0.6) (4.0) (0.6) ( 3.2)

Temporal dummy 1994 1996 1997 1998 1999 2000

0.07 0.19 0.01 0.11 0.28 0.24

(0.8) (2.2) ( 0.1) (1.1) (2.9) (2.1)

0.10 0.16 0.13 0.04 0.17 0.27

(1.2) (1.9) ( 1.3) (0.4) (1.8) (2.5)

Duration (except for the previous employed in a temporary job) 1–6 months 0.07 (2.5) 7–12 months 0.06 ( 2.6) 13–24 months 0.01 (0.6) 25–36 months 0.06 ( 2.4)

0.06 0.06 0.01 0.07

(2.3) ( 2.5) (0.7) ( 3.4)

Duration (for the previous employed in a temporary job) 1–6 months 0.02 (0.7) 7–12 months 0.05 ( 1.9) 13–24 months 0.01 ( 0.6) 25–36 months 0.04 ( 1.3)

0.02 0.05 0.01 0.04

(0.9) ( 2.0) ( 0.6) ( 1.4)

sw

0.24

(68.5)

0.0079

(11.8)

Discount rate r

0.0039

2.1) 4.4) 2.6) 0.2)

Using Reported Reservation Wages

(7.7)

2.4) 4.8) 2.8) 0.2)

Note: t-ratio are in parentheses. Estimation by maximum likelihood. Reference: A man, born in France, junior high school graduate, unemployed in 1995, previously employed in a permanent fulltime job. Source: French sample of the ECHP, Insee, 1994–2000. Sample: Unemployement spells beginning after December 1993.

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404

Figure 3.

Simulated job offers arrival rate, and observed and simulated exit rates ðc2 ¼ 50%Þ

0.3

0.25 Simulated job offers

0.2

Observed exit rate 0.15

Simulated exit rate

0.1

0.05

0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 Note: The arrival rate of job offers and the exit rate are simulated on the sample using the estimated parameters of the job search model with
2=50%. Source: French sample of the ECHP, Insee, 1994−2000. Sample: Unemployment spells beginning after December 1993.

the observed one. The differences between them occur at the very beginning of the spell (the observed hazard rate is increasing and above the simulated one) and after two and a half years of unemployment (the simulated hazard rate remains constant, while the observed one is clearly decreasing). Comparison between the arrival rate of job offers and the exit rate from unemployment reveals that an important share of job offers are rejected by the unemployed (more than 50%). This result is at variance with several estimations (van den Berg, 1990; Garcia-Perez, 2003). For instance, Garcia-Perez (2003) finds that in Spain acceptance probabilities are roughly equal to 1 after 4 months of unemployment, even if, at the beginning of the spell, only 40% of job offers are accepted. How can this difference be interpreted? Van den Berg (1990) or Garcia-Perez (2003) do not know the number of rejected offers. We have this information and, according to the data set, roughly one job offer over two received the previous month has been rejected. Note that this high rejection rate is in line with that found by Cases and Lollivier (1993) and Joutard and Ruggiero (2000) on French data. Finally, Figure 4 presents the exit rate from unemployment simulated for different values of c2 (25%, 50%, 75%, and 90%). The choice of c2 appears to have a very limited impact on the hazard.

405

A structural Nonstationary Model of Job Search 2

Figure 4.

Observed and simulated exit rates ðc ¼ 50%; 25%; 70%; 90%Þ

0.16 0.14 Observed exit rate

0.12 0.1

Simulated exit rate for ψ2=50%

0.08

Simulated exit rate for ψ2=25%

0.06

Simulated exit rate for ψ2=75%

0.04

Simulated exit rate for ψ2=90%

0.02 0 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

Source: French sample of the ECHP, Insee, 1994−2000. Sample: Unemployment spells beginning after December 1993. Expected unemployment duration: 14.01 months for
2=50%; 14.46 months for
2=25%; 14.35
2=75%; 15.04 months for
2=90%.

5. Simulation of different economic policy changes Once the structural parameters of the model have been estimated, we can simulate the effects, on the behavior of job search and on unemployment duration, of different reforms of the unemployment compensation system. Four policy changes are examined: (A) a 14% increase in the amount of unemployment insurance benefits, keeping unchanged the declining time sequence of benefits; (B) the replacement of the declining time sequence of insurance benefits by a constant sequence; (C) the reform B combined with the imposition of punitive sanctions if two job offers are refused;16 and (D) a 3month increase in the duration of UI entitlement. 5.1. Reform A: 14% increase in insurance benefits, keeping unchanged the declining time sequence More generous insurance benefits have a negative, but quite limited, impact on the exit rate from unemployment, resulting in a small increase in duration (Table 5, column 2). More precisely, the expected duration of unemployment goes from

16

Reform B is the soft version of the reform of the unemployment compensation system that has been implemented in France in July 2001, while reform C can be seen as the hard version.

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406

Table 5. Elapsed Duration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Expected Duration

Simulated hazard in the reference situation and with reforms A, B, C, and D for all the unemployed Simulated Hazard Reference

Simulated Hazard (A)

Simulated Hazard (B)

Simulated Hazard (C)

Simulated Hazard (D)

0.114 0.112 0.110 0.105 0.102 0.097 0.090 0.085 0.079 0.072 0.065 0.061 0.058 0.058 0.056 0.055 0.054 0.050 0.050 0.048 0.045 0.045 0.040 0.041 0.040 0.042 0.042 0.041 0.042 0.045 0.045 0.044 0.044 0.044 0.045 0.043 0.045 14.01

0.112 0.110 0.107 0.103 0.100 0.095 0.088 0.083 0.078 0.070 0.064 0.060 0.057 0.056 0.055 0.054 0.053 0.050 0.049 0.047 0.044 0.044 0.039 0.040 0.040 0.042 0.042 0.041 0.042 0.045 0.045 0.044 0.043 0.045 0.045 0.043 0.045 14.35

0.111 0.109 0.106 0.101 0.098 0.092 0.085 0.079 0.074 0.066 0.060 0.056 0.052 0.052 0.051 0.049 0.048 0.045 0.044 0.042 0.039 0.038 0.035 0.035 0.035 0.037 0.038 0.037 0.038 0.041 0.041 0.040 0.040 0.041 0.040 0.040 0.042 15.4

0.119 0.118 0.116 0.113 0.111 0.106 0.099 0.093 0.087 0.078 0.071 0.067 0.063 0.063 0.061 0.059 0.058 0.054 0.053 0.050 0.047 0.047 0.041 0.042 0.040 0.042 0.041 0.040 0.042 0.043 0.043 0.042 0.041 0.043 0.044 0.043 0.043 13.15

0.114 0.112 0.109 0.105 0.102 0.097 0.090 0.084 0.079 0.072 0.065 0.061 0.057 0.058 0.056 0.055 0.054 0.050 0.050 0.048 0.045 0.045 0.040 0.041 0.040 0.042 0.042 0.040 0.042 0.044 0.045 0.044 0.044 0.045 0.045 0.044 0.045 14.07

Notes: The completed duration is computed using the simulated hazard (with c2 ¼ 50%). (A) a 14% increase in UI benefits; (B) the replacement of the declining time sequence of benefits by a constant sequence; (C) the reform B combined with the imposition of punitive sanctions if two job offers are refused; and (D) a 3-month increase in the maximum duration of UI entitlement. Sample: All the umemployed.

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14.01 months to 14.35 months (i.e. +2.42%) when the level of UI benefit is raised by 14%. This result is in line with various studies that find a positive, but small, impact of the generosity of UI benefits on unemployment duration in the 80s (van den Berg, 1990; Atkinson and Micklewright, 1991; Layard et al., 1991). However, evidence on the 90s indicates a larger elasticity of expected duration with respect to benefits. For instance, Carling et al. (2001), using Swedish data in the 90’s, try to assess the effect of a decrease from 80 to 75% in the replacement rate on the exit rate from unemployment. Their estimates suggest that the reform caused an increase in the transition rate of roughly 10%. Compared with this estimation, our simulation suggests a quite smaller impact of UI benefits on unemployment duration. This impact may nevertheless be stronger for some sub-populations. The study by Dormont et al. (2001) suggests that high-wage workers are the more likely to be affected by more generous benefits. For this reason, the simulation exercise is repeated for the top quartile of the UI benefits distribution. Column 2 in Table 6 shows that their expected unemployment duration goes from 16.01 months to 16.7 months (i.e. +4.31%) when the level of UI benefit is raised by 14%. These unemployed are thus more sensitive to the amount of benefits than the rest of the population. 5.2. Reform B: The replacement of the declining time sequence of UI benefits by a constant sequence In the theoretical literature, most papers show that the optimal profile of benefits over the unemployment spell is a declining one (Hopenhayn and Nicolini, 1997; Fredriksson and Holmlund, 2001). Despite the importance of the issue, the empirical evidence is limited. Moreover, most empirical studies are interested in one particular issue, the decrease in the level of benefits that occurs when the unemployed exhaust their entitlement to insurance and become assistance recipients (van den Berg, 1990). One exception is the paper by Dormont et al. (2001). The authors evaluate the effect of the whole profile of UI benefits on unemployment duration in the beginning of the 90s in France. In our structural nonstationary model of job search, we may simulate the replacement of the declining time sequence of UI benefits by a constant one. In this aim, we fix the constant monthly benefit to the value of the ‘‘full-rate benefit’’ in the reference situation. This simulated reform is thus very close to the one implemented in France in July 2001.17 Ex ante (i.e., keeping unchanged job

17

In fact, the rules of eligibility in reform B and in the reform implemented in France differ on only one point. In the implemented reform, to be entitled to UI, a worker must have been employed for a minimal duration of 4 months over the last 18 months, while in the simulated reform, the requirement is of 4 months over the last 8 months.

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Table 6. Elapsed Duration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Expected Duration

Simulated hazard in the reference situation and with reforms A, B, C, and D for the top quartile of UI recipients Simulated Hazard Reference

Simulated Hazard (A)

Simulated Hazard (B)

Simulated Hazard (C)

Simulated Hazard (D)

0.103 0.101 0.098 0.093 0.089 0.083 0.075 0.071 0.072 0.064 0.057 0.051 0.050 0.051 0.050 0.046 0.050 0.048 0.0440.043 0.040 0.041 0.039 0.037 0.038 0.041 0.042 0.043 0.042 0.043 0.047 0.047 0.044 0.046 0.042 0.043 0.043 16.01

0.099 0.097 0.094 0.089 0.084 0.080 0.072 0.068 0.069 0.061 0.054 0.049 0.048 0.049 0.047 0.044 0.048 0.046 0.042 0.041 0.039 0.040 0.038 0.037 0.037 0.041 0.041 0.042 0.041 0.043 0.046 0.046 0.045 0.046 0.041 0.043 0.043 16.7

0.097 0.093 0.089 0.084 0.079 0.073 0.064 0.060 0.060 0.052 0.045 0.041 0.039 0.039 0.037 0.036 0.038 0.035 0.033 0.031 0.030 0.030 0.028 0.028 0.027 0.030 0.032 0.033 0.033 0.037 0.039 0.040 0.037 0.038 0.034 0.035 0.037 19.93

0.114 0.115 0.113 0.110 0.107 0.103 0.092 0.087 0.087 0.076 0.067 0.062 0.061 0.059 0.058 0.053 0.056 0.055 0.049 0.048 0.043 0.044 0.041 0.038 0.036 0.043 0.042 0.041 0.040 0.042 0.044 0.041 0.041 0.041 0.040 0.038 0.043 13.78

0.102 0.100 0.097 0.092 0.088 0.082 0.074 0.071 0.072 0.063 0.057 0.051 0.050 0.050 0.049 0.046 0.050 0.048 0.044 0.043 0.040 0.041 0.038 0.037 0.038 0.041 0.042 0.042 0.041 0.043 0.046 0.046 0.043 0.046 0.041 0.043 0.043 16.17

Note: The completed duration is computed using the simulated hazard (with c2 ¼ 50%). (A) a 14% increase in UI benefits; (B) the replacement of the declining time sequence of benefits by a constant sequence; (C) the reform B combined with the imposition of punitive sanctions if two job offers are refused; and (D) a 3-month increase in the maximum duration of UI entitlement. Sample: the top quartile of the UI benefits distribution (at full rate).

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search behaviors), reform B increases the financing cost for the UI agency by 14%. Therefore, reforms A and B have ex ante the same cost and only differ by the profile of benefits over the spell of unemployment. Comparison of these two reforms will thus allow us to assess the effect of time sequencing of benefits on the expected duration of unemployment. The simulation results are in accordance with the theoretical literature: replacing a declining time sequence by a flat profile lengthens the spell of unemployment. Duration is raised by 1.39 months (+9.92%) with reform B versus 0.34 month (+2.42%) with reform A, for the same ex ante cost (Table 5, column 3). Reform B is also simulated on the top quartile of the UI benefits distribution. A flat profile of benefits has a dramatic impact on this subset. Indeed, the expected duration of unemployment is raised by 3.92 months (+24.48%) (Table 6, column 3). Former high-wage workers are thus far more sensitive to the time sequencing of benefits than the rest of the population. This result is in line with Dormont et al. (2001). 5.3. Reform C: The reform B combined with the imposition of punitive sanctions if two job offers are refused Grubb (2001) shows that sanctions are now an important policy tool. Indeed, in many OECD countries, UI recipients are supposed to comply with guidelines on job search effort that are imposed by the UI agency. If they fail to comply, they may be exposed to a sanction, that may be a temporary or a permanent, a full or a partial reduction in benefits. The empirical evidence seems to show that sanctions substantially raise individual re-employment rate (Abbring et al., 2005, for UI recipients; van den Berg et al., 2004, for social assistance recipients). In principle, the reform of UI implemented in France in July 2001 allows for sanctions. However, in practice, the final decision belongs to the ministry of labor, resulting in a very small number of sanctions. We choose to simulate a ‘‘hard’’ sanction policy: a permanent suppression of UI benefits if two job offers are refused. As in reform B, the declining time sequence of benefits is replaced by a constant sequence. Therefore, reform C can be seen as a ‘‘hard’’ version of the 2001 French reform of UI. Compared with reform B, the imposition of sanctions seems to shorten substantially the expected duration of unemployment, which goes from 15.4 months to 13.15 months (i.e. 14.61%) (Table 5, column 4). The arrival rate of job offers is relatively high. When sanctions are not used, about one job offer over two is accepted. But when the rejection of two job offers results in a sanction, things are quite different, since this event occurs relatively early in the spell of unemployment.18 This explains why sanctions affect in an important way

18

On average for the survivors, the probability of having received two job offers after 6 months of unemployment is close to 47%; this goes to 76% after 1 year and to 92% after 2 years.

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the behavior of job search. Of course, the agents differ from one another in their sensitivity to sanctions. Again, former high-wage workers are the more sensitive. Implementing a sanction policy leads their expected duration of unemployment to decrease from 19.93 months to 13.78 months ( 30.85%) (Table 6, column 4). 5.4. Reform D: A 3-month increase in the duration of UI entitlement A 3-month increase in the maximum duration of UI entitlement has a quite limited impact, since the expected unemployment duration goes only from 14.01 months to 14.07 months (Table 5, column 5). And this time, the top quartile of the UI benefits distribution is not more sensitive to the reform than the rest of the unemployed (Table 6, column 5). This result is clearly below the range of estimates found in the empirical literature. Indeed, studies since the early 1980s suggest that a one-week increase in maximum duration of benefits extends average unemployment spells by between 0.1 and 0.5 weeks (Moffitt 1985; Katz and Meyer 1990; Jurajda and Tannery, 2003). However, it is not very surprising in the French case where a majority of UI recipients are entitled to 30 months of benefits and thus go out of unemployment a long time before exhausting their entitlement. 6. Conclusion As far as we know, this paper is the first one that identifies the respective effects of three sources of nonstationarity in a structural job search model. These three sources are the duration-dependence in benefits, in the arrival rate of job offers, and in wage offers. We find a quite limited duration-dependence in job offers and a slightly more pronounced duration-dependence in wage offers. Another important result is that those formerly employed in temporary jobs are far more sensitive to duration than the other unemployed. We also simulate different reforms of the unemployment compensation system. More generous insurance benefits (+14%) have a negative, but quite limited, effect on the exit rate from unemployment. For the same ex ante cost, the replacement of the declining time sequence of UI benefits by a constant sequence has a stronger effect on unemployment duration. The time sequencing of benefits should thus be carefully chosen. Finally, the imposition of sanctions shortens substantially unemployment duration. In these three cases, former high-wage workers are much more sensitive to the reform than the other unemployed. Acknowledgment We would like to thank Gerard van den Berg, Bas van der Klaauw, Michael Burda, Jean-Marc Robin, and the participants to the Crest seminar in 2002; the CEPR workshop on a Dynamic Approach to Europe’s unemployment problem

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in Berlin (2003); the Econometric Society European Meeting 2003; the SED congress 2003; and the LMM conference 2004 for their useful comments.

References Abbring, J., G. van den Berg and J. van Ours (2005), ‘‘The effect of unemployment insurance sanctions on the transition rate from unemployment to employment’’, Economic Journal, Vol. 115, pp. 602–630. Mimeo, Free University Amsterdam. Atkinson, A. and J. Micklewright (1991), ‘‘Unemployment compensation and labor market transitions: a critical review’’, Journal of Economic Literature, Vol. 29, pp. 1679–1727. Carling, K., B. Holmlund and A. Vejsiu (2001), ‘‘Do benefit cuts boost job findings? Swedish Evidence from the 1990s’’, Economic Journal, Vol. 111, pp. 766–790. Cases, C. and S. Lollivier (1993), ‘‘A structural model of transition from unemployment with multiple issues’’, Working Paper No. 9332, Crest. Dormont, B., D. Fouge`re and A. Prieto (2001), ‘‘L’effet de l’allocation unique de´gressive sur la reprise d’emploi’’, Economie et Statistique, Vol. 343, pp. 3–28. Eckstein, Z. and K. Wolpin (1995), ‘‘Duration to first job and the return to schooling: estimates from a search-matching model’’, Review of Economic Studies, Vol. 62, pp. 263–286. Flinn, C. and J. Heckman (1982), ‘‘New methods for analyzing structural models of labor force dynamics’’, Journal of Econometrics, Vol. 18, pp. 115–168. Fredriksson, P. and B. Holmlund (2001), ‘‘Optimal unemployment insurance in search equilibrium’’, Journal of Labor Economics, Vol. 19, pp. 370–399. Frijters, P. and B. van der Klaauw (2005), ‘‘Job search with non participation’’, Economic Journal, forthcoming. Garcia-Perez, J.I. (2003), ‘‘Non-stationarity when jobs do not last forever: a structural estimation’’, Mimeo, CentrA. Grubb, D. (2001), ‘‘Eligibility criteria for unemployment benefits’’, Labour Market Policies and Public Employment Service, OECD. Hopenhayn, H. and J.P. Nicolini (1997), ‘‘Optimal unemployment insurance’’, Journal of Political Economy, Vol. 105, pp. 412–438. Joutard, X. and M. Ruggiero (2000), ‘‘Recherche d’emploi et risques de re´currence du choˆmage : une analyse des qualifications’’, Annales d’Economie et Statistique, Vol. 57, pp. 239–265. Jurajda, S. and F. Tannery (2003), ‘‘Unemployment durations and extended unemployment benefits in local labor markets’’, Industrial and Labor Relations Review, Vol. 56, pp. 324–348.

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Katz, L. and B. Meyer (1990), ‘‘The impact of potential duration of unemployment benefits on the duration of unemployment’’, Journal of Public Economics, Vol. 41, pp. 45–72. Layard, R., S. Nickell and R. Jackman (1991), Unemployment: Macroeconomic Performance and the Labor Market, Oxford: Oxford University Press. Meyer, B. (1990), ‘‘Unemployment insurance and unemployment spells’’, Econometrica, Vol. 58, pp. 757–782. Moffitt, R. (1985), ‘‘Unemployment insurance and the distribution of unemployment spells’’, Journal of Econometrics, Vol. 28, pp. 85–101. Narendranathan, W., S. Nickell and J. Stern (1985), ‘‘Unemployment benefits revisited’’, Economic Journal, Vol. 95, pp. 307–329. Van den Berg, G. (1990), ‘‘Non stationarity in job search theory’’, Review of Economics Studies, Vol. 57, pp. 255–277. Van den Berg, G., B. van der Klaauw and J. van Ours (2004), ‘‘Punitive sanctions and the transition rate from welfare to work’’, Journal of Labor Economics, Vol. 22, pp. 211–241. Wolpin, K. (1987), ‘‘Estimating a structural search model: the transition from school to work’’, Econometrica, Vol. 55, pp. 801–817.

CHAPTER 17

Can Rent Sharing Explain the Belgian Gender Wage Gap?$ Francois Rycx and Ilan Tojerow Abstract This study investigates, on the basis of a unique combination of two large-scale data sets, how rent sharing interacts with the gender wage gap in the Belgian private sector. Empirical findings show that individual gross hourly wages are significantly and positively related to firm profits-per-employee even when controlling for group effects in the residuals, individual and firm characteristics, industry wage differentials and endogeneity of profits. Our instrumented wage–profit elasticity is of the magnitude 0.06 and it is not significantly different for men and women. Of the overall gender wage gap (on average women earn 23.7% less than men), results show that around 14% can be explained by the fact that on average women are employed in firms where profits-per-employee are lower. Thus, findings suggest that a substantial part of the gender wage gap is attributable to the segregation of women in less profitable firms.

Keywords: wages, profits, rent sharing, gender JEL classifications: D31, J16, J31, J70

$

This paper is produced as part of a TSER programme on Pay Inequalities and Economic Performance financed by the European Commission (Contract nr. HPSE-CT-1999-00040). A preliminary version of this paper has been published in the International Journal of Manpower, Vol. 25 (3/4), 2004. Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75017-3

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

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1. Introduction Since Becker’s (1957) seminal paper on the economics of discrimination, studies on the magnitude and sources of the gender wage gap have proliferated (e.g. Blau and Kahn, 2000). Numerous studies have in particular focused on the relationship between labour market segregation and the gender wage differential (e.g. Groshen, 1991; Fields and Wolff, 1995; MacPherson and Hirsch, 1995; Carrington and Troske, 1998; Rycx and Tojerow, 2002). These papers examine basically to what extent the observed sex wage gap can be explained by occupational and sectoral segregation. Although the evidence is still inconclusive, these studies show that a large fraction of the gender wage gap is accounted for by segregation of women in lower-paying occupations, industries and occupations within establishments. Nevertheless, in contrast to previous research, Bayard et al. (1999) suggest that a substantial part of the sex wage gap remains attributable to the individual’s sex. Besides, there is a growing literature, essentially concentrated on the AngloSaxon countries, showing that firms share rents with their employees (e.g. Abowd, 1989; Carruth and Oswald, 1989; Denny and Machin, 1991; Christofides and Oswald, 1992; Abowd and Lemieux, 1993; Blanchflower et al. 1996; Van Reenen, 1996; Hildreth and Oswald, 1997). In other words, recent findings suggest that Ceteris paribus profitable firms pay higher wages to their employees. For Britain, Canada and the US, the estimated elasticities between wages and profits-per-employee range between 0.04 and 0.2, depending on the quality of instruments used to control for the endogeneity of profits. Notice that weak instrumenting biases downward the effect of profits on worker’s wages. Results for continental Europe, although not very numerous, tend in the same direction. For example, using Swedish matched worker-firm data for 1981 and 1991, Arai (2003) reports that the elasticity of wages with respect to profits-per-employee is of the magnitude 0.01. The existence of rent sharing has also been examined in France and Norway by Margolis and Salvanes (2001) on the basis of linked employer–employee panel data. Considering a large number of statistical and economic explanations, the authors find that the estimated coefficient on profits is not significant in France, while it is small but statistically different from zero in Norway. Nevertheless, let us notice that, using a large cross-section of French manufacturing workers, Fakhfakh and FitzRoy (2002) report a significant wage–profit elasticity of between 0.01 and 0.02. In sum, there is strong evidence supporting that workers’ wages depend upon the firms’ ability to pay. Yet, very little is known about: (i) the relative magnitude of the pay–profit elasticity for male and female workers, and (ii) the contribution of rent sharing to the overall gender wage gap. Recent findings (e.g. Arai and Heyman, 2001; Fakhfakh and FitzRoy, 2002) suggest, however, that the relationship between wages and profits is not neutral with respect to gender. For example, using a large Swedish matched employer–employee data set for 1991 and 1995, Nekby (2002) finds that the wage-profit elasticity is about 30–60%

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lower for women than for men. The author also shows that gender differences in rent sharing account for less than 2% of the overall gender wage gap. The objective of this paper is to examine the interaction between rent sharing and the gender wage gap in the Belgian private sector. The current evidence regarding the level and sources of the gender wage gap in Belgium is still far incomplete. Jepsen (2001) shows, on the basis of the 1994 and 1995 Panel Study of Belgian Households (PSBH), that the gender wage gap between full-time workers stands at around 15% and that only a very small part of it can be explained by gender differences in endowments. In contrast, using the 1995 Structure of Earnings Survey, Plasman et al. (2001) suggest that the wage gap between (all) men and women working in the Belgian private sector reaches almost 22% and that half of it is attributable to gender differences in working conditions, individual and firm characteristics. Moreover, while the existence of rent sharing has been recently highlighted for Belgium by Goos and Konings (2001), its impact on the gender pay differential is still unknown. The present paper aims to partially fill this gap by investigating, on the basis of a unique combination of two large-scale, matched employer–employee data sets (i.e. the Structure of Earnings Survey and the Structure of Business Survey) how rent sharing interacts with the gender wage gap in the Belgian private sector. To do so, we first investigate how individual gross hourly wages are related to firm profits-per-employee when controlling for group effects in the residuals, individual and firm characteristics (e.g. education, prior experience, tenure, sex, occupation, region, autonomy at work, type of contract, firm size, level of wage bargaining), industry wage differentials (Nace 3-digit level) and endogeneity of profits. Secondly, we examine whether these results vary for men and women. Finally, we analyse, on the basis of the Oaxaca (1973) and Blinder (1973) decomposition technique, what proportion of the overall gender wage gap can be attributed to: (i) gender differences in mean profit levels of employing firms, (ii) differences between wage–profit elasticities for men and women, and (iii) differences by gender in all other factors, i.e. intercepts, working conditions, individual and firm characteristics. Empirical findings reported in this paper show that individual gross hourly wages are significantly and positively related to firm profits-per-employee even when controlling for group effects in the residuals, individual and firm characteristics, industry wage differentials and endogeneity of profits. Our instrumented wage–profit elasticity is of the magnitude 0.06 and it is not significantly different for men and women. Of the overall gender wage gap (on average women earn 23.7% less than men), results show that around 14% can be explained by the fact that on average women are employed in firms where profits-per-employee are lower. Thus, findings suggest that a substantial part of the gender wage gap is attributable to the segregation of women in less profitable firms. The remainder of this paper is organised as follows. Section 2 presents the theoretical model. Sections 3 and 4 describe the data set and the empirical findings. Section 5 concludes.

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2. Theoretical framework Two models have become standard in the literature for the analysis of the impact of profits-per-employee on wages in a bargaining framework. These are the right-to-manage and the efficient bargaining models, so-named respectively by Nickell and Andrews (1983) and McDonald and Solow (1981). In the right-tomanage model, firms unilaterally determine employment, while wages are the result of a confrontation between the objectives of the firm and of the employees. In the efficient bargaining model, bargaining takes place with respect to both employment and wages. While both models yield identical wage equations, they differ fundamentally in that in the former employment is endogenous with respect to wages whereas in the latter it is exogenous. Nevertheless, they both suggest that wages are related to the firm’s ability to pay, i.e. to the firm’s profitability.1 In this paper, we rely on the right-to-manage model.2 Hence, suppose a bargaining situation, where a firm’s real profit function is given by: P ¼ RðLÞ

WL

ð1Þ

with P the real profits, R(L) the real revenue, W the real wage and L the employment level. Also consider a risk-neutral group of workers, not necessarily a union, which attempts to maximise the expected utility of a representative member, defined as:   L L U¼ Wþ 1 A ð2Þ N N with N the number of members in the group (0oLpN) and A the outside option (W>A). The outside option is the expected value of real revenue perceived by an individual in the event of redundancy. It depends positively on the unemployment benefit and on the expected real wage that a worker would obtain elsewhere, and negatively on the unemployment rate.

1

See, e.g. Pencavel (1991). Using Belgian aggregate data from 1957 to 1988, Vannetelbosch (1996) has shown that both the right-to-manage and the efficient bargaining models can be rejected in favour of the general bargaining model, developed by Manning (1987). This means that the outcome of the bargaining process is located somewhere between the labour demand curve and the contract curve. Nevertheless, this result must be considered with caution for at least two reasons. First, the estimates are very sensitive to the specification of the reservation wage, and second, the trade union density and the number of strikes are far from ideal as a surrogate for the relative bargaining power of unions. This uncertainty is not very surprising since ‘‘the empirical literature has not yet been able to find an appropriate test to distinguish between the principal models’’ (Booth, 1995, p 141). Also noteworthy is that, while these models have different implications for unemployment and economic welfare, they generate identical wage equations. Hence, for the sake of simplicity, we have chosen to rely on the right-tomanage model. 2

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The model is solved backwards: the profit-maximising firm determines the employment level, given the bargained wage in the first stage of the game. The resulting deal is represented by the maximisation of the generalised Nash bargain. This approach boils down to maximising the weighted product of both parties’ net gain, i.e. the difference between levels of utility in the event of an agreement and in the event of no agreement. For a company, without fixed costs, the level of utility reached when bargaining fails equals zero. Indeed, since we assume that all workers are affiliated to the group, the company will have to cease production if agreement is not reached. The fallback position of a representative member of the group is equal to A. Accordingly, the generalised Nash bargaining problem can be written as follows:3 MaxW U b P ¼ MaxW s:t: R0 ðLÞ ¼ W

L N

ðW

AÞ


b

ðRðLÞ

WLÞ

ð3Þ

with bA[0, 1] the relative bargaining power of the workers in the wage bargain. The first-order condition of this problem is given by: W ¼Aþb

ðRðLÞ

WLÞ L

ð4Þ

Expression (4) suggests that real wages are affected by the outside option, real profits-per-employee and the relative bargaining power of the workers. The corresponding statistical specification, which will serve as a benchmark for our empirical analysis, can be written as follows: W i ¼ d0 þ d1 URs þ d2 SW s þ lðP=LÞf þ
i

ð5Þ

with Wi the logarithm of the gross hourly wage of the individual i (i ¼ 1, y ,N); URs the logarithm of the unemployment rate in sector s; SWs the logarithm of the average individual gross hourly wage in sector s; (P/L)f the logarithm of the profits-per-worker in firm f; d0, d1, d2 and l are the parameters to be estimated and ei a white noise error term. URs and SWs reflect the outside option of a representative worker and (P/L)f captures the firm’s good fortune. According to bargaining theory, an increase in the outside option of a representative worker reduces wage moderation. Therefore we expect a negative sign for d1 and a positive sign for d2. The intuition behind this is that when the sector unemployment rate diminishes, the probability of finding a job elsewhere goes up and therefore wage claims increase. In contrast, a drop in the expected alternative wage mitigates envy effects and wage claims. l measures the relative bargaining power of the workers. The sign of the latter is expected to be positive and some theories suggest that its magnitude may be different for men and women.

3

See Nickell (1999, p. 3) for a discussion on the notation.

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According to the theoretical model developed by Sap (1993), gender differences in l’s may appear if the bargaining position of the union is mainly reflective of the male workers’ interests. In other words, the biased composition of the union may generate a more favourable bargaining outcome for men. At the individual bargaining level, differences in rent sharing can also arise if men are more proficient than women at bargaining over wages (Nekby, 2002). Finally, differences in l’s can be explained by the fact that male workers are more present in capital-intensive firms. This explanation, based on Katz and Summers (1989) argument, implies that workers in higher capital-intensive industries have more power to extort rents during wage bargaining. 3. Description of the data The present study is based upon a unique combination of two large-scale data sets. The first, carried out by Statistics Belgium, is the 1995 Structure of Earnings Survey (SES). It covers the Belgian establishments employing at least 10 workers and whose economic activities fall within Sections C–K of the Nace Rev.1 nomenclature.4 The survey contains a wealth of information, provided by the management of the establishments, both on the characteristics of the latter (e.g. sector of activity, region, size of the establishment, level of wage bargaining) and on the individuals working there (e.g. education, potential experience, seniority, gross hourly wages, bonuses, number of working hours paid, gender, occupation). Unfortunately, it provides no financial information. Therefore, the SES has been merged with the 1995 Structure of Business Survey (SBS). It is a firmlevel survey, conducted by Statistics Belgium, whose coverage differs from the SES in that it includes neither the financial sector (Nace J) nor the establishments with less then 20 employees. The SBS provides firm-level information on financial variables such as sales, value added, value of production, gross operating surplus and value of acquired goods and services. The final sample, combining both data sets, covers 34,972 individuals working for 1,501 firms. It is representative of all firms employing at least 20 workers within Sections C–K of the Nace Rev.1 nomenclature, with the exception of the financial sector. Table 1 sets out the means (standard deviations) of selected variables for the overall sample as well as for men and women. We note a clear-cut difference between the average characteristics of male and female workers. The point is that on average men earn significantly higher wages, are employed in larger firms where profits-per-capita are higher, work a larger number of (paid) hours, and have more (potential) experience and seniority. Also noteworthy is that while the average sectoral unemployment rate is smaller for men, the average sectoral

4

The following sectors are therefore not part of the sample: (i) agriculture, hunting and forestry, (ii) fisheries, (iii) public administration, (iv) education, (v) health and social action, (vi) collective, social and personal services (vii) domestic services, and (viii) extra-territorial bodies.

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Table 1.

Means (standard deviations) of selected variables Overall Sample

Gross hourly wage (BEF)b Profits-per-worker (BEF)c Sector average gross hourly wage (in BEF)d Sector unemployment rate (in %)d,e Prior potential experience (years)f Seniority in the current company (years) Size of the establishmentg Hoursh Female (yes) Overtime paid (yes) Bonuses for shift work, night work and/or weekend work (yes) Number of observations in the sample

543.07 (262.60) 854.492 (1,375.90) 477.17 (76.52) 13.77 (4.65) 9.20 (8.18) 10.14 (8.87) 645.49 (1,331.53) 160.20 (28.74) 0.29 0.09 0.21 34,972

a

Men

Women

581.17 447.74 (281.79) (173.49) 926.916 623.279 (1,342.81) (1,439.54) 485.92 455.26 (71.44) (84.06) 13.28 15.01 (4.33) (5.16) 9.37 8.77 (8.14) (8.26) 10.66 8.84 (9.09) (8.14) 767.74 339.58 (1,494.85) (697.85) 166.65 144.08 (21.10) (37.63) 0.00 1.00 0.12 0.03 0.26 0.10 26,650

8,322

a

The descriptive statistics refer to the weighted sample. Includes overtime paid, premiums for shift work, night work and/or weekend work and bonuses (i.e. irregular payments which do not occur during each pay period, such as pay for holiday, 13th month, profit sharing, etc.). 1 EURO ¼ 40.3399 BEF. c Approximated by the firm annual gross operating surplus per worker. The gross operating surplus corresponds to the difference between the value added at factor costs and the total personnel expenses. This variable is expressed in thousands of BEF. d Sectors are defined at the Nace 2 digit level. e The data relative to the sectoral unemployment rate were taken from the monthly bulletin of labour statistics, published by the ONEM (1995). f Experience (potentially) accumulated on the labour market before the last job. It has been computed as follows: age – 6 – years of education – seniority. g Number of workers. h Number of hours paid in the reference period, including overtime paid. b

wage is lower for women. Finally, Table 1 shows that the proportion of workers being paid a bonus for overtime or shift work, night work and/or weekend work is significantly larger among men (see Table A.1 for a more detailed description). 4. Empirical analysis In the remainder of this paper, we investigate the existence and the magnitude of rent sharing in the Belgian private sector. Our estimation strategy is as follows. Firstly, we present simple OLS estimates for our benchmark specification, i.e.

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Table 2. Models 1

Estimation procedure Specifications

Estimation of our benchmark specification, i.e. Equation (5), by OLS with White (1980) heteroscedasticity consistent standard errors. Model 1+correction for group effects (Greenwald, 1983; Moulton, 1990). Model 2+control for individual characteristics, firm size and the level of wage bargaining. Model 3+control for industry wage differentials. Model 4 estimated by 2SLS. Instruments for profits include all variables contained in Model 4, plus per capita firm-level value added and per capita total amount of goods and services purchased by the firm. Models 4 and 5 estimated separately for men and women.

2 3 4 5

6

Equation (5). Secondly, we consider the possibility of group effects in the covariance matrix, which might bias the estimated standard errors towards zero. Thirdly, we control for industry wage differentials and a wide range of observable individual and firm characteristics. Fourthly, we instrument profitsper-worker since by construction they are endogenous. Finally, we estimate our model separately for men and women, and we use the Oaxaca (1973) and Blinder (1973) decomposition technique in order to determine what proportion of the overall gender wage gap is due profit effects. Table 2 summarises the different stages of our empirical analysis. 4.1. Benchmark specification Our first model (see Table 2) serves as a benchmark for our analysis. In this basic model, we regress the logarithm of the individual gross hourly wages on the logarithms of the sector unemployment rate, the sector average gross hourly wage and the firm profits-per-worker. This is done by using OLS with White (1980) heteroscedasticity consistent standard errors. As shown in the first column of Table 3, all the coefficients have the expected sign and they are significant at the 1% level, both in individual terms and globally. Our estimate of l, i.e. the elasticity between wages and profits-per-worker, is 0.074. This means that on average a doubling of profits-per-worker increases earnings by 7.4%. To evaluate the impact of profits on the distribution of wages, Lester’s (1952) range of pay due to rent sharing can be calculated. This statistic estimates the fraction of the overall wage inequality that is due to the variability in profits-per-worker. It is obtained by applying the following formula: sðX Þ ð6Þ X where l^ is the estimated wage–profit elasticity, X measures the level of firm profits-per-worker, and s (X) and X denote the standard deviation and 4l^

421

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Table 3.

Earnings equations for all workers

Variables/Models

OLS (1)

Intercept Profits-per-worker (ln)b Sector unemployment rate (ln) Sector average wage (ln) Group effectsc Individual characteristics & working conditionsd Firm characteristicse Industry effects (149 dummies) Adjusted R2 F-test Test over identification restrictionsf Lester’s range of wages Number of observations Number of groups

a

(2)

2SLS (3)

(4)

(5)

1.946

1.946

4.367

4.162

(21.35) 0.074 (39.49) 0.035 ( 7.37) 0.624 (43.11) No No

(5.25) 0.074 (8.96) 0.035 ( 1.93) 0.624 (10.43) Yes No

(18.45) 0.036 (8.11) 0.010 ( 0.89) 0.218 (5.70) Yes Yes

(25.80) 0.029 (8.15) 0.129 ( 5.93) 0.305 (11.98) Yes Yes

5.085 (26.36) 0.063 (11.71) 0.148 ( 7.02) 0.139 (4.28) Yes Yes

No No

No No

Yes No

Yes Yes

Yes Yes

0.208 1786 —

0.208 133 —

0.694 270 —

0.719 1491 —

47.7% 34972 —

47.7% 34972 1501

23.2% 34972 1501

18.7% 34972 1501

0.712 1967 3.497 (0.174) 40.6% 34972 1501

a

The dependent variable is the (Naperian) logarithm of the individual gross hourly wages. t-statistics are between brackets. Standard errors have been corrected for heteroscedasticity by the method of White (1980). b Approximated by the firm annual gross operating surplus per worker. c Group effects estimations use the correction for common variance components within groups proposed by Greenwald (1983) and Moulton (1990). d Sex, 6 dummies for education, prior potential experience, its square and its cube, seniority within the current company and its square, a dummy for individuals with no seniority, dummy variable indicating whether the individual supervises other workers, a variable showing whether the individual received a bonus for shift work, night work and/or weekend work, a dummy for overtime paid, 3 dummies for the type of contract, 2 regional dummies and 23 occupational dummies. e The size of the establishment and the level of wage bargaining (2 dummies). f NR2 and associated (p-value). , Significant at 1% and 5%, respectively.

the mean value of X, respectively. On the basis of this formula, it appears that about 48% of the variance in individual wages is due to the variability in profits.5 To put it another way, given that the mean hourly wage stands at BEF 543, rent sharing explains the variation of wages between BEF 284 and BEF 802.6

5 6

Notice that [0.074  4  (1, 375.9/854.492)]  100 is equal to 47.7%. 1 EUR equals 40.3399 BEF.

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Our benchmark regression clearly supports the hypothesis that individual wages are significantly and positively related to the firm’s ability to pay. Nevertheless, caution is required. Indeed, the explanatory power of Model 1 is limited. The adjusted R2 reaches only 0.21. Moreover, results might suffer from various econometric problems, e.g. group effects in the residuals, omitted variable biases, endogeneity of profits. In the next sections, we will therefore try to improve the robustness of our model by controlling for these statistical issues. 4.2. Group effects The first potential trap derives from the simultaneous use of grouped observations and individual data. Indeed, the presence of aggregate explanatory variables in Model 1 can bias the estimated standard errors and as a result distorts the significance of our coefficients. To account for these group effects, Model 2 applies the correction for common variance components within groups, as suggested by Greenwald (1983) and Moulton (1990). This correction transforms the covariance matrix of the errors, but leaves the point estimates and the determination coefficient unaffected. Therefore, our second model has the same estimated coefficients than our benchmark model, but different t-statistics. Findings, reported in column (2) of Table 3, clearly support the overestimation phenomenon demonstrated by Moulton (1990). However, all regression coefficients, except that of unemployment, remain significant at the 1% level. Therefore, it appears that the positive and significant relationship between wages and profits cannot be attributed to group effects.7 4.3. Individual and firm characteristics The omitted variable bias is another important issue that has to be investigated. According to the standard Walrasian (competitive) model of the labour market, where the equilibrium wage is determined through marginal productivity, two agents with identical productive characteristics necessarily receive the same wages. However, so-called compensating differences may occur between similar individuals placed in different working conditions. Indeed, the disutility undergone by one individual following the performance of a task in an unfavourable situation may lead to wage compensation. In accordance with this simple description of the wage determination process, variables reflective of the productivity of the workers and their working conditions have been added to our model. These include seven indicators showing the highest level of education; prior potential experience, its square and its cube; seniority within the current company and its square; a dummy variable controlling for entrants, i.e.

7

Because of its substantial impact on the standard errors of the estimates, the correction for group effects has been applied to all other models presented in this paper.

Can Rent Sharing Explain the Belgian Gender Wage Gap?

423

individuals with no seniority; the number of hours paid; a dummy for extra paid hours; 22 occupational dummies; three dummies for the type of contract; an indicator showing whether the individual is paid a bonus for shift work, night-time and/or weekend work and a dichotomic variable showing whether the individual supervises other workers. Moreover, in order to control for gender and regional wage differentials, which are well documented in the literature (e.g. Farber and Newman, 1989; OECD, 2002), we have also included a dummy for the sex of the individual and two regional dummies indicating whether the establishment is located. Finally, in line with recent labour market theories, supporting the existence of an effect of the employer’s characteristics on wages, the size of the establishment and two dummies showing the level of wage bargaining have also been inserted in our regression. The results relative to this new specification are presented in the third column of Table 3. We find that the wage–profit elasticity is still significant at the 1% level, but that its magnitude decreases from 0.074 to 0.036. As a result, Lester’s (1952) range of pay due to rent sharing drops from 48% to 23%. Let us also notice that the inclusion of individual and firm characteristics has a substantial impact on the explanatory power of the model. Indeed, the adjusted R2 now reaches almost 70%. In sum, results suggest that the positive and significant relationship between wages and profits does not derive from an omitted variable bias. Yet, the absence of specific control variables for the sectoral affiliation of the workers may be problematic.

4.4. Industry wage differentials The empirical debate about the causes of earnings inequalities was reopened at the end of the 1980s by an article by Krueger and Summers (1988). The authors showed that wage disparities persisted in the US among workers with apparently identical individual characteristics and working conditions, employed in different sectors. Since then, similar results have been obtained for numerous industrialised countries (e.g. Vainioma¨ki and Laaksonen, 1995; Hartog et al., 1997; Goux and Maurin, 1999; Rycx, 2002). In light of this literature, it could be argued that rent sharing is simply reflective of industry wage premiums. In other words, it is possible that the positive correlation between wages and profits is generated by sectoral shocks rather than by the division of firms’ rents. For this reason, 173 dummy variables indicating the sectoral affiliation of the workers have been added to our model. The results of this new specification are presented in column (4) of Table 3. As expected, we find that sectors have a significant impact on the wage–profit elasticity. Indeed, the coefficient on profits drops from 0.036 in Model 3 to 0.029 in the current frame. However, rent sharing is still significant at the 1% level and the adjusted R2 reaches now more than 70%. Moreover, from Table A.2, it can be seen that all other coefficients have the expected sign and that the vast majority of them are significant at the 5% level.

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4.5. Endogeneity of profits Although Model 4 seems quite accurate, it is still suffering from a serious problem, namely the endogeneity of profits. Indeed, by construction, wages have a negative impact on profits. Therefore, previous OLS estimates are not only biased but also inconsistent. To bypass this problem, we estimated Model 4 using the method of instrumental variables. This method consists in finding instruments, which are at the same time highly correlated with the endogenous variable and uncorrelated with the error term. We used as instruments for profits all the variables contained in Model 4, plus per capita firm level value added and per capita total amount of goods and services purchased by the firm.8 Results of our 2SLS regression are presented in the last column of Table 3.9 Not surprisingly, we find that the wage-profit elasticity increases from 0.029 to 0.063, which confirms the downward biasness of our previous estimates.10 It follows that Lester’s range of pay is about 42% of the mean wage. Yet, it could be argued that the instruments that have been used are in appropriate. To check for this, Sargan’s (1964) over-identification test has been used. The corresponding test statistic is computed as NR0 2, where N is number of observations and R0 2 the percent of variation explained in the regression of the residuals from the second-stage equation on the instruments and all exogenous variables in the model. This statistic is distributed w2 with degrees of freedom equal to the number of overidentifying restrictions (in this case, df ¼ 2). The results of this test, presented at the bottom of column (5) in Table 3, show that the over identifying restrictions cannot be rejected at the level of 15%. This suggests that our instruments are valid and that Model 5 is well specified. Our findings are significantly smaller than those reported for Belgium by Goos and Konings (2001). Indeed, the latter report a wage–profit elasticity of around 0.10 and a Lester’s range of pay of approximately 60% of the mean wage. Nevertheless, many factors can explain these differences. Firstly, Goos and Konings only include a very limited number of control variables in their analysis, namely the ratio of white- over blue-collar workers and one digit industry dummies. Secondly, they use pooled data regressions covering the period 1987–1994, while the present study relies on a single cross-section relative to 1995. Thirdly, their data set has a larger coverage than ours. In particular, it covers the financial sector, where rent sharing is expected to be relatively high.

8

The sum of both instruments corresponds to the firm level value of production per worker. As suggested by Margolis and Salvanes (2001, p. 16), one might think of sales (or production) per worker, conditional on the sector being correctly identified, as a measure of the firm’s market power. Other instruments used in the literature to control for the endogeneity of profits include: output elasticity, interaction of share of exports with exchange rate and lagged profits. 9 First stage regressions are shown in Table A.2 and detailed results in Table A.4. 10 All coefficients in the first-stage regression are jointly significant at the 1% level. Results are shown in Table A.4.

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425

Fourthly, they instrument profits by their lagged values when we rely on a set of instrumental variables including per capita firm level value added and per capita total amount of goods and services purchased by the firm. Finally, let us also notice that their dependant variable is less precise since it is constructed by dividing annual labour costs by the number of employees in the firm. Be that as it may, both studies suggest that rent sharing exists in Belgium and that a substantial part of the dispersion in wages is due to the variability in profits. 4.6. Rent sharing for men and women So far, the existence and magnitude of rent sharing in the Belgian private sector has been investigated for all workers, independent of their sex. In this section, we analyse if the wage–profit elasticity is significantly different for male and female workers and we decompose the overall gender wage gap to determine what proportion is due rent sharing. 4.6.1. Wage-profit elasticities by gender To investigate the interaction between rent sharing and gender, Models 4 and 5 have been estimated separately for men and women.11 The results from this analysis are reported in Table 4 and Table A.4. As mentioned previously, the OLS estimates are inconsistent because of the endogeneity of profits. Therefore, we focus on the 2SLS results.12 For both sexes, we find that the coefficients on profits-per-worker are significant at the 1% level. In addition, profit effects look approximately 10% higher for men than for women. Indeed, the wage–profit elasticity stands at 0.059 for women and at 0.066 for men. Yet, using a standard t-test, we find no significant difference between the regression coefficients of both sexes.13 Hence, there appears to be no gender difference in remuneration from firm profits. Finally, let us notice that Lester’s (1952) range of pay due to rent sharing equals respectively 38.2% and 56.2% for men and women. 4.6.2. Decomposition of the gender wage gap To complete our analysis, we decomposed the overall gender wage gap in order to assess what proportion is due to: (a) differences between female and male wage–profit elasticities, (b) differences by gender in the average value of firm profits-per-capita and (c) differences by gender in all other factors, i.e. intercepts, interindustry wage differentials, working conditions, individual and firm characteristics.

11

Because of a strong multicollinearity problem, the sector average gross hourly wage and the sector unemployment rate have been not been included in the regression. 12 Let us notice that the p-value relative to the over-identification test for the maleregression equals only 0.069. 13 For the 2SLS regressions (Model 5), t ¼ 0.69.

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Table 4.

Men and women, OLS versus 2SLS estimatesa

Variables/Models

Model 4 (OLS) Men

Women

Model 5 (2SLS) Men

Women

5.738

5.424

5.631

(45.45) 0.030 (8.30) No No Yes Yes Yes Yes

(73.15) 0.025 (4.71) No No Yes Yes Yes Yes

(45.35) 0.066 (12.81) No No Yes Yes Yes Yes

5.270 (61.93) 0.059 (6.68) No No Yes Yes Yes Yes

Adjusted R2 F-test Test of over identification restrictionsf

0.709 1348 —

0.678 1544 —

Lester (1952) range of wages Number of observations Number of groups

17.4% 26650 1475

21.4% 8322 1217

0.702 624 5.330 (0.069) 38.2% 26650 1475

0.669 1506 1.664 (0.435) 50.4% 8322 1217

Intercept Profits-per-worker (ln)b Sector unemployment rate (ln) Sector average wage (ln) Group effectsc Individual characteristics & Working Conditionsd Firm characteristicse Industry effects (149 dummies)

a

The dependent variable is the (Naperian) logarithm of the individual gross hourly wages. t-statistics are between brackets. Standard errors have been corrected for heteroscedasticity by the method of White (1980). b Approximated by the firm annual gross operating surplus per worker. c Group effects estimations use the correction for common variance components within groups proposed by Greenwald (1983) and Moulton (1990). d Sex; 6 dummies for education; prior potential experience, its square and its cube; seniority within the current company and its square, a dummy for individuals with no seniority; dummy variable indicating whether the individual supervises other workers; a variable showing whether the individual received a bonus for shift work, night work and/or weekend work, a dummy for overtime paid, 3 dummies for the type of contract, 2 regional dummies and 23 occupational dummies. e The size of the establishment and the level of wage bargaining (2 dummies). f NR2 and associated (p-value). , Significant at 1% and 5%, respectively.

To do so, we used the decomposition procedure developed by Oaxaca (1973) and Blinder (1973), who showed that the difference between the average hourly wage (in logarithms) of men and women could be broken as follows:   m W  f ¼ ðP=LÞ ðl^ m l^ f Þ þ ðP=LÞ ðP=LÞf l^ m W f f
þ X f ð^um u^ f Þ þ X m X f u^ m ð7Þ or

m W

 f ¼ ðP=LÞ ðl^ m W m þ X m ð^um

 ƒƒƒ l^ f Þ þ ðP=LÞf

u^ f Þ þ X m


X f u^ f

ƒƒƒ  ðP=LÞf þ l~ f

ð8Þ

427

Can Rent Sharing Explain the Belgian Gender Wage Gap?

Table 5.

Decomposition of the gender wage gap

a

Percentage of Overall Wage Gap due to Differences in Non discriminatory wage structure Male wage structure Female wage structure a

Overall Gender Wage Gap m W f W

Average Values of Firm Profits per capita ððP=LÞ ðP=LÞ Þ l^ mðf Þ

Wage-Profits Elasticities ðP=LÞ ðl^ m l^ f Þ

All Other Factors

0.237

14.3%

17.3%

68.4%

0.237

12.7%

18.9%

68.4%

m

f ðmÞ

f

Computation based on the 2SLS estimates reported in Table 4.

 where the indices m and f refer respectively to male and female workers, W represents the average (Naperian logarithm) of the hourly wage, ðP=LÞ is the average value of firm profits-per-capita and X a vector containing an intercept and the average values of the individual characteristics of the workers, their working conditions, their sectoral affiliation, the size of their establishment and the level of wage bargaining therein. u^ and l^ are the 2SLS regression coefficients relative to ðP=LÞ and X reported in Table 4 and Table A.3. Equations (6) and (7) take as a non-discriminatory wage structure that of men and women, respectively. As shown in Table 5, the overall gender wage gap, measured as the difference between mean log wages of male and female workers, stands at 0.237. This means that that the average female worker earns 76.3% of the mean male wage. Moreover, depending on the non-discriminatory wage structure used, results indicate that between 12.7% and 14.3% of the overall gender wage gap can be explained by the fact that on average women are employed in firms where profits-per-employee are lower. Table 5 also suggests that around 18% of the overall gender wage gap derives from differences between wage–profit elasticities for men and women.14 However, the latter result should be interpreted with caution because the wage–profit elasticity is not significantly different for both sexes. 5. Conclusion This paper investigates, on the basis of a unique combination of two large-scale data sets, how rent sharing interacts with the gender wage gap in the Belgian private sector. Empirical findings show that individual gross hourly wages are significantly and positively related to firm profits-per-employee even when controlling for group effects in the residuals, individual and firm characteristics, industry wage differentials and endogeneity of profits. Our instrumented wage–profit elasticity is of the magnitude 0.06 and it is not significantly

14

We have also estimated Blinder and Oaxaca decompositions with and without profits of firms. This allows us to notice a decrease of the explained part by 2.2% points when profits are excluded.

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different for men and women. Of the overall gender wage gap (on average women earn 23.7% less than men), results show that around 14% can be explained by the fact that on average women are employed in firms where profitsper-employee are lower. To put it differently, findings suggest that a substantial part of the gender wage gap is attributable to the segregation of women in less profitable firms. Future research concerning the magnitude of rent sharing in Belgium should rely on matched employer–employee panel data so as to control for the non-observed individual characteristics of the workers. Indeed, these characteristics might modify our results if it emerged that they were not randomly distributed between firms and/or sexes. Unfortunately, at the moment such a data set does not exist. Acknowledgment We are most grateful to Christophe Demunter for his assistance in getting access to the Belgian Structure of Earnings Survey and Structure of Business Survey. The usual disclaimer applies. References Abowd, J.A. (1989), ‘‘The effect of wage bargaining on the stock market value of the firm’’, American Economic Review, Vol. 79(4), pp. 774–800. Abowd, J.A. and T. Lemieux (1993), ‘‘The effects of product market competition on collective bargaining agreements: the case of foreign competition in Canada’’, Quarterly Journal of Economics, Vol. 108(4), pp. 983–1014. Arai, M. (2003), ‘‘Wages, profits and capital intensity: evidence from matched worker-firm data’’, Journal of Labour Economics and Working Paper in Economics, 1999, p. 3, Department of Economics, Stockholm University, Vol. 21(3), pp. 593–618. Arai, M. and F. Heyman (2001), ‘‘Wages, profits and individual unemployment risk: evidence from matched worker-firm data’’, FIEF Working Paper Series, No. 172. Bayard, J.M., J. Hellerstein, D. Neumark and K. Troske (1999), ‘‘New evidence on sex segregation and sex difference in wages from matches employeremployee data’’, NBER Working Paper, No. 7003. Becker, G. (1957), The Economics of Discrimination, Chicago: University of Chicago Press. Blanchflower, D.G., A.J. Oswald and P. Sanfey (1996), ‘‘Wages, profits and rent-sharing’’, Quarterly Journal of Economics, Vol. 111(1), pp. 227–251. Blau, F. and L. Kahn (2000), ‘‘Gender differences in pay’’, Journal of Economic Perspectives, Vol. 14(4), pp. 75–99. Blinder, A. (1973), ‘‘Wage discrimination: reduced form and structural variables’’, Journal of Human Resources, Vol. 8(4), pp. 436–465.

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Booth, A. (1995), The Economics of the Trade Union, Cambridge: Cambridge University Press. Carrington, W. and K. Troske (1998), ‘‘Sex segregation in US manufacturing’’, Industrial and Labor Relations Review, Vol. 51(3), pp. 445–464. Carruth, A.A. and A.J. Oswald (1989), Pay Determination and Industrial Prosperity, Oxford: Clarendon Press. Christofides, L.N. and A.J. Oswald (1992), ‘‘Real wage determination and rentsharing in collective bargaining agreements’’, Quarterly Journal of Economics, Vol. 107(3), pp. 985–1002. Denny, K. and S. Machin (1991), ‘‘The role of profitability and industrial wages in firm-level wage determination’’, Fiscal Studies, Vol. 12, pp. 34–45. Fakhfakh, F. and F. FitzRoy (2002), ‘‘Basic wages and firm characteristics: rent-sharing in French manufacturing’’, Mimeo, ERMES, Universite´ de Paris II. Farber, S. and R. Newman (1989), ‘‘Regional wage differentials and the special convergence of worker characteristics prices’’, Review of Economics and Statistics, Vol. 71(2), pp. 224–231. Fields, J. and E. Wolff (1995), ‘‘Interindustry wage differentials and the gender wage gap’’, Industrial and Labor Relations Review, Vol. 49(1), pp. 105–120. Goos, M. and J. Konings (2001), ‘‘Does rent-sharing exist in Belgium? An empirical analysis using firm level data’’, Reflets et perspectives de la vie e´conomiques, Vol. XL(1–2), pp. 65–79. Goux, D. and E. Maurin (1999), ‘‘Persistence of interindustry wage differentials: a reexamination using matched worker-firm panel data’’, Journal of Labor Economics, Vol. 17(3), pp. 492–533. Greenwald, B.C. (1983), ‘‘A general analysis of bias in the estimated standard errors of least square coefficients’’, Journal of Econometrics, Vol. 22(3), pp. 323–338. Groshen, E.L. (1991), ‘‘The Structure of the female/male wage differential: is it who you are, what you do, or where you work?’’, Journal of Human Resources, Vol. 26(3), pp. 457–472. Hartog, J., R. Van Opstal and C. Teulings (1997), ‘‘Inter-industry wage differentials and tenure effects in the Netherlands and the US’’, De Economist, Vol. 145(1), pp. 91–99. Hildreth, A.K.G. and A.J. Oswald (1997), ‘‘Rent-sharing and wages: evidence from company and establishment panels’’, Journal of Labor Economics, Vol. 15(2), pp. 318–337. Jepsen, M. (2001), ‘‘Evaluation des differentiels salariaux en Belgique: hommefemme et temps partiel- temps plein’’, Reflets et Perspectives de la vie economique, Vol. 40(1–2), pp. 51–63. Katz, L.F. and L.H. Summers (1989), ‘‘Industry rents: evidence and implications’’, Brooking Papers on Economic Activity, Vol. 1989, pp. 209–275. Krueger, A. and L. Summers (1988), ‘‘Efficiency wages and inter-industry wage structure’’, Econometrica, Vol. 56(2), pp. 259–293.

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Lester, R.A. (1952), ‘‘A range theory of wage differentials’’, Industrial and Labour Relations Review, Vol. 5(4), pp. 483–500. MacPherson, D. and B. Hirsch (1995), ‘‘Wages and gender composition: why do women’s jobs pay less?’’, Journal of Labor Economics, Vol. 13(3), pp. 426–471. Manning, A. (1987), ‘‘An integration of trade union models in a sequential bargaining framework’’, Economic Journal, Vol. 97(385), pp. 121–139. Margolis, D.N. and K.G. Salvanes (2001), ‘‘Do firms really share rents with their workers?’’ CREST Working Paper, No. 2001-16. Moulton, B.R. (1990), ‘‘An illustration of a pitfall in estimating the effects of aggregate variables on micro units’’, Review of Economics and Statistics, Vol. 72(2), pp. 334–338. McDonald, I. and R. Solow (1981), ‘‘Wage bargaining and employment’’, American Economic Review, Vol. 74(5), pp. 896–908. Nekby, L. (2002), ‘‘A Note on gender differences in rent sharing and its implications for the gender wage gap’’, Mimeo, Trade Union Institute for Economic Research (FIEF) and Department of Economics, Stockholm University. Nickell, S. (1999), ‘‘Product markets and labour markets’’, Labour Economics, Vol. 6(1), pp. 1–20. Nickell, S. and M. Andrews (1983), ‘‘Unions, real wages and employment in Britain 1951–79’’, Oxford Economic Papers, Vol. 35(supplement), pp. 183–206. Oaxaca, R. (1973), ‘‘Male–female wage differentials in urban labour markets’’, International Economic Review, Vol. 14(3), pp. 693–709. OECD (2002), Employment outlook, OECD, Paris. ONEM (1995), Monthly bulletin, October issue, Brussels. Pencavel, J. (1991), Labor markets under trade unionism. Employment, wages and hours, Cambridge, MA: Blackwell publishers. Plasman, A., R. Plasman, M. Rusinek and F. Rycx (2001), ‘‘Indicators on gender pay equality’’, Cahiers Economiques de Bruxelles, Vol. 45(2), pp. 11–40. Rycx, F. (2002), ‘‘Inter-industry wage differentials: evidence from Belgium in a cross-national perspective’’, De Economist, Vol. 150(5), pp. 555–568. Rycx, F. and I. Tojerow (2002), ‘‘Inter-industry wage differentials and the gender wage gap in Belgium’’, Cahiers Economiques de Bruxelles, Vol. 45(2), pp. 119–141. Sap, J. (1993), ‘‘Bargaining power and wages. A game-theoretic model of gender differences in union wage bargaining’’, Labour Economics, Vol. 1(1), pp. 25–48. Sargan, J.D. (1964), ‘‘Wages and prices in the United Kingdom: a study in econometric methodology’’, in: P.E. Hart, G. Mills and J.K. Whitaker, editors, Econometric Analysis for National Economic Planning, London: Butterworths. Vainioma¨ki, J. and S. Laaksonen (1995), ‘‘Inter-industry wage differentials in Finland: evidence from longitudinal census data for 1975–85’’, Labour Economics, Vol. 2(2), pp. 161–173.

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Vannetelbosch, V. (1996), ‘‘Testing between alternative wage-employment bargaining models using Belgian aggregate data’’, Labour Economics, Vol. 3(1), pp. 43–64. Van Reenen, J. (1996), ‘‘The creation and capture of rents: wages and innovation in a panel of UK companies’’, Quarterly Journal of Economics, Vol. 111(1), pp. 195–226. White, H. (1980), ‘‘A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity’’, Econometrica, Vol. 48(4), pp. 817–830. Appendix Tables A.1, A.2, A.3, A.4

Table A.1.

Description and means (standard deviations) of selected variablesa Overall sample

Gross hourly wage: (in BEF) includes overtime paid, premiums for shift work, night work and/or weekend work and bonuses (i.e. irregular payments which do not occur during each pay period, such as pay for holiday, 13th month, profit sharing, etc.). 1 EURO ¼ 40.3399 BEF. Profits-per-worker: (in BEF) approximated by the firm annual gross operating surplus per worker. The gross operating surplus corresponds to the difference between the value added at factor costs and the total personnel expenses. This variable is expressed in thousands of BEF. Sector average gross hourly wage: (in BEF), sectors are defined at the Nace 2 digit level. Sector unemployment rate:b (in %), sectors are defined at the Nace 2 digit level. Education Primary or no degree: 0–6 years. Lower secondary: 9 years. General upper secondary: 12 years. Technical/Artistic/Prof. upper secondary: 12 years. Higher non-university short type, higher artistic training: 14 years. University and non-university higher education, long type: 16 years. Post-graduate: 17 years or more.

Men

Women

543.07 (262.60)

581.17 (281.79)

447.74 (173.49)

854.492 (1,375.90)

926.916 (1,342.81)

623.279 (1,439.54)

477.17 (76.52) 13.77 (4.65)

485.92 (71.44) 13.28 (4.33)

455.26 (84.06) 15.01 (5.16)

11.4 25.7 16.2 25.9

11.6 25.3 13.0 29.5

11.0 26.7 24.1 17.0

12.4

11.3

15.3

7.9

8.9

5.5

0.5

0.5

0.4

432

Francois Rycx and Ilan Tojerow

Table A.1 (continued ) Overall sample Prior potential experience: (years), experience (potentially) accumulated on the labour market before the last job. It has been computed as follows: age – 6 – years of education – seniority. Seniority in the current company: (years). Size of the establishment: number of workers. Hours: number of hours paid in the reference period, including overtime paid. Female (yes) Overtime paid (yes) Bonuses for shift work, night work and/or weekend work (yes) Supervises the work of other workers (yes) Type of contract: Unlimited-term employment contract Limited-term employment contract Apprentice/Trainee contract Other Region: geographic location of the establishment Brussels Wallonia Flanders Level of wage bargaining Collective wage agreement only at the national and/or sectoral level Collective wage agreement at the company level Other Number of observations in the sample a

Men

Women

9.20 (8.18)

9.37 (8.14)

8.77 (8.26)

10.14 (8.87) 645.49 (1,331.53) 160.20 (28.74) 28.6 9.3 21.0

10.66 (9.09) 767.74 (1,494.85) 166.65 (21.10) 0.0 11.7 25.5

8.84 (8.14) 339.58 (697.85) 144.08 (37.63) 100 3.4 9.7

15.6

18.2

8.9

96.8 2.8 0.1 0.3

97.4 2.2 0.1 0.3

95.4 4.1 0.2 0.4

12.7 20.4 66.9

11.6 19.5 69.0

15.6 22.7 61.7

47.1

44.4

53.8

46.3

49.3

38.7

6.6

6.3

7.5

34,972

26,650

8,322

The descriptive statistics refer to the weighted sample. Descriptive statistics relative to the sectoral affiliation of the workers and their occupations are available on request. b The data relative to the sectoral unemployment rate were taken from the monthly bulletin of labour statistics, published by the ONEM (1995).

433

Can Rent Sharing Explain the Belgian Gender Wage Gap?

Table A.2.

Earnings equations for all workers, detailed results

Variables/Models

Model 4 (OLS)

Intercept

5.655 (76.71) 0.029 (8.15) —

Profits-per-worker (ln)a Sector unemployment rate (ln) Sector average wage (ln) Education Primary or no degree Lower secondary General upper secondary Technical/Artistic/Prof. upper secondary Higher non university short type, higher artistic training University and non-university higher education, long type Post-graduate Prior experience Simple Squared/102 Cubed/104 Individual with no seniority (Yes) Seniority in the company Simple Squared/102 Hours paid (ln) Bonus for shift work, night work and/or weekend work (Yes) Overtime paid (Yes) Contract Unlimited-term employment contract Limited-term employment contract

Model 5 (2SLS) 4.162 (25.80) 0.029 (8.15) 0.129 ( 5.93) 0.305 (11.98)

5.528 (73.40) 0.063 (11.71) —

Reference 0.045 (4.71) 0.143 (12.19) 0.143 (13.43) 0.256 (18.79) 0.417 (22.99) 0.586 (12.20) 0.016 (12.74)

Reference 0.045 (4.71) 0.143 (12.19) 0.143 (13.43) 0.256 (18.79) 0.417 (22.99) 0.586 (12.20) 0.016 (12.74)

Reference 0.046 (4.76) 0.144 (12.07) 0.142 (13.28) 0.252 (18.42) 0.414 (22.88) 0.583 (12.34) 0.016 (12.38)

Reference 0.046 (4.76) 0.144 (12.07) 0.142 (13.28) 0.252 (18.42) 0.414 (22.88) 0.583 (12.34) 0.016 (12.38)

0.039 ( 4.65) 0.002 (1.455) 0.036 ( 1.09)

0.039 ( 4.65) 0.002 (1.455) 0.036 ( 1.09)

0.037 ( 4.36) 0.002 (1.23) 0.045 ( 1.28)

0.037 ( 4.36) 0.002 (1.23) 0.045 ( 1.28)

0.019 (22.47) 0.024 ( 9.72) 0.043 ( 3.35) 0.059 (7.21) 0.021 (2.78)

0.019 (22.47) 0.024 ( 9.72) 0.043 ( 3.35) 0.059 (7.21) 0.021 (2.78)

0.019 (21.62) 0.024 ( 9.30) 0.056 ( 4.48) 0.059 (7.08) 0.016 (1.98)

0.019 (21.62) 0.024 ( 9.30) 0.056 ( 4.48) 0.059 (7.08) 0.016 (1.98)

Reference

Reference

Reference

Reference

0.052 ( 3.77)

0.052 ( 3.77)

0.057 ( 4.11)

0.057 ( 4.11)

—

—

5.085 (73.40) 0.063 (11.71) 0.148 ( 7.02) 0.139 (4.28)

434

Francois Rycx and Ilan Tojerow

Table A.2 (continued ) Variables/Models

Model 4 (OLS)

Model 5 (2SLS)

0.588 ( 3.29) 0.082 ( 3.28) 0.121 ( 17.87) Yes

0.588 ( 3.29) 0.082 ( 3.28) 0.121 ( 17.87) Yes

0.599 ( 3.38) 0.071 ( 2.87) 0.120 ( 17.62) Yes

0.599 ( 3.38) 0.071 ( 2.87) 0.120 ( 17.62) Yes

Reference 0.016 ( 1.25) 0.042 ( 3.72) 0.126 (13.76) 0.038 (9.45)

Reference 0.016 ( 1.25) 0.042 ( 3.72) 0.126 (13.76) 0.038 (9.45)

Reference 0.009 ( 0.69) 0.036 ( 3.03) 0.129 (13.74) 0.037 (9.09)

Reference 0.009 ( 0.69) 0.036 ( 3.03) 0.129 (13.74) 0.037 (9.09)

Reference

Reference

Reference

Reference

0.032 (2.97) 0.009 ( 0.66) Yes

0.032 (2.97) 0.009 ( 0.66) Yes

0.029 (2.76) 0.014 ( 1.04) Yes

0.029 (2.76) 0.014 ( 1.04) Yes

Group effects:

Yes

Yes

Yes

Yes

Test of over-identification restrictionsb

—

—

3.497

3.497

0.719 3728 34972 1501

0.719 1491 34972 1501

(0.174) 0.712 590 34972 1501

(0.174) 0.712 1967 34972 1501

Apprentice/Trainee contract Other employment Contract Female (Yes) Occupation (23 dummies) Region Brussels Wallonia Flanders Supervises the work of coworkers (Yes) Size of the establishment (ln) Level of wage bargaining CA only at the national and/or sectoral level CA at the company level Other Industry effects (149 dummies)

R2 adjusted F-test Number of observations Number of groups

Note: The dependent variable is the (Naperian) logarithm of the individual gross hourly wages. tstatistics are between brackets. White (1980) heteroscedasticity consistent standard errors. Group effects estimations use the correction for common variance components within groups proposed by Greenwald (1983) and Moulton (1990). a Approximated by the firm annual gross operating surplus per worker. b NR2 and associated (p-value). ,,1 Significant at 1%, 5% and 10%, respectively.

435

Can Rent Sharing Explain the Belgian Gender Wage Gap?

Table A.3.

Earnings equations by gender, detailed results Model 4 (OLS)

Variables/ Models Men Intercept Profits-perworker (ln)a Sector unemployment rate (ln) Sector average wage (ln) Education Primary or no degree Lower secondary General upper secondary Technical/ Artistic/Prof. upper secondary Higher non university short type, higher artistic training University and nonuniversity higher education, long type Post-graduate Prior experience Simple Squared/102 Cubed/104 Individual with no seniority (Yes) Seniority in the company Simple

5.738 (45.45) 0.030 (8.30) —

Women

4.697 (10.44) 0.030 (8.30) 0.128 ( 4.03) 0.229 (3.07)

—

Model 5 (2SLS)

Reference

Men

Women

5.424 (73.15) 0.025 (4.71) —

5.563 (7.79) 0.025 (4.71) 0.155 ( 3.57)

5.631 (45.35) 0.066 (12.81) —

5.270 (61.93) 0.059 (6.68) —

—

0.447 (0.35)

—

—

Reference

Reference

0.0201

0.0201

(5.30) 0.142 (10.33) 0.147 (11.84)

(1.70) 0.135 (8.92) 0.122 (7.76)

(1.70) 0.135 (8.92) 0.122 (7.76)

(5.20) 0.142 (10.22) 0.146 (11.70)

(1.77) 0.133 (8.72) 0.121 (7.58)

0.249 (16.58)

0.249 (16.58)

0.258 (13.69)

0.258 (13.69)

0.246 (16.11)

0.250 (13.63)

0.418 (22.18)

0.418 (22.18)

0.396 (14.19)

0.396 (14.19)

0.416 (21.78)

0.388 (14.26)

0.587 (10.41)

0.587 (10.41)

0.554 (8.49)

0.554 (8.49)

0.584 (10.53)

0.550 (8.37)

0.015 (9.70) 0.024 ( 2.50) 0.004 ( 0.197) 0.024 ( 0.73)

0.015 (9.70) 0.024 ( 2.50) 0.004 ( 0.197) 0.024 ( 0.73)

0.020 (9.69) 0.072 ( 5.40) 0.084 (3.42) 0.064 ( 1.26)

0.020 (9.69) 0.072 ( 5.40) 0.084 (3.42) 0.064 ( 1.26)

0.014 (9.29) 0.021 ( 2.14) 0.001 ( 0.50) 0.037 ( 1.11)

0.020 (9.55) 0.074 ( 5.43) 0.010 (3.46) 0.069 ( 1.29)

0.018

0.018

0.020

0.020

0.018

0.020

0.057

0.057

(5.30) 0.142 (10.33) 0.147 (11.84)

0.056

0.0211

436

Francois Rycx and Ilan Tojerow

Table A.3 (continued ) Model 4 (OLS)

Variables/ Models

Model 5 (2SLS)

Men Squared/102 Hours paid (ln) Bonus for shift work, night work and/or weekend work (Yes) Overtime paid (Yes)

(20.16) 0.023 ( 8.82) 0.061 ( 2.86) 0.058 (6.54)

(20.16) 0.023 ( 8.82) 0.061 ( 2.86) 0.058 (6.54)

0.021

0.021

(2.66) Contract Unlimitedterm employment contract Limited-term employment contract Apprentice/ Trainee contract Other employment contract Occupation (23 dummies) Region Brussels Wallonia Flanders Supervises the work of coworkers (Yes) Size of the establishment (ln) Level of wage bargaining CA only at the national and/ or sectoral level

Women

(2.66)

Men

Women

(12.28) 0.023 ( 4.33) 0.019 ( 1.53) 0.039 (3.15)

(12.28) 0.023 ( 4.33) 0.019 ( 1.53) 0.039 (3.15)

(19.79) 0.022 ( 8.52) 0.077 ( 3.70) 0.057 (6.16)

(11.85) 0.024 ( 4.30) 0.028 ( 2.26) 0.051 (4.06)

0.034

0.034

0.016

0.0251

(2.47)

Reference

(2.47)

(2.01)

Reference

(1.69)

Reference

0.051 ( 2.73)

0.051 ( 2.73)

0.049 ( 2.58)

0.049 ( 2.58)

0.059 ( 3.13)

0.052 ( 2.66)

0.612 ( 2.44)

0.612 ( 2.44)

0.560 ( 3.40)

0.560 ( 3.40)

0.619 ( 2.47)

0.573 ( 3.58)

0.073 ( 2.66)

0.073 ( 2.66)

0.09 ( 2.24)

0.09 ( 2.24)

0.062 ( 2.19)

0.083 ( 2.13)

Yes

Yes

Yes

Yes

Reference 0.0231 0.0231 ( 1.65) ( 1.65) 0.029 0.029 ( 2.43) ( 2.43) 0.130 0.130 (13.86) (13.86) 0.039 (10.32)

0.039 (10.32)

Reference

Yes

Yes

Reference 0.011 0.011 ( 0.63) ( 0.63) 0.073 0.073 ( 4.31) ( 4.31) 0.121 0.121 (7.10) (7.10) 0.037 (5.74)

0.037 (5.74)

Reference

Reference 0.019 0.001 ( 1.29) (0.03) 0.025 0.064 ( 1.98) ( 3.54) 0.132 0.125 (13.86) (7.30) 0.038 (9.62)

0.038 (5.76)

Reference

437

Can Rent Sharing Explain the Belgian Gender Wage Gap? Table A.3 (continued ) Model 4 (OLS)

Variables/ Models Men

Model 5 (2SLS) Women

Men

Women

0.021 (2.15) 0.002 ( 0.15) Yes

0.021 (2.15) 0.002 ( 0.15) Yes

0.059 (3.51) 0.123 ( 0.91) Yes

0.059 (3.51) 0.123 ( 0.91) Yes

0.021 (2.09) 0.009 ( 0.58) Yes

0.051 (3.06) 0.015 ( 0.97) Yes

Group effects:

Yes

Yes

Yes

Yes

Yes

Yes

Test of overidentification restrictionsb R2 adjusted F-test Number of observations Number of groups

—

—

—

—

5.330 (0.069)

1.664 (0.435)

0.709 1348 26650

0.709 2959 26650

0.678 1544 8322

0.678 1441 8322

0.702 624 26650

0.669 1506 8322

1475

1475

1217

1217

1475

1217

CA at the company level Other Industry effects (149 dummies)

Note: The dependent variable is the (Naperian) logarithm of the individual gross hourly wages. tstatistics are between brackets. Standard errors have been corrected for heteroscedasticity by the method of White (1980). Group effects estimations use the correction for common variance components within groups proposed by Greenwald (1983) and Moulton (1990). a Approximated by the firm annual gross operating surplus per worker. b NR2 and associated (p-value). ,,1 Significant at 1%, 5% and 10%, respectively.

438

Francois Rycx and Ilan Tojerow

Table A.4.

2SLS earnings equations, first stage regressions

Variables/Models

Model 5 (2SLS)

Model 5 (2SLS)

Aggregate Intercept Value added per worker (ln) Total Amount of goods and services purchased per worker (ln) Sector unemployment rate (ln) Sector average wage (ln) Education Primary or no degree Lower secondary General upper secondary Technical/Artistic/Prof. upper secondary Higher non university short type, higher artistic training University and non-university higher education, long type Post-graduate Prior experience Simple Squared/102 Cubed/104 Individual with no seniority (Yes) Seniority in the company Simple Squared/102 Hours paid (ln) Bonus for shift work, night work and/or weekend work (Yes) Overtime paid (Yes) Contract Unlimited-term employment contract

Men

Women

9.247

9.730

( 72.08) 2.025 (124.35) 0.076

( 69.49) 2.091 (117.64) 0.068

8.352 ( 36.85) 1.859 (53.64) 0.101

(7.96) — —

(6.38) — —

(4.89) — —

Reference 0.015 (0.97) 0.027 ( 1.42) 0.021 ( 1.31) 0.082 ( 3.99) 0.158 ( 6.95) 0.251 ( 5.36)

Reference 0.003 (0.21) 0.046 ( 2.17) 0.0331 ( 1.95) 0.086 ( 3.92) 0.182 ( 7.52) 0.277 ( 5.20)

Reference 0.0561 (1.82) 0.045 (1.25) 0.023 (0.69) 0.035 ( 0.82) 0.049 ( 0.94) 0.123 ( 1.07)

0.009 ( 3.30) 0.038 (2.13) 0.004 ( 1.30) 0.093 (2.06)

0.008 ( 2.80) 0.029 (1.48) 0.002 (0.69) 0.107 (0.27)

0.129 ( 2.56) 0.074 (2.15) 0.001 ( 1.60) 0.193 (2.17)

0.006 ( 3.96) 0.095 ( 0.20) 0.059 ( 3.70) 0.0261 ( 1.85) 0.102 ( 10.21)

0.005 3.18) 0.003 0.68) 0.037 2.08) 0.042 2.97) 0.078 7.98)

( ( ( ( (

( ( (

(

0.007 2.10) 0.000 0.12) 0.064 2.75) 0.083 (1.95) 0.023 9.25)

Reference

Reference

Reference

0.003

0.032

0.025

439

Can Rent Sharing Explain the Belgian Gender Wage Gap? Table A.4 (continued ) Variables/Models

Model 5 (2SLS) Aggregate

Limited-term employment contract Apprentice/Trainee contract Other employment contract Female (Yes) Occupation (23 dummies) Region Brussels Wallonia Flanders Supervises the work of co-workers (Yes) Size of the establishment (ln) Level of wage bargaining CA only at the national and/or sectoral level CA at the company level Other Industry effects (149 dummies) Group effects: R2 adjusted F-test Number of observations Number of groups

Model 5 (2SLS) Men

Women

( 0.16)

( 1.34)

0.185 (3.12) 0.143 ( 2.83) 0.003 ( 0.29) Yes

0.095 (1.08) 0.181 ( 2.74)

0.293 (3.46) 0.060 ( 1.00)

— Yes

— Yes

Reference 0.0311 (1.75) 0.083 (5.59) 0.048 (3.92) 0.044 ( 9.40)

Reference 0.052 (2.43) 0.073 (4.12) 0.051 (3.90) 0.058 ( 12.48)

Reference 0.013 ( 0.45) 0.085 (3.20) 0.018 (0.64) 0.020 ( 2.00)

Reference

Reference

Reference

0.033 (2.94) 0.152 (7.23) Yes Yes 0.820 2651 34972 1501

0.012 (0.96) 0.186 (8.71) Yes Yes 0.806 2046 26650 1475

(0.70)

0.117 (5.10) 0.066 (1.40) Yes Yes 0.844 1595 8322 1217

Note: The dependent variable is the (Naperian) logarithm of the individual gross hourly wages. tstatistics are between brackets. Standard errors have been corrected for heteroscedasticity by the method of White (1980). Group effects estimations use the correction for common variance components within groups proposed by Greenwald (1983) and Moulton (1990). ,,1 Significant at 1%, 5% and 10%, respectively.

This page is left intentionally blank

440

CHAPTER 18

Modeling Individual Earnings Trajectories Using Copulas: France, 1990–2002 Ste´phane Bonhomme and Jean-Marc Robin Abstract We use copulas to construct a flexible dynamic model of individual earnings allowing for both observed and unobserved heterogeneity. We show that the dynamics of earnings ranks is best modeled using Plackett’s (1965) parametric copula. We use discrete mixtures to model unobserved heterogeneity. For estimation, we develop a sequential EM algorithm, which is shown to be root-N consistent and asymptotically normal. This algorithm is simple to implement and fast enough to converge for bootstrapping to be a recommendable procedure to estimate standard errors. We estimate this model using the 1990–2002 French Labour Force Survey data.

Keywords: earnings dynamics, mobility, copulas JEL classifications: D30, D63, J22, J64 1. Introduction None of the many models of earnings dynamics, with unobserved heterogeneity that one finds in the literature, is consistent with the conventional way of describing earnings mobility via matrices of transition probabilities across earnings quantiles.1 In this paper, we show how the statistical concept of copula can be

Corresponding author. 1

For a review of earnings dynamics models with unobserved heterogeneity, see Alvarez et al. (2001).

CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75018-5

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

441

442

Ste´phane Bonhomme and Jean-Marc Robin

used to fill this gap. All models in this literature describe the dynamics of individual earnings levels using linear decompositions of the error terms into deterministic, stationary and nonstationary components.2 By comparison, Markov chains on discrete state spaces of earnings quantiles are flexible ways of describing earnings dynamics, which is not the same throughout the earnings distribution; but it is then parametrically very costly to allow for unobserved heterogeneity. Linear models severely restrict the dynamics of earnings. In a recent paper, Meghir and Pistaferri (2004) developed an autoregressive conditional heteroskedastic (ARCH) panel data model of earnings dynamics and found a strong evidence of sizeable ARCH effects as well as evidence of unobserved heterogeneity in the variances. Our aim in this paper is to develop a model of relative earnings mobility, flexible enough to fit the data well, but also simple enough to allow for unobserved heterogeneity. Intuitively, modeling the dynamics of individual positions (ranks) within marginal earnings distributions is the correct procedure if one does not want to restrict the parameters of earnings dynamics and the parameters of marginal distributions. To make this statement precise, let Xt and X tþ1 be two consecutive earnings. The marginal distribution of X tþ1 is completely determined by the marginal distribution of Xt and the conditional distribution of X tþ1 given Xt. On the other hand, the marginal distributions of Xt and X tþ1 and the joint distribution of corresponding ranks (the marginal cdf’s of Xt and X tþ1 ) are functionally independent objects. The distribution of ranks is called a copula in statistical theory. Transition probability matrices for earnings quantiles are particular specifications of the true underlying earnings copula. Considering various parametric copula families, we find that the single parameter copula due to Plackett (1965) succeeds in describing relative mobility very well. The dynamics of earnings ranks, or relative mobility, has therefore a structure that has not yet been fully recognized by labor economists. We eventually construct a model of individual earnings dynamics using Plackett copulas and log-normal marginal distributions. Observed and unobserved heterogeneity can then be easily incorporated into this simple three-parameter model. We adopt the nonparametric approach to unobserved heterogeneity of Heckman and Singer (1984) and estimate discrete mixtures of the parametric model using a slightly modified version of the Expectation Maximization (EM) algorithm. The layout of the paper is as follows. Section 2 is entirely devoted to the construction of this model. We first introduce the concept of copula and the factorization of multidimensional density functions allowed for. We then review several popular parametric families of copulas, and test their ability to fit earnings data. Section 3 is devoted to the estimation of copula models allowing

2

This empirical approach can be traced back to the works of Lillard and Willis (1978), Lillard and Weiss (1979), Hause (1980), MaCurdy (1982) and Abowd and Card (1989).

Modeling Individual Earnings Trajectories

443

for both observed and unobserved individual heterogeneity. Section 4 develops a particular specification of the model and presents estimates using data drawn from the French Labour Force Survey (LFS), 1990–2002. Section 5 shows how to use the model to construct mobility indices. The last section concludes. 2. Copula models for earnings dynamics Let y ¼ ðyt ; t ¼ 1; . . . ; TÞ be a sequence of one individual’s earnings. We assume that y is the realization of some first-order Markov chain Y ¼ ðY t ; t ¼ 1; 2; :::Þ of continuous real random variables. The density of Y at y can thus be factorized as T Y1 ‘ðyÞ ¼ f 1 ðy1 Þ f t;tþ1 ð ytþ1 jyt Þ t¼1

¼ f 1 ðy1 Þ

T Y1 t¼1

f t;tþ1 ð yt ;ytþ1 Þ , f t ð yt Þ

ð1Þ

where ft(yt) denotes the marginal density of Yt (at point yt), f t;tþ1 ð yt ; ytþ1 Þ the

density of (Yt, Y tþ1 ) and f t;tþ1 ð ytþ1 jyt Þ ¼

f t;tþ1 ð yt ; ytþ1 Þ f t ð yt Þ

the conditional density of

Y tþ1 given Yt. In this section, we intend to show that the statistical concept of copula, now reviewed, offers an appealing alternative to this likelihood factorization. 2.1. Copulas The notion of copula is an old statistical tool that has become popular in the financial econometrics literature but, as far as we know, has not yet been applied in labor economics. A bidimensional copula is a function C:[0, 1]2-[0, 1] with the following properties: (1) C(0, u) ¼ C(u, 0) ¼ 0 and C(1, u) ¼ C(u, 1) ¼ u, for all u A[0, 1] and (2) Cð
;
Þ is bi-increasing (or supermodular), for all u0 >u and v0 >v: Cðu0 ; v0 Þ

Cðu0 ; vÞ  Cðu; v0 Þ

Cðu; vÞ.

Let X and Y be two random variables with cdf’s FX and FY, respectively, and let C be a bidimensional copula, then the function ðx; yÞ7!CðF X ðxÞ; F Y ð yÞÞ is a cdf. Copula functions are thus useful tools to construct ‘‘distributions with given marginals.’’ A theorem, stated by Sklar (1959), shows that the converse is true. For any pair of scalar random variables (X, Y ) such that FX and FY are the marginal cdf’s of X and Y, and F is the cdf of couple (X, Y ), there exists a copula function C:[0, 1]2-[0, 1] such that
 F ðx; yÞ ¼ C F X ðxÞ; F Y ðyÞ ; 8ðx; yÞ.

The copula function C is unique if the marginal cdf’s FX and FY are continuous.

Ste´phane Bonhomme and Jean-Marc Robin

444

One can also define copula densities in a similar way as probability densities are defined. Let the distribution of (X, Y ) be continuous. The differentiated form of Sklar’s theorem splits the joint density of X and Y, f (x, y), into the product of marginal densities fX(x ) and fY( y) and of the copula density, @2 Cðu;vÞ cðu; vÞ  @u@v (using left-differentiation):
 f ðx; yÞ ¼ f X ðxÞ f Y ðyÞc F X ðxÞ; F Y ð yÞ .

Notice that because FX(X ) and FY(Y ) have uniform distributions, C(u, v) is the density of (FX(X ), FY(Y )) at (u, v) and it is also the conditional density of FY(Y ) at point v given FX(X ) ¼ u.3 Let (Yt, Y tþ1 ) be a couple of subsequent earnings. The above theory implies that the joint density of (Yt, Y tþ1 ) can be decomposed into the product of the marginal densities and the copula density:
 f t;tþ1 ðyt ; ytþ1 Þ ¼ f t ðyt Þf tþ1 ðytþ1 Þct;tþ1 F t ð yt Þ; F tþ1 ðytþ1 Þ . ð2Þ The likelihood in (1) can thus be alternatively factored as ‘ðyÞ ¼

T Y

f t ð yt Þ


t¼1

T Y1 t¼1


 ct;tþ1 F t ð yt Þ; F tþ1 ðytþ1 Þ .

ð3Þ

|fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ‘1 ðyÞ

‘2 ðyÞ

Hence, ‘ðyÞ is the product of two terms: ‘1 ðyÞ, the product of marginal densities, and ‘2 ðyÞ, the product of copula densities. Equivalently, ‘2 ðyÞ is the likelihood of the sequence of individual ranks in the corresponding sequence of marginal distributions. 2.2. Transition probability matrices as copula approximations In this subsection, we link the usual approach to relative mobility via matrices of transition probabilities across earnings quantiles to the copula framework introduced above. Most studies of relative earnings mobility indeed start by categorizing earnings into quantile intervals and then compute the probabilities of transition between these quantile intervals. Let Q be the integer that specifies the number of quantiles one wants to use to partition the support of earnings distributions (Q ¼ 4 for quartiles, Q ¼ 10 for deciles, Q ¼ 100 for centiles, y). Let qt ð
Þ be the function categorizing Yt by its quantiles:   qt ðyt Þ  Q
F t ðyt Þ þ 1, 3

The conditions R 1 defining copula functions R 1 also restrict copula densities. Condition 1 is equivalent to assuming that o cðu; vÞdu ¼ 1; 8u, and 0 cðu; vÞdv ¼ 1; 8v. Condition 2 is equivalent to assuming that cðu; vÞ  0;R 8u; v. The copula C(u, v) can then be computed from its density by integration: u Rv Cðu; vÞ ¼ 0 0 cðe u; e vÞde u de v.

Modeling Individual Earnings Trajectories

445

where bxc denotes the integer part of any real x. The transition probability matrix Pt, t+1 is then the Q  Q matrix, which generic (i, j) element (i, j A{1, y , Q) is:   Pt;tþ1 ðjjiÞ  Pr qtþ1 ðY tþ1 Þ ¼ jjqt ðY t Þ ¼ i   Pr qt ðY t Þ ¼ i; qtþ1 ðY tþ1 Þ ¼ j   ¼ Pr qt ðY t Þ ¼ i Z Z f t ðyt Þf tþ1 ðytþ1 Þct;tþ1 ¼Q


qt ðyt Þ¼i

qtþ1 ðytþ1 Þ¼j



n F t ðyt Þ; F tþ1 ðytþ1 Þ dyt dytþ1

¼Q

Z

i Q i 1 Q

Z

j Q j 1 Q

ct;tþ1 ðu; vÞdu dv.

Transition probability matrices thus provide approximations of copula densities. Qu þ1 To see why, first remark that b Qc ! u when Q-N, uniformly for all u A[0,1]. Hence, for any u and v in [0,1] and for any continuous copula density C t;tþ1 ,     b ct;tþ1 ðu; vÞ ¼ Q  Pt;tþ1 Qv þ 1j Qu þ 1 ! ct;tþ1 ðu; vÞ as Q ! 1,

uniformly in u, v. Transition probability matrices are thus nonparametric approximations of the underlying true copula. It is of course possible to smooth these piecewise constant approximations. Bernstein’s polynomial regularization (see, e.g. Drouet–Mari and Kotz, 2001) or bivariate kernel density estimators (e.g. Gagliardini and Gourie´roux, 2002) can be used for this purpose. Unfortunately, the nonparametric estimation of bivariate density functions requires large amounts of data, which may be available for applications in finance but are usually not available in labor economics. Furthermore, the curse of dimensionality makes it almost infeasible to incorporate observed – not to mention unobserved – heterogeneity in practice. For this reason, we decided to adopt a parametric approach. As the most commonly used parametric models are linear models of log-earnings dynamics (ARIMA), we start by discussing the implication of these models regarding the distribution of earnings ranks. 2.3. Linear dynamics of levels and dynamics of ranks If quantile Markov chains are routinely used to model relative earnings dynamics, ARIMA models are undoubtedly the most popular models of absolute earnings mobility. Since we focus in this section on first-order Markov models, let us here consider the AR(1) model: ytþ1 ¼ ryt þ utþ1 ,

ð4Þ

446

Ste´phane Bonhomme and Jean-Marc Robin

with i.i.d. innovation errors ut and rA[ 1,1], including 1 and +1 so that (yt) may be a random walk. One can think of this equation as representing the link between past and current earnings after controlling for observed individual characteristics and time dummies (so that mean earnings are 0 at each date). The autocorrelation coefficient, r, is thus a direct measure of absolute mobility. Suppose that the initial condition (y0) has normal distribution Nð0; s20 Þ and that innovations ut have normal distribution Nð0; s2u Þ. The marginal distribution of yt is Nð0; s2t Þ, with s2t ¼ r2 s2t 1 þ s2u . The rank of yt in the marginal distribution of time t earnings, say rt ¼ Fðsytt Þ 2 ½0; 1Š, obeys the following nonlinear first-order autoregressive process: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 1 ðrtþ1 Þ ¼ ttþ1 F 1 ðrt Þ þ 1 t2tþ1
t , ð5Þ

t and
t ¼ ut =su is a Gaussian strong white noise. Since the where ttþ1 ¼ srs tþ1 marginal distribution of rt is uniform on [0,1], the joint distribution of ðF 1 ðrt Þ; F 1 ðrtþ1 ÞÞ is bivariate normal with zero means, unit variances and covariance tt+1. The copula of the couple of earnings level ( yt, ytþ1 ) is said to be Gaussian. Two remarks are now in order. First, it is clear that ranks rt ¼ Ft ( yt) can satisfy Equation (5) even if Ft is not Gaussian. In this case, Equation (4) cannot be true: using yt instead of F 1(Ft ( yt)) is a source of measurement error inducing spurious dynamics in the residuals.4 Second, the dynamics of ranks does not necessarily follow model (5). In the next section, we examine different choices of copulas.

2.4. Parametric copulas Among the many parametric copula specifications one finds in the statistical literature, does there exist one that fits the dynamics of earnings ranks well? We now investigate this question using earnings data drawn from the French LFS of 1990 and 1991. Appendix A details the copula families we consider. To remain as much agnostic as possible as far as the form of the marginal distributions is concerned, we first regress log-earnings on education and experience separately for 1990 and 1991, and compute the empirical cumulative distribution functions of the residuals. Then, we estimate the different copula parameters by maximizing the likelihood of the empirical ranks. Finally, we categorize the residuals by quintiles and compare the empirical probability transition matrix to the predictions from the different copula estimates. Table 1 shows the actual and predicted transition probability matrices for the Gaussian, Plackett, Frank, Gumbel, Joe, Clayton, FGM (fabric, Gumbel, Morgensterm) and Log-copulas. Several specifications are able to capture the general form of the observed transition matrix. In particular, there should be

4

It is indeed generally found that ut+1 is a moving-average of order one (sometimes two).

Table 1. 0

Observed

0:02

0:02

0

B 0:68 0:26 B B B 0:26 0:45 B B 0:05 0:23 B B @ 0:01 0:05 0:00

0:01

0

B 0:73 0:19 B B B 0:19 0:39 B B 0:05 0:24 B B @ 0:02 0:12 0:01

0:06

0:08 0:22

0:03 0:06

0:47 0:19

0:20 0:53

0:04

0:18

Frankðd ¼ 0:19Þ 0:05 0:23

0:44 0:23 0:05 Claytonðd ¼ 0:50Þ 0:05 0:24

0:30 0:24 0:17

0

0:00 C C C 0:02 C C 0:05 C C C 0:19 A

0:00

0:74

1

0:01 0:00 C C C 0:05 0:01 C C 0:23 0:05 C C C 0:45 0:26 A

0:26

0:68

1

0:02 0:01 C C C 0:12 0:06 C C 0:24 0:17 C C C 0:31 0:31 A 0:31

B 0:66 0:25 B B B 0:25 0:37 B B 0:08 0:26 B B @ 0:01 0:11

0:45

0:01

0

B 0:63 0:26 B B B 0:26 0:39 B B 0:09 0:25 B B @ 0:02 0:09 0:00

0:01

0

B 0:33 0:26 B B B 0:26 0:23 B B 0:20 0:20 B B @ 0:14 0:17 0:07

0:14

Gaussianðd ¼ 0:30Þ 0:08 0:26

0:32 0:26 0:08 Gumbelðd ¼ 0:22Þ 0:09 0:25

0:37 0:25 0:04 FGMðd ¼ 0:83Þ 0:20 0:20

0:20 0:20 0:20

1

0:01 0:00 C C C 0:11 0:01 C C 0:26 0:08 C C C 0:37 0:25 A

0:25

0:66

1

0:02 0:00 C C C 0:09 0:01 C C 0:25 0:04 C C C 0:45 0:19 A 0:19

0:76

1

0:07 C C C 0:14 C C 0:20 0:20 C C C 0:23 0:26 A 0:14 0:17

0:26

0:33

0

B 0:73 0:20 B B B 0:20 0:52 B B 0:04 0:21 B B @ 0:02 0:05 0:01

0:02

0

B 0:46 0:31 B B B 0:31 0:34 B B 0:17 0:24 B B @ 0:06 0:10 0:00

0:01

0

B 0:66 0:25 B B B 0:25 0:42 B B 0:07 0:25 B B @ 0:02 0:07 0:00

0:01

Plackettðd ¼ 0:08Þ 0:04 0:21

0:50 0:21 0:04 Joeðd ¼ 0:41Þ 0:17 0:24

0:32 0:23 0:04

1

0:02 0:01 C C C 0:05 0:02 C C 0:21 0:04 C C C 0:52 0:20 A

0:20

0:06 0:00 C C C 0:10 0:01 C C 0:23 0:04 C C C 0:43 0:18 A

0:18

0:77

Log-Copulaðd ¼ 0:17Þ 0:07 0:25

0:40 0:24 0:04

0:73

1

1

0:02 0:00 C C C 0:07 0:01 C C 0:24 0:04 C C C 0:47 0:20 A

0:20

Modeling Individual Earnings Trajectories

B 0:68 0:21 B B B 0:20 0:50 B B 0:07 0:21 B B @ 0:03 0:06

Fit of some parametric families of couplas to earnings data

1

0:75

447

448

Ste´phane Bonhomme and Jean-Marc Robin

more inertia at the top and bottom of the distributions of ranks than in the middle. However, Plackett’s (1965) family of copulas:  1 1 Cðu; v; ZÞ ¼ Z 1 1 þ Zðu þ vÞ ½ð1 þ Zðu þ vÞÞ2 4ZðZ þ 1ÞuvŠ2 , 2 where Z is a scalar parameter, fits transition probabilities better than all other families. The Euclidian distance d between predicted and actual matrices is 0.08 for this copula, while the Gumbel has d ¼ 0:22, the Frank: d ¼ 0:19, the Gaussian: d ¼ 0:30, and the FGM copula: d ¼ 0:83. We checked the robustness of this result by varying samples and controls. Plackett copulas always provided the best fit. We thus choose to model year-to-year mobility using Plackett copulas. Apart from its ability to fit well-relative earnings dynamics, this family has the advantage of mathematical simplicity. We refer to Appendix B for a presentation of the Plackett copula and its properties. The most remarkable property is that relative mobility decreases as Z increases. So, Z is a useful index of relative inertia. 3. Estimation of mixtures of copula models In the preceding section, we argued that copulas were well adapted to model earnings mobility. With one single parameter, the Plackett copula was shown to fit the dynamics of earnings ranks well. In this new section, we first argue that it is necessary to allow for both observed and unobserved heterogeneity in order to preserve the empirical validity of the first-order Markov assumption. Then, we develop a sequential EM algorithm adapted to the estimation of mixtures of copula models of earnings dynamics. 3.1. Heterogeneity The first-order Markov assumption we have maintained throughout the preceding analysis, attractive though it may be for its low computational cost, is usually found to be rejected by earnings data. This assumption implies that the transition probability matrix of a given discrete mobility process between two nonconsecutive dates t and t+2 is the product of the (t; t þ 1) and (t þ 1; t þ 2) transition matrices. In practice, the product matrix thus obtained usually presents more mobility than the true transition matrix between t and t þ 2.5 We give an illustration using the French LFS data of 1990–1992. The product matrix obtained from the observed mobility between 1990 and 1991, and the

5 See for instance Blumen et al. (1955), Shorrocks (1976) and Singer and Spilerman (1976). Empirical evidence shows that earnings ranks are more persistent than what they should be if they were firstorder Markov processes. For example, using PSID data, Gottschalk (1997) finds that being in the first quintile of the earnings distribution in 1970, there is more than 50% chances of being in the same quintile 20 years later. Guillotin and Bigard (1992), for France, obtain similar orders of magnitude.

449

Modeling Individual Earnings Trajectories

actual transition matrix between 1990 and 1992 are

P201

0

0:57

B B 0:27 B ¼B B 0:11 B @ 0:04

0:01

0:26 0:11 0:05 0:01

1

C 0:38 0:22 0:10 0:03 C C 0:24 0:37 0:22 0:06 C C; C 0:10 0:24 0:42 0:20 A 0:02 0:06 0:21 0:70

0

P02

0:67 0:22 0:08 0:02 0:01

1

C B B 0:24 0:49 0:22 0:04 0:01 C C B B ¼ B 0:06 0:24 0:50 0:18 0:02 C C. C B @ 0:02 0:04 0:18 0:59 0:17 A 0:01 0:01 0:02 0:17 0:79

In the other studies, we find more mobility between quantiles in the predicted matrix than in the actual 1990–1992 matrix of transition frequencies.6 The discrepancy persists whatever control variable we use to condition the 1990–1991 transition probability matrix. Controlling for education, experience, etc., somewhat reduces the discrepancy but not all of it. One way of explaining state dependence, without abandoning the first-order Markov assumption, is to postulate the existence of unobserved heterogeneity as in the mover–stayer model of Blumen et al. (1955). Allowing for unobserved heterogeneity should generate enough additional autocorrelation in individual earnings trajectories to fill the remaining gap. This idea has had a large success in the econometric literature on labor market transitions, and various forms of unobserved heterogeneity have already been modeled: random effects in Lillard and Willis (1978) and Geweke and Keane (2000), fixed effects in Gottschalk and Moffit (1994). For simplicity, and not to depend too heavily on parametric assumptions, we use a discrete heterogeneity model, assuming that there exists at most K types of earnings processes differing in the values of the parameters (see Heckman and Singer, 1984). 3.2. A sequential EM algorithm for discrete mixtures of copulas We now address the issue of estimating a discrete mixture of copulas from a sample of individual earnings trajectories { yn ¼ ( ynt, t ¼ 1, y ,T), n ¼ 1, y , N}. For simplicity, we assume away observed covariates. We suppose that there exists a latent discrete heterogeneity variable zn A{1, y , K }, n ¼ 1, y , N, indicating which heterogeneity group each individual in the sample belongs to. Conditional on zn ¼ k, the density of yn ¼ ( ynt, t ¼ 1, y ,T ) is f ðyn ; bk ; tk Þ ¼

T Y t¼1

f ð ynt ; bk Þ

T Y1 t¼1


 c F ð ynt ; bk Þ; F ð yn;tþ1 ; bk Þ; tk ,

ð6Þ

where f(y; bk) is the marginal density of ynt ¼ y given zn ¼ k and C(u, v; tk) is the density of (F(ynt; bk), F(yn,t+1; bk)).7 Let pk be the proportion of individuals of type k A{1, y, K }, in the population. bk and tk are parameter vectors.

6

In this paper, ‘‘more mobility’’ is to be understood in terms of the concordance ordering "c. See Appendix B for details. 7 The estimation method is here presented in the stationary case for ease of exposition. Notice that the marginal distributions and the copulas can depend on calendar time t.

Ste´phane Bonhomme and Jean-Marc Robin

450

To estimate the parameter vector y ¼ {bk, tk, pk, k ¼ 1, y , K}, we apply the following sequential EM algorithm. (1) (Expectation- or E-stage). For an initial value y(s) of y and all k A{1, y , k}, compute the posterior probability of zn ¼ k given yn and y(s) as pðsÞ kn ¼

ðsÞ ðsÞ pðsÞ k f ðyn ; bk ; tk Þ . K P ðsÞ ðsÞ pðsÞ f ðy ; b ; t Þ n ‘ ‘ ‘

ð7Þ

‘¼1

ðsÞ ðsÞ (2) (Maximization- or M- stage). Successively update bðsÞ k , tk and pk as

¼ arg max bðsþ1Þ k bk

tkðsþ1Þ ¼ arg max tk

¼ pðsþ1Þ k

N X

n¼1 N X n¼1

N 1X pðsÞ . N n¼1 kn

pðsÞ kn pðsÞ kn

T X

t¼1 T 1 X t¼1

ln f ðynt ; bk Þ;

ð8Þ

  i h ln c F ynt ; bkðsþ1Þ ; F yn;tþ1 ; bkðsþ1Þ ; tk ,

ð9Þ ð10Þ

ðsÞ This algorithm differs from the standard EM since bðsÞ k and tk are not PN algorithm, ðsÞ simultaneously updated by maximizing n¼1 pkn ln f ðyn ; bk ; tk Þ jointly with respect to both bk and tk, but sequentially. Proceeding this way renders the maximization stage considerably more tractable. This is because the copula part of ln f(yn; bk, tk) depends on bk through the marginal cdf’s F(ynt; bk) and F(yn,t+1; bk), and therefore, in general, in a very nonlinear way. The algorithm is easier to implement and also much faster.8 This idea has been used many times under various forms to simplify the practical implementation of fixed-point algorithms. In particular, another type of sequential EM algorithm has recently been used in the context of finite mixtures of distributions by Arcidiacono and Jones (2003). Notice that, unlike the standard case considered by Dempster et al. (1977), the sample likelihood does not increase at each new iteration of the sequential algorithm. This caveat has two important consequences. First, the numerical convergence of the sequential estimator is not guaranteed. However, the usual sufficient conditions for convergence are strong (Dempster et al., 1977; Wu, 1983) and rarely verified in practice. Numerical convergence is rather assumed than formally proved in most cases. Second, the sequential estimator differs from the MLE. We show in Appendix C that, under the identifiability and regularity conditions detailed there, if the sequential EM algorithm introduced above numerically converges to a limit yN for infinitely many values of N, then one can

8

The slow numerical convergence of EM is usually thought to be the main drawback of the algorithm (see, e.g. Redner and Walker, 1984). However, we found our sequential modification very fast to converge, even with large datasets.

Modeling Individual Earnings Trajectories

451

extract a subsequence (ys(N)) which is root-N consistent in probability and asymptotically normal. More precisely, we show that if the sequential EM algorithm numerically converges, it converges to the solution of a set of moment conditions different from the optimal ones. The estimator solution to this system is consistent and asymptotically normal under the assumption of i.i.d. individual trajectories within each group of heterogeneity, but it is not efficient. The efficiency loss is the cost to pay for very significant gains: numerical convergence is achieved much faster and the algorithm is very simple to implement.9 4. An empirical model of French individual earnings and employment dynamics, 1990–2002 We now turn to the study of earnings (and employment) dynamics in France, using data covering the period 1990–2002. We first introduce the data and present the empirical specification of our model. We then detail the estimation results and, lastly, we describe how the model fits the data. 4.1. The French Labour Force Survey, 1990– 2002 The data we use come from the 1990–2002 French LFS, conducted by INSEE, the French Statistical Institute. We use the LFS data instead of administrative earnings data (DADS), as in Alvarez, et al. (2001), because the DADS data do not follow individuals across states which are not private jobs. The French LFS is a rotating panel, a third of the sampling units (dwellings) being replaced, every year, by an equivalent number of newly sampled units. Large samples of about 150,000 individuals aged 15 or more, in 75,000 households, can thus be interviewed three times, in March of three subsequent years, about various aspects of their employment histories. Note that a panel length of three years is intuitively barely enough to ensure identification: two years are necessary to characterize the first-order Markov dynamics, one additional year is required in case of unobserved heterogeneity. Yet, the short panel length is compensated by the fact that the LFS provides large, nationally representative samples for every year. As is usually the case for this sort of survey based on individuals’ responses to interviews, hours worked are badly reported. For instance, we noticed that many individuals alternatively reported 39 hours worked in one week (the legal working time in the 1990s), and 40 hours in another. To limit the influence of hour measurement errors on our results, we chose to use monthly salaries, which

9

Note also that maximum likelihood in the M-stage of the algorithm can be replaced by pseudo maximum likelihood, so as to make the algorithm even more user-friendly, without hampering the consistency of the estimator. It is indeed easy to see that the consistency proof works just as fine if the ML programs in the sequential EM algorithm are replaced by Pseudo-ML programs using linearexponential distributions instead of the true distribution. For example, weighted ordinary least squares (OLS) regressions can be used to update group-specific means.

452

Ste´phane Bonhomme and Jean-Marc Robin

is what the questionnaire asks for, rather than hourly wages. Monthly earnings were finally divided by the retail price index. We dropped all observations for students, retired persons and self-employed, and kept only male labor trajectories, removing female trajectories from the sample to limit the role of labor supply as a determinant of earnings dispersion. We also trimmed the data below the first and above the 99th percentiles of the wage distribution. The French LFS being a rotating panel, we had to distinguish three types of trajectories: complete trajectories ( yt, yt+1, yt+2), where t is the date of the first recorded observation, accounting for 53% of the sample; 25% of two-year trajectories ( yt, yt+1,
), where the last earnings observations are missing; and 22% of one-year trajectories ðyt ;
;
Þ, for which no information about mobility can be inferred. There is thus a certain amount of attrition we assume exogenous to the employment-earnings process. 4.2. Model specification In this application, we introduce unemployment as a specific state. This is important for any reasonable description of labor market trajectories in the Euro area, where unemployment rates are between 8 and 10%. We also allow for observed and unobserved heterogeneity. The parametric specification is detailed in Appendix D. Here, we only summarize its main characteristics. 4.2.1. Heterogeneity After trying various specifications, we decided to introduce two latent unobserved heterogeneity variables: one variable, z1 A{1, y , K1}, conditions every parameter of the model, marginal distributions of earnings and employment states as well as copulas, and a second variable, z2 A{1, y , K2}, specifically conditions the mobility process.10 Individual employment-earnings trajectories are also conditioned by a set of time-varying covariates, denoted as xt. In our application, xt only comprises labor market experience (age minus age at the end of school) and squared experience. Lastly, covariates (z1, z2) are determined by some vector z0 of non-time-varying individual attributes:  Education (Ed), classified into five categories according to the highest degree

obtained: ‘‘no degree,’’ ‘‘junior high-school,’’ ‘‘senior high-school,’’ ‘‘less than three years of university’’ and ‘‘more than three years.’’  The year of entry in the labor market (b), defining a ‘‘cohort.’’ Given calendar time t, experience is equal to t–b and xt is thus deterministic conditional on b.

10

We indeed observed that if one does not suppose a specific unobserved heterogeneity variable in the dynamic part of the model, the EM algorithm uses all the mixing possibilities to fit the marginal distributions. This is certainly because there is considerably more variance in the cross-section dimension than in the time-series dimension.

Modeling Individual Earnings Trajectories



453

Because a baccalaure´at (high-school diploma in France) obtained in 1950 had more market value than a baccalaure´at obtained in 1990,11 we also allow for education and cohort interactions (Ed  b).

Therefore, education conditions the earnings process only via the latent heterogeneity variables z1 and z2, which may thus be understood as two different components of the human capital an individual possesses when entering the labor market.12 4.2.2. Model specification Let et A{0,1} denote the employment state at calendar time t (et ¼ 1) if employed, 0 otherwise). Let yt denote the logarithm of employees’ wages. We write the joint density of individual data (yt, yt+1, yt+2, et, et+1, et+2, z1, z2) (with respect to the appropriate measure) conditional on z0 as the following product: f ð yt ; ytþ1 ; ytþ2 ; et ; etþ1 ; etþ2 ; z1 ; z2 jxt ; xtþ1 ; xtþ2; z0 Þ ¼ Prfz1 jz0 gPrfz2 jz1 ; z0 g     Prfet jxt ; z1 g  Pr etþ1 jet ; xt ; z1 ; z2 Pr etþ2 jetþ1 ; xtþ1 ; z1 ; z2 f ð yt jxt ; z1 Þet f ð ytþ1 jxtþ1 ; z1 Þetþ1 f ð ytþ2 jxtþ2 ; z1 Þetþ2
e e c F ð yt jxt ; z1 Þ; F ð ytþ1 jxtþ1 ; z1 Þjxt ; z1 ; z2 t tþ1
e e c F ð ytþ1 jxtþ1 ; z1 Þ; F ð ytþ2 jxtþ2 ; z1 Þjxtþ1 ; z1 ; z2 tþ1 tþ2 ,

ð11Þ

where each component is specified as follows: 

Distribution of unobserved heterogeneity. The probability distribution of the latent variable z1 given z0, Pr{z1|z0}, and the conditional probability of z2 given z1 and z0, Pr{z2|z1, z0}, are modeled as Multinomial LOGIT.  Cross-sectional distributions. The cross-section log-wage density, f ( yt|xt, z1), is assumed to be Gaussian conditional on experience xt and heterogeneity z1. We let both the cross-section mean and the cross-section variance depend on xt as in Moffit and Gottschalk (2002). We also allow the intercept and the returns to experience in both first- and second-order moments to be group-specific (z1 dependent). The unconditional unemployment probability, Pr {et ¼ 0|xt, z1}, follows a PROBIT model. With unobserved heterogeneity, it is important to specify this probability in order to compute the likelihood component corresponding to the first observation period of each individual trajectory.

11

In 1980–1985, only 35% of a student cohort would attain an education level assimilable to a high school diploma. Between 1985 and 1995, a voluntarist educational policy doubled that number: since 1995 about 70% of a student cohort reach grade 12 (terminale)–80% of them obtaining the baccalaure´at. 12 We believe this methodological approach sensible in the absence of any ‘‘natural’’ instrument for education in labor force surveys like the French LFS.

Ste´phane Bonhomme and Jean-Marc Robin

454

 Mobility. We model the conditional unemployment-to-unemployment prob-

ability, i.e. Pr{et+1 ¼ 0|et ¼ 0, xt, z1, z2}, and the conditional employment-tounemployment probability, i.e. Pr{et+1 ¼ 0|et ¼ 1, xt, z1, z2}, using two PROBIT models. Lastly, we use the Plackett copula to model the likelihood of earnings ranks, conditional on employment, c(u, v|xt, z1, z2). All parameters of the PROBIT and Plackett models specifically depend on unobserved heterogeneity (z1, z2). 4.3. Estimation methodology We use the sequential EM algorithm developed in subsection 3.2 to estimate our empirical model, and we bootstrap the estimation sample 500 times to compute standard errors. The details of the implementation of the EM algorithm for estimating the previous specification are given in Appendix D. Here, we only rapidly describe the estimation strategy. We estimate the model assuming that every three-year panel identifies a specific set of cross-section and mobility parameters. However, we force the parameters of the distribution of unobserved heterogeneity (z1, z2) conditional on z0 (Pr{z1|z0} Pr {z2|z1, z0} to remain the same across the different panels. We impose this constraint because we believe that there is too little information in three years of panel data to identify both the distribution of earnings processes given heterogeneity (z1, z2) and the distribution of unobserved heterogeneity (z1, z2) given observed heterogeneity z0 (education and cohort). The way the parameters of marginal distributions and copulas vary across three-year panels (1990–1992 to 2000–2002) thus tells us how much of the changes in earnings distributions is not explained by observed composition effects (i.e. changes in the distribution of education and cohort) and by experience, assuming that there is no unobserved composition effect. Note that, by proceeding this way, we allow three individuals from three different three-year panels to face different parameter sets at the same calendar year. This may seem undesirable but it is well known that time, cohort and experience effects are not separately identified. Alternatively, we could estimate parameters assuming they remained fixed over a certain period of time. For example, we could contrast the periods 1990–1993, 1994–1998 and 1999–2002. This is the strategy followed by Gottschalk and Moffit (1994) who estimate different models for the 1970s and the 1980s. Our approach avoids arbitrary choices of different periods. 4.4. Estimation results In this section, we present the estimates of the distribution of heterogeneity and the estimates of the cross-section and mobility parameters corresponding to the 1990–1992 three-year panel, as the results for the other three-year panels are qualitatively similar. The number of heterogeneity groups is likely to influence the interpretation. We used two popular criteria based on penalized likelihood to select these

455

Modeling Individual Earnings Trajectories

Figure 1.

Components of the cross-sectional log-earnings density in 1990 (K 1 ¼ 1)

density

1

0.5

0 7

8

9 log wage total group 2

10

11

group 1

numbers. We found (K1 ¼ 2, K2 ¼ 2) optimal using the Bayesian Information Criterion (BIC) and (K1 ¼ 3, K2 ¼ 2) Akaike’s (1969) criterion.13 For expositional convenience, we shall present the parameters for the minimal number of groups, (K1 ¼ 2, K2 ¼ 2). We shall then analyze the fit for (K1 ¼ 3, K2 ¼ 2). 4.4.1. Cross-sectional earnings distribution and unemployment risk The first group of cross-section heterogeneity (z1 ¼ 1) collects about 25% of the whole population. It gathers individuals with, on average, higher (aggregate mean of 9.33 vs. 8.75) and more dispersed (aggregate standard deviation of 0.39 vs. 0.28) monthly earnings than those in the second group. Figure 1 brings a visual confirmation that one component of the earnings mixture is both to the left of the other one and less dispersed.14 The probability of being unemployed is found significantly lower for Group 1 than for Group 2 (aggregate probability of 3.1% vs. 11.2%). When a third group of heterogeneity is allowed for, then a small third group of even poorer and more often unemployed workers is selected out of the second group (about 15% of them). The other two components of the earnings distribution mixture do not change much (Figure 2).

13 Let L be the log-likelihood of a given sample of size N, and K be the number of parameters in a 1 given model. BIC maximizes N1 L K lnðNÞ K N2 . Obviously, N with respect to K. Akaike maximizes N L the optimal number of parameters is larger for Akaike than for BIC. 14 In this figure, the area below each density component is equal to the group’s relative frequency, so that the sum of all components is the aggregate density.

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Figure 2.

Components of the cross-sectional log-earnings density in 1990 (K 1 ¼ 3) 1.5

density

1

0.5

0 7

8

9 log wage total group 2

10

11

group 1 group 3

It can be seen from the estimated parameters displayed in Table 2 that the experience-earnings profile is concave. The effect of experience on unemployment probabilities is U-shaped (see Table 2). 4.4.2. Cross-section heterogeneity Table 3 gives a full account of the estimates of the parameters of the (LOGIT) probability of z1 ¼ 1 (vs. z1 ¼ 2) given the set z0 of individual characteristics (education, cohort and interactions). To make the interaction of the cohort and education variables more easily interpretable, we compute in Table 4 the proportion of Type-1 individuals by education and cohort (date of entry into the labor market). First, one remarks that there is no direct cohort effect: all three cohorts have about the same probability of being in the first group (25%). By contrast, higher education unambiguously increases the probability of belonging to Group 1. But this effect is not uniform: only the top education group (‘‘college +,’’ i.e. more than three years of university) is guaranteed to belong to the top earnings group whatever the date of entry into the labor market. For the other education groups, older means better. This result is consistent with the accelerated democratization of the French education system between 1985 and 1995. The value of an education degree such as the baccalaure´at (high school diploma) was much higher 20 or 30 years ago because bacheliers (high school graduates) were much fewer. This explains why 70% of the HS graduates of the oldest cohort belong to Group 1 when only 11% of the youngest HS graduates are classified in Group 1. It is useful to know how the different individuals in the sample are reallocated when one allows for a third group of cross-section heterogeneity. For a

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Modeling Individual Earnings Trajectories

Table 2.

Cross-sectional earnings distribution – Parameter estimates

Earnings

Mean Experience Squared experience Intercept Variance Experience Squared experience Intercept Unemployment probability Experience Squared experience Intercept

Cross-Section Heterogeneity z1 ¼ 1

z1 ¼ 2

0.034 (0.00091) 0.00068 (0.000020) 8.99 (0.16)

0.030 (0.00046) 0.00053 (0.0000097) 8.43 (0.0045)

0.0041 (0.00060) 0.00011 (0.000013) 0.16 (0.063)

0.00095 (0.00026) 0.000015 (0.0000053) (0.058) (0.0027)

0.042 (0.0060) 0.0012 (0.00012) 1.77 (0.066)

0.084 (0.0032) 0.0017 (0.000069) 0.046 (0.029)

given value of the postulated number of groups (K1) and each individual in the sample, the posterior probability of belonging to one group given the individual’s observations can be estimated. The group with the highest probability is the best predictor of individual types. Comparing the classification obtained for K1 ¼ 2 to the one obtained for K1 ¼ 3, ordering groups according to mean earnings, 83% of the individuals classified in Group 1 for K1 ¼ 2 remain classified in this group when a third group is allowed for and that about 16% move to Group 2. Only 1% move to Group 3. Moreover, 75% of Type-2 individuals remain in Group 2, but 17% move to the third group and 8% to the first group. 4.4.3. Earnings mobility Table 5 displays the estimates of the group-specific parameters (Zz1 z2  expðx0t wz1 z2 Þ) of the Plackett copula used to model the transition process of marginal earnings ranks. The sign of the experience coefficient implies that wage mobility decreases with experience among low-wage earners (z1 ¼ 2) and increases with experience among high-wage earners (z1 ¼ 1). To give a better idea of the magnitude of wage mobility differentials across worker types, we display in Table 6 the predicted quintile-to-quintile probability transition matrices for the four combinations of (z1, z2) and three levels of experience: 5, 20 and 35 years.

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Table 3.

Distribution of cross-section heterogeneity given exogenous controls

Independent Variable

Estimate

Intercept

90 (8.4)

Education Junior high-school

72 (9.4) 116 (11.1) 76 (14.6) 39 (34.0) 0.048 (0.0043)

Senior high-school Some college College+ Cohort (year of entry into the labor market) Cohort * Education Junior high-school

0.036 (0.0048) 0.057 (0.0056) 0.035 (0.0074) 0.024

Senior high-school Some college College+

(0.017) 0.53

Pseudo-R2

Table 4. Probability of being a high-wage earner (z1 ¼ 1) by education and cohort (model with two groups of cross-section heterogeneity) Education Cohort

None

Junior HS

Senior HS

Some College

College +

Overall

o1970 1970–1985 >1985

0.03 0.01 0.01

0.11 0.05 0.01

0.70 0.38 0.11

0.90 0.86 0.60

0.99 0.95 0.97

0.24 0.25 0.27

Overall

0.01

0.06

0.30

0.72

0.97

0.25

The six featured matrices unambiguously designate individuals with z2 ¼ 1 as a group of ‘‘stayers’’ and those with z2 ¼ 2 as a group of ‘‘movers’’: the numbers in the first diagonal of the matrices corresponding to the second group are indeed smaller and the off-diagonal ones larger, irrespective of the cohort of origin. Note that there is less heterogeneity among high-wage individuals (i.e. those with z1 ¼ 1) than among low-wage individuals z1 ¼ 2. Moreover, the effect of experience is more pronounced for low-wage workers than for high-wage workers.

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Modeling Individual Earnings Trajectories

Table 5.

Parameter Z of the earnings copula (greater if less mobility) conditional on experience and unobserved heterogeneity Mobility Heterogeneity

z1 ¼ 1 Experience Intercept

z1 ¼ 2 Experience Intercept

z2 ¼ 1

z2 ¼ 2

0.015 (0.0025) 6.44 (0.14)

0.051 (0.0056) 5.58 (0.16)

0.049 (0.0050) 3.43 (0.092)

0.048 (0.0065) 0.87 (0.32)

4.4.4. Employment dynamics. Table 8 presents the parameters of the PROBIT probabilities of unemployment at t þ 1 conditional on employment at t and of unemployment at t þ 1 conditional on unemployment at t, and Table 7 below shows the implied levels of these transition probabilities for the four combinations of ðz1 ; z2 Þ and the same three levels of experience as before. Compared to individuals with z2 ¼ 2 (movers), individuals with z2 ¼ 1 (stayers) clearly have very little chance of becoming unemployed. Yet, when they become unemployed they tend to remain unemployed longer. Being both a low-wage worker (z1 ¼ 2) and a mover ðz2 ¼ 2Þ, maximizes the chance of becoming unemployed. Older workers tend to remain unemployed longer while the risk of becoming unemployed decreases with experience.15 4.4.5. Mobility-specific heterogeneity Table 9 describes the probability of z2 ¼ 1 (vs. z2 ¼ 2), conditional on crosssection heterogeneity z1 and other covariates in z0. As stated before, these parameters are more easily interpreted after computing the predicted probability of z2 ¼ 1 by education, cohort and cross-section heterogeneity (Table 10). There are slightly fewer stayers among high-wage earners ðz1 ¼ 1Þ than among lowwage earners: about 50% and 60%, respectively. Moreover, younger cohorts of high-wage earners are more mobile than older ones. The opposite conclusion applies to low-wage earners. Notice that the pseudo R2 of the PROBIT regressions of unobserved heterogeneity variables z1 and z2 on observed covariates z0

15

For all but high-wage workers ðz1 ¼ 1Þ. But, high-wage workers have a very low risk of unemployment.

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Table 6.

Earnings mobility by experience and unobserved heterogeneity z1 ¼ 1

Experience (years)

z2 ¼ 1

z2 ¼ 2

5

0.92 0.07 0.01 0.00 0.00

0.07 0.83 0.09 0.01 0.00

0.01 0.09 0.82 0.09 0.01

0.00 0.01 0.09 0.83 0.07

0.00 0.00 0.01 0.07 0.92

0.87 0.11 0.01 0.01 0.00

0.11 0.74 0.13 0.01 0.01

0.01 0.13 0.72 0.13 0.01

0.01 0.01 0.13 0.74 0.11

0.00 0.01 0.01 0.11 0.87

20

0.91 0.08 0.01 0.00 0.00

0.08 0.82 0.09 0.01 0.00

0.01 0.09 0.80 0.09 0.01

0.00 0.01 0.09 0.82 0.08

0.00 0.00 0.01 0.08 0.91

0.82 0.15 0.02 0.01 0.00

0.15 0.65 0.17 0.02 0.01

0.02 0.17 0.62 0.17 0.02

0.01 0.02 0.17 0.65 0.15

0.00 0.01 0.02 0.15 0.82

35

0.90 0.09 0.01 0.00 0.00

0.09 0.80 0.10 0.01 0.00

0.01 0.10 0.78 0.10 0.01

0.00 0.01 0.10 0.80 0.09

0.00 0.00 0.01 0.09 0.90

0.75 0.19 0.04 0.01 0.01

0.19 0.56 0.20 0.04 0.01

0.04 0.20 0.52 0.20 0.04

0.01 0.04 0.20 0.56 0.19

0.01 0.01 0.04 0.19 0.75

z1 ¼ 2 z2 ¼ 1

z2 ¼ 2

5

0.74 0.20 0.04 0.02 0.00

0.20 0.53 0.21 0.04 0.02

0.04 0.21 0.50 0.21 0.04

0.02 0.04 0.21 0.53 0.20

0.00 0.02 0.04 0.20 0.74

0.41 0.26 0.16 0.10 0.07

0.26 0.27 0.22 0.15 0.10

0.16 0.22 0.24 0.22 0.16

0.10 0.15 0.22 0.27 0.26

0.07 0.10 0.16 0.26 0.41

20

0.81 0.16 0.02 0.01 0.00

0.16 0.63 0.18 0.02 0.01

0.02 0.18 0.60 0.18 0.02

0.01 0.02 0.18 0.63 0.16

0.00 0.01 0.02 0.16 0.81

0.51 0.26 0.12 0.07 0.04

0.26 0.32 0.23 0.12 0.07

0.12 0.23 0.30 0.23 0.12

0.07 0.12 0.23 0.32 0.26

0.04 0.07 0.12 0.26 0.51

35

0.86 0.12 0.01 0.01 0.00

0.12 0.72 0.14 0.01 0.01

0.01 0.14 0.70 0.14 0.01

0.01 0.01 0.14 0.72 0.12

0.00 0.01 0.01 0.12 0.86

0.60 0.25 0.09 0.04 0.02

0.25 0.39 0.23 0.09 0.04

0.09 0.23 0.36 0.23 0.09

0.04 0.09 0.23 0.39 0.25

0.02 0.04 0.09 0.25 0.60

(50% vs. less than 4%; see Tables 3 and 8) indicate that education and cohort do not determine z2 as much as they determine z1. 4.4.6. Summary of the main results We isolate two main types of cross-section earnings distributions. One component has both higher mean and higher variance than the other one. High-wage individuals (about 25% of the population) have a low probability of being

461

Modeling Individual Earnings Trajectories

Table 7.

Employment mobility by experience and unobserved heterogeneity z1 ¼ 1 z2 ¼ 1

Experience (years)

z2 ¼ 2

Empl. (at t+1)

Unempl.

Empl.

Unempl.

5

Empl. (at t) unempl.

1.00 0.73

0.00 0.27

0.97 0.99

0.03 0.01

20

Empl. Unempl.

1.00 0.02

0.00 0.98

0.97 0.87

0.03 0.13

35

Empl. Unempl.

0.99 0.00

0.01 1.00

0.97 0.47

0.03 0.53

z1 ¼ 2 z2 ¼ 1

z2 ¼ 2

Empl.

Unempl.

Empl.

Unempl.

5

Empl. Unempl.

0.89 0.36

0.11 0.64

0.91 0.91

0.09 0.09

20

Empl. Unempl.

0.99 0.19

0.01 0.81

0.93 0.68

0.07 0.32

35

Empl. Unempl.

1.00 0.08

0.00 0.92

0.94 0.33

0.06 0.67

Note: Empl., Employment; Unempl., Umemployment.

Table 8.

Conditional employment–unemployment transition probabilities Empl. to Unempl.

z1 ¼ 1 Experience Intercept

z1 ¼ 2 Experience Intercept

Unempl. to Unempl.

z2 ¼ 1

z2 ¼ 2

z2 ¼ 1

z2 ¼ 2

0.043 (0.0044) 3.82 (0.13)

0.00017 (0.0043) 1.89 (0.11)

0.178 (0.011) 1.50 (0.27)

0.079 (0.54) 2.71 (23.4)

0.084 (0.0039) 0.82 (0.057)

0.0070 (0.0018) 1.34 (0.048)

0.033 (0.0026) 0.20 (0.059)

Note: Empl., Employment; Unempl., Umemployment.

0.060 (0.028) 1.66 (0.070)

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Table 9.

Distribution of cross-section heterogeneity given exogenous controls

Independent Variable

Cross-Section Heterogeneity z1 ¼ 1

Intercept Education Junior high-school Senior high-school Some college College+ Cohort Cohort*Education Junior high-school Senior high-school Some college College+

27 (16.7)

91 (3.4)

38 (18.5) 50 (17.4) 38 (18.8) 35 (19.8)

22 (4.4) 65 (8.2) 71 (40.7) 14 (95.4)

0.014 (0.0085)

0.046 (0.0018)

0.019 (0.0094) 0.025 (0.0089) 0.019 (0.0096) 0.018 (0.10)

0.011 (0.0022) 0.033 (0.0042) 0.036 (0.021) 0.0062 (0.0048)

0.025

0.036

Pseudo–R2

Table 10.

z1 ¼ 2

Conditional probability of being a stayer (z2 ¼ 1)

Cohort

Education None

Junior HS Senior HS

Some College

College+

Overall

0.56 0.53 0.31 0.50

0.73 0.58 0.47 0.55

0.60 0.53 0.45 0.50

0.61 0.54 0.44 0.52

0.60 0.65 0.69 0.67

0.72 0.52 0.65 0.63

n.s. n.s. n.s. n.s.

0.49 0.61 0.71 0.62

Cross-Section Heterogeneity: z1 ¼ 1 o1970 1970–1985 >1985 Overall

n.s. n.s. n.s. n.s.

0.58 0.40 0.33 0.50

Cross-Section Heterogeneity: z1 ¼ 2 o1970 1970–1985 >1985 Overall

0.44 0.60 0.75 0.60

0.50 0.61 0.73 0.61

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Modeling Individual Earnings Trajectories

unemployed, and belonging to the high-wage group is strongly positively correlated with higher education. We also isolate two types of earnings and employment dynamics, one being more stable than the other. Movers (about 45%) face more earnings mobility (both upwards and downwards) and also more employment mobility, leaving both employment and unemployment faster. Education and cohort have little influence on the probability of being a mover. Experience protects from unemployment risk while, at the same time, unemployment lasts longer for older workers. 4.5. Model fit In this subsection, we analyze the ability of the model to fit the data for the choice of the pair (K1, K2) ¼ (3,2). Again, we focus on the years 1990–1992, because the results are very similar for the other periods. 4.5.1. Cross-sections Usual autoregressive specifications of earnings dynamics lack the flexibility of our model because the estimation of the marginal densities interferes with the estimation of the dynamic parameters. The predicted cross-section distributions are thus often unable to precisely fit the data. For instance, the however rich model of Geweke and Keane (2000) underestimates the mode of the marginal distributions. Our copula-based model, which independently specifies cross-section and dynamic parameters and uses a bivariate unobserved heterogeneity distribution, gives unlimited flexibility to describe the marginal distributions of the underlying process. Figure 3 shows that a three-component mixture of normal distributions already achieves a remarkable fit. 4.5.2. Earnings mobility We then analyze the capacity of the model to fit earnings mobility as characterized by the yearly dynamics of earnings quintiles. To this end, we use the estimated model to simulate individual states at dates t+1 and t+2 conditional on the observed state at t ( ¼ 1990). The Appendix details our simulation procedure. We then compare actual and predicted aggregate transition probability matrices. The actual aggregate 1990–1991 transition probability matrix and the predicted matrix are as follows: 0

P01

0:72

0:19

0:06

0:02 0:01

B B 0:21 B ¼B B 0:06 B @ 0:01 0:00

0:54 0:21 0:05 0:01

0:19 0:53 0:20 0:02

0:05 0:18 0:61 0:14

1

C 0:01 C C 0:02 C C; C 0:13 A 0:83

b01 P

0

0:70

B B 0:21 B ¼B B 0:06 B @ 0:02 0:01

0:20 0:07

0:02 0:01

0:54 0:21 0:04 0:01

0:04 0:18 0:62 0:14

0:20 0:53 0:18 0:02

1

C 0:01 C C 0:02 C C. C 0:14 A 0:82

These two matrices are very close indeed, confirming the descriptive power of the Plackett family to fit empirical earnings data. Over two years, unobserved heterogeneity is also found to account properly for most of the non-first-order Markov state dependence. The actual 1990–1992

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Figure 3.

Cross-sectional log-earnings fit for (K 1 ¼ 3)

1.5

density

1

0.5

0 7

8

9 log wage actual

10

11

predicted

transition probability matrix and the predicted one are, respectively: 0

P02

1

0:67

0:22

0:08

0:02

0:01

B B 0:24 B ¼B B 0:06 B @ 0:02 0:01

0:49 0:24 0:04 0:01

0:22 0:50 0:18 0:02

0:04 0:18 0:59 0:17

C 0:01 C C 0:02 C C; C 0:17 A 0:79

b02 P

0

0:63 0:24

B B 0:24 B ¼B B 0:08 B @ 0:04 0:01

0:43 0:24 0:08 0:01

1

0:09 0:03

0:01

0:22 0:43 0:22 0:04

C 0:02 C C 0:04 C C. C 0:16 A 0:77

0:09 0:21 0:50 0:17

The fit is slightly worse than before but considerably better than with the homogeneous model (i.e. when predicting P02 by P201 ; see subsection 3.1). The reader will have noticed that our description of earnings dynamics goes well beyond the standard evaluation of the one-year-ahead prediction quality the R2 of an autoregressive regression measures. Our parsimonious parametric specification succeeds in describing one-year and two-year-ahead entire conditional distributions. 4.5.3. Three-year averages of individual earnings In this paragraph, we test the ability of the model to fit the distribution of individual log-earnings averaged over the three panel dates (we here only use the balanced panel of individuals with non missing earnings at all three dates). This is a way of analyzing the capacity of the model to fit the dynamics of earnings levels instead of earnings ranks. We simulate individual trajectories over years t+1 and t+2 conditional on year t earnings. The actual distribution of three-year log-earnings means and its prediction using simulated data are displayed in Figure 4. The model fit is only slightly worse than for one single cross-section (Figure 3).

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Modeling Individual Earnings Trajectories

Figure 4.

Effect of the mobility on the distribution of three year log-earnings means (K 1 ¼ 3) 2

density

1.5

1

0.5

0 7

8

9 mean log wage

actual predicted, independent

10 predicted

Moreover, to show the importance of taking mobility into account when simulating earnings sequences, we also provide in this figure the distribution of three-year earnings means simulated under the assumption of independence.16 Omitting earnings dynamics has dramatic effects on the distribution of three-year averages. With a single cross-section, the different components of the earnings mixture are not separate enough to create multiple modes. When one follows individuals over time, however, low-wage and high-wage earners accumulate in different intervals with sufficient densities for different modes to appear. 5. Relative earnings mobility In this section, we show how the copula model that we have estimated can be used to analyze relative earnings mobility. Among the many possible mobility indicators proposed in the literature,17 we selected Spearman’s rho for being both natural in the current framework and simple to compute. 5.1. Spearman rho Let X and Y be two random variables with marginal cdf’s FX and FY, then the Spearman rho between X and Y is the correlation coefficient between ranks

16 This corresponds to the use of the independent copula: C?(u, v) ¼ uv, which is a particular case of the Plackett copula with parameter Z approaching zero. See Appendix B. 17 See Fields and Ok (1999) for a comprehensive survey on mobility indices.

Ste´phane Bonhomme and Jean-Marc Robin

466 X

X

U ¼ F (X ) and V ¼ F (Y ), i.e. CovðU; V Þ rS ðX ; Y Þ ¼ rðU; V Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . VarðU ÞVarðV Þ

We also write rs ðX ; Y Þ  rðC Þ, where C(u, v) is the cdf of (U, V ) or the copula of (X, Y ). By Hoeffding’s formula (e.g. Drouet-Mari and Kotz, 2001), one has: Z Z ½Cðu; vÞ uvŠdu dv. rS ðX ; Y Þ ¼ 12EðUV Þ 3 ¼ 12 So Spearman’s rho is proportional to the integral of the difference between the copula C and the copula for independent variables: C?(u, v) ¼ uv. Spearman’s rho thus measures how much more dependence there is in a given process of ranks than in an independent process. Now, let Cð
;
jzÞ denote the copula of ( yt, yt+1) conditional on some variable Z ¼ z. Cð
;
jzÞ is the cdf of the couple of r.v.’s F( yn,t|z), F( ynt+1|z)). To estimate rðCð
;
jzÞÞ from a sample (yn) of i.i.d. observations, we use Hoeffding’s formula:
 rðCð
;
jzÞÞ ¼ 12E F ð ynt jzÞF ðyn;tþ1 jzÞjZ ¼ z 3,

and we replace the conditional expectation by the sample mean of F( yn,t|z) F( ynt+1|z). If z is unobserved heterogeneity, then we draw a value of z for each individual in the sample using the posterior probability that Zn ¼ z conditional on the individual observation data. These posterior probabilities are obtained as a by-product of the estimation of the empirical model by the EM algorithm.18 5.2. Results As even a small set of variables with a small set of attributes quickly generates a large set of interactions, in practice, we compute a Spearman rho for each value of the heterogeneity vector z ¼ (z0, z1, z2). Then, we average Spearman rho values over all individuals with same education, same z1, same z2, or same experience (i.e. a given cohort).

18

Let yn denote the complete set of observations for individual n and let the posterior probability be jzÞPðzÞ pz ðyn Þ ¼ f ðyfn ðy Þ . Then, n

N P

plimN!1 n¼1

F t ð ynt jzÞF tþ1 ð yn;tþ1 jzÞpz ðyn Þ N P

n¼1

¼ pz ðyn Þ

R

pz ðyn Þf ðyn Þdyn .

R

F t ð ynt jzÞF tþ1 ðyn;tþ1 jzÞpz ðyn Þf ðyn Þdyn R pz ðyn Þf ðyn Þdyn
 ¼ E F t ðY nt jzÞF tþ1 ðY n;tþ1 jzÞjZ ¼ z ,

¼

as PðzÞ ¼ E½pz ðYn ފ ¼


 E F t ðY nt jZ ÞF tþ1 ðY n;tþ1 jZÞpz ðYn Þ
 E pz ðYn Þ

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Modeling Individual Earnings Trajectories

Table 11.

Relative mobility measured by Spearman rho 1990

1993

1995

1997

2000

0.903 0.906 0.904 0.895 0.893 0.915 0.875 0.904 0.913

0.915 0.911 0.919 0.915 0.900 0.918 0.886 0.912 0.934

0.914 0.909 0.919 0.913 0.913 0.902 0.880 0.913 0.929

0.890 0.890 0.885 0.898 0.895 0.887 0.850 0.885 0.917

0.861 0.866 0.859 0.907 0.794 0.844 0.876 0.866

0.866 0.863 0.868 0.960 0.717 0.836 0.889 0.879

0.875 0.848 0.887 0.911 0.823 0.852 0.888 0.882

0.847 0.835 0.851 0.886 0.790 0.819 0.866 0.894

0.828 0.851 0.825 0.700 0.885 0.752 0.779 0.837 0.823

0.829 0.860 0.832 0.567 0.950 0.662 0.716 0.823 0.863

0.844 0.839 0.863 0.627 0.894 0.772 0.753 0.847 0.855

0.812 0.822 0.822 0.630 0.859 0.742 0.746 0.828 0.838

Observed Heterogeneity Model Overall Education Education Education Education Education Experience Experience Experience

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

1 2 3 4 5 5 20 35

0.861 0.857 0.854 0.879 0.870 0.867 0.826 0.854 0.889

Unobserved Heterogeneity Model, K1 ¼ 2 Overall z1 ¼ 1 z1 ¼ 2 z2 ¼ 1 z2 ¼ 2 Experience ¼ 5 Experience ¼ 20 Experience ¼ 35

0.798 0.827 0.707 0.919 0.631 0.822 0.804 0.846

Unobserved Heterogeneity Model, K1 ¼ 3 Overall z1 ¼ 1 z1 ¼ 2 z1 ¼ 3 z2 ¼ 1 z2 ¼ 2 Experience ¼ 5 Experience ¼ 20 Experience ¼ 35

0.759 0.803 0.757 0.554 0.900 0.589 0.771 0.774 0.820

Table 11 shows that the dispersion of rS essentially originates from differences in unobserved heterogeneity (z1, z2). Using education instead of (z1, z2), as a marker for individual heterogeneity, generates very little mobility differences. Experience generally has an increasing effect on rS of moderate size when compared to unobserved heterogeneity, but of significant size when compared to education. Besides, Table 9 confirms the analysis of subsection 4.4. Higher z2 is unambiguously associated with being a mover, and thus with more mobility. The effect of z1 is less clear as there are more movers among high-wage individuals ðz1 ¼ 1Þ, but these individuals are less mobile than the movers with ðz1 ¼ 2Þ. The evolution of Spearman rho’s over the 1990s displays a hump-shaped form, which means that there is less earnings mobility in business cycle busts than in booms.

468

Ste´phane Bonhomme and Jean-Marc Robin

6. Conclusion In this paper, we construct a model of earnings dynamics with unobserved heterogeneity which is consistent with the literature on earnings mobility as we model the dynamics of individuals’ positions or ranks within cross-section distributions instead of the dynamics of earnings levels. To make this approach tractable in the presence of unobserved heterogeneity, we use the statistical tool of copula. Since we model unobserved heterogeneity in a discrete way, the EM algorithm becomes a natural device for estimation. We estimate the model using this technique on data drawn from the French LFS. Copula models seem a promising area to shed a new light on earnings dynamics. In the future, we plan to investigate the possibility that the ranks process would not be first-order Markov. An important question we could not address in this paper, both by lack of space and because we need longer panel data to do so, is whether the MA(1) error usually detected in autoregressive models of logearnings levels comes from misspecifying the dynamics of earnings (wrongly assuming linearity in log-earnings levels) or whether it really is in the data. Other applications of the copula approach, such as the use of multivariate extensions of some parametric families (Gaussian, Frank, Plackett y), could be well-suited for this purpose.

References Abowd, J. and D. Card (1989), ‘‘On the covariance structure of earnings and hours changes’’, Econometrica, Vol. 57, pp. 411–445. Akaike, H. (1969), ‘‘Fitting autoregressive models for prediction’’, Annals of the institute of statistical Mathematics, Vol. 21, pp. 243–247. Arcidiacono, A. and J. Jones (2003), ‘‘Finite mixture distributions, sequential likelihood and the EM algorithm’’, Econometrica, Vol. 71, pp. 933–946. Alvarez, J., Browning, M., and M. Ejrnæs (2001), Modeling income processes with lots of heterogeneity, Mimeo. Blumen, I., M. Kogan and P.J. McCarthy (1955), The Industrial Mobility of Labor as a Probability Process, Ithaca: Cornell University Press. Dempster, A., N. Laird and D. Rubin (1977), ‘‘Maximum likelihood from incomplete data via the EM algorithm’’, Journal of Royal Statistical Society Series B, Vol. 39, pp. 1–38. Drouet-Mari, D. and S. Kotz (2001), Correlation and Dependence, London: Imperial College. Fields, G. and E. Ok (1999), ‘‘The Measurement of Income Mobility: an Introduction to the Literature’’, pp. 611–645 in: and J. Silber, editors, Handbook on Income Inequality Measurement, Vol. 69(3), Amsterdam: Kluver. Fre´chet, M. (1935), ‘‘Ge´ne´ralisations du The´ore`me des Probabilite´s Totales’’, Fundamenta Mathematicac, Vol. 25, pp. 379–387.

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Gagliardini, P. and C. Gourie´roux (2002), ‘‘Constrained nonparametric copulas’’, CREST Working Paper. Genest, C. and R.J. MacKay (1986), ‘‘Copules Archime´diennes et Familles de Lois Bidimensionnelles dont les Marges sont Donne´es’’, Canadian Journal of Statistics, Vol. 14, pp. 145–159. Geweke, J. and M. Keane (2000), ‘‘An empirical analysis of income dynamics among men in the PSID: 1968–1989’’, Journal of Econometrics, Vol. 96, pp. 293–356. Gottschalk, P.T. (1997), ‘‘Inequality, income growth and mobility: the basic facts’’, Journal of Economic Perspectives, Vol. 11(2), pp. 21–40. Gottschalk, P.T. and R. Moffit (1994), ‘‘The growth of earnings instability in the US labor market’’, Brookings Papers on Economic Activity, Vol. 2, pp. 217–272. Guillotin, Y. and A. Bigard (1992), ‘‘La Mobilite´ Hie´rarchique des Salaires en France de 1967 a` 1982’’, Economie et Pre´vision, Vol. 102-103, pp. 189–204. Hause, J.C. (1980), ‘‘The fine structure of earnings and the on-the-job training hypothesis’’, Econometrica, Vol. 48(4), pp. 1013–1029. Heckman, J.J. and B. Singer (1984), ‘‘A method for minimizing the impact of distributional assumptions in econometric models for duration data’’, Econometrica, Vol. 52, pp. 271–320. Joe, H. (1997), Multivariate Models and Dependence Concepts, London: Chapman & Hall. Lillard, L. and Y. Weiss (1979), ‘‘Components of variation in panel earnings data: American scientists 1960-70’’, Econometrica, Vol. 47, pp. 437–454. Lillard, L. and R. Willis (1978), ‘‘Dynamic aspects of earnings mobility’’, Econometrica, Vol. 46, pp. 985–1012. MaCurdy, T. (1982), ‘‘The use of time series processes to model structure of earnings in a longitudinal data analysis’’, Journal of Econometrics, Vol. 18, pp. 83–114. Meghir, C. and L. Pistaferri (2004), ‘‘Income variance dynamics and heterogeneity’’, Econometrica, Vol. 72, pp. 1–32. Moffit, R. and P.T. Gottschalk (2002), ‘‘Trends in the transitory variance of earnings in the US’’, The Economic Journal, Vol. 112, pp. C68–C73. Nelsen, R.B. (1999), An introduction to copulas, New York: Springer-Verlag. Plackett, R.L. (1965), ‘‘A class of bivariate distributions’’, Journal of American Statistical Association, Vol. 60, pp. 516–522. Redner, R. and H. Walker (1984), ‘‘Mixture densities, maximum likelihood and the EM algorithm’’, SIAM Review, Vol. 26, pp. 195–239. Shorrocks, A. (1976), ‘‘Income mobility and the markov asumption’’, The Economic Journal, Vol. 86, pp. 566–578. Shorrocks, A. (1978), ‘‘The measurement of mobility’’, Econometrica, Vol. 46, pp. 1013–1024.

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Singer, B. and S. Spilerman (1976), ‘‘Some methodological issues in the analysis of longitudinal surveys’’, Annals of Economic and Social Measurement, Vol. 5, pp. 447–474. Sklar, A. (1959), ‘‘Fonctions de Repartition a` n Dimensions et leurs Marges’’, Public Institute of Statistics of the University of Paris, Vol. 8, pp. 229–231. Wu, C.F.J. (1983), ‘‘On the convergence properties of the EM algorithm’’, The Annals of Statistics, Vol. 11, pp. 95–103. Appendix A: Parametric copulas In this Appendix, we review the copulas mentioned in subsection 2.4. We refer the interested reader to Joe (1997), Nelsen (1999) and Drouet-Mari and Kotz (2001) for further reading. Genest and MacKay (1986) introduce the Archimedean family of copulas. Let f be a convex decreasing function from [0,1] to R+ such that f(1) ¼ 0. Then, the bivariate function: Cðu; vÞ ¼ f 1 ðfðuÞ þ fðvÞÞ

is a copula. Such Archimedean copulas are useful, to the extent that they reduce the study of bivariate distributions to the research of univariate functions, their generator f. Below, we list the Archimedean copulas we use in the paper:  Frank:



1 fðu; tÞ ¼ ln 1

e e

 ; tu t

t40.

 Gumbel:

fðu; tÞ ¼ ð lnðuÞÞt ;

t  1.

 Clayton:

 t 1 fðu; tÞ ¼ u

1;

t40.

ð1

uÞt Þ;

 Joe:

lnð1

fðu; tÞ ¼

t  1.

 Log-copula:

fðu; tÞ ¼



1

lnðuÞ t1 t2

t1 þ1

1;

t ¼ ðt1 ; t2 Þ;

t1 40;

t2 40.

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Modeling Individual Earnings Trajectories

This is the only two-parameter family we present in this paper. We also consider three non-Archimedean copulas: 

Gaussian:  Cðu; v; tÞ ¼ F2 F 1 ðuÞ; F 1 ðvÞ; t ,

where F2 ðx; y; tÞ is the cdf of a couple of normal random variables     0 1 t N ; . 0 t 1 

FGM (Farlie, Gumbel, Morgenstern): Cðu; v; tÞ ¼ uvð1 þ tð1



uÞð1

vÞÞ;

1  t  1.

Plackett: n 1 Cðu; v; tÞ ¼ t 1 1 þ tðu þ vÞ 2

½ð1 þ tðu þ vÞÞ2

o 4tðt þ 1ÞuvŠ1=2 ;

t

1.

Appendix B: The Plackett copula Plackett (1965) generalizes the independence condition for contingency tables. He shows that, if U and V are uniform r.v. on [0,1], then the following equation: PðU  u; V  vÞPðU4u; V 4vÞ ¼ Z þ 1 8ðu; vÞ, PðU  u; V 4vÞPðU4u; V  vÞ

ðB:1Þ

where Z>–1 is a given constant, has one single solution. This solution is a copula, and writes: n o 1 Cðu; v; ZÞ ¼ Z 1 1 þ Zðu þ vÞ ½ð1 þ Zðu þ vÞÞ2 4ZðZ þ 1ÞuvŠ1=2 . 2 From (B.1), Z is a natural mobility index. More precisely, let us define the following ordering "c on copulas, called the concordance ordering (e.g. Joe, 1997): C 1 "c C 2

iff

C 1 ðu; vÞ  C 2 ðu; vÞ;

8ðu; vÞ.

"c is the first-order stochastic dominance ordering. It measures relative mobility: the ranks process governed by copula C1 is said to be more mobile than the ranks process governed by C2 if C1"cC2. Moreover, the concordance ordering possesses a lower and an upper bound (Fre´chet, 1935). The lower bound CL satisfies: CL(u, v) ¼ max(u+v 1,0). The upper bound CU satisfies: CU(u, v) ¼ min(u, v).

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472

The Plackett copula satisfies the following properties (Joe, 1997): (1) (2) (3) (4)

Cð:; :; Z1 Þ"c Cð:; :; Z2 Þ for all Z2>Z1. Cð:; :; ZÞ ! C L when Z- 1. Cð:; :; ZÞ ! C U when Z-N. Cð:; :; ZÞ ! C ? when Z-0, where C?(u, v) ¼ uv is the independent copula.

Therefore, the Plackett copula is mobility decreasing with respect to its parameter, and the Plackett family covers the whole range of bivariate dependence, from immobility (CU) to independence (C?) and perfect mobility (CL). These properties are shared by the Gaussian and the Frank copulas, for instance, which makes these families good candidates for modeling empirical ranks processes. Appendix C: Asymptotic properties of the sequential EM algorithm for copula models We consider finite discrete mixtures of latent distributions for first-order Markov earnings trajectories (yn ¼ (ynt, t ¼ 1, y , T), n ¼ 1, y , N). Let znA{1, y , K} be a latent variable indicating which group individual n belongs to. The density of yn given zn ¼ k is f ðyn ; bk ; tk Þ ¼

T Y t¼1

f ðynt ; bk Þ


T Y1 t¼1


 c F ðynt ; bk Þ; F ðyn;tþ1 ; bk Þ; tk .

We denote as pk the proportion of individuals in Group k in the population. We start by making the following assumptions: Assumptions (1) The parameter space, i.e. the set of possible values of b and s, is compact. (2) True weights are positive, p0k 40, for all k ¼ 1, y , k. (3) The true values of the parameters, b ¼ ðb01 ; . . . ; b0K Þ , t0 ¼ ðt01 ; . . . ; t0K Þ and p0 ¼ ðp01 ; . . . ; p0K Þ, belong to the interior of the parameter space. (4) The marginal densities f(y; b) are continuously differentiable with respect to b and so are the copula densities c(u, v; t) with respect to t. (5) The unknown latent parameters b ¼ (b1, y , bK) of the marginal densities f(ynt; bk) are identified, up to a permutation, from cross-section clusters, i.e. for all k, f ðy; b0k Þ ¼ f ðy; b1k Þ; 8y ) b0k ¼ b1k . (6) The whole set of unknown latent parameters b ¼ (b1, y , bK) and t ¼ (t1, y , tK) and the weights p ¼ (p1, y , pK) are identified, up to a

473

Modeling Individual Earnings Trajectories

permutation, from couples of cross-sections, i.e. K X k¼1

p0k f ðy; b0k ; t0k Þ ¼

K X k¼1

p1k f ðy; b1k ; t1k Þ;

8y ¼ ðy1 ; :::; yT Þ

) b 0 ¼ b1 ; t 0 ¼ t 1 ; p 0 ¼ p1 .

The numerical limit of the algorithm The sequential EM algorithm writes as a set of first-order equations:   N T  P P ðsþ1Þ @ pk yn ; yðsÞ lnf ðy ; b Þ ¼0 nt k @b n¼1 N P

n¼1

t¼1

  TP1 pk yn ; yðsÞ t¼1

pðsþ1Þ ¼ N1 k where



N P

pk yn ; yðsÞ

n¼1

 pk yn ; yðsÞ ¼

@ @t lnc



  h i ðsþ1Þ ðsþ1Þ ðsþ1Þ F ynt ; bk ; F yn;tþ1 ; bk ; tk ¼0

ðsÞ ðsÞ pðsÞ k f ðyn ; bk ; tk Þ . K P ðsÞ ðsÞ ðsÞ pm f ðyn ; bm ; tm Þ

ðC:1Þ

m¼1

ðsÞ ðsÞ We suppose that the sequence yðsÞ ¼ ðbðsÞ k ; tk ; pk ; k ¼ 1; . . . ; KÞ converges nuN N N merically to the sequential estimator y ¼ ðbN k ; tk ; pk ; k ¼ 1; . . . ; KÞ, solution to the following system of equations: ! T N  X 1X @ N N ln f ðynt ; bk Þ ¼ 0, pk yn ; y ðC:2Þ N n¼1 @b t¼1 N 1X N n¼1

pN k ¼

T 1 X  N
 @ N ln c F ynt ; bN pk yn ; y k ; F yn;tþ1 ; bk ; tk @t t¼1



N

N  1X pk yn ; yN . N n¼1

!

¼ 0,

ðC:3Þ ðC:4Þ

Pseudo-true value The observations are i.i.d.. By the Weak Law of Large Numbers, the sample averages in (C.2), (C.3) and ((C.4) therefore converge to their theoretical expectation analogs uniformly in the parameters. The set of parameters to which yN belongs is compact. One can thus extract from (yN) subsequence which converges to some limit yN ¼ (bN, sN, pN), solution to the limiting system of

Ste´phane Bonhomme and Jean-Marc Robin

474

equations: ! T X @ 1 lnf ðynt ; bk Þ ¼ 0, E pk yn ; y @b t¼1

ðC:5Þ

! T 1
 1  X @ 1 1 lnc F ynt ; bk ; F yn;tþ1 ; bk ; tk ¼ 0, E pk yn ; y @t t¼1

ðC:6Þ


  1 . p1 k ¼ E pk yn ; y

ðC:7Þ



1



1

Denoting as y0 the true set of DGP parameters, we now prove the consistency of yN by proving that yN ¼ y0. Root-N consistency and asymptotic normality will follow from standard arguments once consistency has been proven. We shall thus not develop this point further.

Consistency Because we shall not be concerned here by nonparametric identification issues, we take T ¼ 2 for all trajectories. Allowing for greater values of T only makes the estimation more efficient, but does not affect consistency given the identifying assumptions we have made. From assumption (6), we deduce that the true value y0 ¼ (b0, s0, p0) of the parameters is the unique maximizer of the incomplete log-likelihood: E ln

K X k¼1

pk f ðY 1 ; Y 2 ; bk ; tk Þ

!

ðC:8Þ

P subject to the constraint K k¼1 pk ¼ 1. The likelihood being continuously differentiable and the true value y0 belonging to the interior of the parameter space, the first-order conditions of the Lagrangian are necessary for b0, t0, p0 to be the optimum by writing the condition for p0k we obtain: 1

0

B p0 f ðY ; Y ; b0 ; t0 Þ C 1 2 k k C B p0k ¼ EB K k C A @P 0 pk f ðY 1 ; Y 2 ; b0k ; t0k Þ k¼1

¼ Epk ðY 1 ; Y 2 ; b0 ; t0 ; p0 Þ

ðC:9Þ

Modeling Individual Earnings Trajectories

and differentiating (C.8) with respect to tk yields

 @lnf ðY 1 ;Y 2 ; b0k ;t0k Þ E pk ðY 1 ; Y 2 ; b0 ; t0 ; p0 Þ ¼0 @tk

m

 @lnc ½F ðY 1 ; b0k Þ;F ðY 2 ; b0k Þ;t0k Š ¼ 0: E pk ðY 1 ; Y 2 ; b0 ; t0 ; p0 Þ @tk

475

ðC:10Þ

It follows that s0 and p0 satisfy conditions (C.6) and (C.7) if b0 ¼ bN. The first-order condition for bk: ! 0 0 0 0 0 @lnf ðY 1 ; Y 2 ; bk ; tk Þ E pk ðY 1 ; Y 2 ; b ; t ; p Þ ¼0 @bk does not yield a similar equation as condition (C.5) as c½F ðY 1 ; b0k Þ; F ðY 2 ; b0k Þ; t0k Š also depends on b0k : @lnf ðY 1 ; Y 2 ; b0k ; t0k Þ @lnf ðY 1 ; b0k Þ @lnf ðY 2 ; b0k Þ þ . a @bk @bk @bk But, for t ¼ 1, 2: Z  0  @lnf ðyt ; b0k Þ 0 0 0 @lnf ðY t ; bk Þ dy1 dy2 E pk ðY 1 ; Y 2 ; b ; t ; p Þ ¼ p0k f ðy1 ; y2 ; b0k ; t0k Þ @bk @bk Z @lnf ðyt ; b0k Þ ¼ p0k f ðyt ; b0k Þ dyt ¼ 0. @bk Hence, "

2 X @lnf ðY t ; b0k Þ E pk ðY 1 ; Y 2 ; b0 ; t0 ; p0 Þ @bk t¼1

#!

¼ 0.

ðC:11Þ

Moreover, by assumption (5), b0k is the unique solution to (C.11) if p0k a0. We have thus shown that b0, s0, p0 are the unique solution to the limiting Equations (C.5), (C.6) and (C.7). The sequential EM estimator is therefore consistent. Appendix D: Detailed estimation procedure We here consider the problem of estimating the empirical model over each threeyear panel separately. The data consist of N independent observations of employment states en ¼ (ent, t ¼ 1, y , T), earnings yn ¼ (ynt, t ¼ 1, y , T) (with ynt ¼
if ent ¼ 0), time-varying exogenous attributes xn ¼ (xnt, t ¼ 1, y , T) and observed heterogeneity z0n, for n ¼ 1, y , N.

476



Ste´phane Bonhomme and Jean-Marc Robin

The parameters to be estimated are: a ¼ ða1 ; ::: ; aK 1 Þ, the parameters of the probability distribution of latent variable z1: Prfz1 ¼ k1 jz0 g  pk1 ðz0n ; aÞ ¼

expz00n ak1 ; K1 P 0 expz0n a‘

k1 2 f1; ::: ; K 1 g,

‘¼1

with the normalization: a1 ¼ 0.  For every k1 A{1,y,K1}, bk ¼ ðb1jk ; . . . ; bK jk Þ, the parameters of the 1 1 2 1 probability distribution of variable z2, conditional on z1 ¼ k1: Prfz2 ¼ k2 jz1 ¼ k1 ; z0 g  pk2 jk1 ðz0n ; bk1 Þ ¼

expz00n bk2 jk1 K2 P

‘¼1

;

k2 2 f1; ::: ; K 2 g,

expz00n b‘jk1

with the normalization: b1jk1 ¼ 0. mk1 ; ok1 , k1 2 f1; . . . ; K 1 g the parameters of cross-section earnings distributions, !  yt x0t mk1 . F t yt jxt ; z1 ¼ k1 ¼ F pffiffiffiffiffiffiffiffiffiffiffi x0t ok1 

dk1 , k1 2 f1; ::: ; K 1 g the parameters of the unconditional unemployment probability,  Prfet ¼ 0jxt ; z1 ¼ k1 g ¼ F x0t dk1 .



wik1 k2 , i A{0,1}, k1 A{1, y , K1}, k2 A{1, y , K2}, the parameters of the conditional unemployment probability,    Pr etþ1 ¼ 0jet ¼ i; xt ; z1 ¼ k1 ; z2 ¼ k2 ¼ F x0t wik1 k2 .



Zk1 k2 , k1 A{1, y , K1}, k2 A{1, y , K2}, the parameters of the Plackett copula densities:   cðu; vjxt ; z1 ¼ k1 ; z2 ¼ k2 Þ  c u; v; exp x0t Zk1 k2


 2 
   32 4exp x0t Zk1 k2 1 þ exp x0t Zk1 k2 uv ¼ 1 þ ðu þ vÞexp x0t Zk1 k2
 
   1 þ exp x0t Zk1 k2 1 þ ðu þ v 2uvÞexp x0t Zk1 k2 :

Note that we restrict the parameter of the Plackett copula, i.e. expðx0t Zk1 k2 Þ, to be higher than zero, to exclude mean reversion.19

19

On the issue of negative dependence, see Shorrocks (1978).

477

Modeling Individual Earnings Trajectories

Let y gather all parameters in one single vector. If all heterogeneity variables were observed one would maximize the following conditional likelihood (omitting the individual index n): f ðe; yjx; z0 ; z1 ; z2 ; yÞ ¼ Prfe1 ¼ 0jx1 ; z1 g1 

T Y  e f t yt jxt ; z1 t

e1

½1

Prfe1 ¼ 0jx1 ; z1 gŠe1

t¼1



T Y1 t¼1

 1 Pr etþ1 ¼ 0jet ; xt ; z1 ; z2

etþ1


 e  1 Pr etþ1 ¼ 0jet ; xt ; z1 ; z2 tþ1 T Y1
     c F t yt jxt ; z1 ; F tþ1 ytþ1 jxtþ1 ; z1 jxt ; z1 ; z2 et etþ1 . t¼1

The estimation algorithm updates the step-s value y(s) of the parameter vector y by going through the following steps: (1) For all k1 A{1, y, K1}, compute the posterior probability of z1n ¼ k1 given yn, xn, z0n and y(s) as K2   X pzk11 yn ; xn ; z0n ; yðsÞ ¼ pzk11;z;‘2 yn ; xn ; z0n ; yðsÞ , ‘¼1

where  pzk11;z;k22 yn ; xn ; z0n ; yðsÞ

ðsÞ ðsÞ ðsÞ ðsÞ pðsÞ k2 jk1 ðz0n ; bk1 Þpk1 ðz0n ; a Þ f ðen ; yn jxn ; z0n ; z1n ¼ k 1 ; z2n ¼ k 2 ; y Þ ¼ K K P1 P2 ðsÞ ðsÞ ðsÞ ðsÞ p‘jk ðz0n ; bðsÞ k Þpk ðz0n ; a Þ f ðen ; yn jxn ; z0n ; z1n ¼ k; z2n ¼ ‘; y Þ k¼1 ‘¼1

is the posterior probability of z1n ¼ k1 and z2n ¼ k2 given yn, xn, z0n and y(s). (2) Update parameters mk1 , k1 A{1, y, K1}, by regressing ynt on xnt using OLS and weighting each observation by pzk11 ðyn ; xn ; z0n ; yðsÞ Þ. Then, update ok1 , by regressing squared residuals on xnt , again using weighted least squares. (3) Update dk1 , k1 A{1, y, K1}, as in the preceding step: estimate the PROBIT model of whether i1 ¼ 0 or not conditional on x1, weighting each observation by pzk11 ðyn ; xn ; z0n ; yðsÞ Þ. (4) Update the parameters wik1 k2 , i A{0,1}, k1 A{1, y, K1}, k2 A{1, y, K2}, of the conditional unemployment transition probabilities, Prfetþ1 ¼ 0jet ¼ i; xt ; z1 ¼ k1 ; z2 ¼ k2 g ¼ Fðx0 t wik1 k2 Þ, by running two PROBIT estimations weighting each observation by pzk11;z;k22 ðyn ; xn ; z0n ; yðsÞ Þ.

Ste´phane Bonhomme and Jean-Marc Robin

478

(5) Update Zk1 k2 as:

Zkðsþ1Þ 1 k2

8 N h  TP1 P z1 ;z2  ðsþ1Þ ðsþ1Þ > ðsÞ > y jx ; z ¼ k ; m ; o y ; x ; z ; y p ln c F > t t 1 1 n 0n t n k1 k1 > n¼1 k1 ;k2 > > t¼1 > > > > < et a0; ¼ arg max Zk1 k2 > > > > etþ1 a0 > > > >

o > > : y jx ; z ¼ k ; mðsþ1Þ ; oðsþ1Þ ; expðx0 Z Þ : F tþ1

tþ1

tþ1

1

1

k1

k1

t k1 k2

Due to the mathematical simplicity of the Plackett copula, this maximization is easy to perform. (6) Update a by solving the program K1 N P P

aðsþ1Þ ¼ arg max a

n¼1 k¼1

s:t:

K1 P

k¼1

 pzk1 yn ; xn ; z0n ; yðsÞ lnpk ðz0n ; aÞ pk ðz0n ; aÞ ¼ 1

In practice, we did not solve this ML program. To benefit from standard algorithms available in most statistical (as STATA), we instead simulated ðDÞ for each n a set of D zð1Þ 1n ; :::; z1n of the latent group indicator z1n from the multinomial distribution ðsÞ ðsÞ z1 MðPz1 1 ðyn ; xn ; z0n ; y Þ; . . . PK 1 ðyn ; xn ; z0n ; y ÞÞ

and computed the standard ML estimate: aðsþ1Þ ¼ arg max a

K1 N X D X

 X 1 zðdÞ ¼ k lnpk ðz0n ; aÞ. K1 n¼1 d¼1 k¼1

Various experiments showed that D ¼ 1 already gave a good approximation of the exact ML estimator. (7) Update bk1 , for all k1, by solving the program ¼ arg max bkðsþ1Þ 1

K2 N P P

bk1 n¼1 ‘¼1

s:t:

K2 P

‘¼1

 pzk11;z;‘2 yn ; xn ; z0n ; yðsÞ lnp‘jk1 ðz0n ; bk1 Þ p‘jk1 ðz0n ; bk1 Þ ¼ 1;

by using the same simulation method

IV. MACROECONOMETRIC PAPERS

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CHAPTER 19

Amplification of Productivity Shocks: Why Don’t Vacancies Like to Hire the Unemployed?$ E´va Nagypa´l Abstract In this paper, I study a new amplification mechanism in search models that arises when workers can choose to search on the job and when, for endogenous reasons, employers reap higher benefits from contacting employed searchers. The motivation for on-the-job search in the model is job shopping, where workers look for jobs they find appealing and the appeal of a job to the worker is not observed by the firm. In equilibrium, workers arriving from unemployment are more likely to leave a job for a more appealing one. Knowing this, firms prefer to contact alreadyemployed searchers. Employers’ preference for contacting already-employed searchers introduces a new amplification mechanism into search models. Vacancies in this type of model respond more to aggregate shocks than in standard search models: the probability that a newly encountered searcher is employed rises in a boom, thereby making it more attractive for firms to post vacancies. Using simulations of the proposed model, I explore the extent to which this new amplification mechanism helps in explaining the volatility of unemployment and vacancies over the business cycle. The calibration results show that, for standard parameter values, this mechanism can generate eight times more amplification in response to productivity shocks compared to the baseline model. Moreover, the proposed model, unlike the standard search model, predicts that unemployment and vacancies respond with different signs to shocks in the rate of exogenous separation. This brings back these type of shocks as plausible driving forces behind labor-market fluctuations over the business cycle.

$

This paper was previously titled ‘‘Worker Reallocation over the Business Cycle: The Importance of Job-to-Job Transitions, Part 2: Theory.’’ CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75019-7

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

481

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Keywords: search, matching, U.S. labor-market, vacancies, labor-market flows, business cycles JEL classifications: E24, E32, J32, J63

1. Introduction Close to half of all labor market transitions are job-to-job transitions – movement of workers from one employer to another without any intervening unemployment (Fallick and Fleischman, 2004; Nagypa´l, 2005b). Despite their magnitude, such job changes have often been ignored by macroeconomists in the formal modeling of aggregate labor-market dynamics over the business cycle. In this paper, I consider a model of on-the-job search with the novel feature that firms reap higher benefits from contacting already-employed workers, meaning that firms prefer a pool of searchers with more employed workers. While most recruitment professionals are aware of this preference, macroeconomists have not yet considered ways to incorporate such a feature into models of the aggregate labor market. In fact, all of the existing models of on-the-job search implicity feature a preference by firms for contacting the unemployed. I show that this novel feature gives rise to a new amplification mechanism in search models. The model I consider builds on the search and matching framework. In the last two decades, search and matching models have gained wide popularity in the analysis of aggregate labor markets. This is largely due to their ability to explain several labor-market phenomena that the standard, neo-classical growth model cannot tackle, such as the existence of equilibrium unemployment. Several authors have asserted that these models can also quantitatively explain the cyclical variation in key labor-market variables. Recently, however, this view has been challenged, and the search and matching approach has been criticized for its lack of amplification (Shimer, 2005). Shimer argues that earlier works were wrong to declare success, since they did not consider either the magnitude of the exogenous driving forces (Blanchard and Diamond, 1989; Mortensen and Pissarides, 1994; Cole and Rogerson, 1999) or the highly negative correlation between unemployment and vacancies (Merz, 1995; Andolfatto, 1996; Ramey and Watson, 1997; Gomes et al., 2001). He shows that, in response to shocks to the productivity of employment relationships, the standard search model results in elasticities of unemployment and of vacancies that are approximately a 10th of the elasticities observed in the data. In response to shocks to the separation rate, the standard search model results in a positive correlation between unemployment and vacancies, while the data show a strong negative correlation. A commonly adopted simplification in models of search is that there are no job-to-job transitions (this holds, for example, in all the works mentioned

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above), even though this has long been recognized as a serious shortcoming (Tobin, 1972). There are a handful of models in the literature that explicitly model on-the-job search and the resulting job-to-job transitions, and these provide many important insights into the ways in which on-the-job search alters labor-market outcomes. These models, though, either do not study cyclical fluctuations (Burdett and Mortensen, 1998; Burdett et al., 2004), or do not consider the extent of amplification generated by the model (Mortensen, 1994; Pissarides, 1994; Barlevy, 2002). In this paper, I construct a tractable search model with job-to-job transitions that is capable of generating more amplification than the standard search model and that can match both the magnitude and the cyclicality of job-to-job transitions. Existing frameworks with on-the-job search (Mortensen, 1994; Pissarides (1994); Burdett and Mortensen, 1998; Barlevy, 2002) do not help in resolving the amplification puzzle because, in all of them, the expected payoff to employers is higher when contacting unemployed searchers than when contacting employed searchers.1 This exacerbates the lack of amplification, since, during recessions, the extent of on-the-job search declines, the composition of the searching pool shifts toward the unemployed, and the pool of searchers changes in favor of vacancy creation. For example, even though Barlevy (2002) does not report the elasticity of unemployment to productivity shocks in his model, it can be calculated from the numbers that he reports to be hundred times smaller than the elasticity observed in the data. There are two channels through which preference for contacting unemployed searchers arises in existing models of on-the-job search. First, since unemployed searchers face worse alternatives on average, they are more likely to accept a match of a given quality or productivity. Second, employed searchers have better outside options, which means that they might command higher wages than unemployed searchers, depending on the nature of wage setting. In a standard search model, firms prefer higher acceptance rates and lower wages, meaning that they prefer unemployed searchers. I argue that models where employers benefit more from contacting employed searchers are more plausible and provide scope for amplification. In this case,

1

It should be noted, however, that the emphasis of these authors has not been the role of on-the-job search in amplifying productivity shocks. A notable exception is Shimer (2003), who studies amplification with on-the-job search. The mechanism in his model is very different from the one I study in this paper, since he departs in several ways from the standard search model. In his model, a higher expected benefit from contacting unemployed searchers is what helps in resolving the amplification puzzle.It should be noted, however, that the emphasis of these authors has not been the role of onthe-job search in amplifying productivity shocks. A notable exception is Shimer (2003), who studies amplification with on-the-job search. The mechanism in his model is very different from the one I study in this paper, since he departs in several ways from the standard search model. In his model, a higher expected benefit from contacting unemployed searchers is what helps in resolving the amplification puzzle.

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there is a complementarity between vacancies and employed searchers in the matching market. Increased search activity from either of them during a boom encourages search activity from the other party, leading to increased amplification. I also argue, based on the work of Eriksson and Lagerstro¨m (2004), that there is compelling evidence that employers indeed expect to reap a higher payoff from contacting employed searchers. What changes in the model could allow firms to reap higher benefits from contacting an employed searcher? While Nagypa´l (2005a) discusses this issue in a general framework, here I adopt a specific set of assumptions that generates this feature. The key component is that firms do not always prefer to contact searchers with higher acceptance rates, because some matches lead to a negative payoff to the firm. Such a negative payoff at the time of creation of the employment relationship is not included in the standard search model; in that model all costs of creating a vacancy are borne prior to meeting a worker through vacancy creation cost. It is natural to assume, however, that the firm has to expend some additional resources (on training, relocation, and other match-specific investments) at the time the match is formed. Of course, in order for firms to be willing to enter matches that lead to a negative payoff, it is necessary that the firm has less information about the match than the worker. This is naturally the case when the quality of the match is observed only by the worker and directly enters only into the utility function of the worker. Given this asymmetric-information setup, the basic mechanism of the model is simple. Workers can undertake job shopping at a cost both while unemployed and employed, where job shopping simply means searching for a match with a higher idiosyncratic value to the worker. Unemployed searchers are ‘‘desperate:’’ they are willing to accept any idiosyncratic value above some minimum threshold. Employed searchers, on the other hand, are more selective, and accept only matches that have a value above that of their current match. Turnover declines with the idiosyncratic value of the match for two reasons. First, the probability of finding a better match declines. Second, as a consequence, the incentives to search for a better job also decline, leading to lower endogenous search effort. This means that the expected turnover of previously unemployed workers is higher than that of previously employed workers, making them less attractive candidates for firms to hire. Of course, this higher turnover has to be weighed against the higher acceptance rate of unemployed workers. As Nagypa´l (2005a) shows, the turnover effect can outweigh the acceptance-rate effect when there is a possibility that the firm will make negative profits in a match. Amplification is a direct result of this mechanism. Vacancies that reap higher expected payoffs from contacting employed searchers respond more to aggregate shocks than in the standard search model: the probability of contacting an employed searcher increases in a boom, making it more attractive for firms to post vacancies during these good times.

Amplification of Productivity Shocks

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The qualitative and quantitative results of the calibration exercise are promising. They show that on-the-job search and job-to-job transitions vary positively with aggregate productivity shocks when comparing stationary equilibria for different aggregate productivity levels. They also show that the amplification mechanism embedded in the model shows up clearly when considering the response of the unemployment and the vacancy rate to changes in aggregate productivity. In the standard model, as Shimer (2005) has shown, the elasticity of the vacancy–unemployment ratio with respect to labor productivity (which, in the standard model, is equal to market tightness) is below 2 for reasonable parameter values. In my model, the elasticity of the vacancy–unemployment ratio is 12.5 for the same parameter values, due both to the decline in unemployment (elasticity of –3.9) and to the increase in vacancy rate (elasticity of 8.6). Moreover, my results show that the vacancy rate varies negatively and the unemployment rate varies positively with job-destruction shocks. Shimer (2005) argues that destruction shocks induce a positive correlation between vacancies and unemployment rate in the standard model, in stark contrast to the data. In my model, on the other hand, a higher destruction rate discourages vacancy creation, since it shifts the composition of searchers toward the unemployed. Higher destruction shocks thus lead to higher exogenous turnover through separations into unemployment and to lower endogenous turnover via job-to-job transitions, consistent with the data (Nagypa´l, 2005b). This brings destruction shocks back into the picture as a plausible source of business-cycle variation in vacancies and unemployment. 2. Environment The model is set in continuous time with an infinite horizon. There is a unit measure of infinitely lived workers, who are ex-ante identical. Workers can be either employed or unemployed, and the objective of workers is to maximize Z 1 e rt ut dt, ð1Þ t¼0

where ut ¼



wt þ mt b

cðst Þ

cðst Þ if employed;

if unemployed:

ð2Þ

Here, wt is the wage received when employed at time t; mt denotes the attractiveness or appeal to employed workers of their current employment match (the utility they derive from having a job they ‘‘like’’); and st denotes the search effort of the worker at time t. The appeal, m, of a job to the worker (which I also refer to below as match quality) is determined upon meeting a potential employer and is drawn from the distribution F(
), where

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 ! ½0; 1Š is a continuous, twice differentiable, strictly increasing disF : ½m; mŠ tribution function, m 2 R [ f 1g, and m 2 R [ f1g. In addition, workers can choose to engage in search at a flow cost of c(s t), where  c0 þ c^ðst Þ if st 40; cðst Þ ¼ ð3Þ 0 if st ¼ 0; where c^ð
Þ is a strictly increasing, strictly convex, twice continuously differentiable function with c^ð0Þ ¼ 0. This means that there is both a fixed cost and a variable cost of searching, so that, when the incentives become sufficiently low, the worker stops searching altogether. Below, I will assume that the variable cost function takes the particular functional form c^ðsÞ ¼ c1 s1þr . Finally, b denotes the constant utility flow that a worker receives while unemployed (derived from leisure and from unemployment-insurance benefits). There is a large measure of ex-ante identical firms. The objective of the firms is to maximize Z 1 e rt ðpt Kxt Þdt, ð4Þ t¼0

where

pt ¼

8 > <0 > :

p

if the firm is inactive; if the firm is active with a vacant job;

k wt

if the firm is active with a filled job;

and xt is the Dirac delta function, indicating whether a match was created at time t. This means that, as in the standard search model, any firm can enter the market and become active by posting a vacancy at flow cost k. If a firm posts a vacancy then it participates in the matching market for creating new matches. When a firm contacts a worker and the worker–firm pair decides to create a match, the firm needs to pay a one-time match-specific start-up cost of K, in addition to having paid the vacancy creation costs. Subsequently, the firm receives a flow profit of (p wt) until the match dissolves, where p is the output of a match, assumed to be the same for all matches. It is the existence of the fixed cost of vacancy creation that ensures that some matches with sufficiently high turnover will lead to a negative expected payoff to the firm. If the firm could distinguish these matches from matches with longer expected duration, it would choose not to form them.2

2

It is important to note that the optimal allocation in this environment would command that the fixed cost be paid by the worker, the informed party, and not the firm. This is ruled out due to borrowing constraints or adverse selection problems.It is important to note that the optimal allocation in this environment would command that the fixed cost be paid by the worker, the informed party, and not the firm. This is ruled out due to borrowing constraints or adverse selection problems.

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Employment matches dissolve for exogenous reasons at rate d and for endogenous reasons when the worker decides to form a new employment relationship as a consequence of on-the-job search. There is a single matching market with a meeting function determining the number of meetings, mt, as a function of the total amount of search effort of workers, st, and the number of vacancies posted, vt mt ¼ mðst ; vt Þ,

ð5Þ

such that ms ðs; vÞ40, mð0; vÞ ¼ 0 for any v, and mvs ðs; vÞ40, mðs; 0Þ ¼ 0 for any s. I assume that m(s,v) has constant returns to scale, so that the meeting rate per unit of search effort for workers can be written as   mðst ; vt Þ vt lt ¼ lðyt Þ ¼ ¼ m 1; ¼ mð1; yt Þ, st st

ð6Þ

where yt ¼ vt =st is the appropriately defined market tightness in the model at time t. Similarly, the meeting rate for firms can be written as

Zt ¼ Zðyt Þ ¼

mðst ; vt Þ mðst ; vt Þ=st lðyt Þ ¼ ¼ . vt =st vt yt

ð7Þ

The timing of match formation is as follows. If a worker and a firm meet, the worker observes the appeal of the potential match and can decide whether or not to form the match. The firm does not observe neither the appeal of the match to the worker nor whether the worker was previously unemployed or employed. As for wages, they are assumed to be set as a fixed fraction c 2 ð0; 1Þ of output p; i.e., the worker and the firm split the output of the match in fixed proportions. This is, of course, a stark assumption regarding wage setting (also adopted for its simplicity in other works with on-the-job search, such as Shimer, 2001; Burdett et al., 2004). The reason for choosing it is twofold. First, we know from Shimer (2005) that, in the standard model, the elasticity of wages to productivity shocks is very close to unity, while the elasticity of wages to separation-rate shocks is close to zero. Thus, adopting this simple wage setting gives quantitatively similar results regarding wages to those given by the standard model. Any amplification in the model considered will thus come from the proposed mechanism and not from the Wage-setting mechanism. Second, there is little agreement in the literature about how to resolve the problem of wage setting with on-the-job search (for related work, see Postel-Vinay and Robin, 2002; Shimer, 2005), let alone with asymmetric information. The question of

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how to address wage setting in this environment in a way that has more microfoundation is an interesting topic for future study.3 One thing that the current wage-setting mechanism does not allow for is wage differences between previously employed and unemployed workers. In a bargaining setting, such differences could arise through two channels. First, if the outside option of employed searchers were treated as their previous employment match and no subsequent renegotiation were allowed, then employed searchers would command a higher wage. Second, the expected match quality of employed searchers is higher than that of unemployed searchers. This means that in a surplus-sharing framework they would be willing to accept a lower wage, since they would receive a higher fraction of their payoff from a good match quality. These two forces would affect the wages of employed and unemployed searchers in different directions, leaving open the possibility that the firm would still prefer to contact employed searchers.

3. Equilibrium 3.1. Definition of stationary equilibrium For the sake of simplicity, and to keep the analysis tractable, I consider the stationary equilibrium of the above model. I let the value of unemployment be U, the value of a worker employed in a match of quality m be W(m), the value for the firm of a vacancy be V, and the value of an employment match of tenure t be Jt.

3

In a previous version of the paper, I assumed that wages were determined upon the formation of the match by splitting the expected surplus such that the worker received b fraction of the surplus, where the outside option of the worker was assumed to be unemployment, i.e., there was no recall. Wages were assumed to be subsequently renegotiated only if otherwise the participation constraint of the parties would have been violated conditional on the worker not being able to credibly communicate the existence of outside offers. It was possible to show in that environment that wages would never be renegotiated after the match was formed, resulting in a constant wage in steady state. This alternative wage setting mechanism generated very similar results to what is reported below. Since there is no axiomatic foundation for surplus sharing with on-the-job search (see Shimer (2005)), I opted for the simpler Wage-setting mechanism described in the text.In a previous version of the paper, I assumed that wages were determined upon the formation of the match by splitting the expected surplus such that the worker received b fraction of the surplus, where the outside option of the worker was assumed to be unemployment, i.e., there was no recall. Wages were assumed to be subsequently renegotiated only if otherwise the participation constraint of the parties would have been violated conditional on the worker not being able to credibly communicate the existence of outside offers. It was possible to show in that environment that wages would never be renegotiated after the match was formed, resulting in a constant wage in steady state. This alternative Wage-setting mechanism generated very similar results to what is reported below. Since there is no axiomatic foundation for surplus sharing with on-the-job search (see Shimer, 2005), I opted for the simpler Wage-setting mechanism described in the text.

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Definition 1. A recursive stationary search equilibrium is unemployment rate u, vacancy rate v, asset values {U,V,W(m),Jt}, wage w, unemployed searchers’ search effort su, employed searchers’ search effort function s(m), and a distribution of employed workers G(m), such that 

    

U and W(
) are the value of unemployment and of working for workers making optimal searching and matching decisions, given u,v, w, and G(
). Su and S(
) are the corresponding optimal search efforts. V and Jt are the value of a vacancy and of a filled job of tenure t for firms making optimal vacancy creation decisions, given u,v, w, and G(
). Agents update their beliefs rationally. There is free entry of vacancies. Wages are set as a fraction c of output. The distribution G(
), the unemployment rate u, and the vacancy rate v are consistent with the decisions of the agents in the economy.

3.2. Worker side The Bellman equation characterizing the value of being a worker with quality m is ( Z m rW ðmÞ ¼ max m þ w cðsÞ þ lðyÞs max½W ðm0 Þ W ðmÞ; 0ŠdF ðm0 Þ s0

þdðU

W ðmÞÞ

m

)

ð8Þ

The flow payoff from working is the utility derived from being in a match of quality m and from the wage w. An employed worker needs to choose her search effort at cost c(s). If she encounters a new firm, she needs to decide whether to form the new match, given its quality m0 drawn from the distribution F. Moreover, the worker suffers a loss of asset value due to exogenous separation at rate d. The Bellman equation characterizing the value of being an unemployed worker is ( ) Z m 0 0 rU ¼ max b cðsÞ þ lðyÞs max½W ðm Þ U; 0ŠdF ðm Þ . ð9Þ s0

m

An unemployed worker also needs to choose her search effort at cost c(s). If she encounters a firm, she needs to decide whether to form the new match given its quality m0 drawn from the distribution F. Equation (8) defines a contraction, and therefore the Contraction Mapping Theorem implies that W(m) is continuously increasing in m and, given the assumptions on F(
) and c(
), differentiable except at the points where the search decision changes discontinuously. This in turn implies that, as intuition would

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dictate, acceptance decisions have the reservation property, with the quality of the current match being the reservation match quality. Taking derivatives with respect to m on both sides of the worker’s asset equation and rearranging gives the following for the derivative of the function W, where it exists: 1 W 0 ðmÞ ¼ . ð10Þ r þ d þ lðyÞsðmÞF ðmÞ I next turn to studying the worker’s search decision. Given the structure of the search cost and using the reservation property of acceptance decisions, the worker’s decision problem can be rewritten as follows:   rW ðmÞ ¼ max m þ w þ dðU W ðmÞÞ; max m þ m cðsÞ s40  Z m þlðyÞs ½W ðm0 Þ W ðmÞ dF ðm0 Þ þ dðU W ðmÞÞ ð11Þ m

where the search decision has been broken down into two steps: a decision of how much to search if searching and a decision whether to search at all. I assume that, if the worker is indifferent between searching and not searching, then she chooses to search. The first-order condition characterizing the first of these maximization problems is given by Z

m

c0 ðsðmÞÞ ¼ lðyÞ

m

¼ lðyÞ

Z

½W ðm0 Þ

m

W ðmފ dF ðm0 Þ

W 0 ðm0 ÞF ðm0 Þ dm0 ;

ð12Þ

m

where the second equality follows from integration by parts, and F ¼ 1 F is the survival function of the distribution F. Clearly, the right-hand side of Equation (12) is declining in m. Given the strict convexity of c, then search effort is strictly declining in m. Turning to the second maximization problem: since the payoff from search is declining with m, the optimal policy regarding whether to search has the reservation property. This means that there exists a ms above which the worker will choose not to search at all and below which she will choose to search. At ms, the condition of optimality states that Z m ½W ðm0 Þ W ðmފdF ðm0 Þ cðsðms ÞÞ ¼ lðyÞsðms Þ ms m

¼ lðyÞsðms Þ

Z

ms

W 0 ðm0 ÞF ðm0 Þdm0 ,

ð13Þ

491

Amplification of Productivity Shocks

where again the second equality follows from integration by parts. From these two optimality conditions, given the properties of c(
), the following proposition follows. (This, together with all other propositions, is proven in the appendix.) Proposition 1. There exists a ms such that, for all m4oms , sðmÞ ¼ 0. At ms, the search effort, s(ms), of the worker is a constant s40, which is determined by the parameters characterizing the cost of search function. The optimal search effort, s(m), of the worker is continuous and strictly declining in m for all moms . In addition, ms and s(m) are increasing in l(y) for all moms . Notice that the discontinuous jump in the search effort introduces a kink in the function W(m) at ms, since there is a positive difference between its derivative from the left ð1=ðr þ d þ lðyÞsðms ÞF ðms ÞÞÞ and its derivative from the right ð1=ðr þ dÞÞ. We can characterize the optimal search decision further by using the particular functional form for the variable search cost function introduced above. With this functional form, Equation (12), evaluated at m ¼ ms , and Equation (13) imply that  1=1þr c0 . ð14Þ sðms Þ ¼ s ¼ c1 r This, in turn, together with Equation (12), evaluated at m ¼ ms , implies that 

c0 c1 ð1 þ rÞ c1 r

r=1þr

¼ lðyÞ

Z

m

W 0 ðm0 ÞF ðm0 Þdm0

ms

¼

lðyÞ rþd

Z

m

F ðm0 Þdm0 ,

ð15Þ

ms

where the derivative of W has been substituted from Equation (10). This gives an equilibrium condition that determines ms as a function of l(y) and of exogenous parameters. Substituting Equation (10) into the optimality condition for search for moms, and using the functional form for the search function gives r

Z

F ðm0 Þ dm0 0 0 m r þ d þ lðyÞsðm ÞF ðm Þ Z lðyÞ m  0 F ðm Þ dm0 . þ rþd m

ð1 þ rÞc1 sðmÞ ¼ lðyÞ

ms

ð16Þ

Taking derivatives on both sides with respect to m gives s0 ðmÞ ¼

lðyÞF ðmÞsðmÞ1 r . ð1 þ rÞrc1 ðr þ d þ lðyÞsðmÞF ðmÞÞ

ð17Þ

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This differential equation together with the boundary condition in (14) fully characterizes the search decision of workers as a function of the quality of their match, and can be solved numerically for a given value of l(y). Finally, given that W(m) is increasing as argued above, an unemployed worker will clearly adopt a reservation match quality policy when searching for a job. Hence, ( ) Z m 0 0 rU ¼ max b cðsÞ þ lðyÞs ½W ðm Þ UŠdF ðm Þ , ð18Þ s0

mm

where mm is an unemployed worker’s reservation match quality implicitly defined by W ðmm Þ ¼ U.

ð19Þ

Comparing the asset equation of a worker at match quality mm and that of an unemployed worker, it follows that su ¼ sðmm Þ,

ð20Þ

and mm ¼ b

w¼b

ð21Þ

cp.

These results imply that an unemployed searcher takes any job where the total payoff, (w þ m), compensates for the foregone flow payoff, b, from unemployment. Moreover, a worker in a marginal match searches with the same intensity as an unemployed searcher. These results follow from the fact that unemployed and employed searchers have access to the same search technology, so that there is no option value of search lost or gained when switching employment status (unlike in Burdett, 1978). 3.3. Firm side The value of being a firm with a match of tenure t is Z m Jt ¼ JðmÞdH t ðmÞ,

ð22Þ

m

where Ht(m) is the distribution of match quality across matches of tenure t and J(m) the value of having a match with a worker of quality m. Recall that m is not observable to the firm, which is why expectations need to be taken with respect to the distribution of match quality. Also, given that the firm does not take any actions after the match is formed, it is sufficient to focus on J0. The Bellman equation characterizing J(m) is rJðmÞ ¼ p

w þ lðyÞsðmÞF ðmÞðV

JðmÞÞ þ dðV

JðmÞÞ.

ð23Þ

The flow payoff of a match to the firm is (p w). In addition, the firm needs to take into account that the match might end for exogenous reasons at rate d and

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for endogenous reasons if the worker decides to move to another job. The latter happens at rate lðyÞsðmÞF ðmÞ. Since endogenous turnover is decreasing with m (and becomes zero once m4ms ), the value of a match to the firm increases in m. Given that free entry drives the value of a vacancy to zero, (i.e., V ¼ 0), and given the search policy of the worker, we can write the above as 8 p w > if m  ms < r þ d þ lðyÞsðmÞF ðmÞ . ð24Þ JðmÞ ¼ > :p w if m4ms rþd The Bellman equation characterizing the value of a vacancy can be expressed as " Z m

rV ¼

k þ ZðyÞ g

ðJðmÞ

K

V ÞAe ðmÞdF ðmÞ

m

þð1

gÞ

Z

m

ðJðmÞ

K

#

V ÞAu ðmÞdF ðmÞ ,

m

ð25Þ

where g is the probability that the contacted worker is employed, Ae(m) the probability that an employed searcher accepts a match of type m, and Au(m) the probability that an unemployed searcher accepts a match of type m. Given free entry and the fact that unemployed workers accept all matches above the threshold mm, this can be rewritten as Z m k ¼g ðJðmÞ K ÞAe ðmÞdF ðmÞ ZðyÞ m Z m þ ð1 gÞ ðJðmÞ K ÞdF ðmÞ. ð26Þ mm

3.4. Equilibrium distribution of workers In order to determine Ae(m) and g, I turn to the derivation of G(m), which is the stationary measure of employed workers below match quality m, and u, which is the stationary unemployment rate. Clearly, given the acceptance policy of the  with GðmÞ  ¼ 1 u. searchers the support of G is ½mm ; mŠ, The stationary measure of unemployment can be derived from equating the flow into and out of unemployment: ulðyÞsðmm ÞF ðmm Þ ¼ dð1

uÞ.

ð27Þ

The flow out of unemployment is the term on the left, which takes into account the search intensity decision, su ¼ sðmm Þ, of the unemployed and the fact that unemployed workers accept all matches above mm. The flow into unemployment

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is the term on the right, which is simply the result of exogenous separations at rate d. Therefore, u¼

d . d þ lðyÞsðmm ÞF ðmm Þ

ð28Þ

To determine the distribution G(m), one can equate the flow into and out of G(m) (just as in the Burdett and Mortensen (1998) model). The flow into the pool of employed workers with match quality m or lower is ulðyÞsðmm ÞðF ðmm Þ

F ðmÞÞ,

ð29Þ

while flow out of the pool of employed workers with match quality m or lower is dGðmÞ þ lðyÞF ðmÞ

Z

minðm;ms Þ

mm

sðm0 ÞdGðm0 Þ.

ð30Þ

The inflow clearly consists of those unemployed workers (searching at intensity su ¼ sðmm Þ), who find a match above mm but below m. The outflow consists of workers that separate exogenously and workers that find a match that is better than m, where one has to take into account that only workers below match quality ms are searching and workers with different match quality search with different intensity. Equating these two flows when mrms gives ulðyÞsðmm ÞðF ðmm Þ F ðmÞÞ F ðmÞ Z m dGðmÞ sðm0 ÞdGðm0 Þ ¼ þ lðyÞ F ðmÞ mm

ð31Þ

Differentiating both sides with respect to m and rearranging gives G 0 ðuÞ ¼

ulðyÞsðmm ÞF ðmm Þ dGðmÞ f ðmÞ d þ lðyÞsðmÞF ðmÞ F ðmÞ

ð32Þ

For m>ms, the same steps give G 0 ðuÞ ¼

ulðyÞsðmm ÞF ðmm Þ d

dGðmÞ f ðmÞ F ðmÞ

ð33Þ

These differential equations together with the boundary condition Gðmm Þ ¼ 0 fully characterize the distribution of workers for a given value of l(y) and s(m).

Amplification of Productivity Shocks

495

Given the distribution G, the firm’s initial belief that a match of quality m is accepted by an employed searcher can be expressed as 8 1 if m4ms ; > > R > < m sðm0 Þ dGðm0 Þ m ð34Þ Ae ðmÞ ¼ R mms if mm  m  ms ; 0 0 > > mm sðm Þ dGðm Þ > : 0 if m4mm ;

Also, the fraction of employed searchers in the searching population is R ms 0 0 mm sðm Þ dGðm Þ R ms g¼ . usðmm Þ þ m sðm0 Þ dGðm0 Þ

ð35Þ

m

Using these results and substituting in the value of J(m) from Equation (24) and using the wage setting condition, w ¼ cp, the free entry condition can be rewritten as  Z ms  k pð1 cÞ ¼ K ZðyÞ r þ d þ lðyÞsðmÞF ðmÞ mm   pð1 cÞ K , ð36Þ AðmÞ dF ðmÞ þ F ðms Þ rþd where AðmÞ ¼

usðmm Þ þ

R minðm;ms Þ mm

usðmm Þ þ

R ms

mm

sðm0 Þ dGðm0 Þ

sðm0 Þ dGðm0 Þ

ð37Þ

3.5. Characterization of equilibrium To understand better the mechanism of the model, it is useful to show two more results. One simply follows from the positive selection of employed workers into different match qualities and formally shows that the expected turnover of unemployed searchers is higher than that of employed searchers. This is one of the crucial elements necessary to generate higher profits to the firm from contacting an employed searcher. Proposition 2. The probability that a previously unemployed worker with tenure t leaves a job is always higher in a stationary equilibrium than the same probability for a previously employed worker with the same tenure. This proposition means that increasing the fraction of employed workers has two effects: it decreases turnover conditional on match formation, but it decreases the probability that a match is formed, since employed searchers are less likely to form a newly contacted potential employment match. The first has a positive effect on vacancy creation, while the second has a negative effect. The

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crucial question is which of these two effects dominate, that is under what conditions will the firm reap higher benefits from contacting an employed searcher than from contacting an unemployed searcher. Proposition 3. Let Pu be the expected profit of a firm from contacting an unemployed searcher and Pe be the expected profit of a firm from contacting an em^ ployed searcher. There exists a K^ such that for K  K, PuZPe, while for K4K, PuoPe. This proposition shows that whether the firm prefers to contact an employed or an unemployed worker depends crucially on the size of the fixed cost K that the firm has to expend in order to create the employment relationship. The larger this cost is, the worse turnover will be for the firm, and the less it will benefit from the high acceptance rate of unemployed searchers. 4. Calibration In this section, I calibrate the model and look at what happens when I compare stationary equilibria with different values of the aggregate productivity parameter, p, and the exogenous job destruction rate, d. In the calibration exercise, I follow the literature and use a Cobb–Douglas matching function mðst ; vt Þ ¼ m0 sat vt1 a . 1 a

ð38Þ 1 a

and ZðyÞ ¼ m0 y , so that the expected time for a firm to Then lðyÞ ¼ m0 y make a contact with a worker is 1=ZðyÞ ¼ 1=m0 ya ¼ 1=m0 ðlðyÞ=m0 Þa=ð1 aÞ ¼ 1=ða 1Þ m0 lðyÞa=ð1 aÞ . For the numerical implementation of the model, it is useful to note that two parameters, c1 and m0, can be eliminated from the equilibrium conditions. In other words, two normalizations are possible. This can be achieved by letting 1=ð1þrÞ 1=ð1 rÞ 1=ð1 aÞ 1=ð1 rÞa=ð1 aÞ ^ s^ðmÞ ¼ sðmÞc1 , lðyÞ ¼ lðyÞc1 , and k^ ¼ km0 c1 . These normalizations are possible, because the arrival rate of contacts for workers is determined by both l(y) and s(m). An economy with high l(y) and low s(m) is not distinguishable from one with low l(y) and high s(m) as long as k is adjusted appropriately, since that is the parameter that determines the relative costliness of posting a vacancy compared to the parameter, c1, determining the marginal cost of search for the worker. Also, the model cannot identify whether few matches are created because the cost of vacancies is high or because the matching function has a low scale parameter, so a second normalization of the cost of ^ vacancy posting is possible, which gives the expression for k. For the choice of the calibrated parameters, I follow Shimer (2005) as closely as possible, to facilitate direct comparison of my results with his. In particular, the following parameters are set to be the same as his. The aggregate productivity can be normalized without loss of generality to be p ¼ 1. The model is set to generate monthly series so r is chosen to be 0.4%, giving an annual discount

Amplification of Productivity Shocks

497

rate of 4.8%, while the exogenous job destruction rate is set to 3.33%. The value of leisure or of unemployment insurance, b, is set to 0.4 at 40% of the match output, while the elasticity of the matching function with respect to search effort a is set to 0.72. Unlike in Shimer (2005), the cost of posting a vacancy is set at 5% of monthly output. This gives a vacancy–unemployment ratio of around 0.5, reported as the empirical value in Faberman (2005). Somewhat surprisingly, the model-generated data are not very sensitive to this cost of vacancy posting, unlike in the standard search model. In my model, most of the costs of creating an employment relationship are the cost borne once the match is formed (K). Vacancy-creation cost constitute a small part of creating an employment match, and hence matter little for the simulation results. I next turn to the calibration of the new parameters in this model. For the distribution of match qualities, I use a normal distribution with mean zero and standard deviation s. There is no good empirical counterpart that could guide this choice. In choosing s, I aim to keep the relative payoff that a worker gets in equilibrium from the appeal of a job to be low compared to the payoff from the wage w. This is important, since it would be unappealing to study a model, where most payoffs to workers come from some unobserved amenity of jobs. My choice of s ¼ 0:2 implies that, in terms of the total payoff from production, the probability of drawing a match quality that is half as large as output is below 1%. Of course, due to job-to-job transitions, the endogenous distribution of match qualities first-order stochastically dominates the distribution of the initial draw of match qualities, so the question of how important match quality is compared to wages is determined endogenously. The parameters of the cost of search function are set to generate job-to-job transitions close to what is observed in the data (2.73% on a monthly basis, as reported by Nagypa´l, 2005b). To achieve this target the parameter r is set equal to 0.5 and the fixed cost of search for a worker is set close to zero at 0.1% of output, though even such a small fixed cost leads to no more search once the 86th percentile of the quality distribution is reached. (The extent of job-to-job transitions is much more sensitive to r than to c0, as long as c0 is small enough.) A crucial element in the calibration is the fraction of output, c, that is paid out in wages and the fixed cost, K, of creating a match. Based on measures of markup, I set c ¼ 0:9. This is significantly lower than the wage that would result in a standard search model, but this model includes additional costs that the firm has to bear after the match is formed. Given c, I choose K to match an unemployment rate of 6%. The resulting K is 1.505, roughly one and a half months’ output of a match. There is no clear empirical counterpart to this number, although the calibrated value does seem reasonable. It is clear from Proposition 4 that a relatively high value is necessary for this model to generate amplification. As the model is very sensitive to these two parameters, I will report sensitivity analysis to these parameters below. For the calibrated parameter values, the equilibrium values of interest are reported in Table 1. As can be seen, the unemployment rate is exactly at its

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Table 1.

Variables of interest generated by the model for the baseline calibration

Unemployment rate Vacancy rate Job-finding rate of unemployed Job-to-job transition rate Total separation rate Fraction of employed workers searching Prob. of contacting employed searcher Lowest accepted match quality Search threshold match quality Average match quality

6.00% 2.96% 51.7% 2.68% 5.98% 65.8% 60.8% 0:5 ðF ¼ 99:38%Þ 0:222 ðF ¼ 13:35%Þ 0.166

calibrated value, while the vacancy rate and the job-to-job transition rate are very close. The average match quality is 16.6% of output, which I find a plausible value. Even though there are many more employed searchers than unemployed searchers, due to the higher search effort of the unemployed, a firm has nearly a 40% chance of contacting an unemployed searcher. It would be interesting to know how this number relates to relevant empirical estimates. Also, we can see that unemployed workers in the model are ‘‘desperate:’’ they accept the vast majority of jobs offered to them. They then keep searching on the job until a match quality in the upper 13th percentile of the distribution is found. In Figure 1, I plot the search effort chosen by workers with different match qualities, the distribution of initial match quality for unemployed searchers, and the endogenous distribution of employed workers across match qualities in equilibrium. Due to job-to-job transitions, the second distribution first-order stochastically dominates the first, just as in Burdett and Mortensen (1998). With endogenous search intensity, for a given level of job-to-job transitions, there is an even larger shift in the distribution than in the Burdett and Mortensen (1998) model. This is because it is exactly workers in low-quality matches who choose a high search effort, and hence are most likely to make a job-to-job transition. 4.1. Comparative statics – aggregate productivity Next, I allow aggregate productivity to vary. As stated earlier, I rely on comparisons of stationary equilibria to assess the response of the model to aggregate shocks. In the standard search model, such a comparative static exercise invariably gives results that are very close to the dynamic response of the full stochastic model. In that model transition dynamics are very fast, due to the forward-looking and instantaneous adjustment of market tightness and the calibrated high job-finding rate. Due to the presence of on-the-job search, the full stochastic version of my model is much more complex. In particular, the state space includes the complete distribution of match qualities across employed workers and the stochastic dynamic equilibrium is thus more difficult to

499

Amplification of Productivity Shocks

Figure 1. Search effort as a function of match quality, the distribution of initial match quality for unemployed searchers, and the endogenous distribution of employed workers across match qualities in equilibrium in the baseline calibration 1.2 1

search effort

0.8

equilibrium distribution of employed workers

0.6

density

search effort

0.4 0.2 0 -0.50 -0.40 -0.30 -0.20 -0.10

distribution of initial match quality

0.00 0.10 0.20 0.30 0.40

0.50 0.60 0.70

match quality

characterize. Intuition would suggest, though, that the dynamics of the model become more gradual. With on-the-job search, the market tightness does not adjust instantaneously to its new long-run value. While the job-finding rate of unemployed workers is as high as in the standard model, the job-to-job transition rate of employed workers is much lower, and the adjustment toward the new steady state could therefore be much more prolonged. In Table 2, I report how the variables of interest change across stationary equilibria in response to a 1% increase in aggregate productivity. I also report the implied elasticities and compare these to the elasticities that Shimer (2005) finds in the standard model and in the data. The amplification mechanism of the model is clearly evident in these results. In the standard model, the vacancy–unemployment ratio (which is, in that model, equal to market tightness) has an elasticity with respect to labor productivity below 2, for reasonable parameter values. In my model, the elasticity of the vacancy–unemployment ratio is 12.54, which is due both to the decline in unemployment (elasticity of 3.89) and to the increase in the vacancy rate (elasticity of 8.65). These elasticities are short of those observed in the data reported in Table 2, but are much closer than the ones generated by the standard model. To demonstrate more clearly why my model is generating more amplification, I plot in Figure 2 the payoff to a firm from creating matches of different quality, and the probability that these different matches are accepted by unemployed and employed searchers. Clearly, there is a strong complementarity

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Table 2. Response of the variables of interest across stationary equilibria to a 1% increase in productivity, their implied elasticity, and the corresponding elasticities in the standard model and the data as calculated from Shimer’s results as a ratio of standard deviations of the log of the variables taking into account their reported correlations 1% Increase in Production

Unemployment rate Vacancy rate Job-finding rate of unemployed Job-to-job transition rate Total separation rate Fraction of employed workers searching Probability contacting employed searcher Lowest accepted match quality Search threshold match quality Average match quality

Implied Elasticity

Standard Elasticity of Shimer

Data Elasticity of Shimer

Baseline Equilibrium

New Equilibrium

6.00% 2.96% 51.7%

5.77% 3.21% 53.9%

3.89 8.65 4.3

0.45 1.35 0.5

9.5 10.1 5.9

2.68%

2.73%

1.98

—

—

5.98% 65.8%

6.03% 65.9%

0.89 0.15

— —

— —

60.8%

61.5%

1.15

—

—

0.5

0.51

1.8

—

—

0.222

0.225

1.11

—

—

0.166

0.168

1.4

—

—

Figure 2. Firm payoff from forming matches of different quality and the probability that different quality matches are accepted by unemployed and employed searchers 1.5

1 0.9

probability

0.7

1 probability of acceptance by unemployed searcher

0.5

0.6 0.5 0.4

0

probability of acceptance by employed searcher

-0.5

0.3 0.2

expected payoff to firm from creating match

0.1 0 -0.50

-1 -1.5

-0.30

-0.10

0.10 match quality

0.30

0.50

payoff

0.8

501

Amplification of Productivity Shocks

Figure 3. The sensitivity of the calibrated value of the one-time match-creation cost and of the elasticity of unemployment with respect to productivity changes to the choice of the wage/sharing rule 2 fixed cost of creating an employment match

1 0 0.90 -1

0.91

0.92

0.93

0.94

0.95 wage

0.96

0.97

0.98

0.99

-2 -3 -4

elasticity of unemployment to productivity

-5

between vacancies and employed searchers, since it is exactly the matches that generate negative payoffs to the firm that employed searchers are likely to reject, while unemployed searchers accept jobs indiscriminately. This gives firms a preference for contacting an employed worker. Since the probability of contacting an employed worker rises with a drop in unemployment, firms are willing to wait much longer to fill a vacancy in these markets. This allows the contact rate for workers to rise eight times more than in the standard model.4 With regards to the response of the other variables, I find that the job-to-job transition rate is strongly procyclical, although only about half as much as the job-finding rate of the unemployed. This translates into a procyclical total separation rate. As for the average match quality, it is also strongly procyclical due to the higher level of job-to-job transitions. Of the above elasticities, the most important one is that of the unemployment rate. To study its robustness with respect to the choice of the variables c and K, I calculate its sensitivity to c, while correspondingly varying K to keep the unemployment rate in the baseline case equal to 6%. The results are plotted in Figure 3. As the variable c increases, the corresponding value of K declines significantly. The elasticity of unemployment (in absolute value) also decreases;

4

The elasticity of l(y) is 2.49 in this model, while, in the standard model, it can be shown that, for the Wage-setting mechanism assumed in this paper, it would be exactly ð1 aÞ=a ¼ 0:39.

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however, even for c ¼ 0:99, the elasticity is well above 2. (It is worth noting that, at c ¼ 0:999, there is no value of K that generates a 6% unemployment rate, and, for K ¼ 0, the elasticity is very low.) This means that the reported results are fairly robust to the choice of c and K.

4.2. Comparative statics – destruction rate Next, I allow the destruction rate to vary. In Table 3, I report how the variables of interest change across stationary equilibria in response to a 1 % increase in the destruction rate. I also report the implied elasticities and compare these to the elasticities that Shimer (2005) finds in the standard model and in the data. Here, the principle difficulty in the standard model is not with regards to magnitudes; in fact, the magnitude of the elasticities generated by the standard model is closer to the data with respect to destruction-rate shocks than with respect to productivity shocks. A more significant problem is the sign of the elasticities. The standard model predicts that both unemployment and vacancies vary positively with destruction-rate shocks, thereby introducing a strong positive correlation between them, greatly at odds with the data. Table 3. Response of the variables of interest across stationary equilibria to a 1% increase in the destruction rate, their implied elasticity, and the corresponding elasticities in the standard model and the data as calculated from Shimer’s results as a ratio of standard deviations of the log of the variables taking into account their reported correlations 1% Increase in Destruction Rate

Unemployment rate Vacancy rate Job-finding rate of unemployed Job-to-job transition rate Total separation rate Fraction of employed workers searching Probability contacting employed searcher Lowest accepted match quality Search threshold match quality Average match quality

Standard Elasticity of Shimer

Data Elasticity of Shimer

Baseline Equilibrium Value

New Equilibrium Value

6.00% 2.96% 51.7%

6.18% 2.82% 50.6%

2.94 4.71 2.07

0.87 0.79 0.5

2.68%

2.65%

0.9

—

—

5.98% 65.8%

5.99% 65.7%

0.15 0.13

— —

— —

60.8%

60.1%

1.12

—

—

0.5

0.5

0

—

—

0.222

0.219

1.09

—

—

0.166

0.163

1.85

—

—

Implied Elasticity

2.53 2.69 1.57

Amplification of Productivity Shocks

503

This model, unlike the standard model, predicts that the vacancy rate decreases in response to an increase in the destruction rate. A high destruction rate discourages vacancy creation, since it shifts the composition of searchers toward the unemployed. This brings destruction shocks back into the picture as a plausible source of business-cycle variation in vacancies and unemployment. If anything, the model overestimates the response of unemployment and of vacancies to a change in the separation rate, since the implied elasticities are larger than those empirically found by Shimer. An interesting feature of the model is that, in response to changes in the destruction rate, the total separation rate is nearly acyclical. A higher destruction rate discourages job-to-job transitions, since it reduces the payoff from finding a high-quality match. Thus, in response to the higher destruction rate, the rate of separations into unemployment increases, but the rate of separations that lead to job-to-job transitions declines, in qualitative agreement with the data (Nagypa´l, 2005b). The lower rate of job-to-job transitions also implies that workers have a lower chance of securing high-quality matches, so that the average match quality also declines.

5. Conclusion Since the work of Shimer (2005), it has become well-known that the textbook search model does not generate nearly enough volatility of the correct sign in response to shocks to labor productivity or to the rate of separations. A vibrant recent body of work explores how this failure can be remedied through different generalizations. Most of these papers focus on the role of wage setting in generating volatility (Hall, 2005; Kennan, 2005; Menzio, 2005). While these explanations certainly provide important insights, this paper offers a complementary explanation of how the search model can generate more amplification. This explanation is based on the idea that, if there is complementarity between vacancies and employed searchers in the creation of employment matches, then in a model with on-the-job search a recession is a bad time to create a vacancy. This is because the extent of on-the-job search and of job-to-job transitions is low during these bad times. In order to introduce such a complementarity between vacancies and employed searchers, I introduced two new elements. One is that firms cannot observe how much a worker likes a potential employment match. Firms anticipate, however, that unemployed searchers are ‘‘desperate’’ and are willing to take any job, while employed searchers have better existing options and are thus more selective. This element by itself is not enough, though, to overcome the fact that unemployed searchers are much more likely to accept a job meaning that it is easier, thus more profitable for a firm to hire them. In addition, it is necessary that the firm make negative profits from very short matches. To achieve this, I add the second element, a cost that the firm has to pay when an employment

504

E´va Nagypa´l

relationship is formed, such as a cost of training. This natural addition means that firms prefer lower turnover individuals even if they are initially more difficult to attract. Given this complementarity between vacancies and employed searchers, calibration of the proposed model provides important results. First, this model generates a much larger response than the standard model in unemployment and vacancies to changes in labor productivity. The extent of amplification is still not enough to generate the variability observed in the data, but the results are much closer than those from the standard model. Second, this model generates responses in vacancies to an increase in the separation rate of the correct sign, unlike the standard model. In my model a high destruction rate discourages vacancy creation, since it shifts the composition of searchers toward the unemployed. In other word, changes in the destruction rate are a plausible source of business-cycle fluctuations. As Nagypa´l (2005a) shows, the proposed mechanism is not the only one capable of generating complementarity between vacancies and employed searchers. Another large class of models can give the same result using the idea that workers are of different, unobserved qualities, and higher quality workers are more likely to be employed. In such models, unemployed workers are more likely to be ‘‘lemons,’’ and unemployment therefore carries a stigma (as in the model of Vishwanath, 1989). These two alternatives give similar predictions regarding the amplification of labor productivity and destruction shocks. More work will be required to distinguish the two.

References Andolfatto, D. (1996), ‘‘Business cycles and labor-market search’’, American Economic Review, Vol. 86(1), pp. 112–132. Barlevy, G. (2002), ‘‘The sullying effect of recessions’’, Review of Economic Studies, Vol. 69(1), pp. 65–96. Blanchard, O.J. and P. Diamond (1989), ‘‘The Beveridge curve’’, Brookings Papers on Economic Activity, Vol. 1, pp. 1–60. Burdett, K. (1978), ‘‘A theory of employee job search and quit rates’’, American Economic Review, Vol. 68(1), pp. 212–220. Burdett, K., R. Imai, and R. Wright (2004), ‘‘Unstable relationships’’, B.E. Journals in Macroeconomics: Frontiers of Macroeconomics, Vol. 1(1), Article 1. http://www.bepress.com/bejm/frontiers/vol1/iss1/art1 Burdett, K. and D. Mortensen (1998), ‘‘Equilibrium wage differentials and employer size’’, International Economic Review, Vol. 39(2), pp. 257–274. Cole, H. and R. Rogerson (1999), ‘‘Can the Mortensen–Pissarides matching model match the business cycle facts?’’, International Economic Review, Vol. 40(4), pp. 933–959.

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Eriksson, S. and J. Lagerstro¨m (2004), ‘‘Competition between employed and unemployed job applicants: Swedish evidence’’, Working Paper, IFAU Institute for Labor Market Policy Evaluation. Faberman, R.J. (2005), ‘‘Studying the labor market with the job openings and labor turnover survey’’, Mimeo, Bureau of Labor Statistics. Fallick, B. and C.A. Fleischman (2004), ‘‘The importance of employer-toemployer flows in the U.S. labor market’’, Mimeo, Federal Reserve Bank Board of Governors. Gomes, J., J. Greenwood and S. Rebelo (2001), ‘‘Equilibrium unemployment’’, Journal of Monetary Economics, Vol. 48(1), pp. 109–152. Hall, R.E. (2005), ‘‘Employment fluctuations with equilibrium wage stickiness’’, American Economic Review, Vol. 95(1), pp. 50–65. Kennan, J. (2005), ‘‘Private information, wage bargaining, and employment fluctuations’’, Mimeo, University of Wisconsin-Madison. Menzio, G. (2005), ‘‘High frequency wage rigidity’’, Mimeo, Northwestern University. Merz, M. (1995), ‘‘Search in the labor market and the real business cycle’’, Journal of Monetary Economics, Vol. 36(2), pp. 269–300. Mortensen, D.T. (1994), ‘‘The cyclical behavior of job and worker flows’’, Journal of Economic Dynamics and Control, Vol. 18(6), pp. 1121–1142. Mortensen, D.T. and C.A. Pissarides (1994), ‘‘Job creation and job destruction in the theory of unemployment’’, Review of Economic Studies, Vol. 61(3), pp. 397–415. Nagypa´l, E. (2005a), ‘‘Job-to-job transitions and labor market fluctuations’’, Mimeo, Northwestern University. Nagypa´l, E. (2005b), ‘‘Worker reallocation over the business cycle: the importance of job-to-job transitions’’, Mimeo, Northwestern University. Pissarides, C.A. (1994), ‘‘Search unemployment with on-the-job search’’, Review of Economic Studies, Vol. 61(3), pp. 457–475. Postel-Vinay, F. and J.-M. Robin (2002), ‘‘Equilibrium wage dispersion with worker and employer heterogeneity’’, Econometrica, Vol. 70(6), pp. 2295–2350. Ramey, G. and J. Watson (1997), ‘‘Contractual fragility, job destruction, and business cycles’’, Quarterly Journal of Economics, Vol. 112(3), pp. 873–911. Shimer, R. (2001), ‘‘The impact of young workers on the aggregate labor market’’, Quarterly Journal of Economics, Vol. 116(3), pp. 969–1008. Shimer, R. (2003), ‘‘Dynamics in a model of on-the-job search’’, Mimeo, University of Chicago. Shimer, R. (2005), ‘‘The cyclical behavior of equilibrium unemployment and vacancies’’, American Economic Review, Vol. 95(1), pp. 25–49. Tobin, J. (1972), ‘‘Inflation and unemployment’’, American Economic Review, Vol. 62(1), pp. 1–18. Vishwanath, T. (1989), ‘‘Job search, stigma effect, and escape from unemployment’’, Journal of Labor Economics, Vol. 7(4), pp. 487–503.

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Appendix Proof of Proposition 1 As stated in the text, the reservation property of search decisions simply follows R m from the fact that the payoff from a unit of search ðlðyÞ m W 0 ðm0 ÞF ðm0 Þ dm0 Þ is decreasing in m. Given the specification of the search cost function in Equation (3), it follows from Equations (12) and (13), evaluated at ms, that sðms Þ ¼ s is a solution to the equation c^0 ðsÞs ¼ c0 þ c^ðsÞ

ðA:1Þ

Given the assumptions on c^, the right-hand side exceeds the left-hand side at s ¼ 0 and the left-hand side is increasing faster than the right-hand side. This implies that the above equation has a unique solution s40, which is independent of the other model parameters. From Equation (12), the optimal search effort for moms can be expressed as   Z m 0 0  0 0 0 1 sðmÞ c W ðm ÞF ðm Þ dm . lðyÞ ðA:2Þ m

Given the assumptions on c, c0 1 is a continuous and strictly increasing function. Given that its argument is continuously decreasing in m, this means that s(m) is continuous and strictly declining in m for all moms . The result that ms is increasing in l(y) simply follows from the fact that, given the constancy of s(ms), the left-hand side of Equation (12) evaluated at ms is a constant. The right-hand side is increasing in l(y) and decreasing in ms, hence to maintain equilibrium a higher l(y) must result in a higher level of ms. Finally, the result that s(m) is increasing in l(y) follows from totally differentiating the optimality condition in Equation (12).

Proof of Proposition 2 Let the initial distribution of previously unemployed workers across match qualities at tenure 0 be H u0 ðmÞ and let the initial distribution of previously employed workers across match qualities at tenure 0 be H e0 ðmÞ. Given the search and acceptance decisions of the different type of workers, these distributions can be expressed as H u0 ðmÞ ¼ H e0 ðmÞ

¼

F ðmÞ F ðmm Þ . F ðmm Þ R minðm;ms Þ mm

sðm0 ÞðF ðmÞ F ðm0 ÞÞ dGðm0 Þ R ms sðm0 ÞF ðm0 Þ dGðm0 Þ mm

ðA:3Þ

ðA:4Þ

Amplification of Productivity Shocks 0

0

F ðmÞ F ðm0 Þ . F ðm0 Þ

For m, m 2 ½m; mŠ, define dðm; m Þ ¼ F, dðm; m0 Þ is strictly decreasing in m0 since f ðm0 ÞF ðm0 Þ þ f ðm0 ÞðF ðmÞ F ðm0 Þ2 f ðm0 ÞF ðmÞ o0 ¼ F ðm0 Þ2

@dðm; m0 Þ ¼ @m0

507

Clearly, given the assumptions on

F ðm0 ÞÞ

ðA:5Þ

Then,

H e0 ðmÞ

¼

¼

R minðm;ms Þ mm

sðm0 ÞðF ðmÞ F ðm0 ÞÞdGðm0 Þ R ms 0  0 0 m sðm ÞF ðm Þ dGðm Þ

R minðm;ms Þ

m

sðm0 Þ dðm; m0 ÞF ðm0 Þ dGðm0 Þ R ms 0  0 0 mm sðm ÞF ðm Þ dGðm Þ R minðm;ms Þ 0 sðm Þ F ðm0 Þ dGðm0 Þ m  dðm; mm Þ ¼ H u0 ðmÞ, odðm; mm Þ m R ms sðm0 ÞF ðm0 Þ dGðm0 Þ mm

ðA:6Þ

mm

hence the initial distribution of the employed first-order stochastically dominates the distribution of the unemployed. As for the distributions H ut ðmÞ and H et ðmÞ for t40, these can be derived from u H 0 ðmÞ and H e0 ðmÞ by applying the same out-flow rates to them as a function of m, which is lðyÞsðmÞF ðmÞ. Since these rates do not depend on whether the worker was previously employed or unemployed, the resulting H et ðmÞ first-order stochastically dominates H ut ðmÞ.

Proof of Proposition 3 Define PD ¼ Pu Pe. Notice that given the workers’ search and acceptance decisions, Pu ¼ ¼

Z

Z

m

ðJðmÞ

KÞAu ðmÞ dF ðmÞ

m ms 

pð1 cÞ r þ d þ lðyÞsðmÞF ðmÞ mm  pð1 cÞ dF ðmÞ þ F ðms Þ rþd

 K  K

ðA:7Þ

E´va Nagypa´l

508

and Pe ¼

Z

m

ðJðmÞ

m

Z

KÞAe ðmÞ d F ðmÞ

ms 

pð1 cÞ ¼ r þ d þ lðyÞsðmÞF ðmÞ mm   pð1 cÞ  K , þ F ðms Þ rþd

K



Ae ðmÞd F ðmÞ ðA:8Þ

hence PD can be expressed as PD ¼ Pu

Pe ¼

Z

ms 

mm

ð1

pð1 cÞ r þ d þ lðyÞsðmÞF ðmÞ

Ae ðmÞÞ dF ðmÞ

K

 ðA:9Þ

Given that 1 Ae(m) is always positive and is strictly positive for some mA[mm,ms], PD is strictly decreasing in K. Moreover, PD is positive for K ¼ 0 since co1, and is negative for K ¼ pð1 cÞ=r þ d þ lðyÞsðms ÞF ðms Þ. Therefore, there exists a ^ PD is positive, and, for K4K, ^ PD is strictly negative. K^ such that, for K  K,

CHAPTER 20

Evaluating the Performance of the Search and Matching Model$ Eran Yashiv Abstract Does the search and matching model fit aggregate U.S. labor market data? While the model has become an important tool of macroeconomic analysis, recent literature pointed to some failures in accounting for the data. This paper aims to answer two questions: (i) Does the model fit the data, and, if so, on what dimensions? (ii) Does the data ‘‘fit’’ the model, i.e. what are the data which are relevant to be explained by the model? The analysis shows that the model does fit certain specifications of the data on many dimensions, though not on all. This includes capturing the high persistence and high volatility of most of the key variables as well as the negative co-variation of unemployment and vacancies. These findings differ from the cited literature mostly because of the use of convex rather than linear hiring costs and because of the role given to the separation rate in discounting match asset values. The paper offers a workable, empirically grounded version of the model for the analysis of aggregate U.S. labor market dynamics, and provides macroeconomists guidance concerning the relevant ‘‘building block’’ for modeling the labor market, both in terms of the model and in terms of the data.

Keywords: search, matching, U.S. labor market, vacancies, labor market flows, business cycles JEL classifications: E24, E32, J32, J63

$

Prepared for the conference in honor of Dale Mortensen at Sandbjerg, Denmark, August 15–18, 2004. I am indebted to Dale for inspiration and encouragement over the years. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75020-3

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

509

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Eran Yashiv

1. Introduction The importance of the role played by labor market frictions in labor market dynamics and in macroeconomic fluctuations is increasingly recognized. The key modeling tool in this context is the aggregate search and matching model developed by Diamond, Mortensen and Pissarides (see Mortensen and Pissarides (1999a,b) and Yashiv (2005b) for surveys and Pissarides (2000) for a detailed exposition). However, recent papers have questioned the model’s empirical performance on U.S. data. This literature poses doubts with respect to the model’s ability to account for the observed behavior of key variables – unemployment, vacancies, worker flows, wages, and the duration of unemployment.1 The aims of this paper are to answer two questions: (i) Does the model fit the data, and, if so, on what dimensions? (ii) Does the data ‘‘fit’’ the model, i.e. which data are relevant in the context of the model? The analysis produces a workable, empirically grounded version of the model that may be used to analyze U.S. data and study policy questions. The basic motivation is to try to understand whether the model – beyond its theoretical appeal – is indeed useful for the analysis of the U.S. labor market. The idea is to make precise the dimensions on which the model does well and those on which it does not do well or even fails. When coming to undertake such analysis a number of fundamental problems arise: (i) How to model the driving shocks? It is known from the business cycle literature that fluctuations often mimic the pattern of the shocks and that the model itself may lack propagation mechanisms. A similar problem may be the case here too. (ii) How to avoid the ‘‘contamination’’ of labor market analysis by misspecification of other parts of the macroeconomy? General equilibrium models have well-known problems that are yet to be resolved. Studying the empirical performance of the search and matching model in conjunction with other parts of the macroeconomy could thus lead to rejection of the model due to the aforementioned ‘‘contamination.’’ (iii) How to model the frictions so as to determine what role they play in explaining the data? Many studies in the literature are highly stylized, use short cuts, or employ reduced-form modeling that is useful for theoretical exposition but is problematic when attempting an empirical examination of the model. It is then difficult to see how the degree of frictions matters in accounting for the data. For a model that puts frictions at center stage this is highly problematic. Thus, in cases where the model is shown not to fit the data, it is not always clear whether a particular specification of labor market frictions is responsible for the difficulty or whether this reflects a more essential problem.

1

See Yashiv (2005b) for a detailed discussion.

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(iv) How to determine the equilibrium effects of shocks? There is a need for a structural model that can trace out the effects of shocks on equilibrium dynamics and determine what is the role of frictions in this process. (v) What are the proper data to use for the key elements of the model – the pool of searching workers and the stock of searching jobs (vacancies)? Questions regarding these data have been raised a number of times by different authors but have not been systematically treated in the current context. Hence, it is unclear to which extent the model-data fit is subject to problems with the data used rather than to problems with the model. The paper takes the following approach to address these problems: to deal with (i) and (ii), it uses a partial equilibrium model and a reduced-form VAR of the actual data to specify the driving shocks. This ‘agnostic’ approach precludes the possibility that labor market dynamics will be affected by misspecifications in other parts of a more general macroeconomic model. Thus, it does not take a particular stand on the sources of the driving shock processes nor does it formulate an explicit structure for the rest of the macroeconomy. To deal with (iii) and (iv) it uses a structural approach: it specifies agents’ objectives and constraints, their optimal behavior, and the dynamic paths of key variables in equilibrium. The structural parameters quantify the degree of frictions. The fifth issue is treated by looking at alternative formulations of the pool of searching workers, taking into account also non-employed workers outside the official unemployment pool. The paper also uses newly available gross worker flow data that are compatible with the model’s formulations, rather than vacancy data, which are shown to be inconsistent with the concepts of the model. The model is a stochastic version of the search and matching model. It characterizes firms’ optimal search behavior, deriving a relationship that equates marginal hiring costs with the present value of a hire – the ‘‘asset value’’ of the job–worker match. The firm’s decision on vacancy creation feeds into a matching function, which generates a flow of hires given the stocks of job vacancies and unemployment. The wage is determined by the Nash bargaining solution, which splits the surplus from the job–worker match. The resulting model is a partial equilibrium model, with employment (and unemployment), vacancies, and the wage defined by the equilibrium solution. The non-linear equilibrium solution of the model is log-linearized using a first-order Taylor approximation to obtain a system of linear difference equations. The system formalizes the dynamics of the key variables in terms of exogenous, stochastic variables and in terms of structural parameters that quantify the frictions. The model is then calibrated. As mentioned, a reduced-form VAR determines the relevant parameters for the exogenous shock processes. These include labor productivity shocks, shocks to the match separation rate, and shocks to the discount rate. Other calibrated values are set using historical data averages or econometric estimates. The model’s implied second moments are compared to U.S. data in terms of persistence, co-movement, and volatility.

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I find that for the most part the model fits U.S. labor market data relatively well. This includes capturing the high persistence and high volatility of most of the key variables, as well as the negative co-variation of unemployment and vacancies (the ‘Beveridge curve’). The paper makes several contributions: first, it offers an empirically grounded model of aggregate U.S. labor market dynamics as implied by the search and matching model. Doing so it provides macroeconomists some guidance concerning the relevant ‘‘building block’’ for modeling the labor market. Second, by calibrating the model and evaluating it against alternative formulations of the data, it is able to show on what dimensions the model fits the data and which data series are the relevant ones. Third, particular specifications of the model are able to replicate key empirical regularities in U.S. data which other models have been unable to capture; the reasons for this improved performance are discussed. In terms of the calibration–simulation methodology, there is an innovation in the ‘‘agnostic’’ approach taken with respect to the formulation of shocks. The paper proceeds as follows: Section 2 presents the search and matching model and derives the equilibrium dynamics and steady state, highlighting the role of frictions. At the end of this section, I point out the contributions of Dale Mortensen in the current, macroeconomic context. Section 3 discusses the data series, presents the properties of the data that are to be matched, and proposes alternative formulations of the data to be used. It also reports the results from recent literature that found problems in the fit of the model. Section 4 calibrates the model and evaluates the model-data fit. Section 5 examines the mechanism underlying the results, deriving the modeling and data lessons for using the model to study the U.S. labor market. Section 6 concludes. 2. The search and matching model In this section, I briefly present a stochastic, discrete-time version of the prototypical search and matching model.2 An important addition to the standard analysis is a convex formulation for the hiring costs function, which has the standard linear formulation as a special case. This in turn necessitates taking into account the effect of employment on wages in the firm’s optimality condition and in the wage solution. 2.1. The basic set-up There are two types of agents: unemployed workers (U) searching for jobs and firms recruiting workers through vacancy creation (V). Firms maximize their

2

A detailed exposition may be found in Pissarides (2000). The stochastic, discrete time formulation presented here follows Yashiv (2004).

Evaluating the Performance of the Search and Matching Model

513

intertemporal profit functions with the choice variable being the number of vacancies to open. Each firm produces a flow of output (F), paying workers wages (W) and incurring hiring costs (G). Workers and firms are faced with different frictions such as different locations leading to regional mismatch or lags and asymmetries in the transmission of information. These frictions are embedded in the concept of a matching function which produces hires (M) out of vacancies and unemployment, leaving certain jobs unfilled and certain workers unemployed. Workers are assumed to be separated from jobs at a stochastic, exogenous rate, to be denoted by d. The labor force (L) is growing with new workers flowing into the unemployment pool. The setup, whereby search is costly and matching is time-consuming, essentially describes the market as one with trade frictions. Supply and demand are not equilibrated instantaneously, so at each date t there are stocks of unemployed workers and vacant jobs. The model assumes a market populated by many identical workers and firms. Hence, I shall continue the discussion in terms of ‘‘representative agents.’’ Each agent is small enough so that the behavior of other agents is taken as given. As is well known this creates various externalities. In particular, more search activity creates a positive trading externality for the trading partner and a congestion externality for similar agents (see the discussion in Pissarides, 2000, Ch. 7). 2.2. Matching A matching function captures the frictions in the matching process; it satisfies the following properties: e t; V tÞ M t;tþ1 ¼ MðU e e @M @M 40; 40 ð1Þ @U @V Empirical work (see the survey by Petrongolo and Pissarides, 2001) has shown that a Cobb–Douglas function is useful for parameterizing it: M t;tþ1 ¼ mU st V 1t

s

ð2Þ

where m stands for matching technology. From this function the job-finding rates – P, the worker probability of finding a job and Q, the firm’s probability of filling the vacancy – are derived:  1 s M t;tþ1 Vt Pt;tþ1 ¼ ð3Þ ¼m Ut Ut   s M t;tþ1 Vt ð4Þ ¼m Qt;tþ1 ¼ Vt Ut The parameter s reflects the relative contribution of unemployment to the matching process and determines the elasticity of the hazard rates with respect to market tightness.

514

Eran Yashiv

2.3. Firms Firms maximize the expected, present value of profits (where all other factors of production have been ‘‘maximized out’’): ! 1 t X Y max E 0 bj ½F t W t N t Gt Š ð5Þ fV g

t¼0

j¼0

where bj ¼ 1=ð1 þ rt 1;t Þ. This maximization is done subject to the employment dynamics equation given by N tþ1 ¼ ð1

dt;tþ1 ÞN t þ Qt;tþ1 V t

ð6Þ

The Lagrangean of this problem is (where L is the discounted Lagrange multiplier): ! 1 t X Y bj ½F t W t N t Gt þ Lt fð1 dt;tþ1 ÞN t þ Qt;tþ1 V t N tþ1 gŠ L ¼ E0 t¼0

j¼0

The FOC are: @Gt ¼ Qt;tþ1 E t Lt @V t  @F tþ1 Lt ¼ E t btþ1 @N tþ1 þ E t ð1

N tþ1 ¼ ð1

ð7Þ

ð8Þ @Gtþ1 @N tþ1

W tþ1

@W tþ1 N tþ1 @N tþ1

dtþ1;tþ2 Þbtþ1 Ltþ1





@F T @N T

ð9Þ

dt;tþ1 ÞN t þ Qt;tþ1 V t

ð10Þ

and the transversality condition: " ! T Y1 bj lim E t T!1



j¼0

WT

ð@GT

1 Þ=ð@LT 1 Þ

bT

1

NT

@W T 1 @N T

1 1



 NT ¼ 0

ð11Þ

The first, intratemporal condition (Equation (8)) sets the marginal cost of hiring @Gt =@V t equal to the expected value of the multiplier times the probability of filling the vacancy. The second, intertemporal condition (Equation (9)) sets the multiplier equal to the sum of the expected, discounted marginal profit in the next period   @F tþ1 @Gtþ1 @W tþ1 W tþ1 N tþ1 E t btþ1 @N tþ1 @N tþ1 @N tþ1 and the expected, discounted (using also d) value of the multiplier in the next period E t ð1 dtþ1;tþ2 Þbtþ1 Ltþ1 . Note that because I postulate that G depends on N (see below), the net marginal product for the firm depends on N. This marginal product is part of the match surplus bargained over, and therefore part of the wage solution

Evaluating the Performance of the Search and Matching Model

515

discussed below. Hence the term, @W tþ1 =@N tþ1 , usually absent, is not zero in this formulation. For production I assume a standard Cobb–Douglas function F t ¼ At K at N 1t

a

ð12Þ

where A represents technology and K the capital. Hiring costs refer to the costs incurred in all stages of recruiting: the cost of advertising and screening – pertaining to all vacancies (V), and the cost of training and disrupting production – pertaining to actual hires (QV). For the functional form I use a general power function formulation which encompasses the widely used linear and quadratic functions as special cases. This functional form emerged as the preferred one – for example as performing better than polynomials of various degrees – in structural estimation of this model reported in Yashiv (2000a,b) and in Merz and Yashiv (2004). The former study used an Israeli data-set that is uniquely suited for such estimation with a directly measured vacancy series that fits well the model’s definitions. The latter study used U.S. data. Formally this function is given by Gt ¼

  Y fV t þ ð1 fÞQt V t gþ1 Ft 1þg Nt

ð13Þ

Hiring costs are function of the weighted average of the number of vacancies and the number of hires. They are internal to production and hence are proportional to output. The function is linearly homogenous in V, N and F. It encompasses the cases of a fixed cost per vacancy (i.e. linear costs,g ¼ 0) and increasing costs (g40). When g ¼ 1 I get the quadratic formulation ðY=2ððfV t þ ð1 fÞQt V t Þ=N t Þ2 F t Þ which is analogous to the standard formulation in ‘‘To-bin’s q’’ models of investment where costs are quadratic in I=K. Note that Y is a scale parameter, f the weight given to vacancies as distinct from actual hires, and g expresses the degree of convexity.3 2.4. Wages In this model, the matching of a worker and a vacancy against the backdrop of search costs, creates a joint surplus relative to the alternatives of continued search. Following Diamond (1982) and Mortensen (1982a), the prototypical search and matching model derives the wage (W) as the Nash solution of the

3

Note that its derivatives, used below, are given by  g @Gt fÞQt Þ fV t þð1N tfÞQt V t NF tt @V t ¼ Yðf þ ð1  gþ1 h i fV t þð1 fÞQt V t @Gt Ft 1 a 1 @N t ¼ Y N t 1þg Nt

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Eran Yashiv

bargaining problem of dividing this surplus between the firm and the worker (see the discussion in Pissarides, 2000, Ch. 1 and 3). Formally this wage is W t ¼ arg maxðJ N t N

U

x F JU t Þ ðJ t

1 JV t Þ

x

ð14Þ

where J and J are the present value for the worker of employment and unemployment respectively; JF and JV are the firm’s present value of profits from a filled job and from a vacancy respectively; and 0oxo1 reflects the degree of asymmetry in bargaining. Using the Stole and Zwiebel (1996) approach as implemented by Cahuc et al. (2004) to solve (14) taking into account the fact that @W tþ1 =@N tþ1 a0, the wage is given by4  a   Kt 1 fV t þ ð1 fÞQt V t gþ1 þ Yð W t ¼ x ð1 aÞA Þ 1 ax Nt Nt  aþg  þPt;tþ1 Lt ð1 þ gÞ ð1 aÞ ð1 xð1 þ a þ gÞÞ þ ð1

xÞbt

ð15Þ

where b is the income of the unemployed, such as unemployment benefits. I assume that bt is proportional to Wt, i.e. bt ¼ tWt, so t may be labeled the ‘‘replacement ratio.’’ Hence  a   x Kt 1 fV t þ ð1 fÞQt V t gþ1 ð1 aÞA þ Yð Wt ¼ Þ 1 ð1 xÞt 1 ax Nt Nt  aþg  þPt;tþ1 Lt ð1 þ gÞ ð1 aÞ ð1 xð1 þ a þ gÞÞ

Denoting x=ð1 ð1 xÞtÞ by Z and formulating the wage in terms of the labor share in income (by dividing this wage by the average product) I get Ft W t ¼ st Nt "   1 fV t þ ð1 fÞQt V t gþ1 þY st ¼ Z ð1 aÞ 1 ax Nt   aþg  ð16Þ þ Pt;tþ1 lt ð1 þ gÞ ð1 aÞ ð1 xð1 þ a þ gÞÞ

This wage solution has the following properties: (a) Wages are proportional to productivity F t =N t but the proportionality coefficient st is time varying.

4

The solution entails postulating the asset values of a filled job and of a vacant job for the firm and the asset values of employment and unemployment for the worker in (14). See the technical appendix at http://www.tau.ac.il/yashiv/research.html for details of the derivation.

Evaluating the Performance of the Search and Matching Model

517

(b) Wages and the wage share increase with worker bargaining power (x) or replacement ratio (t) as expressed by Z. (c) Wages and the wage share are a positive function of the future value given by Pt;tþ1 lt . Thus wages are positively related to the asset value of the match. 2.5. Equilibrium The stocks of unemployment and employment and the flow of hiring emerge as equilibrium solutions. Solving the firms’ maximization problem yields a dynamic path for vacancies; these and the stock of unemployment serve as inputs to the matching function; matches together with separation rates and labor force growth change the stocks of employment and unemployment. Formally ! 1 t X Y V t ¼ arg max E 0 bj ½F t W t N t Gt t¼0



þ Lt ð1

U t ¼ Lt N t e t; V tÞ M t;tþ1 ¼ MðU N tþ1 ¼ ð1

j¼0

dt;tþ1 ÞN t þ Qt;tþ1 V t

dt;tþ1 Þ N t þ M t;tþ1

W t ¼ arg max ðJ N t

x F JU t Þ ðJ t

1 JV t Þ

N tþ1

x



ð17Þ

This dynamic system may be solved for the five endogenous variables V, U, M, N, and W given initial values U0, N0 and given the path of the exogenous variables. The latter include b, d, and L, variables included in the firms’ profit function and variables included in the workers unemployment value function JU. As noted above, this is a partial equilibrium model. The exogenous variables include the worker’s marginal product, the discount factor, and the separation rate. If the production function is CRS and if the capital market is perfect – as I shall assume – the capital–labor ratio will be determined in equilibrium at the point where the marginal product of capital equals the interest rate plus the rate of depreciation. This in turn will determine production and the marginal product of labor. This setup is consistent with several different macroeconomic models. For example, Merz (1995) and Andolfatto (1996) have shown that a special case of this model may be combined with traditional elements of RBC models to yield a dynamic general equilibrium model. In these models, the interest rate equals the marginal rate of intertemporal substitution in consumption. In the following subsection I solve explicitly for a stochastic, dynamic equilibrium using a stochastic structure for the exogenous variables. 2.6. The dynamics and the steady state I use a log-linear approach, transforming the non-linear problem into a first-order, linear, difference equations system through approximation and then solving the

518

Eran Yashiv

system using standard methods. More specifically, I undertake the following steps: (i) I postulate a stochastic setup for the exogenous variables. As noted in the introduction, I use a reduced-form VAR procedure to characterize the shocks rather than imposing a structural formulation. (ii) Using the FOC, I characterize the non-stochastic steady state. (iii) The deterministic version of the FOC of the firm’s problem, including the flow equation for employment, is linearly approximated in the neighborhood of this steady state, using a first-order Taylor approximation. (iv) This yields a first-order, linear, difference equations system, whose solution gives the dynamic path of the control and the endogenous state variables as functions of sequences of the exogenous variables. (v) Working from a certainty equivalence perspective, the deterministic sequences for the exogenous variables are then replaced with the conditional expectations at time t for the afore-cited stochastic processes. Basically I am interested in exploring the dynamics of the labor market relative to its growth trend. Growth comes from two sources: productivity growth and population growth. To abstract from population growth, in what follows I cast all labor market variables in terms of rates out of the labor force Lt, denoting them by lower case letters. Productivity growth is captured by the evolution of A, which enters the model through the dynamics of F t =N t so I divide all variables by the latter. This leaves a system that is stationary and is affected by shocks to labor force growth, to productivity growth, as well as to the interest rate and to the job separation rate, to be formalized below. I now further develop the FOC in order to get a stationary representation with fewer variables. I begin with (8). Transforming the equation into a stationary one through the afore-mentioned divisions the equation becomes Yðf þ ð1 þ fÞQt Þ ð

fV t þ ð1 fÞQt V t g Þ ¼ Qt;tþ1 E t lt Nt

ð18Þ

where I define lt 

Lt F t =N t

ð19Þ

Inserting into (9) and dividing throughout by Ft+1/Nt+1:    fV tþ1 þ ð1 jÞQtþ1 V tþ1 gþ1 lt 1 þYð Þ ¼ E t btþ1 ð1 ZÞ ð1 aÞ X Nt 1 ax G tþ1   ð20Þ aþg  ZPtþ1; tþ2 ltþ1 ð1 þ gÞ ð1 aÞ ð1 xð1 þ a þ gÞÞ þE t ð1 dtþ1; tþ2 Þ btþ1 ltþ1

Evaluating the Performance of the Search and Matching Model

519

where the (gross) rate of productivity growth is given by GX tþ1 

F tþ1 =N tþ1 F t =N t

ð21Þ

Dividing (6) by Lt throughout I get ntþ1 G Ltþ1 ¼ ð1

dt; tþ1 Þnt þ Qt; tþ1 ut

ð22Þ

where I define 1+the growth rate of the labor force as G Ltþ1 

Ltþ1 Lt

ð23Þ

The model has four exogenous variables. These are productivity growth (GX 1, see Equation (21)), labor force growth (GL 1, see Equation (23)), the discount factor (b), and the separation rate (d). Empirical testing reveals that GL can be modeled as white noise around a constant value. When I tried to add it as a stochastic variable to the framework below, the results were not affected. I thus treat it as a constant. It is the other three variables that inject shocks into this system. These variables are basically the variables that make up the firm’s future marginal profits, with higher GX increasing these profits and higher b or d decreasing them. As mentioned, I do not formulate the underlying shocks structurally. Instead, I postulate that they follow a first-order VAR (in terms of log deviations from their non-stochastic steady-state values): 2 X3 3 X G^ tþ1 G^ 6 t 7 7 6 7 ¼ P6 b^ 7 þ S 6 b^ 5 4 5 4 2

tþ1

t

d^ tþ1

d^ t

ð24Þ

In the empirical section below, I use reduced-form VAR estimates of the data to quantify the coefficient matrix P and the variance–covariance matrix of the disturbances S. Thus the current model is consistent with both RBC-style models that emphasize technology shocks as well as with models that emphasize other shocks. In the non-stochastic steady state the rate of vacancy creation is given by   fV þ ð1 fÞQV g GX b Yðf þ ð1 þ fÞQÞ ¼Q p ð25Þ N ½1 ð1 dÞG X bŠ The LHS are marginal costs and the RHS is the match asset value. It is probability of filling a vacancy (Q) times the marginal profits accrued in the steady state. The latter are the product of per-period marginal profits p and a discount factor G X b=ð1 GX bð1 dÞÞ that takes into account the real rate of interest, the rate of separation, and productivity growth.

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Eran Yashiv

Labor market flows are given by ðd þ G L

1Þ ¼

m Qv ¼ n n

ð26Þ

This expression equates the rate of increase in employment through matching with the sum of the rates of separation and increase in the labor force. From this equation the rate of unemployment in equilibrium is given by u¼

d þ ðG L 1Þ d þ ðG L 1Þ þ P

ð27Þ

I log-linearly approximate the deterministic version of the FOC in the neighborhood of the steady state. For each variable Y, I use the notation Y^ t ¼ ðY t Y Þ=Y  ln Y t ln Y ; where Y is the steady-state value, so all variables are log deviations from steady state. Approximating the intratemporal condition (18), the intertemporal condition (20), and the dynamic equation for employment (22) and combining all of them I get 2 X 3 2 X 3 G^ tþ1 G^     6 t 7 6 7 n^ tþ1 n^ t 6 7 7 ð28Þ ¼ W ^ þ R6 5 4 b^ tþ1 5 þ Q4 b^ t lt l^ tþ1 d^ t; tþ1 d^ tþ1;tþ2

where W, R, Q are matrices which are functions of the parameters and steadystate values. This system is a first-order, linear, difference equation system in the state X ^ and d. ^ The variable n^ and the co-state l^ with three exogenous variables, G^ , b, matrices of coefficients are defined by the parameters a, Y, g, f, m, s, and Z and by the steady-state values of various variables. The solution of this system enables us to solve for the control variable – vacancies, and for other variables of interest, such as unemployment, hires, the matching rate, and the labor share of income. 2.7. The contributions of Dale Mortensen Dale Mortensen has made fundamental contributions to the model presented above. The first one, in the celebrated 1970 ‘‘Phelps volume’’ (Mortensen, 1970), introduced the flow approach to the labor market and incorporated search costs. In that paper, the firm’s intertemporal choice was shown to be akin to investment with adjustment costs. The papers which followed introduced all the key ingredients of the model: bilateral matching and surplus division in Mortensen (1978), aggregate matching and surplus division using the Nash solution in Mortensen (1982a); the last paper also discussed the incentives of agents to engage in search. Efficiency aspects of the matching and bargaining situations were analyzed in Mortensen (1982b). Finally, the model was expanded to

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521

incorporate idiosyncratic productivity and endogenous separations in Mortensen and Pissarides (1994). 3. U.S. labor market data In coming to relate the model to U.S. data, a number of important issues arise. The following discussion shows that the different variables have multiple representations in the data and some are not consistent with the concepts of the model. The idea is to select those series that do match these concepts and to examine alternative representations wherever relevant. I proceed as follows: in 3.1 I discuss the relevant pool of searching workers, who are defined as u, unemployed, in the model; in 3.2 I conduct a similar discussion for job vacancies v; in 3.3 I consider the flow of hires m and in 3.4 the relevant wage data s; in 3.5 I briefly discuss other data series; finally, in 3.6 I present summary statistics, including second moments, and discuss the properties of the data in terms of persistence, volatility, and co-movement. I end the section with a discussion of the findings of recent literature that question the model’s ability to fit these data. 3.1. The relevant pool of unemployment In order to see how the model relates to the data, a key issue that needs to be resolved is the size of the relevant pool of searching workers. The question is whether this pool is just the official unemployment pool or a bigger one. The model speaks of two states – employment and unemployment; in the model matches are flows from unemployment to employment and separations are flows from employment to unemployment.5 In the actual data – taken from the Current Population Survey (CPS) – several important issues arise: (i) Flows between the pool out of the labor force and the labor force, including flows directly to and from employment, are sizeable. Blanchard and Diamond (1990) report that in worker flows data – adjusted according to the methodology of Abowd and Zellner (1985) and covering the period 1968–1986 – unemployment to employment flows are only slightly bigger – at 1.6 million workers per month – than out of the labor force to employment flows, at 1.4 million per month. More recent data, computed using the methodology of Bleakley et al. (1999) for the period 1976–2003, indicate that unemployment to employment flows are on average 1.9 million workers per month, while out of the labor force to employment flows are 1.5 million workers per month on average.

5

Additionally, labor force growth (with new participants joining the unemployment pool) is an exogenous variable.

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(ii) Clark and Summers (1979) have argued that there is substantial misclassification of unemployment status and that ‘‘many of those not in the labor force are in situation effectively equivalent to the unemployed’’ (p. 29), providing several measures to substantiate this claim. One implication that emerges from their study is that including such misclassified workers in the unemployment pool generates a longer average duration for unemployment and hence a smaller job finding (P in the model) relative to the official unemployment pool. Data collectors were aware of this issue: following the recommendations of the Gordon committee, which recognized that there could be some form of ‘‘hidden unemployment,’’ beginning in January 1967 the CPS included questions on out of the labor force people who could potentially be defined as unemployed. This generated a quarterly series on people that responded affirmatively to the question if they ‘‘wanted a job now’’ (I report this series below). Castillo, 1998 (who offers a detailed discussion and additional references concerning the CPS procedure) reports, using 1994 data, that about 30% of this pool actually searched for work in 12 months prior to the survey, and 47% were in the prime age category (25–54). (iii) Working on the re-designed CPS data in the period 1994–1998, Jones and Riddel (2000) further demonstrate the importance of these distinctions. Key results include estimates of the job-finding rates for three worker groups: the unemployed, the marginally attached, and the unattached, the last two being officially classified as out of the labor force. Their monthly job-finding rates are 0.20–0.35 for the unemployed, 0.10–0.20 for the marginally attached, and below 0.05 for the unattached. Various tests indicate that these are indeed three distinct states. (iv) The out of the labor force flows exhibit markedly different cyclical properties relative to the flows between employment and unemployment: the unemployment to employment flows are counter-cyclical or acyclical, while the out of the labor force to employment flows are pro-cyclical.6 Given this evidence, it appears natural to consider pools of workers outside the labor force when coming to study labor market dynamics and worker flows. The question is how to add the unobserved ‘unemployment’ pool from out of the labor force to the ‘official’ pool of unemployment. Blanchard and Diamond (1989, 1990) suggest that the relevant additional pool is made up of the people

6

There are two major data sets on worker flows: Blanchard and Diamond (1990, see in particular Figures 1,10, and 11) and Bleakley et al. (1999). The unemployment to employment flows are negatively correlated with the rate of employment ( 0.91 for the 1968–1986 of data Blanchard and Diamond and 0.50 for the 1976–2003 data of Bleakley et al.) and negatively correlated or uncorrelated with GDP per capita ( 0.61 for the 1968–1986 data and 0.12 for the 1976–2003 data). The out of the labor force to employment flows are positively correlated with the rate of employment (0.56 for the 1968–1986 data of Blanchard and Diamond and 0.70 for the 1976–2003 data of Bleakley et al.) and with GDP per capita (0.56 for the 1968–1986 data and 0.76 for the 1976–2003 data).

Evaluating the Performance of the Search and Matching Model

Figure 1.

523

Alternative unemployment series (rates)

.18 .16 .14 .12 .10 .08 .06 .04 .02 1970

1975

1980

1985

1990

1995

2000

official official+want a job matching based

described in point (ii) above. Another method to estimate this pool is to compute the number of people within the out of the labor force group that would generate matching rates (M/U) identical to those that are observed for unemployment to employment flows.7 Figure 1 shows the resulting two new series, as well as the official unemployment rate, for the period in which the series exist.

7

This estimate – based on the relationship P ¼ M=U – is computed as follows: first note that

M UE U where MUE is the unemployment to employment flow and U the official pool. Then, assuming PUE holds true, and using data on out of the labor force to employment MNE flow, we compute PUE ¼

 UE    M UE þ M NE M þ M NE M NE ¼ U 1 þ ¼ U PUE M UE M UE The official rate is given by U/(N+U), while the new rate is given by U*/(N+U*). The relation between the two series in rates is thus Un ¼

  U n =ðN þ U n Þ U n N þ U M NE 1 þ U N ¼ ¼ 1 þ n U N þ Un U=ðN þ UÞ M UE 1 þ UN

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The official rate, the lower line in the figure, has a sample mean of 6.3%. The series that adds workers who said they want a job (the longer series at the top) has a mean of 10.4% and the second series, computed as explained, has a mean of 10.9% of the relevant labor force. As can be seen in the figure, the series are highly correlated: the new series have a 0.97 correlation between them and high correlations with the official rate (0.98). These series, however, are likely to be lower bounds on the relevant unemployment pool: the first series was constructed in reference to indicating a desire for a job now; this precludes other out of the labor force groups who wish to work at a future date, for example people in training or in school. In terms of our two-state model these people could be subsequently flowing directly to the employment pool and thus have to be classified as unemployed. As to the second series, the data support the proposition that the matching rate (P ¼ M/U) used to generate it is an upper bound on the relevant rate, as persons out of the labor force are likely to experience higher unemployment durations. Hence the resulting unemployment pool is a lower bound on the relevant pool. For example, analysis of 1994 data in Castillo (1998) shows that while matching rates for the officially unemployed were 0.53, they were only 0.31 for the ‘‘wanted a job’’ pool. The issue, then, is how many non-employed workers does one add from out of the labor force to the already expanded pools shown in Figure 1. The strategy I use in the empirical work is as follows. I look at the three ‘‘natural’’ candidate series: (i) official unemployment, (ii) official unemployment and the ‘‘want a job’’ category (depicted above), and (iii) the entire working age population. As the relevant pool may lie between the second and third cases and as no measured pool is available, I look at two additional specifications that try to approximate that pool. Thus, starting from case (ii) I gradually add parts of the remaining workers from the out of the labor force pool. Table 1 provides sample statistics of these five series. Two features stand out: while the mean of the series evidently rises with the expansion of the pool, the volatility hardly changes going from the official +want a job pool to the larger pools; the series are highly correlated (though the correlation slightly declines as the pool expands). In what follows, I look at the properties of the data under these different specifications. In Section 5 below, I compare the performance of the calibrated model against these alternative specifications and decide on a benchmark specification for the subsequent analysis.8

8

Previous papers have taken on a variety of approaches to this issue, which are special cases of the current examination: Blanchard and Diamond (1989) have essentially considered the relevant pool to be the first series reported in Figure 1, Merz (1995) took the official pool of unemployment, and Andolfatto (1996) considered the entire working age population as unemployed in the model.

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Table 1.

Stochastic properties of alternative measures of the pool of searching workers (Quarterly Data)

Pool of Workers

0 ¼ official pool U1 1 ¼ official+want a job 2 ¼ official+want a job+0.15(POP N U1) 3 ¼ official+want a job+0.3(POP N U1) 4 ¼ POP N

Mean

0.06 0.10 0.16 0.21 0.39

Std.

0.01 0.02 0.02 0.02 0.02

Correlations 0

1

2

3

4

1 0.98 0.96 0.94 0.856

1 0.99 0.98 0.91

1 0.996 0.95

1 0.98

1

Notes: 1. POP, working age population; N civilian employment. 2. See data appendix for sources.

3.2. Job vacancies The relevant concept of vacancies in the model is the one relating to those vacancies that are to be filled with workers from outside the employment pool (I shall denote it by VUN). But the available and widely used data series pertains to another concept, which also includes vacancies that are subsequently filled with 9 workers moving from job to job (to be denoted by VNN t ). Simply this can be expressed as follows: UN þ V NN V tot t ¼ Vt t

The Vtot t series in the U.S. economy has two representations: one is the index of Help Wanted advertising in newspapers published by the Conference Board; this series was analyzed and discussed in Abraham (1987). A newer series is the job openings series available from the Bureau of Labor Statistics (BLS) since December 2000 using the Job Openings and Labor Turnover Survey (JOLTS).10 The two series have a correlation of 0.88 over 37 monthly observations. Figure 2 plots these two vacancies series as well as a third series: the gross flows of workers from outside employment (unemployment and out of the labor force) to employment, which can be taken to represent Q  VUN. The latter was recently

9

This should not be taken to mean that firms post two types of vacancies. The idea is just to say that some vacancies are ex-post filled by previously unemployed workers and the rest by previously employed workers moving directly from job to job. 10 A job is ‘‘open’’ only if it meets all three of the following conditions: (i) a specific position exists and there is work available for that position; the position can be full-time or part-time, and it can be permanent, short-term, or seasonal; (ii) the job could start within 30 days, whether or not the establishment finds a suitable candidate during that time; and (iii) there is active recruiting for workers from outside the establishment location that has the opening.

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Figure 2.

Hiring vs. the two vacancy measures (normalized)

5 4 3 2 1 0 -1 -2 -3 1980

1985 help wanted

1990

1995 JOLTS

2000 hires

compiled at the Boston Fed based on CPS data (see the appendix below and Bleakley et al., 1999). The figure shows all three series normalized. The smooth line is the Help Wanted series while the less persistent series is the gross flows series, in the dashed line. The JOLTS series is much shorter and can be seen only from late 2000 in between the two other series. The hiring flows series is negatively correlated with the two vacancy series: 0.27 with the JOLTS series and 0.36 with the Help Wanted ads series. The flows series also appears to be much less persistent than the vacancies series. Table 2 presents the coefficient of variation (or standard deviation) and autocorrelation for the two vacancy series and for the hiring series. It does so first for their respective individual samples and then for the common sample period. It reports the variables in levels, in logged HP-filtered form, and subsequently in terms of their ratios to unemployment, which can be taken as a measure of market tightness. Compared to the volatility of the Help Wanted Index, hiring volatility is about a seventh in levels, a fifth in HP-filtered terms, and about a third in ratios to unemployment terms. It is similarly less volatile than the JOLTS data series. In terms of persistence, hiring is substantially less persistent in levels and even more so in HP-filtered terms; it has similar persistence in market tightness terms. This comparison – between Vtot and Q  VUN – suggests that Vtot may either be very different from VUN or that the behavior of Q generates these discrepancies. Without direct measures of VUN it is not possible to determine which explanation holds true but there is indirect evidence. Using the data on Q  VUN

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Table 2.

Stochastic properties of vacancies and hires (Quarterly Data) Individual Samples

Common Sample

Sample

Std/Mean

AR

Std/Mean

AR

1951:I–2003:IV 2001:I–2003:IV 1976:II–2003:IV

0.34 0.24 0.05

0.98 0.61 0.61

0.27 0.24 0.03

0.56 0.61 0.17

Levels V V QV

Help wanted index JOLTS job openings Hires Logs, HP-filtered

V V QV

Help wanted index JOLTS job openings Hires

Std. 1951:I–2003:IV 2001:I–2003:IV 1976:II–2003:IV

Ratio to unemployment V U V U QV U

Help wanted index JOLTS job openings Hires

0.14 0.28 0.03

0.89 0.60 0.05

Std/Mean 1951:I–2003:IV 2001:I–2003:IV 1976:II–2003:IV

0.44 0.35 0.16

Std. 0.95 0.66 0.94

0.47 0.35 0.15

0.62 0.66 0.64

Notes: 1. See data appendix for sources. 2. AR indicates autocorrelation.

shown above one can run a regression of the matching function as follows: !     Qt; tþ1 V UN Ut Vt t ^ ln ln ¼ ln m^ þ s^ ln þ ð1 sÞ Nt Nt Nt Previous estimates of the matching function – as surveyed by Petrongolo and Pissarides (2001) and as exemplified by the Blanchard and Diamond (1989) analysis – indicate that s^ is around 0.4–0.5. This means that the regression should yield around 0.4–0.5 for the coefficient of ln U t =N t and around 0.6–0.5

for the coefficient of ln V t =N t . Using the Help Wanted index for V one gets an

but only 0.1 for the =N estimate of around 0.3 for the coefficient of ln U t t

coefficient of ln V t =N t , irrespective of the estimation technique (OLS, TSLS or GMM) or the instruments used. The extremely low value for the latter coefficient implies that hiring rates (from outside employment, i.e. Qt;tþ1 V UN t =N t ) do not relate well to the vacancy rate when the latter is measured by the Help Wanted Index. This is consistent with the interpretation that the behavior of the Help Wanted Index, which relates to the broader Vtot, is different from the behavior of the relevant series here, VUN. Another exercise is to compute a ‘‘reconciling Q’’ to be denoted Qsim t; tþ1 , i.e. Qsim t; tþ1 ¼

M UN t;tþ1 V tot t

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Eran Yashiv tot

UN

which makes the series (V , V ) compatible and then see whether it reasonably fits with the model. The model predicts the following relationship:   s Vt Qt;tþ1 ¼ m Ut Note that the ‘‘reconciling,’’ simulated Q can be written as Qsim t; tþ1 ¼

UN M UN t;tþ1 UN V t tot ¼ Qt; tþ1 Vt V tot t

sim UN tot Note that as V UN t =V t ! 1, then Qt; tþ1 ! Qt; tþ1 . Running a regression based on the model’s formulation, i.e.   Vt ln Qt;tþ1 ¼ ln k a ln Ut

using Qsim for Q and the Help Wanted Index for V, I get an estimate of a of about 0.7, higher than the 0.4–0.5 range cited above for s. In other words, the simulated Q is more responsive to variations in market tightness than usual estimates of s indicate. This is likely to occur the more different are VUN and t 11 Vtot t . Another body of evidence is to be found in recent work on gross worker flows. Fallick and Fleischman (2004), using CPS data in the period 1994–2003, find that employment to employment flows are large; in fact 2/5 of new jobs represent employer changes. They also show that the cyclical properties of employment to employment flows are very different from those of the flows into and out of employment. Hence V NN ¼ V tot V UN appears to be of substantial magnitude. 11

The regression postulates

 tot  Vt Ut The model posits

a

Qsim t; tþ1 ¼ k

M UN t;tþ1 V tot t  UN  Vt ¼m Ut

Qsim t; tþ1 ¼

Hence

s

V UN t V tot t

 tot  a  UN  s UN Vt Vt Vt ¼m Ut Ut V tot t Taking logs and re-arranging I get k

tot ln m ln k ð1 sÞ ln ðV UN t =V t Þ tot tot ln ðV t Þ ln ðV t =U t Þ tot Only when V UN t =V t ¼ 1 and when m ¼ k will the estimate a equal s.

a¼s

Evaluating the Performance of the Search and Matching Model

529

Pissarides (1994) offers one possible explanation for the higher volatility of vacancies that relate to both job to job movements (VNN in the above notation) and to movements from out of employment (VUN): these ‘‘broader’’ vacancies (i.e. Vtot) are more responsive to productivity changes and hence are more volatile. The reason for the increased responsiveness is that employed workers, as well as unemployed workers, change their search activities following productivity changes, thereby affecting the incentives of firms in opening job vacancies. The conclusion from this discussion is the following: the present model, being an aggregate, representative firm-type of model, does not deal with job to job movements (as is also the case for the recent literature cited below), and thus the Help Wanted Index is not the relevant series to use as it probably does not behave like the relevant series (VUN). Given that the latter is unobserved, I resort to focusing on the observed worker flow series (i.e. on Q  VUN) whenever comparing the model to the data (though the model will generate predictions also with respect to V). The above discussion also implies that care must be taken when using or discussing vacancy data in the U.S. economy. 3.3. The flow of hires and the worker job-finding rate The discussion on the relevant pool of unemployment makes it clear that it is important to analyze the flow of matches or hires using out of the labor force to employment flows as well as unemployment to employment. This implies that in addition to different unemployment pools U, there will be different matching flows M and consequently different worker job-finding rates P ¼ M/U. Figure 3 presents these different rates. The 0 and 1 series – which are very close in the figure – represent the ones derived from the official rate and from the official+want a job series, respectively, both depicted in Figure 1. They imply unemployment durations of 17.4 and 16.5 weeks, respectively. In comparison, official BLS data on duration indicate 15.1 weeks on average (with 2.5 weeks standard deviation) in the same period. The other series – specifications 2,3, and 4 – evidently imply higher durations: 28.0, 39.0, and 91.8 weeks, respectively. All rates are highly correlated with the notable exception of specification 4, which is almost uncorrelated with the 0 and 1 specifications (correlations of 0.10 and 0.03, respectively) and weakly to moderately correlated with the 2 and 3 specifications (0.23 and 0.53, respectively). Table 3 compares these estimates, based on aggregate worker flow and nonemployment stock data, to the estimates of the afore-cited Jones and Riddell (2000) micro-based, CPS 1994–1998 study: The official pool has a slightly higher job finding than the upper bound of the Jones and Riddel estimate (i.e. 0.80 compared to 0.73). From the three intermediate pools, specification 2 has a job-finding rate (0.47) that is consistent with a mixture of the unemployed and the marginally attached. The largest pool, that includes all non-employed workers (specification 4 here), has a job-finding rate

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Figure 3.

Alternative specifications of the job-finding rate P

1.2 1.0 0.8 0.6 0.4 0.2 0.0 1980

1985

1990

1995

0 1 2

3 4

2000

(0.14) that seems to be consistent with a mixture of the unattached with the other two groups. 3.4. Wages The existence of diverse data series for wages with different cyclical properties was noted by several papers (see in particular Abraham and Haltiwanger, 1995, Abraham et al., 1999, and Krueger, 1999). The discussion in these papers does not lead to any definite conclusion as to which series is the most appropriate. Figure 4 illustrates one aspect of this issue by plotting BEA series of the labor share s ¼ W/(F/N), once using total compensation12 and once using wages. The series are correlated 0.83 but have a number of important differences: the wage series declines more over time, is lower by 10 percentage points on average, and displays much more variation (coefficient of variation of 0.037 relative to 0.016 for the other series). It should also be noted that both series have very weak correlation with the cycle: the compensation series has 0.05 correlation with the employment rate and the wage series has a 0.12 correlation. In what follows I use the compensation series as it takes all firm’s wagerelated costs, which are the relevant concept in the model.

12

Defined as total compensation of employees relative to GDP; this includes, beyond wages and salaries, supplements such as employer contribution for employee pension and insurance funds and employer contribution for goverment social insurance.

Evaluating the Performance of the Search and Matching Model

Table 3. Specification

Worker job finding rates P, quarterly 1994–1998

Average

0 1 2 3 4

531

0.80 0.82 0.47 0.39 0.14

Jones and Riddell (2000) Range

Specification Unemployed

0.49–0.73

Marginally attached

0.27–0.49

Unattached

Around 0.10, always below 0.14

Notes: 1. The five specifications – columns (0)–(4) correspond to the definitions in Table 1. 2. The Jones and Riddel (2000) numbers in the last column are taken from their discussion on pp. 10–11 of their Figure 1; they report monthly rates, which have been converted to quarterly here.

Figure 4.

The labor share s

0.60

0.56

0.52

0.48

0.44 70

75

80

85 compensation

90

95

00

wages

3.5. Other data series Figure 5 shows the other data series to be used in the empirical work below: productivity growth GX (gross rate), the discount factor b, and the separation rate d (separately for the official unemployment specification d0 and for the others d). For productivity growth I take the rate of change of GDP per worker; for the discount factor I take b ¼ 1/(1+r), where r is the cost of firm finance (weighted average of equity finance and debt finance); for the separation rate I take the flow from employment divided by the stock of employment.

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Figure 5.

GX, b, d0, d

1.03

1.3

1.02

1.2

1.01

1.1

1.00

1.0

0.99

0.9 0.8

0.98 70

75

80

85

90

95

00

70

75

80

85

G_X

90

95

00

95

00

BETA

0.07

0.11

0.06

0.10

0.05 0.09 0.04 0.08

0.03 0.02

0.07 70

75

80

85

90

DELTA0

95

00

70

75

80

85

90

DELTA

3.6. Data properties Table 4 reports the first two moments of all relevant data series, including measures of persistence and co-movement. When looking at these moments it is important to recall that the business cycle is most clearly manifested in the labor market – there is high correlation between employment and output and their volatility is similar. The table describes the key moments across the different specifications of the unemployment pool (and the labor force). Column 0 is the benchmark specification – it uses the official unemployment pool. Column 1 is the ‘lower bound’ specification with unemployment including the official pool and those workers out of the labor force indicating that they ‘‘want a job.’’ Columns 2 and 3 add to column 1 a fraction – 1 5% and 30%, respectively – of the remaining persons out of the labor force. Column 4 considers the entire working age population as the labor force. The following major properties can be said to characterize the data: Persistence. All the main labor market variables are persistent: the rate of unemployment (and thus employment), hiring, separation, and the wage share all

Evaluating the Performance of the Search and Matching Model

Table 4.

Data properties

533

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Eran Yashiv

Table 4. (Continued )

Notes: 1. All data are quarterly for the period 1970:I–2003:IV, except for hires and separations which begin in 1976:I and end in 2003:III. 2. The different columns differ with respect to the specification of the unemployment pool. I defiine e and L ¼ N þ U; where U0 is the official pool and U e are the following additions: U ¼ U0 þ U Column 1 adds workers out of the labor force indicating that they ‘‘want a job.’’ Columns 2 and 3 add to the latter column 15% and 30% of the remaining persons out of the labor force. Column 4 considers the entire working age population as the labor force. Column 0 is the official unemployment pool.

exhibit high persistence. There is some variation across the different cases: unemployment persistence increases slightly as the unemployment pool is expanded and the reverse is true for hiring and separation. At the same time productivity growth and the discount factor are not persistent at all. Note that one of the three driving shocks – the separation rate – is persistent. Volatility. (i) The table indicates differential volatility across variables and across specifications of the pool of searching workers. With only the officially unemployed considered searching (column 0), the volatility of the unemployment rate is the highest at 0.22 in terms of log levels; less volatile are the hiring and separation rates, which have comparable volatility at about 0.13–0.16; the discount factor has a volatility of 0.06 in the same terms; the volatility of the rate of employment (n ¼ N/L) is of the same order of magnitude as that of the wage share at around 0.015 (in the same log terms); productivity growth at 0.007 has even lower volatility. Note that this is somewhat akin to investment behavior: the capital stock (here the employment stock) is much less volatile than investment (here the hiring flow). Note too the relatively high volatility of the rate of separation and of the discount factor, which are driving factors. (ii) When

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535

moving across columns to a broader specification of the pool of searching workers some patterns change: in relative terms, hiring and separation become more volatile (at 0.07–0.10 in column 4) than unemployment (at 0.07). In absolute terms the employment rate becomes more volatile while unemployment, hiring, and separation (in rates) all become less volatile. Co-movement. (i) Hiring rates (m) co-vary positively with the unemployment rate, while the worker job-finding rate usually co-varies negatively with the same variable. This means that hiring flows (in rates) are counter-cyclical (see the discussion of the various unemployment pools above),13 but job-finding rates P ¼ m/u are pro-cyclical. Note that in recessions U rises and P falls; hence M=L ¼ PU =ðN þ UÞ rises as the effect of a rising U dominates the fall in P and any change in L. (ii) Separation rates (d) are counter-cyclical. Note that hiring and separation rates move together, unlike the widely known negative correlation between job creation and job destruction in the manufacturing sector. (iii) The labor share in income (s) varies between acyclicality and counter-cyclicality (with respect to employment) according to the specification of the labor force. Throughout it covaries positively with the hiring rate. This means that when F/N rises in booms, the wage (W) rises by less and thereby the labor share (s) declines. (iv) There is low correlation between employment and productivity growth, a fact which has received considerable attention in the RBC literature. Note that the data here are in terms of the log of the employment rate and the first differences of the log of productivity but that they display essentially the same co-movement pattern that is discussed in the business cycle literature in terms of HP-filtered and logged productivity and employment. (v) Employment has low co-variation with the discount factor. Note some non-intuitive aspects of these data moments: in booms hiring and separation rates fall and the labor share either does not change or falls. Hiring is strong at the same time as the share of wages is high. Recently, some authors have questioned the ability of the model to explain these data. Cole and Rogerson (1999) found that the model can account for business cycle facts only if the average duration of unemployment is relatively high (9 months or longer) and substantially longer than average duration in BLS data on the unemployed. The intuition for this critique is the following: the data point to a negative correlation between job destruction and job creation. Job destruction leads to higher unemployment thereby raising the incentives for job creation. In order for job creation to be contemporaneously negatively correlated with job destruction there has to be a mitigating factor and that role is played by the relatively low worker job-finding rate, which is the inverse of unemployment duration.

13

This is also true for logged and HP-filtered variables.

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Shimer (2005) showed that in the standard matching model productivity shocks of empirically plausible magnitude cannot generate the observed, large cyclical fluctuations in unemployment and vacancies. The key reason for this result is that the standard model assumes that wages are determined by Nash bargaining, which in turn implies that wages are ‘‘too flexible.’’ Thus, for example, following a positive productivity shock, wages increase, absorbing the shock, thereby dampening the incentives of firms to create new jobs. In particular, while the model predicts roughly the same volatility of the vacancy to unemployment ratio and of productivity, the data indicate this ratio is actually 18 times more volatile. Veracierto (2002) has shown that the model fails to simultaneously account for the observed behavior of employment, unemployment and out of the labor force worker pools. Analyzing the RBC model with search and matching that makes an explicit distinction between these states, he finds that the model has serious difficulties in reproducing the labor market dynamics observed in U.S. data. In particular, employment fluctuates as much as the labor force while in the data it is three times more variable, unemployment fluctuates as much as output while in the data it is six times more variable, and unemployment is acyclical while in the data it is strongly counter-cyclical. An underlying reason is that search decisions respond too little to aggregate productivity shocks. Fujita (2004) conducted empirical tests showing that vacancies are much more persistent in the data than the low persistence implied by the model. Taken together these studies cast doubt on the model’s ability to fit the data on two key dimensions: the volatility and duration of unemployment, and the volatility and persistence of vacancies. They also point to serious shortcomings in accounting for aggregate wage behavior. I turn now to the calibration of the model under the different data specifications discussed above and the evaluation of its performance against the data.

4. Calibration and model-data fit In this section I calibrate the model and examine its performance, taking into account the alternative formulations of the pool of searching workers discussed above. I begin by discussing calibration values in Section 4.1. I then (4.2) examine the performance of the model. 4.1. Calibration Calibration of the model requires the assignment of values to the parameters of the production, hiring costs, and matching functions as well as to the wage bargain, to the steady-state values of the exogenous variables (labor force growth, productivity growth, the discount factor and the separation rate) and to the steady-state values of the endogenous variables. To do so I use, wherever possible,

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537

results from econometric studies and average values of the longest sample period available. I then solve for all other values using the steady state relations (Equations (25) and (26)). There are three structural parameters that are at the focal point of the model and that reflect the operation of frictions. These are the matching function parameter s (elasticity of unemployment), the wage parameter Z, and the hiring function convexity parameter g. For s I use Blanchard and Diamond’s (1989) estimate of 0.4. Structural estimation of the model using U.S. corporate sector data in Merz and Yashiv (2004) indicates a value of g, the convexity parameter of the hiring cost function, around 2, i.e. a cubic function (g þ 1 ¼ 3) for hiring costs. These costs fall on vacancies and on actual hires, with f being the weight on the former. I follow the estimates in Yashiv (2000a) and set it at 0.3. The wage parameter Z depends on the asymmetry of the bargaining solution (x) and on the replacement ratio (t). This is obviously a difficult case for calibration as x is not directly observed and t depends not only on the value of benefits relative to wages but also on actual take-up rates. Following Anderson and Meyer (1997) I postulate a value of 0.25 for the latter; to test for robustness I tried a far higher value, finding that this change has a very small effect on the resulting moments. For x rather than imposing it, I solve it out of the steady state relations. For the production function parameter a I use a fairly traditional value of 0.68, which is also the structural estimate of this parameter in Merz and Yashiv (2004). For the values of the exogenous variables, I use sample average values as follows: in quarterly terms the sample average of the rate of productivity growth (GX 1) is 0.35%; the sample average rate of labor force growth (GL 1) varies according to the definition of the unemployment pool between 0.43% and 0.36%; the stochastic discount factor b – defined as the weighted price of capital (equity and debt financed) – has a sample average of 0.993, which is close to the value used in many business cycle studies; the sample average for the separation rate d is 8.5%.14 For the steady-state values of the endogenous variables, given the discussion above on the relevant unemployment pool, I modify n and u according to the specifications used above. Calibration of Q, the matching rate for vacancies, is problematic as there are no wide or accurate measures of vacancy durations for the U.S. economy. Using a 1982 survey, Burdett and Cunningham (1998) estimated hazard functions for vacancies both parametrically and semi-parametrically finding that the general form of the job hazard function within the quarter is

14

For the case of the official unemployment pool I use d ¼ 4% reflecting the rate of separation from employment to this pool only.

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non-mononotic; based on their estimates the quarterly hazard rate should be in the range of 0.8 – 1. I thus take Q ¼ 0:9 which is also the value used by Merz (1995) and Andolfatto (1996). This implies a certain steady-state value for the vacancy rate (v). I use the average of the labor share in income s which is 0.58. With the above values, I solve the steady-state relations (25–26) for the steady-state vacancy rate v, the hiring cost scale parameter Y, the matching function scale parameter m, and the wage parameter x. I can then solve for the steady-state values of market tightness v/u, the worker job-finding rate P, perperiod profits p, and the match asset value l. Table 5 summarizes the calibrated values for the different specifications of the unemployment pool. Note two features of the implied results: (i) The implied wage parameter Z, encompassing the worker bargaining strength and the replacement ratio, varies between 0.4 and 0.6 across specifications. (ii) Across specifications 1–4, per-period profits (p) are around 0.07–0.13 (in average output terms) and the asset value of the match (l) is around 0.7–1.5. Given that s ¼ 0:58 this means that asset values are about 1.2–2.6 the labor share in income. In other words the match is worth around 1–2.5 quarters of wages in present value terms. This quantification of asset values is based on the values of panel (a). The latter include three values which have not been much investigated in the literature: the vacancy match rate Q and the parameters f and g of the hiring cost function. When varying the latter values within reasonable ranges, there was no significant change in the results. Hence, the steady state calibration gives a sense of the magnitude of hiring costs or match asset values. I return to this point below. As to the stochastic shocks, in order to get numerical values for the coefficient matrix P and for the variance–co-variance matrix of S, I estimate a first-order VAR in labor productivity growth, the discount factor, and the rate of match separation. I estimate this VAR with the relevant data discussed above (see the appendix for definitions and sources).

4.2. Model-data fit I now turn to examine the performance of the model.15 Table 6 shows the moments implied by the model and those of the data (repeating the moments reported in Table 4).

15

I use a modified version of a program by Craig Burnside in Gauss to solve the model (see Burnside, 1997).

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Evaluating the Performance of the Search and Matching Model

The following conclusions can be drawn: Persistence. The model captures the fact that across all specifications u,m, and s are highly persistent. The model tends to somewhat overstate this persistence. Volatility. Column 1 captures very well the volatility of employment and unemployment. Hiring volatility is understated by the model; best performing is column 0 which captures three quarters of this volatility. As to the labor share, the model overstates its volatility with column (4) being the closest to the data. No single specification produces a high model-data fit for all variables. Co-Movement. Under most specifications of the model, the counter-cyclical behavior of hiring and the pro-cyclical behavior of the worker job-finding rate are well captured. The behavior of the labor share is not captured: while in the data it is acyclical to counter-cyclical (across specifications of the unemployment pool) and co-varies moderately with the hiring rate, it is strongly pro-cyclical in the model and has a strong negative relationship with hiring, expect for column 4 where it is positive but overstated. Overall fit. The model captures the persistence, volatility and some of the comovement in the data. The major problem concerns the labor share in income which is not well captured. Column 1 of the model fits the data in terms of employment and unemployment behavior and has reasonable but limited success in fitting hiring flows (fits persistence and cyclicality, understates volatility). Column (4) seems to be providing the better fit for the labor share, doing relatively well on volatility, reasonably well on persistence, moderately well on comovement with hiring and doing badly in terms of cyclical behavior. The model also generates predictions with respect to the behavior of v and q which are unobserved in U.S. data as explained in Section 3.2 above. Strictly speaking the moments involving these variables cannot be compared to the data. But some predictions look reasonable based on the theory and on experience in

Table 5. Baseline calibration values (a) Parameters, exogenous variables, and steady-state values Parameter/Variable

symbol

Production Matching Hiring Hiring II productivity growth labor force growth discount factor separation rate Unemployment Labor share Vacancy matching rate

1 a s g f GX 1 GL 1 b d u s Q

(0)

(1)

0.004296

0.004199

0.0404 0.063

0.104

(2)

(3)

0.68 0.4 2 0.3 0.003536 0.004078 0.003974 0.9929 0.0854 0.164 0.217 0.579 0.9

(4)

0.003631

0.395

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Table 5. (Continued ) (b) Implied values

Matching Hiring Wage bargaining parameter Wage parameter Vacancy rate Market tightness Workers’ hazard Profits Asset Value

m Y x Z v v u

P p l

(0)

(1)

(2)

(3)

(4)

0.80 465 0.37 0.44 0.047 0.74 0.67 0.05 1.02

0.85 82 0.41 0.48 0.089 0.86 0.77 0.07 0.73

0.69 109 0.45 0.53 0.083 0.51 0.46 0.09 0.96

0.60 127 0.49 0.56 0.078 0.36 0.32 0.10 1.12

0.42 171 0.56 0.63 0.060 0.15 0.14 0.13 1.49

Notes: 1. The implied values of v, m, Y and Z are solved for using the steady-state relationships as follows: Matching hazard: v s Q¼m u Beveridge curve:  v s  v  1Þ ¼ m u 1 u Vacancy creation: ðd þ GL

Yðf þ ð1


 fV þ ð1 fÞQV g ¼ Q fÞQÞ N 1

Wage solution: s ¼ Z ð1

aÞ

"

1

1

GX b p ð1 dÞGX b


 fV þ ð1 fÞQV gþ1 þY ax N ð1 þ gÞð1

ð1Þ

# ! aþg þ Pl aÞð1 xð1 þ a þ gÞÞ

The other variables are solved for using P¼

Qv u 0

2

p¼

B B B B ZÞBð1 B B @

l¼ 1

GX b p ð1 dÞGX b

ð1

1

6 1 ax 6
 6 fV þ ð1 fÞQV gþ1 aÞ6 þY 6 N ð1 þ gÞð1 6 4 1 þ ZP 1

GX b ð1 dÞGX b

31 7C 7C 7C aþg 7C C aÞð1 xð1 þ a þ gÞÞ7 7C 5C A

2. The five specifications — columns (0) – (4) correspond to the definitions in the notes to Table 4.

Evaluating the Performance of the Search and Matching Model

Table 6.

541

Model evaluation: alternative specifications

(a) Model vs. Data

^ u^ t 1 Þ rðu; ^ t; m ^ t 1Þ rðm rð^st ; s^t 1 Þ stdðn^ t Þ stdðu^ t Þ ^ tÞ stdðm stdð^st Þ ^ tÞ rðu^ t ; m rðn^ t ; P^ t Þ rðn^ t ; s^t Þ ^ t ; s^t Þ rðm

Data Model Data Model Data Model Data Model Data Model Data Model Data Model Data Model Data Model Data Model Data Model

(0)

(1)

(2)

(3)

(4)

0.962 0.989 0.907 0.991 0.884 0.982 0.015 0.020 0.218 0.298 0.125 0.097 0.016 0.089 0.920 0.997 0.909 0.999 0.060 0.995 0.272 0.985

0.971 0.983 0.853 0.986 0.884 0.976 0.022 0.021 0.188 0.182 0.085 0.051 0.016 0.056 0.810 0.997 0.933 1.000 0.160 0.997 0.451 0.989

0.980 0.990 0.844 0.991 0.884 0.988 0.024 0.032 0.124 0.165 0.083 0.035 0.016 0.052 0.860 0.998 0.802 1.000 0.300 0.999 0.445 0.996

0.985 0.993 0.836 0.993 0.884 0.992 0.028 0.042 0.100 0.151 0.080 0.022 0.016 0.049 0.880 0.999 0.591 1.000 0.390 1.000 0.440 0.999

0.989 0.996 0.805 0.992 0.884 0.996 0.042 0.072 0.069 0.110 0.073 0.018 0.016 0.038 0.870 0.982 0.242 1.000 0.520 1.000 0.414 0.984

(b) Model predictions

rð^vt ; v^t 1 Þ rðq^ t ; q^ t 1 Þ stdð^vt Þ stdðq^ t Þ rðu^ t ; v^t Þ rð^vt ; q^ t Þ rð^vt ; s^t Þ

(0)

(1)

(2)

(3)

(4)

0.952 0.988 0.040 0.135 0.956 0.966 0.981

0.961 0.981 0.037 0.088 0.985 0.990 0.995

0.986 0.989 0.052 0.087 0.998 0.999 0.999

0.992 0.992 0.064 0.086 1.000 1.000 1.000

0.996 0.996 0.103 0.085 0.999 1.000 0.999

Note: The different specifications correspond to the definitions in Table 4.

other economies: the negative correlation of u and v (the ‘Beveridge curve’) and persistent vacancy rates. No single specification matches the data on all dimensions. Focusing on the behavior of unemployment, employment, and hiring, it looks as though specification 1 (the official unemployment pool and the ‘want a job’ category) is the most fitting, though specification 0 (official unemployment pool) cannot be ruled out. I turn now to look at the mechanisms driving this model-data fit.

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5. The underlying mechanism The discussion up till now has shown which data specification is best explained by the model. The natural question to ask now is what underlies the fit. In order to understand the essential mechanism in operation, it is best to consider the following steady-state equation which elaborates on the formulation in (25) and combines it with (26):   Y ~ gþ1 ðv=1 uÞg GX b ¼ Q m ðv=uÞ s 1 G X bð1 dÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} F 0 2 2 6ð1 6 6 6 6 6 6 6 4

B ZÞ@ð1

6 aÞ4

þY



1 1 ax

fV þð1 fÞQV N

h 1 þ ZP 1

gþ1

aþg ð1þgÞð1 aÞð1 xð1þaþgÞÞ

GX b GX bð1 dÞ

i

313

7C7 5A7 7 7 7 ð29Þ 7 7 7 5

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} p

where Q~ ¼ f þ ð1 fÞQ Equation (29) shows the vacancy creation decision as an optimality condition equating the marginal costs of hiring with the asset value of the match. The LHS are marginal vacancy creation costs. They take into account Q, the probability of filling the vacancy. It is clear that the responsiveness of vacancies (v) depends on the two elasticity parameters g (of the hiring cost function) and s (of the matching function). The higher is each of these, the less responsive is vacancy creation. Many studies have assumed this is a linear function (g ¼ 0), thereby imposing a particular shape on the marginal cost function. The RHS is the asset value of the match. This value can vary because perperiod profits p vary or because the discount factor F varies. The former may vary because of changes in the surplus itself or changes in the sharing of the surplus, with a key parameter being Z. Any policy change in the replacement ratio t, for example, will change Z and consequently the sharing of the match surplus. Changes in the discount factor F can happen because of changes in productivity growth (GX), changes in the discount factor (b), or changes in match dissolution rate (d). Hence the essential mechanism is the following: changes in the long run (nonstochastic steady state) and in the stochastic dynamics are generated by changes in the surplus, in surplus sharing, in productivity growth, in the discount rate, and in the match dissolution rate, which change the match asset values (profitability in present value terms).

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543

The following ingredients are therefore essential: (i) The shape of the hiring costs function determining the LHS of Equation (29), i.e. the marginal cost function. In this context g and s are key parameters. (ii) The formulation of the match surplus – this depends both on the data used and on all key parameters of the model. (iii) the surplus sharing rule, where Z is the key parameter. (iv) the discounting of the match surplus – here the data used (for GX, b, and d) and their stochastic properties are key. What is important and what is less important out of these elements of the model? One key element is the convexity of hiring costs. Hence there is an important role for the hiring costs convexity parameter g. This parameter is important because it is the main determinant of the elasticity of vacancies with respect to the present value of the match. Values of g determine the persistence and volatility of vacancy creation. The latter then influences the second moments of matching, and consequently the moments of unemployment and employment. Another key element is the role played by the match dissolution or separation rate d. This is a variable with a relatively high mean and is therefore the main determinant of the relevant discount factor; as it has relatively high volatility and persistence it makes the present value of the match volatile and persistent. This in turn engenders the volatility and persistence of vacancies, hiring and unemployment. Thus, the performance of the model hinges to a large extent on the formulation of g and on the stochastic properties of d. This point is elaborated and reinforced in Yashiv (2005a). The lack of fit in part of the literature is due to the use of a linear hiring cost function (g ¼ 0) instead of a convex one (g ¼ 2 here) and due to the incomplete incorporation of the separation rate d. Other elements of the model play a smaller role. The wage parameter Z basically has a scale effect on per-period profits and hence on the scale of asset values. It therefore affects the value of the variables at the steady state but does not affect the dynamics, as it does not affect the response of vacancy creation to asset values. The matching function parameter s, that does have this ‘elasticity’ type of affects, has a range of possible variation that is much smaller than the variation in values of g. For example, a reasonable change in s would be 0.1 or 0.2 relative to the benchmark value (which is 0.4), but a move from linear (g ¼ 0) to cubic (g ¼ 2) costs is a change of 2 in the value of g. The interest rate and the rate of productivity growth in their turn play a much smaller role than the separation rate in discounting future values. While d has a sample mean of 8.6% and a standard deviation of 0.8%, the rate of productivity growth (GX 1) has a sample mean of 0.4% and standard deviation of 0.6% and the rate of interest (1/b 1) has a sample mean of 1.4% and standard deviation of 5.5%. Further insight can be derived from examination of the time series of match asset values, or, equivalently, marginal hiring costs, i.e. the time variation in the LHS of (8). Figure 6 plots these time series, computed using the parameter

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Eran Yashiv

Figure 6.

Asset values or marginal hiring costs

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1980

1985

1990

1995

2000

values of the preferred specification (column 1 of Table 5), showing NBER recessions dates in shaded areas. Table 7 reports key statistics of this series. Mean asset values or marginal costs are close to the steady-state value reported in Table 5 and are equivalent to about one quarter of wage payments. The median is slightly lower due to some skewness. The series is highly volatile, comparable to the volatility of unemployment (in log terms), as reported in Table 4. It is also highly persistent. This is in line with the above explanation, whereby fluctuations in match asset values can explain the fluctuations in key labor market variables; as the latter are volatile and persistent, it is not surprising to see that asset values are as well. Both the figure and the table indicate that it is counter-cyclical: marginal costs are highly (positively) correlated with the rate of unemployment and are negatively correlated with the worker jobfinding rate and with two cyclical indicators (GDP and employment, logged and HP-filtered). Thus recessions are times of rising asset values or marginal costs. This is consistent with the counter-cyclicality of hiring rates, discussed above, with which asset values are highly correlated. The afore-going discussion implies the following modeling lessons for the aggregate U.S. labor market: (i) In terms of the model formulation, two ingredients are important: convexity of the hiring costs function and allowing separation costs to affect the dynamics, in addition to, and contemporaneously with, any other shock, such as productivity shocks. The latter turn out not to play a big role. (ii) In terms of U.S. data, from amongst the alternative pools of searching workers, the one consisting of official unemployment and the ‘want a job’ category seems to be the most consistent with the model.

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Evaluating the Performance of the Search and Matching Model

Table 7. Yðf þ ð1

fÞQt Þ

(a) Univariate

Asset value or marginal hiring costs key stochastic properties  g fV t þ ð1 fÞQt V t Nt

Levels Mean Median Standard deviation Auto-correlation Logs Mean Median Standard deviation Auto-correlation

0.61 0.59 0.13 0.89 0.52 0.53 0.21 0.89

(b) Co-movement with key labor market variables Correlations m n

0.998 0.87 0.63 0.40

u p wn f

(c) Co-movement with cyclical indicators Correlations, logged, HP filtered MC/Asset Value t 4 t 3 0.04 0.13 GDP Ft 0.17 0.23 Employment Nt

t

2 0.20 0.34

t

1 0.32 0.46

t 0.46 0.52

tþ1 0.54 0.56

tþ2 0.48 0.45

tþ3 0.32 0.29

tþ4 0.22 0.19

Notes: 1. For Y, f and g the values used are those of column 1 in Table 5. 2. For data sources, see the appendix. 3. In panel (c) all variables are logged and HP filtered.

(iii) Hiring rates and estimated asset values are counter-cyclical, volatile, and persistent. This implies that it is important to look at asset values (i.e. at present values), rather than at contemporaneous cyclical variables, in order to understand cyclical fluctuations in the labor market.

6. Conclusions The paper has formulated a model of the U.S. aggregate labor market using loglinear approximation of the search and matching model. It has looked at alternative formulations of the data that would be consistent with the concepts of the model. Using a VAR of the actual data it injected driving shocks into the model. Comparing the resulting moments implied by the model to the moments

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in U.S. data, it has shown that it can account for much of observed labor market fluctuations. In particular, the model fits the data on persistence and volatility of most variables, on the negative relationship between vacancies and unemployment, and on the pro-cyclicality of the workers’ job-finding rate. For the same formulation of the data, it fails to capture the acyclicality of the labor share and its moderate positive co-variation with hiring, and it understates the latter’s volatility; it is, however, able to come closer to capturing these features given a different configuration of the data. The analysis has produced an empirically grounded version of the search and matching model – complete with parameter values and data series – that can be used to study the U.S. labor market, including policy questions. The paper raises a number of issues for further research. A key one is the need to further explain the mechanisms in operation. In particular, one may ask what is the role of search and matching frictions in the dynamics; for example, how would different degrees of frictions lead to different outcomes. This issue is the subject of current research (see Yashiv, 2005a). Other issues are: do the results carry over to other economies and, if so, what are the cross-country differences in parameter values and in the dynamics? For example, it would be of interest to see whether such differences can explain the different U.S.–European unemployment experiences. Acknowledgment I am grateful to Jordi Gali, Martin Lettau, Fabien Postel-Vinay, and Etienne Wasmer for useful conversations, to Craig Burnside for advice, data, and software; to Jeffrey Fuhrer, Hoyt Bleakley, Ann Ferris, and Elizabeth Walat for data; and to Darina Waisman for excellent research assistance. Any errors are mine. References Abowd, J. and A. Zellner (1985), ‘‘Estimating gross labor-force flows’’, Journal of Business and Economics Statistics, Vol. 3, pp. 254–293. Abraham, K. (1987), ‘‘Help-wanted advertising, job vacancies and unemployment’’, Brookings Papers on Economic Activity, Vol. 1, pp. 207–248. Abraham, K.G. and J.C. Haltiwanger (1995), ‘‘Real wages and the business cycle’’, Journal of Economic Literature, Vol. 33(3), pp. 1215–1264. Abraham, K.G., J.R. Spletzer and J.C. Stewart (1999), ‘‘Why do different wage series tell different stories?’’, American Economic Review, AEA Papers and Proceedings, Vol. 89(2), pp. 34–39. Anderson, P.M. and B.D. Meyer (1997), ‘‘Unemployment insurance takeup rates and the after tax value of benefits’’, Quarterly Journal of Economics, Vol. CXII, pp. 913–937.

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Andolfatto, D. (1996), ‘‘Business cycles and labor market search’’, American Economic Review, Vol. 86, pp. 112–132. Blanchard, O.J. and P. Diamond (1989), ‘‘The beveridge curve’’, Brookings Papers on Economic Activity, Vol. 1, pp. 1–60. Blanchard, O.J. and P. Diamond (1990), ‘‘The cyclical behavior of the gross flows of U.S. workers’’, Brookings Papers on Economic Activity, Vol. 2, pp. 85–155. Bleakley, H., A.E. Ferris and J.C. Fuhrer (1999), ‘‘New data on worker flows during business cycles’’, New England Economic Review, July–August, pp. 49–76. Burdett, K. and E.J. Cunningham (1998), ‘‘Toward a theory of vacancies’’, Journal of Labor Economics, Vol. 16(3), pp. 445–478. Burnside, A.C. (1997), ‘‘Notes on the linearization and GMM estimation of real business cycle models’’, Mimeo. Cahuc, P., F. Marque and E. Wasmer (2004), ‘‘A theory of wages and labor demand with intrafirm bargaining and matching frictions’’, Mimeo. Castillo, M.D. (1998), ‘‘Persons outside the labor force who want a job’’, Monthly Labor Review, Vol. 121(7), pp. 34–42. Clark, K.B. and L.H. Summers (1979), ‘‘Labor market dynamics and unemployment: a reconsideration’’, Brookings Papers on Economic Activity, Vol. 1, pp. 13–60. Cole, H.L. and R. Rogerson (1999), ‘‘Can the Mortensen–Pissarides matching model match the business cycle facts?’’, International Economic Review, Vol. 40(4), pp. 933–960. Diamond, P.A. (1982), ‘‘Wage determination and efficiency in search equilibrium’’, Review of Economic Studies, Vol. 49, pp. 761–782. Fallick, B., and C.A. Fleischman (2004), ‘‘Employer to employer flows in the U.S. labor market: the complete picture of gross worker flows’’, Mimeo. Fama, E.F. and K.R. French (1999), ‘‘The corporate cost of capital and the return on corporate investment’’, Journal of Finance, Vol. LIV, pp. 1939–1967. Fujita, S. (2004), ‘‘Vacancy persistence’’, FRB of Philadelphia Working Paper No. 04-23. Jones, S.R.G. and W.C. Riddell (2000), ‘‘The dynamics of U.S. labor force attachment’’, Mimeo. Krueger, A.B. (1999), ‘‘Measuring labor’s share’’, American Economic Review, AEA Papers and Proceedings, Vol. 89(2), pp. 45–51. Merz, M. (1995), ‘‘Search in the labor market and the real business cycle’’, Journal of Monetary Economics, Vol. 36, pp. 269–300. Merz, M. and E. Yashiv (2004), ‘‘Labor and the market value of the firm’’, CEPR Discussion Paper no. 4184. Mortensen, D.T. (1970), ‘‘A theory of wage and employment dynamics’’, in: E.S. Phelps, A.A. Alchian and C.C. Holt, editors, The Microeconomic Foundations of Employment and Inflation Theory, New York: Norton.

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Mortensen, D.T. (1978), ‘‘Specific capital and labor turnover’’, Bell Journal of Economics, Vol. 9, pp. 572–586. Mortensen, D.T. (1982a), ‘‘The matching process as a non-cooperative bargaining game’’, in: J.J. McCall, editor, The Economics of Information and Uncertainty, New York: University of Chicago Press. Mortensen, D.T. (1982b), ‘‘Property rights and efficiency in mating, racing and related games’’, American Economic Review, Vol. 72, pp. 968–979. Mortensen, D.T. and C.A. Pissarides (1994), ‘‘Job creation and job destruction in the theory of unemployment’’, Review of Economic Studies, Vol. 61, pp. 397–415. Mortensen, D.T. and C.A. Pissarides (1999a), ‘‘Job reallocation, employment fluctuations, and unemployment’’, pp. 1171–1228 in: John B. Taylor and Michael Woodford, editors, Handbook of Macroeconomics Ch. 18, Vol. 1B, Amsterdam: North-Holland. Mortensen, D.T. and C.A. Pissarides (1999b), ‘‘New developments in models of search in the labor market’’, in: Orley Ashenfelter and David Card, editors, Handbook of Labor Economics Ch. 39, Vol. 3B, Amsterdam: North-Holland. Petrongolo, B. and C.A. Pissarides (2001), ‘‘Looking into the black box: a survey of the matching function’’, Journal of Economic Literature, Vol. 39(2), pp. 390–431. Pissarides, C.A. (1994), ‘‘Search unemployment with on-the-job search’’, Review of Economic Studies, Vol. 61, pp. 457–475. Pissarides, CA. (2000), Equilibrium Unemployment Theory, 2nd edition, Cambridge: MIT Press. Shimer, R. (2005), ‘‘The cyclical behavior of equilibrium unemployment and vacancies: evidence and theory’’, American Economic Review, Vol. 95(1), pp. 25–49. Stole, L.A. and J. Zwiebel (1996), ‘‘Intrafirm bargaining under non-binding contracts’’, Review of Economic Studies, Vol. 63, pp. 375–410. Veracierto, M. (2002), ‘‘On the cyclical behavior of employment, unemployment and labor force participation’’, WP 2002-12, Federal Reserve Bank of Chicago. Yashiv, E. (2000a), ‘‘The determinants of equilibrium unemployment’’, American Economic Review, Vol. 90(5), pp. 1297–1322. Yashiv, E. (2000b), ‘‘Hiring as investment behavior’’, Review of Economic Dynamics, Vol. 3, pp. 486–522. Yashiv, E. (2004), ‘‘Macroeconomic policy lessons of labor market frictions’’, European Economic Review, Vol. 48, pp. 259–284. Yashiv, E. (2005a), ‘‘Forward looking hiring behavior and the dynamics of the aggregate labor market’’, available at www.tau.ac.il/~yashiv/ Yashiv, E. (2005b), ‘‘Search and matching in macroeconomics’’, European Economic Review, forthcoming.

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Appendix: Data (Sources and definitions) Data sample All data are quarterly U.S. data for the period 1970:I–2003:IV, except for hires and separations which begin in 1976:I and end in 2003:III. Fundamental variables Variable

Symbol

Source

Unemployment – official pool Workers out of l.f. who ‘‘want a job’’ Unemployment – additional poolg Employment (total), household survey Vacancies – Index of Help Wanted ads Hires Separations Working age populationd

U0

CPS, BLS series id: LNS13000000 CPS, BLS series id: LNU05026639 CPS, BLS

Labor sharee Productivity Cost of finance (equity and debtf)

WAJ U~ N V QV dN POP s ¼ WN F

F N

r

CPS, BLS series id: LNS12000000 Conference Boarda CPS-based datab CPS-based datab CPS, BLS series id: LNU00000000 Table 1.12. NIPA, BEAc BLS Tables 1.1.5; 1.1.6 NIPA, BEA

BLS series are taken from http://www.bls.gov/cps/home.htm. a Data were downloaded from Federal Reserve Bank of St. Louis http://research.stlouisfed.org/fred2/ series/HELPWANT/10. b Boston Fed Computations, see Bleakley et al. (1999) for construction methodology. I thank Jeffrey Fuhrer and Elizabeth Walat for their work on this series. c http://www.bea.doc.gov/bea/dn/home/gdp.htm. d Total civilian noninstitutional population 16 years and older. e Total compensation of employees divided by GDP. f This is a weighted average of the returns to debt, rbt , and equity, ret constructed as follows: rt ¼ ot rbt þ ð1

ot Þret

with yt rbt ¼ ð1 tt ÞrCP t e cf b ret ¼ t þ e s yt e st

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Eran Yashiv

Fundamental variables (Continued ) where (i) ot is the share of debt finance as reported in Fama and French (1999). (ii) The definition of rbt reflects the fact that nominal interest payments on debt are tax deductible. is Moody’s seasoned Aaa commercial paper rate. The commercial paper rate for the first rCP t month of each quarter represents the entire quarter. The tax rate is t as discussed above. (iii) y denotes inflation and is measured by the GDP-deflator of NFCB GDP. (iv) For equity return I use the CRSP Value Weighted NYSE, Nasdaq and Amex nominal returns deflated by the inflation rate y.

g

e has five permutations as follows: The additional pool of unemployment U

Permutation number 0 1 2 3 4

e U

0 Want Want Want POP

a job a job +0.15(POP—N—U1) a job +0.3(POP—N—U1) N

Transformations of variables Variable

Symbol

Unemployment – total pool Labor force Employment rate

U ¼ U 0 þ U~ L ¼ N+U n ¼ NþUN0 þU~ u¼1 n m ¼ QV N

Unemployment rate Hiring rate Gross rate of labor force growth Discount factor

þN tþ1 GL ¼ U tþ1 U t þN t

1 b ¼ 1þr

CHAPTER 21

Productivity Growth and Worker Reallocation: Theory and Evidence Rasmus Lentz and Dale T. Mortensen Abstract Dispersion in labor and factor productivity across firms is large and persistent, large flows of workers move across firms, and worker reallocation is an important source of productivity growth. The purpose of the paper is to provide a formal explanation for these observations that clarifies the role of worker reallocation as a source of productivity growth. Specifically, we study a modified version of the Schumpeterian model of growth induced by product innovation developed by Klette and Kortum (2004). More productive firms are those that supply higher quality products in the model. We show that more productive firms grow faster and the reallocation of workers across continuing firms contribute to aggregate productivity growth if and only if current productivity predicts future productivity. We provide evidence in support of the hypothesis that more productive firms become larger in Danish data. In addition, we provide estimates of the distribution of productivity at entry and the parameters of the cost of investment in innovation function and other structural parameters that all firms are assumed to face by fitting the model to observations on value added, employment, and wages drawn from a panel of Danish firms for the years 1992–1997.

Keywords: labor productivity growth, worker reallocation, firm dynamics, firm panel data estimation JEL classifications: E22, E24, J23, J24, L11, L25, O3, O4

Corresponding author. CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 275 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)75021-5

r 2006 ELSEVIER B.V. ALL RIGHTS RESERVED

551

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Rasmus Lentz and Dale T. Mortensen

1. Introduction In their review article on firm productivity, Bartelsman and Doms (2000) draw three lessons from empirical studies based on longitudinal plant and firm data: First, the extent of dispersion in relative productivity across production units, firms or establishments, is large. Second, productivity rank of any unit in the distribution is highly persistent. Third, a large fraction of aggregate productivity growth is the consequence of worker reallocation. In their recent study of wage and productivity dispersion trends in U.S. Manufacturing, Dunne et al. (2004) found that wage differences in wages across plants is an important and growing component of total wage dispersion, most of the between plant increase in wage differences is within industries, and wage and productivity dispersion between plants has grown substantially in the recent past. Although the explanations for productive heterogeneity across firms are not fully understood, economic principles suggest that wage and productivity dispersion should induce worker reallocation from less to more productive firms as well as from exiting to entering firms. Indeed, workers should move voluntarily to capture wage gains while more productive employer have an incentive to expand production. There is ample evidence that workers do flow from one firm to another frequently. As Davis et al. (1996) and others document, job and worker flows are large, persistent, and essentially idiosyncratic in the U.S. Recently, Fallick and Fleischman (2001) and Stewart (2002) found that job- to-job flows without a spell of unemployment in the U.S. represent at least half of the separations and is growing. In their analysis of Danish matched employer–employee IDA data, Frederiksen and Westergaard-Nielsen (2002) report that the average establishment separation rate over the 1980–1995 period was 26%. About two-thirds of the outflow represents the movement of workers from one firm to another. Using firm-level data based on the same source, Christensen et al. (2005) document considerable cross firm dispersion in the average wage paid. Furthermore, they show that separation rates decline steeply with a firm’s relative wage suggesting that workers do move from lower to higher paying jobs. Baily et al. (1992) found a strong positive correlation between productivity and wages paid across plants in U.S. Manufacturing and Bartelsman and Doms (2000) reported that the finding is present in similar studies. Mortensen (2003) argues that dispersion in wages paid for observably equivalent workers is hard to explain unless they reflect differences in firm productivity. To the extent that wage dispersion reflects differences in firm-specific labor productivity, direct voluntary flows of workers from lower to higher paying firms as well as indirect flows through unemployment improve the overall allocation of labor in the economy. As noted earlier, the studies cited by Bartelsman and Doms (2000) document that labor reallocation of this form is a major contributor to aggregate productivity growth. The purpose of this paper is to clarify the role of worker reallocation in the growth process. The model developed by Klette and Kortum (2004), which itself

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553

builds on the endogenous growth model of Grossman and Helpman (1991), is adapted for this purpose. Their version of the model is designed to be consistent with stylized facts about product innovation and its relationship to the dynamics of firm size evolution and the distribution of firm size. In the model, firms are monopoly suppliers of differentiated products viewed as inputs in the production of a final consumption good. Better quality products are introduced from time to time as the outcome of R&D investment by both existing firms and new entrants. As a theoretical result, we show that more productive firms, those that have developed higher quality products in the past, tend to grow larger by developing more product lines in the future only if a firm’s future product quality is positively correlated with its past innovation success. If product quality were i.i.d across innovations, then investment in R&D would be independent of a firm’s current productivity. Interestingly, the qualitative relationship between employment size and labor productivity is ambiguous in the first case and is negative in the second because innovations are labor saving in the sense that fewer workers are required to produce higher quality products. If more productive firms do grow faster, then aggregate productivity growth reflects the fact that workers flow from less to more productive employers as well as from exiting to entering firms. The model developed in the paper provides a useful framework for interpreting empirical growth decomposition exercises, such as those reviewed in Foster et al. (2001). When output weights are used as required by our model, they find that about 34% of productivity growth in U.S. Manufacturing in the 1977–1987 time period can be attributed to entry while 24% is due to worker reallocation across continuing establishments. Our model implies that the latter figure is zero when firms do not differ with respect to the expected productivity of future innovations. We find support for the hypothesis that more productive firms grow faster in Danish firm data in the sense that value added is positively associated with value added per worker across firms but employment size is not. By fitting the moments implied by the model to those derived from panel observations of value added, employment, and wages for Danish firms during the period 1992–1997, we also obtain meaningful estimates of the initial distribution of productivity across firms at entry as well as the parameters of the model. These include the overall rate of creative destruction as well as the parameters of the cost of innovation function that all firms are assumed to face. The remainder of the paper is composed of five sections. In Section 2, an adaptation of the Klette–Kortum model of product creation and destruction is introduced. The implication of productive heterogeneity for differences in average firm size and the composition of aggregate productivity growth are developed in Section 3. A full general equilibrium model with a competitive labor market is sketched in Section 4. Existence of at least one equilibrium solution to the model for the aggregate rate of creative destruction and the wage rate is demonstrated for the case of heterogeneous firms. The empirical evidence and

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estimation results based on Danish firm data are presented in Section 5. The paper concludes with a brief review of the paper’s contributions. 2. A model of creative destruction As is well known, firm employment growth is roughly independent of labor force size; Gibrat’s law holds at least as an approximation. Klette and Kortum (2004) constructed a stochastic market equilibrium model of firm innovation and growth that was consistent with this and other stylized facts regarding firm growth and the size distribution of firms. Although they allow for productive heterogeneity across firms, firm productivity and growth are unrelated because costs and benefits of growth are both proportional to firm productivity in the model. Although we do not make this assumption in our version of the model, the independent of current firm productivity and expected future firm growth is a special case of a more general formulation in which future and current productivity may or may not be correlated. 2.1. Household preferences Following Grossman and Helpman (1991), households consume a continuum of different goods indexed by jA[0,1]. Households are identical and live forever. Intertemporal utility of the representative household at time t is Z 1 Ut ¼ ln C s e rðs tÞ ds; ð1Þ t

where r represents the discount rate and Z 1 ln C t ¼ ln½xt ð jÞzt ð jފ dj 0

ð2Þ

is instantaneous unity of consumption where xt(j) is the service flow of good j at time t, zt(j) represents the quality of good j at date t. For each good type, the quality level develops through a series of product innovations such that t ðjÞ zt ð jÞ ¼ PJi¼1 qi ð jÞ

ð3Þ

where Jt(j) is the number of innovations up to date t and qi(j)>1 denotes the quantitative improvement in the quality of innovation i over the previous version of good j. Households can borrow and lend at nominal interest rate r. The household’s intertemporal budget constraint is implicit in the following law of motion for interest bearing assets: Z 1 Z 1 da ¼ ra þ pt ð jÞxt ð jÞ dj: pt ð jÞ dj þ wt ‘ dt 0 0

Productivity Growth and Worker Reallocation: Theory and Evidence

555

In this equation, a represents the net asset position of the household, pt( j ) the profit earned by supplying the jth good, pt( j ) its price, and wt the wage earned by employed participants at time t, and ‘ the fixed labor endowment. A household’s demands for goods are time paths that maximize intertemporal utility subject to the intertemporal budget constraint and the constraint on the available supply of labor. As households are identical, the only interest rate consistent with equilibrium in the asset market and the necessary transversality condition for intertemporal optimality is the discount rate. Of course, total expenditure by each household is constant when r ¼ r. Given the form of the utility function, the household spreads it expenditure evenly over the continuum of market good types. Following Grossman–Helpman, total aggregate expenditure is set equal to unity by an appropriate choice of the numeraire. This normalization implies that the marginal utility of income is also unity. Hence, the expenditure flow on each commodity is unity. 2.2. The value of a firm Each individual firm is the monopoly supplier of the products created in the past that have survived to the present. The price it can charge for each is limited by the ability of suppliers of previous version to provide a substitute. In Nash– Bertrand equilibrium, any innovator takes over the market for its good type by setting the price just below that at which consumers are indifferent between the higher quality product supplied by the innovator and an alternative supplied by the previous supplier of the product type. The price charged is the product of the relative quality improvement and the previous producer’s marginal cost of production. Given the symmetry of demands for the different good types and the assumption that future quality improvements are independent of the type of good, one can drop the good subscript without confusion. Labor service is the only factor of production and output per worker is normalized at unity for every product type. Hence, p ¼ qw is the price of the good in terms of the numeraire as well as the value of labor productivity where w represent the marginal cost of production of the previous supplier and q>1 is the step up in quality of the innovation. As total expenditure is normalized at unity and there is a unit measure of product types, it follows that total revenue per product type is also unity, i.e., px ¼ 1. Hence, product output and employment are both equal to x¼

1 1 ¼ p wq

ð4Þ

and the gross profit associated with supplying the good is 14p ¼ px

wx ¼ 1

1 40. q

ð5Þ

Following Klette and Kortum (2004), the discrete number of products supplied by a firm, denoted as k, is defined on the integers. Its value evolves over time as a

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Rasmus Lentz and Dale T. Mortensen

birth–death process reflecting product creation and destruction. In their interpretation, k reflects the firm’s past successes in the product innovation process as well as current firm size. New products are generated by R&D investment. The firm’s R&D investment flow generates new product arrivals at frequency gk. The total R&D investment is wc(g)k where c(g)k represents the labor input required in research and development process. The function c(g) is assumed to be strictly increasing and convex. According to the authors, the implied assumption that the total cost of R&D investment is linearly homogenous is the new product arrival rate and the number of existing product, ‘‘captures the idea that a firm’s knowledge capital facilitates innovation.’’ In any case, this cost structure is needed to obtain firm growth rates that are independent of size as typically observed in the data. The market for any current product supplied by the firm is destroyed by the creation of a new version by some other firm, which occurs at the rate d. Below we refer to g as the firm’s creation rate and to d as the common destruction rate faced by all firms.1 As product gross profit and product quality are one-to-one, the profits earned on each products reflect a firm’s current labor productivity. The firm chooses the creation rate g to maximize the expected present value of its future net profit flow conditional on information that is relevant for predicting the product profits of future innovations. Let the parameter y summarize past profit realizations. We assume that this indicator is a sufficient statistic for predicting the distribution of the next innovation’s profit rate. For example, the product quality sequence might be a first-order Markov process, in which case y is the profit on the last product innovation. Alternatively, we might think of the problem as one in which a firm’s product profitability is initially unknown but can be learned over time by observing the past realization. In Jovanovic’s original normal–normal case the sufficient statistic is a pair which include both the current estimate of the mean and its precision. In general, y will be updated in response to the realized profitability of any new product. Let Pk ¼ ðp1 ; p2 ; . . . ; pk Þ denote the firm’s vector of profits for the products currently supplied, let Pkþ1 ¼ ðPk ; p0 Þ represent the profits of the k+1 products where pkþ1 ¼ p0 , and let Pkhii denote Pk excluding element iA{1, y , k}. In terms of this notation, the current value of the firm is a function of its state characterized by Pk and y. It solves the Bellman equation ( k X 
  k pi wcðgÞk þ gk E V kþ1 ððPk ; p0 Þ; y0 Þjy V k ðPk ; yÞ rV k ðP ; yÞ ¼ max g0

þd

1

i¼1

"

k X i¼1

V k 1 ðPkhii ; yÞ

k

V k ðP ; yÞ

#)

,

ð6Þ

These are in fact the continuous time job creation and job destruction rates respectively as defined in Davis et al. (1996).

Productivity Growth and Worker Reallocation: Theory and Evidence

557

where E f
jyg is the expectation operator conditional on information about the quality of the firm’s future products and y0 is the updated value of y given the realized profit of the next innovation, denoted p0 . Notice that no information about future profitability is gained or lost when a product line is destroyed although the firm’s scale as reflected in the number of product supplied fall by one unit. The first term on the right side is current gross profit flow accruing to the firms product portfolio less current expenditure of R&D. The second term is the expected capital gain associated with the arrival of a new product line. Finally, because product destruction risk is equally likely across the firm’s current portfolio, the last term represents the expected capital loss associated with the possibility that one among the existing product lines will be destroyed. Consider the conjecture that the solution takes the following additively separable form V k ðPk ; yÞ ¼

k X pi þ Rk ðyÞ. r þ d i¼1

ð7Þ

That is, we suppose that the value of the firm is the sum of the expected present value of the rents accruing to the firm’s current products plus the value of R&D activities. The latter depends only on expectations about the profitability of future innovationsPand the current number of product lines. Since pi p0 V kþ1 ððPk ; p0 Þ; y0 Þ ¼ ki¼1 rþd þ Rkþ1 ðy0 Þ under the conjecture, Equation (6) þ rþd can be rewritten as k X pi þ rRk ðyÞ rþd i¼1   0 k X p ¼ pi þ k max gE þ Rkþ1 ðy0 Þ g r þ d i¼1

rV k ðPk ; yÞ ¼ r

d

k X pi þ dk½Rk 1 ðyÞ r þ d i¼1

Rk ðyÞjy



 wcðgÞ

Rk ðyފ.

Because the term on the left cancels with the two terms on the right that involve the profits of the products currently supplied, the conjecture holds for any sequence of functions Rk(y), k ¼ 1; 2; . . . that satisfies the difference equation   0   p þ Rkþ1 ðy0 Þ Rk ðyÞjy wcðgÞ rRk ðyÞ ¼ k max gE g rþd þ dk½Rk 1 ðyÞ Rk ðyފ. ð8Þ In words, the return on the value of the R&D department is the expected gain in future profit associated with the next innovation plus the expected capital gains and losses to the R&D operation associated with the possibility of product creation and destruction. In general, these terms are non-zero because a new

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Rasmus Lentz and Dale T. Mortensen

innovation changes expectations about the profitability of future innovation and because a change in scale affects future returns to and costs of R&D. Note that Equation (8) can be rewritten as o 8 n 0 9 p
Because the right-hand side satisfies Blackwell’s sufficient conditions for a contraction that maps the set of non-negative functions defined on the product of the non-negative reals and non-negative integers into itself, a unique solution exists. If the uncertain profit of the next innovation, p0 , is stochastically increasing in expected profitability as summarized by y, the unique solution is an increasing function of y for every value of k by the same argument. Similarly, the fact that the right-hand side is strictly increasing in k, Rkþ1 ðy0 Þ and Rk 1{y} also implies that the contraction maps the functions increasing in k into itself. In sum, the solution has the properties y0 4y ) Rk ðy0 Þ  Rk ðyÞ  Rk ðyÞ and Rkþ1 ðpÞ4Rk ðpÞ. As an implication of (8), a firm’s optimal product creation rate maximizes the expected net return to R&D activity:     V k ðPk ; yÞ wcðgÞ gðyÞ ¼ arg max gE V kþ1 ððPk ; p0 Þ; y0 Þjy g   0   p þ Rkþ1 ðy0 Þ Rk ðyÞjy ¼ arg max gE wcðgÞ . ð9Þ g rþd By implication, the expected growth rate, the difference between the chosen creation rate g and the market determined destruction rate d, is independent of the firm’s current productivity and size if the profitability of the next innovation is independent of past realization of product quality. When past successes have no consequence for future prospects, there is no incentive for firms that are currently more profitable to grow faster and to become larger.

3. Deterministic productive heterogeneity In this section, we explore the implications of the case of deterministic productivity dispersion. These are compared with the alternative hypothesis that all firms are exante identical in the sense that a firm’s product qualities are i.i.d across innovations. 3.1. Product creation We restrict the analysis to the case of deterministic heterogeneity in product quality indexed by p ¼ 1 q 1 . Namely, assume that the profitability of the

Productivity Growth and Worker Reallocation: Theory and Evidence

every innovation is p with probability one. Since    p þ Rkþ1 ðpÞ Rk ðpÞ rRk ðpÞ ¼ k max g g rþd þ dk½Rk 1 ðpÞ Rk ðpފ

wcðgÞ

559



from (8) in this case, it follows that the solution for Rk(p) is proportional to k. Namely, Rk ðpÞ ¼ kDRðpÞ where by substitution  p  grþd wcðgÞ DRðpÞ ¼ max ð10Þ g0 rþd g is the value of R&D per product line for a firm of type p. From Equation (9), an interior solution for the firm’s creation rate choice, denoted g(p), satisfies the following first-order condition: wc0 ðgÞ ¼

p p wcðgÞ þ DRðpÞ ¼ max . g0 r þ d rþd g

ð11Þ

Obviously, the optimal creation rate is a strictly increasing function of the firm’s profit rate. We conjecture that the latter conclusion also holds when expected profitability is positively correlated with past realization as in the case of learning but we do not have a formal proof. 3.2. The distribution of firm size As the set of firms with k products at a point in time must either have had k products already and neither lost nor gained another, have had k 1 and innovated, or have had k+1 and lost one to destruction over any sufficiently short time period, the equality of the flows in and out of the set of firms of type p with k>1 product requires gðpÞðk

1ÞM k 1 ðpÞ þ dðk þ 1ÞM kþ1 ðpÞ ¼ ðg þ dÞkM k ðpÞ

for every p, where Mk(p) is the steady-state mass of firm of type p that supply k products.2 Because an incumbent dies when it loses its last product, but entrants flow into the set of firms with a single product at rate Z, fðpÞZ þ 2dM 2 ðpÞ ¼ ðgðpÞ þ dM 1 ðpÞ, where as defined above f(p) is the fraction of the new entrant flow that realize profit p. Birth must equal deaths in steady state and only firms with one product

2

This equation does not hold in the general case in which an individual firm’s type is transitory. In that case, one must also account for type identity switches that occur as new innovations arrive.

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Rasmus Lentz and Dale T. Mortensen

that lose it die. Therefore, fðpÞZ ¼ dM 1 ðpÞ and   k 1 fðpÞZ gðpÞ k 1 gðpÞM k 1 ¼ M k ðpÞ ¼ k dk d

ð12Þ

by induction. The size distribution of firms conditional on type can be derived using Equation (12). Specifically, the total mass firms of type p is   1 1 X fðpÞZ X 1 gðpÞ k 1 M k ðpÞ ¼ MðpÞ ¼ d k¼1 k d k¼1   Z d dfðpÞ . ¼ ln d d gðpÞ gðpÞ if finite. Hence, the fraction of type p firm with k product is   1 gðpÞ k M k ðpÞ k d . ¼  d MðpÞ ln d gðpÞ

ð13Þ

This is the logarithmic distribution. Note that the firm size distribution is well defined if and only if the creation rate g(p) is less than the overall destruction rate d. Later we show that this condition must hold in any meaningful market equilibrium. Consistent with the observations on firm size distributions, that implied by the model is highly skewed to the right. Furthermore, the mean of distribution conditional on firm profitability,

E fkjpg ¼

gðpÞ d gðpÞ , ¼  d MðpÞ ln d gðpÞ

1 X kM k ðpÞ k¼1

is increasing in g(p). Formally, because ð1 þ aÞlnð1 þ aÞ4a40, the expected number of products is increasing in firm profitability,   @E fkjpg ð1 þ aðpÞlnð1 þ aðpÞÞ aðpÞ dg0 ðpÞ ¼ 40 ð14Þ 2 @p ð1 þ aðpÞÞln ð1 þ aðpÞ ðd gðpÞÞ2 where aðpÞ ¼ gðpÞ=d

gðpÞ, if and only if g0 ðpÞ40.

4. Market equilibrium In this section, we complete the specification of the market model and establish existence of an equilibrium solution in the case of deterministic productive heterogeneity.

Productivity Growth and Worker Reallocation: Theory and Evidence

561

4.1. Firm entry and labor market clearing The entry of a new firm requires an innovation. The cost of entry is the expected cost of the R&D effort required of a potential entrant to discover and develop a new successful product. Hence, if a potential entrant obtains ideas for new products at frequency h per period, the expected opportunity cost of her effort per innovation is w/h, the expected earnings forgone during the required period of R&D activity. As no entrant knows the profitability of its product a priori but all know its distribution, new firms enter if and only if the expected value of a new product given no entry exceeds the cost. Assuming that the condition holds, the endogenous equilibrium product destruction rate, d, adjusts though entry to equate the expected cost and return. Given that product quality at entry is uncertain but that its distribution is common knowledge, the equality of the expected return and cost of entry require that   X X p wcðgÞ w ð15Þ V 1 ðp; yÞfðpÞ ¼ max fðpÞ ¼ g0 r þ d g h p p

from Equations (7) and (11), where f(p) is fraction of entrants with product quality q ¼ ð1 pÞ 1 .3 Because the new product arrival rate of a firm of type p with k products is g(p)k and the measure of such firms is Mk(p), the aggregate rate of destruction is the sum of the entry rate and the creation rates of all the incumbents given that the mass of products is fixed. That is   1 1 XX XX fðpÞZ gðpÞ k 1 d¼Zþ gðpÞ gðpÞkM k ðpÞ ¼ Z þ d d p k¼1 p k¼1 ! !   1 X fðpÞgðpÞ X X dfðpÞ gðpÞ k 1 ¼Z 1þ , ¼Z d d d gðpÞ p p k¼1

where the second equality follows from (12) and the last equality requires that the aggregate rate of creative destruction exceeds the creation rate for every firm type. Using the assumption that the measure of firms is unity, a direct derivation of the same relationship follows:  1  1 X fðpÞ X ZfðpÞ X XX gðpÞ k 1 . ð16Þ ¼Z 1¼ kM k ðpÞ ¼ d k¼1 d d gðpÞ p p p k¼1 Note that d g(p)>0 for all p in the support of the entry distribution from (16) if and only if entry is the entry rate, Z, is positive. Below, we will seek a market solution that satisfies this property. In general, restrictions on fundamental parameters are required to insure that the condition holds.

3

For simplicity, we assume that the number of different product qualities is finite.

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There is a fixed measure of available workers, denoted by ‘, seeking employment at any positive wage. In equilibrium, these are allocated across production and R&D activities, those performed by both incumbent firms and potential entrants. Since the number of workers employed for production purposes per product of quality q is x ¼ 1=wq ¼ ð1 pÞ=w from Equations (4) and (5), the total number demanded for production activity by firms of type p with k products is ‘x ðk; pÞ ¼ kð1 pÞ=w40. The number of R&D workers employed by incumbent firms of type p with k products is ‘R ðk; pÞ ¼ kcðgðpÞÞ. Because a potential entrant innovates at frequency h, the total number so engaged in R&D is ‘E ¼ Z=h given entry rate Z. Hence, the equilibrium wage satisfies the labor market clearing condition ‘¼

1 XX ½‘x ðk; pÞ þ ‘R ðk; pފM k ðpÞ þ ‘E p

k¼1

1  XX 1

 Z þ cðgðpÞÞ kM k ðpÞ þ ¼ w h p k¼1     1 X 1 p fðpÞZ X gðpÞ k þ cðgðpÞÞ ¼ w d k¼1 d p !   1 X 1 p fðpÞ ¼Z þ þ cðgðpÞÞ , h w d gðpÞ p p

1

þ

Z h ð17Þ

where again the last equality is implied by Equation (12) and the requirement that d>g(p) for all p. 4.2. Existence Definition. A steady state market equilibrium with positive entry is a triple composed of a labor market clearing wage w, a positive entry rate Z, and positive creative destruction rate d that satisfy Equations (15)–(17). From (15), the free entry condition is   X p wcðgÞ w fðpÞ ¼ max g0 rþd g h p

ð18Þ

By using Equation (16) to eliminate the positive entry rate Z and Equation (18) to eliminate w/h in Equation (17), one can write the result as ! !  X X1 p þ wcðgðpÞÞ d p wcðgÞ w‘ fðpÞ ¼ d þ max fðpÞ g0 r þ d d gðpÞ d gðpÞ g p p   X d p wcðgÞ 1 r max fðpÞ, ¼ g0 r þ d d gðpÞ g p

Productivity Growth and Worker Reallocation: Theory and Evidence

563

where the first equality is implied by the fact that Sp fðpÞ ¼ 1 and the second is a consequence of the fact that g(p) is the optimal choice of the creation rate for a type p firm. Hence,   P p wcðgÞ fðpÞ r maxg0 r þ d g d gðpÞ 1 ¼ w‘ þ p . ð19Þ P fðpÞ gðpÞ p d

Since total value added is unity by choice of the numeraire, this expression is the income identity. Namely, the total wage bill plus the return on the values of all the operating firms in the economy is equal to value added. In order to focus on the case in which incumbents invest in R&D, we assume their cost, c(g), is strictly convex and that cð0Þ ¼ c0 ð0Þ ¼ 0. Under these restrictions, the optimal creation rate for each type conditional on the market wage and rate of creative destruction is uniquely determined by the following first order condition state as Equation (11). Since the optimal creation rate is strictly increasing in productivity and strictly decreasing in the market wage, a necessary and sufficient condition for the optimal choice to be less than the rate of creative destruction, g(p)od, at any point (w, d) is wc0 ðdÞ4

p

wcðdÞ p 3w4 0 8p. r rc þ cðdÞ

ð20Þ

Of course, the entry rate, Z, is positive only in the union of these regions from Equation (16).  the upper The boundary of the admissible set, that defined by (20) with p ¼ p, support of the given distribution of profit at entry, is labeled BB in Figure 1.  for All pairs above BB satisfy the requirement that g(p)od because gðpÞ  gðpÞ  As illustrated, the wage on the boundary is positive, tends to infinity as all p  p. d tend to zero, is strictly decreasing in d, and tends to zero as d tends to infinity given the assumed properties of the R&D cost function. An equilibrium is any (w, d) pair satisfying Equations (18) and (19) provided that d  gðpÞ on the support of the distribution of profits at entry. Let w ¼ E p ðdÞ represent the locus of points defined by max g0

p wcðgÞ w ¼ rþd g h

and let w ¼ Lp ðdÞ represent solution to   p wcðgÞ 1 ¼ w‘ þ r max g0 r þ d g

ð21Þ

ð22Þ

in the region defined by (20). Since E 0p ðdÞo0 and L0p ðdÞ40, at most one solution exists to both given p. Furthermore, because the solution to (21) for w is monotone increase in h and because both (20) and (22) are independent of h, a solution exists in the required region for all values of h above some critical value.

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Figure 1.

Equilibrium wage and creative-destruction rates

w B L E L E

L E E L 0

B


As Equations (18) and (19) collapse to (21) and (22) respectively when there is a single-firm type, we have established sufficient conditions for both existence and uniqueness in this case. In the case of firm heterogeneity, the same argument implies existence. Specifically, because Sp fðpÞ ¼ 1 where f(p) is the fraction of entrants of type p and because Ep(d) is increasing in p from (21), the locus of point that satisfy the entry condition (18) is bounded above by E p ðdÞ and below by EpðdÞ where p is the lower and p is the upper support of the type distribution at entry. Similarly, because Lp(d) is decreasing in p from (22), the solution to the labor market clearing condition (19) is bounded above by Lp ðdÞ and below by Lp ðdÞ. In Figure 1, the curves LL and LL represent w ¼ Lp ðdÞ and w ¼ LpðdÞ respectively. Similarly, w ¼ E p ðdÞ and w ¼ E p ðdÞ are represented as EE and EE. It follows that any joint solution to the entry and labor market clearing conditions must lie in the shaded area. Given continuity of the relationship, at least one common solution exists to (18) and (19) in that region. Finally, the shaded area lies above BB in the figure for all sufficiently large values of h because Ep(d) is monotone increasing in h and both (20) and (22) are independent of h. Indeed, the critical value is that for which the intersection of w ¼ Lp ðdÞ and w ¼ E p ðdÞ lies on the boundary. Since w^ ¼ h=ðr þ h‘Þ at any joint solution to Equations (21) and (22), the critical value of h, denoted h, band the associated rate of creative ^ are the unique solutions to destruction at the intersection, d, 1 ‘

p h^ p ¼ ¼ w^ ¼ . 0 ^ ^ ^ ^ r þ h‘ ec ðdÞ þ cðdÞ cðdÞ

^ hÞ ^ exists under the hypothesis to the following result. ^ d; A unique triple ðw;

Productivity Growth and Worker Reallocation: Theory and Evidence

565

Proposition 1. If the cost of R&D function, c(g), is strictly convex and ^ In the c0 ð0Þ ¼ cð0Þ ¼ 0, then a steady-state market equilibrium exists for all h4h. case of a single firm type, there is only one. 5. Evidence and estimation If product quality is a permanent firm characteristic, then differences in firm profitability are associated with differences in the product creation rates chosen by firms. Specifically, more profitable firms grow faster, are more likely to survive in the future, and supply a larger number of products on average. Hence, a positive cross firm correlation between current gross profit per product and sales volume should exist. Furthermore, worker reallocation from slow-growing firms that supply products of lesser quality to more profitable fast-growing firms will be an important sources of aggregate productivity growth. On the other hand, if product quality is i.i.d across innovations and firms, all firms grow at the same rate even though persistent differences in profitability exist as a consequence of different realizations of product quality histories. In the section, we demonstrate that firm-specific differences in profitability are required to explain Danish interfirm relationships between value added, employment, and wages paid. In the process of fitting the model to the data, we also obtain estimates of the investment cost of innovation function that all firms face as well as the sampling distribution of firm productivity at entry. 5.1. Danish firm data If more productive firms grow faster in the sense that g0 ðpÞ40, then (14) implies that more productive firms also supply more products and sell more on average. However, because production employment per product decreases with productivity, total expected employment, nEk where n ¼ ð1 pÞ=w þ cðgðpÞÞ, need not increase with p in general and decreases with p when growth is independent of a firm’s past product quality realizations. These implications of the theory can be tested directly. Danish firm data provide information on the relationships among productivity, employment, and value added. The available data set is an annual panel of privately owned firms for the years 1992–1997 drawn from the Danish Business Statistics Register. The sample of approximately 6,700 firms is restricted to those with 20 or more employees in either 1992 or 1995. The variables observed in each year include value added (Y), full-time equivalent employment (N), and the total wage bill (W). The model is estimated on an unbalanced panel of 5,254 firms drawn from the firm panel. The panel is constructed by selecting all existing firms in 1992 and following them through time, while all firms that enter the sample in the subsequent years are excluded. Furthermore, the top and bottom 1% of the firms in the value added distribution for 1992 are censored from the panel to ease numerical challenges in the estimation and to avoid extreme

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observation bias. The censoring means a loss of roughly 110 firms. The main source of loss of firms from 6,700 to 5,254 is the exclusion of entrants. One can in principle include the entrants in the sample for identification purposes. It will then be necessary to correct for the considerable bias in the sampling due to the fact that only firms with greater than 20 workers in 1995 will be registered as entrants. While the estimation method we are using along with the structural model would allow us to correct for this bias, the sample of incumbents is sufficient for identification and we have chosen to rely on this more robust information, only. Thus, in the estimation, the model is simulated to replicate the data sample procedure, and simulated entrants are therefore ignored. Figure 2 presents non-parametric estimates of the distributions of two alternative measures labor productivity. The first measure is value added per worker (Y/N) while the second is value added per unit of quality adjusted employment (Y/N*). The first measure misrepresents cross firm productivity differences to the extent that labor force quality differs across firms. However, if more productive workers are compensated with higher pay as would be true in a competitive labor market, one can use a wage-weighted index of employment to correct for this source of cross firm differences in productive efficiency. Formally, the constructed quality adjusted employment of firm j is defined as W N j ¼ w j where w ¼ Sj W j =ðSj N j Þ is the average wage paid per worker in the market. Although correcting for wage differences across firms in this manner does reduce the spread and skew of the implied productivity distribution somewhat, both distributions have high variance and skew and are essentially the same shape. Figure 3 illustrate non-parametric regressions of value added and employment size on the two productivity measures. The top and bottom curves in the Figure 2.

Value added per worker and per standardized worker pdfs

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Productivity Growth and Worker Reallocation: Theory and Evidence

Figure 3.

Kernel regressions of value added per worker against firm size (1992) Y/N vs. Y

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figures represent a 90% confidence interval for the relationship. Hence, these results strongly reject the hypothesis that firm growth is independent of the firm’s profitability in favor of the alternative that the sales of more productive firms grow larger. 5.2. Model estimation The following identifies the deterministic permanent firm types case of the model. An observation in the panel is given by cjt ¼ ðY jt ; W jt ; N jt Þ, where Yjt is real value added, Wjt the real wage bill, and N jt quality adjusted labor force size of firm j in year t. Let cj be defined by, cj ¼ ðcj1 ; . . . ; cjT Þ Table 1 provides a selected number of moments for the panel. The model is estimated by use of a simulated minimum distance estimator as described in for example Gourieroux et al. (1993), Hall and Rust (2003), and Alvarez et al. (2001). First, define a set of sample auxiliary parameters, G(c1, y , cJ), which in this case takes a cross-section form for each time period. Specifically, 10 data moments are generated for each year: number of surviving firms, E [Y ], Std [Y ], E [W ], Std [W ], Corr[Y, W ], Corr[Y/N*, Y |Y>0], Corr[Y/N*, N*|Y>0], Median[Y |Y>0], Median[W |W>0]. Thus, mGð
Þ consists of 60 moments.

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Table 1.

Survivors E[Y] E[Y|Y>0] Median [Y|Y>0] E[W] E[W|>0] Median [Y|Y>0] Std [Y] Std [W] Corr [Y, W] Corr Y/N*, Y|Y>Q] Corr [Y/N*, N*|Y>0]

Data moments

1992

1993

1994

1995

1996

1997

5,254.00 19,395.84 19,395.84 12,092.39 9,962.15 9,962.15 6,546.79 20,319.85 10,328.97 0.89 0.24 0.06

4,861.00 17,955.18 19,406.81 12,109.85 9,156.43 9,896.70 6,539.92 21,617.13 10,389.14 0.88 0.25 0.03

4,405.00 17,720.47 21,135.84 12,847.01 8,615.13 10,275.58 6,717.87 24,998.44 10,332.56 0.82 0.44 0.02

3,963.00 16,797.64 22,269.69 13,606.83 8,351.38 11,071.96 7,240.28 26,341.06 11,420.99 0.83 0.27 0.02

3,599.00 15,531.19 22,673.21 14,048.41 7,804.45 11,393.32 7,517.31 24,820.29 11,472.95 0.91 0.24 0.02

3,167.00 14,444.30 23,962.84 14,731.58 7,149.29 11,860.55 7,827.89 25,210.02 11,190.37 0.91 0.22 0.02

Second, ðcs1 ðoÞ; . . . ; csJ ðoÞÞ is simulated from the model for a given set of model parameters o. The model simulation is initialized by assuming that the economy is in steady state in the first year and consequently that firm observations are distributed according to the o-implied steady-state distribution. Alternatively, one can initialize the simulation according to the observed data in the first year, (c11, y ,c1J). The assumption that the economy is initially in steady state provides additional identification in that (c11, y ,c1J) can be compared to the model-implied steady-state distribution ðcs11 ðoÞ; . . . ; cs1J ðoÞÞ. The simulated auxiliary parameters are then given by, S 1X ^ Gðcs1 ðoÞ; . . . ; csJ ðoÞÞ, GðoÞ ¼ S s¼1

where S is the number of simulations. The estimator is then the choice of parameters that minimizes the weighted distance between the sample auxiliary parameters and the simulated auxiliary parameters,    0 ^ ^ ^ ¼ arg min GðoÞ Gðc1 ; . . . ; cJ Þ , o Gðc1 ; . . . ; cJ Þ A 1 GðoÞ o2O

^ is the where A is some positive definite matrix. If A is the identity matrix, o equally weighted minimum distance estimator (EWMD). If A is the covariance ^ is the optimal minimum distance matrix of the auxiliary parameter vector Gð
Þ, o estimator (OMD). The OMD estimator is asymptotically more efficient than the EWMD estimator. However, Altonji and Segal (1996) showed that the estimate of A as the second moment matrix of Gð
Þ may suffer from serious small sample bias. Horowitz (1998) suggests an alternative estimator of A based on bootstrap methods. The analysis will adopt the Horowitz (1998) estimator of the covariance matrix A.

Productivity Growth and Worker Reallocation: Theory and Evidence

569

5.3. Model simulation To fit the data, the model simulation produces time paths for value added (Y), the wage sum (W), and labor force size (N) for J firms. Rather than normalize the total consumer expenditure for each product at unity, the expenditure for each product is set at Z. Hence, the demand for each good is xj ¼ Z=pj . Denote by kjt the number of products of firm j at time t. Let the type of firm j be represented by its quality improvement qj. To properly capture the labor share in the data, a capital cost k  K=Z is added to the model. K is the capital associated with the production of a given product. k is the capital cost relative to product expenditure. This modifies the pricing of the intermediary goods. Now, providing an intermediary good at price p yields operational profits, Z(1 w/p) K. Thus, the price of the intermediary goods for which firm j is the quality leader is, pj ¼ qj w=ð1 kÞ. Firm j’s total profits at time t is given by, h  i Pjt ¼ kjt pj xj wxj K wc gðpj Þ " #  Z ¼ kjt Z ð1 kÞ K wc gðpj Þ qj
  ¼ kjt Z pj k w~c gðpj Þ , where pj  1

kÞ=qj . The value added of firm i at time t(Yjt) is given by,

ð1

log Y jt ¼ log kjt þ log Z þ
y ,

ð23Þ

where eY is a noise term which can be interpreted as measurement error and/or demand side shocks. ey is assumed i.i.d with E½
y Š ¼ 0 and Var½
y Š ¼ s2
y . The wage bill of firm j at time t(Wjt) is given by, !  Z ^ jt ¼ kjt w ð1 kÞ þ wc gðpj Þ W wqj
  ¼ kjt Z 1 pj þ w~c gðpj Þ , where c~ðgÞ ¼ cðgÞ=Z. Define the labor share of firm j by,  aj  1 pj þ wc~ gðpj Þ .

Firm j’s wage bill at time t is then given by, log W jt ¼ log kjt þ log Z þ log aj þ
W ,

ð24Þ

where eW is another i.i.d noise term with E½
W Š ¼ 0 and Var½
W Š ¼ s2
W By (11), firm j’s choice of creation rate solves, gðpj Þ ¼ arg min g

pj

k wc~ðgÞ . rþd g

ð25Þ

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Rasmus Lentz and Dale T. Mortensen

Specify the cost function c~ðgÞ ¼ c0 g1þc1 . Then the first order condition for the optimal creation rate choice is, wð1 þ c1 Þc0 gc1 ðr þ d

gÞ ¼ pj

k

wc0 g1þc1 .

Substituting the first order condition into the definition of the labor share yields,  r þ d gðpj Þ wð1 þ c1 Þc0 gc1 . ð26Þ aj ¼ 1 k

Equations (23) and (24) provide the foundation for the model simulation. It then remains to simulate product paths for all firms. The simulation is initialized by the assumption of steady state. Let G(p) be the unknown steady-state distribution of firm types. To simplify matters discretize the support of the type distribution to (p1, y , pM). By (13), the steady-state product size distribution conditional on survival is given by,  k 1 gðpÞ k d . Prðk ¼ kjpÞ ¼  ð27Þ d ln d gðpÞ First, firm j’s type, pj, is determined according to Gð
Þ. Then, the initial product size of a firm j(kj1) is determined according to (27). With a given initial product size kj1, simulation of the subsequent time path requires knowledge of the transition probability function Prðkj2 ¼ kjkj1 ; pj Þ. Denote by pp,n(t) the probability of a type p firm having product size n at time t. As shown in Klette and Kortum (2004), pp,n(t) evolves according to the ordinary differential equation system, p_ p;n ðtÞ ¼ ðn 1ÞgðpÞpp;n 1 ðtÞ þ ðn þ 1Þdpp;nþ1 ðtÞ ðd þ gðpÞÞpp;n ðtÞ; 8n  1 p_ p;0 ðtÞ ¼ dpp;1 ðtÞ Hence, with the initial condition,  1 if n ¼ kj1 pp;n ð0Þ ¼ 0 otherwise:

ð28Þ ð29Þ

One can determine Prðkj2 ¼ kjkj1 ; pj Þ by solving the differential equation system in (28) for ppj, k(1). Solving for ppj, k(1) involves setting an upper reflective barrier to bound the differential equation system. It has been set sufficiently high so as to avoid biasing the transition probabilities. Based on the transition probabilities Prðkjtþ1 ¼ kjkj1 ; pj Þ one can then iteratively simulate product size paths for each firm. 5.4. Identification The set of model parameters to be identified (o) is given by,   o ¼ c0 ; c1 ; d; k; Zðp1 ; . . . ; pM Þ; ðp1 ; . . . ; pM Þ 2 O,

Productivity Growth and Worker Reallocation: Theory and Evidence

571

where pm ¼ Prðp ¼ pm Þ. and O is the feasible set of model parameters choices. The interest rate will be set at r ¼ 0:05 and the noise processes governing eY and eW will be taken as given. The wage w is immediately identified as the average worker wage in the sample w ¼ 221:73. Since SM m¼1 pm ¼ 1, this implies that the estimation will be identifying 2M+4 parameters. Notice that to simulate product size paths and generate csj according to (23) and (24), it is necessary and sufficient to know   c ¼ d; Z; ðg1 ; . . . ; gM Þ; ða1 ; . . . ; aM Þ; ðp1 ; . . . ; pM Þ ,

which is 3M+1 parameters. The choice of o maps into c according to (25) and (26). Denote the mapping by, c ¼ C(o). The dimension of o is strictly greater than c if Mr2. Thus, in the case where there are less than 3 distinct productivity types, there may be multiple o choices that map into the same c which suggests a fundamental identification problem in these cases. Suppose Mr2 and there exists a o0 2 O different from o00 2 O such that c ¼ Cðo0 Þ ¼ Cðo00 Þ. In this case, the simulated data is the same for o0 and o00 , that is s s s s 00 00 0 0 ðc1 ðo Þ; . . . ; cJ ðo ÞÞ ¼ ðc1 ðo Þ; . . . ; cJ ðo ÞÞ, and the distance criterion for the Simulated Minimum Distance (SMD) estimator will be the same for o0 and o00 . The example suggests a potential for failure of identification for Mr2.4 When MZ3, the dimension of o is greater or equal to the dimension of c. While a choice of MZ3 resolves the identification problem associated with the mapping between o and c, it remains necessary that there is enough identifying variation in the data to identify the 2M+4 model parameters. This is the standard identification problem and increasing M will all else equal strain identification on this dimension. The model is estimated for M ¼ 3 and turns out to be identified under this choice. 5 5.5. Estimation results

The model parameter estimates are given in Table 2. The creation rates gm and labor shares am for each type are derived from the model parameter estimates. The interest rate has been set at r ¼ 0:05 and the wage level is identified as the average worker wage in the data for 1992. The lower and upper bounds of the double sided 90% confidence interval are generated by naive bootstrapping. Table 3 produces a comparison of the data moments and the simulated moments associated with the model parameter estimates. First of all, it is seen that the model is quite successful in capturing the overall characteristics of the data. However, the model tends to over estimate the mean and median of the Y

4 Indeed, experimentation with estimation of the model with M ¼ 2 resulted in serious identification problems. The estimation pointed to a region of parameter values, but failed to identify an actual point estimate. 5 Experimentation with M ¼ 4 suggests that M ¼ 3 is a restrictive choice and that data will identify more than three separate types.

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Rasmus Lentz and Dale T. Mortensen

Table 2.

Model parameter estimates

Point Estimate

Lower Bound

Upper Bound

17.039 4.615 0.440 10,577.301 0.181 0.446 0.666 0.709 0.381 0.250 0.369

16.084 4.421 0.435 10,332.222 0.164 0.439 0.654 0.705 0.350 0.240 0.350

23.413 4.698 0.451 10,982.988 0.185 0.466 0.686 0.736 0.403 0.276 0.383

g1 g2 g3 a1 a2 a3

0.055 0.132 0.138 0.559 0.536 0.530

0.033 0.117 0.125 0.548 0.526 0.520

0.069 0.136 0.143 0.564 0.540 0.534

W R

221.735 0.050

c0 c1 k Z d p1 p2 p3 Pr(p1) Pr(p2) Pr(p3)

Table 3.

Survivors E [Y ] Med[Y ]Y>0 E [W ] Med [W ]W>0 Std [Y ] Std [W] Cor[Y, W ]
 Cor NY ; Y


 Cor NY ; N 

Data Sim Data Sim Data Sim Data Sim Data Sim Data Sim Data Sim Data Sim Data Sim Data Sim

— —

— —

Data momemts and model fit

1992

1993

1994

1995

1996

1997

5,254 5,254 19,395.84 19,513.48 12,092.39 12,267.49 9,962.15 10,358.15 6,546.79 6,194.64 20,319.85 18,550.46 10,328.97 9,552.85 0.89 0.95 0.24 0.20 0.06 0.04

4,861 4,709 17,955.18 18,549.46 12,109.85 12,654.26 9,156.43 9,807.16 6,539.92 6,317.59 21,617.13 20,062.22 10,389.14 10,141.98 0.88 0.96 0.25 0.21 0.03 0.01

4,405 4,211 17,720.47 17,253.82 12,847.01 13,132.75 8,615.13 6,717.87 6,463.39 24,998.44 20,998.44 10,332.56 10,595.94 0.82 0.95 0.44 0.18 0.02 0.03 0.04

3,963 3,772 16,797.64 16,547.72 13,606.83 13,952.35 8,351.38 8,748.91 7,240.28 6,669.49 26,341.06 20,973.27 11,420.99 10,748.86 0.83 0.96 0.27 0.20 0.20 0.02

3,599 3,448 15,531.19 15,717.26 14,048.41 14,610.08 7,804.45 8,267.32 7,517.31 6,841.44 24,820.29 21,587.77 11,472.95 10,908.26 0.91 0.96 0.24 0.20 0.02 0.01

3,167 3,127 14,444.30 14,634.14 14,731.58 15,357.57 7,149.29 7,691.72 7,827.89 7,135.53 25,210.02 21,646.84 11,190.37 10,957.03 0.91 0.96 0.22 0.18 0.02 0.02

573

Productivity Growth and Worker Reallocation: Theory and Evidence

and W distributions somewhat. The revenue per product parameter (Z) is central in this respect. It can be shown that irrespective of firm type, the mode of the product size distribution is equal to 1 in the model and consequently the mode of the Y distribution is equal to Z. As is seen in Figure 4, the model estimation fails to perfectly match the mode in the observed Y distribution. The model estimated distribution is shifted somewhat to the right. The reason for the right shift is found in the under estimation of the standard deviation of Y and W. Again, Z is an important determinant of this moment. The higher the value of Z, the greater the variance in Y and W. While Z is not the only determinant, it seems that the estimation has sacrificed some of the first moment and median fits to improve the fit to the second moments. The firm-type distribution also affects the second moment fit, though, and one might suspect that allowing more types in the distribution support could introduce more variance and consequently allow for a lower Z estimate to bring the model estimates a bit more in line with the observed first moments and medians. Figure 4 compares the observed and estimated distribution of value added. The dashed lines depict the value added distribution associated with one of the three possible firm types. The estimated distribution of value added is a mixture of the three single-type distributions. The higher the profit of a type, the greater the variance in the distribution. Figure 5 shows the change in the distribution of value added from 1992 to 1997 in the sample. It is seen that the observed distribution shifts to the right and is more spread out. The model captures this change and explains it as a change in firm type composition over time. The low profit types create new products at a Figure 4.

Observed and estimated value added pdfs

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Rasmus Lentz and Dale T. Mortensen

Figure 5.

Change in observed and estimated value added distribution from 1992 to 1997 Estimated Value Added 8

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lower rate than high profit types and consequently will tend to reduce in size at a greater rate than the high profit types. Therefore, the composition of firm types will switch toward high profit types over time and generate the increased spread in the distribution of value added. This particular source of variation in the data turns out to be an important identifier of d and the degree of firm type heterogeneity. Experimentation with estimation of the model for fixed, lower values of d results in less change in the survival conditional mean and median values of Y and W over time, thus forcing the model to over estimate the means and medians of Y and W early on in the sample and under estimate them in the later years. The lower value of d implies a smaller difference in survival probabilities across firm types and consequently a slower rate of change in the firm type composition over time. Therefore, a lower value of d results in a slower rate of change in the estimated conditional means and medians of Y and W over time. Also, a right shift of the size distribution over time is only feasible with a sufficient amount of firm type heterogeneity. Without type heterogeneity, the size distribution can only shift to the left over time. Figure 6 shows the model fit relative to the correlation between worker productivity and firm size (as measured by either value added or labor force size). It is seen that the model captures the relationships quite successfully. There is not quite enough noise in the model to generate as much spread in the worker productivity distribution support as is seen in data. Furthermore, the model estimates a somewhat flatter relationship between Y/N* and Y than the observed relationship. The model also over estimates the wage share slightly resulting in a small over estimate of the labor force size. The model is designed to satisfy Gibrat’s law in the sense that an individual firm’s innovation rate is independent of its size. This does not however mean

575

Productivity Growth and Worker Reallocation: Theory and Evidence

Figure 6.

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0 1200

Y/N*

that the unconditional relationship between firm size and firm growth will have zero slope for the simulated model. In fact, because of sample selection, larger firms will tend to over-represent higher type firms and consequently one should expect a positive relationship between firm size and firm growth in the current version of the estimation. This is contrary to data, where one finds a slightly negative relationship. However, it is reasonable to suspect that the negative relationship in data can be a result of other factors, such as demand fluctuation and/or measurement error. Note that the wage share is estimated roughly around 55%, which is consistent with the wage share in data and is a little below the aggregate share of factor incomes in Denmark at around 60%. The choice of moments is primarily focused on cross-section moments. It should be noted, however, that the current estimation method implicitly addresses a number of dynamic aspects of the model. The data are sampled such that the 1992 reflects a steady-state cross-section. However, since the data do not include new entries, the following cross-sect ions will increasingly over-represent higher survival probability firms. Thus, as the estimation attempts to match the trend in cross-section moments over time, it implicitly matches the model to dynamic features of the data. 6. Concluding remarks Large and persistent differences in firm productivity and size exist. Evidence suggests that the reallocation of workers across firms and establishments is an important source of economic growth. In the paper, we explore the Schumpeterian model of aggregate growth and firm evolution developed by Grossman and Helpman (1991) and by Klette and Kortum (2004). We find that firms with higher measurable labor productivity will grow larger in the future and that worker reallocation from the less to more productive will

576

Rasmus Lentz and Dale T. Mortensen

contribute to growth only if current productivity predict future productivity in the model. Specifically, there is no relationship between current productivity and expected future firm sales and there will be no contribution to growth of worker reallocation across existing firms if profits are independently distributed over the sequence of new product innovations. Furthermore, one should find a negative relationship between employment size and current productivity measures in this case. However, if some firms consistently develop better products, then profit maximization will imply that the more productive will innovate more frequently and can expect to enjoy larger future sales as a consequence. Existing studies that provide an empirical decomposition of aggregate productivity growth provide strong evidence for the importance of worker reallocation from exiting to entering firms and establishments. The evidence for the importance of reallocation across continuing firms is less clear. However, if gross output weights are used in constructing the productivity index as our model would require, then the two sources of growth are equally important and together explain over half of the productivity growth in the U.S. Manufacturing sector during the 1977–1987 period according to Foster et al. (2001). Our own evidence from Danish firm-level data support the conclusion that more productive firms grow faster. Specifically, the hypothesis that there is no relationship between size as measured by value added and labor productivity is clearly rejected in favor of a positive association between the two. Furthermore, a structural version of our model in which there are three types of firms that vary with respect to the quality of their products does an excellent job of explaining the moments of panel data observations on value added, employment size, and firm survival rates.

References Altonji, J.G. and L.M. Segal (1996), ‘‘Small-sample bias in GMM estimation of covariance structures’’, Journal of Business and Economic Statistics, Vol. 14, pp. 353–366. Alvarez, J., M. Browning and M. Ejrnæs (2001), Modelling income processes with lots of heterogeneity, Copenhagen: Working Paper, University of Copenhagen. Baily, M., C. Hulton and D. Campbell (1992), ‘‘Productivity dynamics in manufacturing plants’’, Brookings Papers on Economic Activity, Microeconomics: 187–249. Bartelsman, E.J. and M. Doms (2000), ‘‘Understanding productivity: lessons from longitudinal microdata’’, Journal of Economic Literature, Vol. 38(3), pp. 569–594. Christensen, B.J., R. Lentz, D.T. Mortensen, G. Neumann and A. Werwatz (2005), ‘‘On the job search and the wage distribution’’, Journal of Labor Economics, Vol. 23(1), pp. 31–58.

Productivity Growth and Worker Reallocation: Theory and Evidence

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Davis, S.J., J.C. Haltiwanger and S. Schuh (1996), Job Creation and Destriction, Cambridge: MIT Press. Dunne, T., L. Foster, J. Haltiwanger and K.R. Troske (2004), ‘‘Wage and productivity dispersion in U.S. manufacturing: the role of computer investment’’, Journal of Labor Economics, Vol. 22(2), pp. 397–431. Fallick, B.C. and C.A. Fleischman (2001), ‘‘The importance of employer-toemployer flows in the U.S. labor market’’. Federal Reserve Board Finance and Economics Discussion Paper, 2001–18. Foster, L., J. Haltiwanger and C.J. Krizan (2001), ‘‘Aggregate productivity growth: Lessons from microeconomic evidence’’, in: C.R. Hulten, E.R. Dean and M.J. Harper, editors, New Developments in Productivity Analysis, Chicago: University of Chicago Press. Frederiksen, A. and N. Westergaard-Nielsen (2002), ‘‘Where did they go?’’ Aarhus School of Business Working Paper. Gourieroux, C., A. Monfort and E. Renault (1993), ‘‘Indirect inference’’, Journal of Applied Econometrics, Vol. 8(0), pp. S85–S118. Grossman, G.M. and E. Helpman (1991), Innovation and Growth in the Global Economy, Cambridge, MA: MIT Press. Hall, G. and J. Rust (2003), Simulated minimum distance estimation of a model of optimal commodity price speculation with endogenously sampled prices, Maryland: Working Paper, University of Maryland. Horowitz, J.L. (1998), ‘‘Bootstrap methods for covariance structures’’, Journal of Human Resources, Vol. 33, pp. 39–61. Klette, T.J. and S. Kortum (2004), ‘‘Innovating firms and aggregate innovation’’, Journal of Political Economy, Vol. 112(5), pp. 986–1018. Mortensen, D.T. (2003), Wage Dispersion: Why Are Similar Workers Paid Differently?, Cambridge, MA: MIT Press. Stewart, J. (2002), ‘‘Recent trends in job stability and job security: evidence from the march CPS’’. U.S. BLS Working Paper, No. 356.

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Subject Index absorbing state 383 advertising 331, 333, 334, 336, 351, 362, 515, 525 age age-at-entry distribution 255, 256 distribution 256, 261 age-composition 183, 184, 189, 199, 200, 207 age-earning profile 200, 208 aggregate shocks 58, 481, 484, 498 Albrecht-Axell model 63, 70 amplification mechanism 482, 485, 499 of productivity shock 10, 481 Armed Forces Qualification Test (AFQT) 231 arrival rate job offers 220, 382–384, 391, 392, 395, 397, 399–401, 404, 409, 410 wage offers 391, 410 asymmetric-information 484 attractiveness 9, 365, 371, 374, 485

blocks 7, 169–172, 174, 176, 177, 179, 180, 512 block assignments 7, 169–171, 175, 179, 180 bootstrap estimated covariance matrix 568 BLS 525, 529, 535 Burdett-Mortensen model 4, 6, 8, 67, 69, 134, 218, 494, 498 business cycle 10, 165, 304, 393, 467, 482, 485, 503, 504, 510, 532, 535, 537 calibrated 104, 496–498, 501, 511, 524, 538 calibration 485, 496–503, 512, 536–539 capital firm-specific 190, 203, 208, 218, 220, 231, 241, 242, 312, 318, 321 circular model 5, 6, 117, 120, 121, 128, 129 Cobb-Douglas 162, 179, 192, 496, 513, 515 cohort 184, 185, 272, 307, 368, 452–454, 456, 458–460, 463, 466 co-movement 512, 521, 532, 535, 539 compensation 63, 83, 84, 98, 99, 102, 237–239, 242–244, 246, 256, 257, 261, 262, 382, 383, 385, 386, 391, 405, 410, 422, 530 compensating differences 422 competitive auctions dynamic model 151 control-specific shock 354 coordination frictions 146, 147, 149, 153, 165 contact rate 122, 501 contracts 38, 260, 299, 312

balanced-budget-constraints 554, 555 bargaining game 3, 4, 15, 17, 25, 37–39, 41–43, 45–52 bargaining strength 179, 195, 200, 538 BEA 530 Bellman equation 41, 64, 65, 71–73, 75, 77, 79, 123, 489, 492, 493, 556 best response 22, 45, 46, 66, 69, 72, 76 Beveridge curve 159, 512, 541 BHPS 307 bidding game 149–150 bilateral monopoly 41 blinder decomposition 415, 420, 426

579

580

Subject Index

copulas discrete mixture 449 Plackett 442, 448 cost of search 3, 4, 17, 18, 28, 30–32, 70, 86, 126–129, 186, 311, 361, 490, 491, 496, 497, 506 of vacancy creation 10, 127, 484, 486, 497, 542 cost elasticity 127 counter-cyclical 522, 535, 536, 539, 544, 545 cross-cohort (intergenerational) redistribution 184, 185 cross section (earnings) distribution 270, 277, 278, 281–283, 302–304, 361, 453, 455, 457, 460, 463, 468, 476 CRS 67, 517 curse of degeneracy 9, 350, 353–356 curse of determinacy 9, 353, 346, 349, 350, 354, 360 curse of dimensionality 332 DADS 248, 251–254, 261, 266, 451 Danish firm data, see IDA density unemployed workers 122 Diamond-equilibrium 62–64, 70, 73, 118, 366, 482, 510 discrimination 5, 83, 84, 86, 107, 414 dispersion = distribution employment 452 firm size 553, 559–560 firm type 573 productivity 52, 55, 56, 138, 141, 142, 144, 552, 558, 566, 574 wage 3–5, 8, 16, 38, 44, 62, 66, 68, 70, 73, 77, 80, 148, 159, 270, 333, 552 distribution of employed worker 361, 489, 498, 499 distribution of wage offers 278, 283, 303, 304, 311, 361, 382, 383, 391, 392 distribution of ranks 442, 448 displaced worker 151, 152, 156 157, 187, 188, 190, 191, 196, 197, 203, 205, 325

duration-dependence 255, 274, 278, 292, 293, 298–300, 370, 382–384, 391, 392, 397, 400, 401, 410 duration of unemployment 221, 383, 384, 391, 392, 397, 405, 409, 410, 510, 535, 536 dynamic general equilibrium 207, 517 ECHP 9, 270, 272, 306, 307, 383, 385, 388–390, 400, 403–405 early retirement benefits 188 earning distribution 10, 134, 140, 219–222, 225, 295, 442, 444, 452, 454, 455, 457, 460, 476 earnings dynamics ARCH models 442 discrete heterogeneity model 449 fixed effects 449 linear models 442, 445 level 442, 446, 464, 468 Markov chains 442, 445 random effects 449 relative 442, 445, 448, 518 earnings test 191, 197 education 9, 83, 219–221, 231, 235, 244, 245, 247, 251–254, 256, 257, 259, 260, 261, 322, 325, 369, 370, 372, 374, 379, 388, 398–403, 415, 418, 419, 421, 426, 431–433, 435, 438, 446, 449, 453, 454, 456, 458–460, 462, 463, 466, 467 efficient bargaining models 416 EM, see expectation maximization algorithm empirical income distribution 133 employment spell 245, 289, 306 employed workers 10, 16–18, 27, 28, 39, 57, 103, 108, 122, 135–137, 147, 152, 172, 187, 210, 272, 273, 276, 278–282, 284, 288, 295, 302, 303, 361, 482, 485, 489, 493–495, 498, 499, 511, 524, 529 employer-employee data 8, 9, 58, 238, 261, 313, 414, 415 employment probability 5, 84–86, 89, 97, 98, 100, 103–108, 207, 213 endogenous growth model 11, 553 encounter function 26

Subject Index

equally weighted minimum distance estimator 568 equilibrium decentralized 155 flow condition 124 heterogeneity 80 multiple 32 quality 16 solution 4, 368, 511, 553, 560 unemployment (rate) 153, 482, 520 unique 6, 25, 38, 50, 57, 86, 91, 101, 126 wage distribution 4, 85, 103, 121 equilibrium dynamic model 333 equilibrium search (models) 6, 62, 119, 133, 134, 142, 220, 223–225, 230, 232, 242, 260 European Commission 310 European Community Household Panel, ECHP 9, 270, 272, 306, 307, 383, 385, 388–390, 400, 403–405 EWMD, see equally weighted minimum distance estimator expectation maximization algorithm 442, 448–451, 454, 466, 468, 472, 473, 475 expected lifetime earnings 87, 107 expected lifetime utility 191, 201, 205 expected payoff 19, 20, 24–28, 32, 150, 151, 162, 483, 486 return 17–20, 23, 28, 391, 561 wage 5, 84, 85, 88, 90, 92, 93, 97–99, 102–104, 106, 108, 175, 319 finite stage random offer game 16 firm size 136, 137, 316, 415, 420, 553, 556, 559, 560, 567, 574, 575 Flemming-Harrington test 369 financial market 184, 366 firm profit-per-employee 415, 420, 427 flow of hires 512, 521, 529 flow of workers 63, 138, 270, 281, 288, 289, 333, 510, 521, 522, 528, 551–553, 565, 575, 576 French administrative earnings data, see DADS 248–249, 261, 451 French Labour Force Survey 306, 443, 446, 448, 451, 452, 468

581

free entry 70, 73, 120, 122–125, 152, 154, 155, 158, 194, 198, 202, 489, 493, 495, 562 friction 5, 7, 11, 16, 18, 62, 84, 98, 118, 121, 129, 146–149, 151, 153, 159, 165, 184, 193, 207, 260, 270, 271, 282, 285, 302, 332, 361, 366, 510–513, 537, 546 gender wage gap 9, 413–415, 420, 425, 427, 428 general equilibrium (models) 146, 188, 189, 207, 232, 380, 510, 517, 553 Gibrat’s law 554, 574 government budget constraints 203–205 grouped observations 422 GSOEP data 308 hazard (job re-accession) rate 274, 275, 278, 280, 282, 291–293, 307, 372, 375–378, 393, 401, 404, 513, 538 hazard function 291, 293, 537 integrated 392 rate 274, 275, 280, 282, 291–293, 307, 372, 375–378, 393, 401, 404, 513, 538 help wanted index 526–529 Heterogenous buyers 147, 148, 153, 159, 160 firms (productivity, employers) 39, 40, 42, 52–57, 134, 171, 553 job searchers, workers 147 jobs 147, 163, 169, 170, 180 sellers 158 worke(rs)=individual 70, 85, 86, 163, 164, 169–171, 179, 180 heterogeneity discrete 449 ex ante 64, 70, 73, 80, 260 ex post 64, 71, 73, 79, 80, 160 firm 8, 52, 165, 237, 238, 260, 261, 314, 564, 574 labor 117 latent variable 449, 452, 453 productivity 6, 62, 552–554, 558, 560 two-sided 5, 117

582

Subject Index

unobserved 238, 256, 299, 315, 382, 401, 441, 442, 445, 448, 449, 451–454, 459–461, 463, 466–468 hierarchical search model, hierarchical assignment model 6, 118, 120, 121, 128, 129 hierarchical sorting (search model) 117 higher productivity jobs 148 hiring costs convex 512, 543 linear 511, 512, 515, 542, 543 volatility 511, 512, 536, 543, 544 high persistence 10, 512, 534 high-productivity workers 85, 95, 97, 98, 104, 106, 108 high volatility 512, 534, 543 homogeneous buyers 147, 149, 160 jobs 179 firms 17, 19, 38, 39, 42, 53, 57, 106, 239 sellers 159, 160 workers 16, 39, 57, 62–64, 85, 86, 179, 222 Hosios rule 147, 155 HP-filter 526, 535, 544 human capital firm-specific 190, 203, 208, 218, 220, 231, 312, 318, 321, 453 IDA 8, 56, 295, 313, 315, 361, 552–554, 565–567 imperfect control 333, 340, 350–354, 358 imperfect information 148, 161–163 income permanent 374 transitory 365 increasing return to scale 121–123, 371 incumbent 312, 323, 325, 327, 559, 562 induced retirement 186, 196–199, 201–203, 207 industry wage differentials 415, 420, 423, 427 INSEE 248, 451 intertemporal utility 554, 555 interval identification 9 involuntary separations 188

Jensens’s inequality 48 job characteristics 171, 172, 180 complex 10, 117, 165, 240, 242, 498 creation 7, 156, 185, 208, 535 destruction (rate) 220, 288, 496, 497 duration 224, 238, 239, 285–287, 292, 298, 311 ladder 7, 217, 218, 232 mobility 142, 218, 286, 301, 311, 321, 327 offers 9, 42, 152, 220, 260, 278–280, 282, 381–385, 389–393, 395, 397, 400, 401, 404, 405, 409, 410 spell 221, 224, 239, 241, 273, 274, 282, 284, 289, 291, 292, 298, 300, 307 job-finding rate 498, 499, 501, 529, 530, 535, 538, 539, 544, 546 job openings and labor turnover survey 525, 526 job quality distribution 129 job search technology quadratic 122 linear 122 job search equilibrium 239 separation 8, 108, 159, 188, 278, 382, 487, 503 job seekers 122, 187, 202, 185, 199 job separations 10, 108, 219, 247, 274, 278, 518 job-to-non-employment transition 273, 284 job-to-job transition 38, 231, 284, 300, 302, 308, 498, 499, 501, 552 job-to-job turnover 273, 274, 288, 290, 296, 298 job vacancies 152, 153, 156–159, 184, 188, 203, 205, 208, 511, 521, 525–529 JOLTS, see job opening and labor turnover survey Kaplan-Meier 274, 291, 292, 369 labor demand 122

labor force 10, 38, 103–107, 184, 187, 189, 190, 194, 195, 197, 200, 202, 205, 207, 271, 272, 513, 517–525, 529, 532, 535–537, 554, 566, 567, 569, 574, 575

Subject Index

labor force growth 517–519, 536, 537 labor market entry 191, 192, 198, 221, 261, 452, 453, 456 flows 520 frictions 84, 98, 270, 271, 282, 285, 510 participation 184, 188, 196, 202, 205, 208 segregation 414 labor supply 122, 219, 452 labor turnover 8, 164, 270, 271, 332, 525 law of one price 61–63, 80 law of two prices 63, 80 LFS, see French Labour Force Survey life cycle 184, 188, 189, 193, 195, 207, 208, 316 likelihood 10, 223, 247, 249, 261, 263–265, 283–285, 298, 299, 301, 302, 332, 333, 339–348, 350, 352, 353, 355, 360, 361, 366, 370, 372–376, 378, 380, 393–397, 443, 444, 446, 450, 453, 454, 474, 477 LFS data 306, 448, 451 longitudinal plant and firm data 552 lump sum benefit 190 market equilibria 3, 15, 18, 26–32, 39, 40, 42–52, 54–58, 98, 124, 554, 560–565 marketing 163, 333–338, 340, 342, 343, 351, 355, 358, 360, 362 Markov process 165, 174, 175, 556 marriage duration 369, 372–374, 376, 379 hazard rate 372, 376 market 7, 9, 16, 120, 372, 374, 379 match quality 370, 485, 488, 490, 492–494, 497–499, 501, 503 shock 75, 241, 246, 355, 358, 511, 544 specific productivity 241 surplus 6, 50, 123, 184, 185, 193, 514, 542, 543 match indicator 121, 128 matched 8, 9, 248, 261, 361, 414, 415, 428, 552 matching assortative 163, 366

583

frictions 98, 184, 207, 546 inefficiency 148, 184 random 145, 146, 163, 379 stochastic matching technology 187, 192 technology 67, 146, 147, 151, 152, 154, 155, 158, 160, 164, 187, 192, 198, 200, 513 two-sided 5, 9, 163, 365, 366, 379 matching function CRS 67, 517 Cobb-Douglas 68, 179, 496, 513 linear 200, 511, 517, 542, 543 matching set 118, 120, 367, 368 match-specific capital 313 measurement error 218, 246, 332, 333, 340, 346, 347, 349, 350, 352–354, 358, 360, 361, 373, 385, 393, 394, 396, 397, 401, 446, 569, 575 mobility in earnings year-to-year 448 mobility indices 443 Mortensen efficiency rule 6, 7, 146, 147, 153–156 multiple equilibria 4, 16, 31, 32, 68, 76 Nash bargaining 17, 38, 39, 123, 165, 190, 193, 195, 196, 202, 205, 242, 417, 536 bargaining rules 146 bilateral bargaining 56 equilibrium 66, 151, 366, 368 solution 511 Nash-Bertrand equilibrium 555 NBER 544 NLSY data 7, 218, 219 NNI 248 non-convex payoffs 37, 39, 43–45 non-cooperative game 145 n-point wage distribution 39 OECD, see organization for economic co-operation and development offer arrival rate 221, 281, 361

584

Subject Index

offer distribution 8, 54, 57, 58, 128, 129, 135, 220, 221, 223, 225, 226, 295, 297, 304, 359, 361, 393–395 OG, see overlapping generations OMD, see optimal minimum distance estimator on-the-job search 16, 17, 28, 30, 37–41, 50, 51, 56, 62, 64, 108, 118, 120, 129, 135, 147, 156–159, 163, 165, 221, 227, 233, 240, 279, 361, 390, 483 on-the-job wage growth 218, 219, 221, 224, 321 optimal minimum distance estimator 572 organization for economic co-operation and development 185, 188, 199, 286, 310, 409 overlapping generations 7, 185–187, 191, 207 overlapping labor market 180 overlapping matching set 377 Oaxaca decomposition 415, 420, 426, 427 panel study of Belgian households 415 Pareto (distribution, families) 6, 56, 140 Pareto efficient 43 Pareto frontier 45 Pareto law weak 133, 134, 140–142, 144 partial equilibrium 199–201, 371 Partial equilibrium model 271, 511, 517 pay-as-you-go 7, 184, 185 payroll taxes 199, 203, 204, 248 pension 7, 184, 186, 191, 195, 202, 316, 530 pension benefit 183, 184, 186, 196, 202, 207 pension program 7, 184, 185, 188, 190, 193, 199, 201, 203 persistence 10, 511, 512, 521, 526, 532, 534, 536, 539, 543, 546 Pools searching workers 511, 521, 525, 534, 536, 544 workers 522, 524, 536 population growth 186, 518 post wage 3, 16, 38, 64, 71, 84, 106, 135, 163

posting vacancy 17, 18, 28, 69, 189, 198, 486, 496, 497 private pension payments 195 product arrival rate 556, 561 destruction rate 561 innovation 553, 554, 556, 576 Productivity differential 84, 85, 97, 98, 103, 106, 107, 190 factor 186, 514, 559 firm 62, 220, 361, 552, 554, 565, 575 firm-specific 241 growth 11, 232, 233, 518, 531, 534, 542, 551, 553, 565, 576 labor 62, 136, 185, 485, 499, 503, 504, 511, 538, 552, 555, 566 Profit 40, 45, 50, 64, 71, 74, 75, 79, 90, 94, 137, 152, 157, 175, 178, 198, 239, 334, 414, 420, 424, 427, 514, 519, 556, 574 propagation mechanisms 510 PSBH, see panel study of Belgian households PSID data 272, 448 publicly funded pension program 207 public pension benefit 184 quality adjusted employment 566 random utility 333, 340, 354–356, 358 random utility shock 354, 355 rapid tail 139, 141 RBC 517, 519, 535, 536 regular tail 139, 141 R&D investment 553, 556 reduced form estimation 382 reduced-form VAR model 511, 518, 519, 538 reforms of unemployment compensation system 382, 383, 405, 410 rent sharing 314, 414, 415, 418, 423, 425–427 replacement rates 127, 208, 407 reservation policy 367, 368 reservation property 65, 74, 76, 80, 490, 506 reserve price 146, 149–151, 153, 154

Subject Index

reservation wage 20, 64, 70, 74, 76, 79, 127, 171, 175, 177, 179, 240, 242, 279, 360, 383, 389, 392, 396, 401 retention 261, 262 retention policies 238, 244, 246, 255 returns to education 309 experience 245, 246, 260, 453 seniority 238, 243, 257, 260, 261, 277, 247 tenure 257, 260 retire 188, 190, 196, 199, 203, 207 right-hand tail 133–135, 137, 138, 140–143 right-to-manage model 416 risk averse 38, 260 risk-neutral 40, 186, 367, 416 SBS, see Statistics Belgium, structure of business survey school-to-work transition 221 search costs 63, 73, 80, 135, 194, 197, 211, 359, 515, 520 disutilities 186 effort 3, 10, 409, 484, 485, 487, 489–491, 497–499, 506 equilibrium 5, 32, 33, 118, 239, 489 frictions 62, 84, 118, 121, 129, 184, 260, 271, 285, 302 intensity 10, 16, 293, 380, 493, 498 model 7, 8, 106, 108, 118, 120, 121, 128, 129, 134, 135, 142, 146, 220, 221, 223–225, 227–230, 232, 233, 260, 270, 271, 295, 299, 304, 311, 332, 333, 358, 360–362, 383, 389,410, 482–484, 486, 497, 498, 503 sequential 191 wage 3, 4, 17, 20, 33 search and matching model 10, 366, 510, 511–521, 545, 546 search models competitive 160 directed 5, 85, 99, 146 price-posting 147, 160, 165 undirected 146, 208

585

wage-posting 4, 38, 40, 46, 55, 57, 58, 147 segmentation 119, 120 separation (rates) 8, 10, 108, 159, 376, 482, 487, 501, 503, 504, 511, 517–520, 531, 534–537, 543, 552 shock job destruction 287, 485, 279 productivity 10, 244, 246, 279, 304, 483, 485, 487, 502, 511, 536, 544 reallocation 8, 279–282, 286 transitory 75–77 simulated minimum distance estimator 567 SIREN 248 socially efficient 84–86, 97, 98, 102, 106 social loss 17 social optimum 97 social planner 85, 98, 155, 156, 163 social security 7, 184–186, 200, 203, 207, 208 social security tax 195 social surplus 191 spell duration 271, 272, 283, 284, 292 start-up cost 486 statistics Belgium structure of business survey 418 steady-state equilibrium 4, 29–31, 189, 196, 202, 205, 368 market equilibrium 29, 565, 562 wage distribution 135, 219, 295 stigma 504 stochastic dynamic equilibrium 498, 517 stochastic market equilibrium model 554 stochastic job match model 120 stochastic job quality model 121 structural model 11, 271, 283, 288, 292, 303, 382, 398, 511, 566 structural nonstationary model of job search 383, 385, 389, 407 subgame perfect equilibrium 39, 43, 45–47, 49, 50, 53, 58

586

Subject Index

surplus 6, 38, 41, 47, 50, 52, 56, 57, 70, 71, 76, 78, 123, 147, 152, 154, 155, 177, 184, 185, 193–195, 202, 210, 418, 488, 511, 514–516, 520, 542, 543 surplus of a match 6, 147, 152, 154, 155, 543 survivor function 138, 394 temporary jobs 384, 397, 401, 410 tenure 7, 8, 184, 190, 208, 218, 255–257, 260, 274, 282, 313, 316, 317, 321–323, 325, 415, 488, 489, 492, 495, 506 trade union 312, 325, 326 transformed worker index 118 transitory shocks 75–77 turnover high 8, 56, 255, 260, 261, 273, 310, 486 intensity 288 involuntary 8, 288, 313 low 8, 255, 273, 274, 289 voluntary 288 types of workers 64, 84–87, 89, 91–96, 99–104, 106, 108, 110, 112, 114, 180, 190, 276 unemployment benefits 134, 135, 188, 359, 383, 385, 386, 516 duration 360, 384, 385, 394, 397, 405, 407, 410, 535 insurance 173, 190, 325, 383, 387, 391, 405, 486, 497 spell 8, 284, 290, 291, 325, 382–386, 389, 391–395, 407, 552 volatility 544 wage 383, 385, 393 unemployment pool 511, 513, 521, 522, 524, 532, 534, 537–539, 541 U.S. labor market 306, 510, 512, 521–536, 544, 546 U.S. Labor market dynamics 512 US national longitudinal survey of mature men 311 utility shock 354–358

vacancy creation 10, 127, 199, 200, 483–486, 489, 495 497, 503, 504, 511, 512, 519, 542, 543 creation costs 10, 127, 484, 486, 497, 542 distribution 120, 125, 126, 129 posting cost 198, 496, 497 value added 418, 424, 425, 553, 563, 565–567, 569, 573–576 value of the firm 556, 557 volatility 10, 503, 511, 512, 524, 526, 529, 532, 534, 536, 539, 543, 544, 546 wage accepted 284, 313, 360, 394, 395 bargained 17, 18, 27, 28, 31, 39, 40, 42, 43, 45, 52, 417 bargaining 41, 42, 52, 56, 58, 191, 208, 325, 415, 418, 423, 427 compensation 422 competition 18, 230 contracting model 58 determination 3–5, 193, 194, 204, 208, 422 differential 5, 84–87, 97, 98, 100, 104, 106–108, 327, 414 dispersion 3–5, 8, 16, 38, 44, 62, 66, 68, 70, 73, 77, 80, 148, 159, 270, 333, 552 distribution 4, 38–41, 43–49, 51–55, 57, 58, 62, 63, 80, 83, 85, 103, 104, 106, 121, 128, 134, 135, 141, 142, 219, 220, 222, 227, 230, 231, 233, 277, 282, 295, 312, 313,361, 452 dynamics 8, 278 equation 243–247, 249, 250, 252, 254–257, 261, 263, 264, 266 equilibrium 4, 31, 32, 50, 53, 55, 77, 85, 103, 106, 121, 134, 141, 159, 160, 223, 311, 422, 562, 564 firm-specific 243, 244, 246, 249, 256, 263 function 52, 54–56, 186, 194, 195, 313, 316, 322 growth 8, 9, 217–222, 224, 225, 228, 229, 231–233, 311, 313, 315, 318, 321–323, 325, 327

Subject Index

high 8, 38, 45, 46, 58, 63, 68, 77, 86, 105, 106, 142, 208, 257, 259, 260, 261, 282, 293, 321, 323, 325, 384, 407, 409, 410, 457–460, 463, 465, 467 wage-offer distribution 8, 54, 57, 58, 128, 135, 220, 221, 223, 225, 230, 293, 295, 297, 304, 361, 393–396 low 8, 45, 46, 68, 85, 105, 106, 231, 257, 259–261, 293, 457–459, 465 market 49, 50 offered 18, 22, 136, 233, 394, 395 offers 9, 45, 53, 58, 135, 136, 223, 278, 283, 293, 303, 304, 311, 313, 315, 359, 382–384, 389, 391, 392, 396, 400, 401, 410 paid 5, 6, 16, 18, 20, 21, 30, 52, 202, 298, 552, 566 posted 92, 72, 76, 84, 147 posting 4, 6, 38, 40, 46, 55, 57, 58, 62, 65, 69, 73, 147

587

post-unemployment 383, 385, 393, 396 premium 311, 314, 315 rejected 45, 383, 389, 390, 395, 396 residual 98, 315, 322 setting 4, 39, 41, 42, 123, 128, 483, 487, 488, 495, 503 starting 222, 224, 227, 243–245, 247, 249, 255–257, 260, 261, 263, 265, 266 trajectories 7, 8, 304, 313, 314, 327 wage-profit elasticity 414, 415, 420, 423–425, 427 Walrasian economy 145, 160 Walrasian equilibrium 123, 127 Wilcoxson test 369 worker flows 270, 281, 288, 289, 333, 510, 521, 522, 528, 552 young workers 184, 186, 188, 194, 195, 197, 202, 203, 208, 221 young jobseekers 185, 187

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588