Exchangeability

The Economics of Search

The economics of search is a prominent component of economic theory, and it has a richness and elegance that underpins a host of practical applications. In this book Brian and John McCall present a comprehensive overview of the economic theory of search, from the classical model of job search formulated forty years ago to the recent developments in equilibrium models of search. The book gives decision-theoretic foundations to seemingly slippery issues in labor market theory, estimation theory and economic dynamics in general, and surveys the entire field of the economics of search, including its history, theory, and econometric applications. Theoretical models of the economics of search are covered as well as estimation methods used in search theory, and topics covered include job search, turnover, unemployment, liquidity, house selling, real options and auctions. The mathematical methods used in search theory such as dynamic programming are reviewed as well as structural estimation methods and econometric methods for duration models. The authors also explore the classic sequential search model and its extensions in addition to recent advances in equilibrium search theory. This book is highly relevant to all graduate economics courses in search theory. It is also ideal as a reference book for economists interested in search theory, mathematical economics and economic theory and philosophy. John McCall is Professor Emeritus of Economics at the University of California Los Angeles and University of California Santa Barbara. Brian McCall is Professor of Education & Professor of Economics at the University of Michigan, USA.

Routledge Advances in Experimental and Computable Economics Edited by K. Vela Velupillai, National University of Ireland, Galway

and Francesco Luna, International Monetary Fund (IMF), Washington, USA

1. The Economics of Search Brian and John McCall

Other books in the series include: Economics Lab An intensive course in experimental economics Alessandra Cassar and Dan Friedman

The Economics of Search

Brian P. McCall and John J. McCall

First published 2008 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Simultaneously published in the USA and Canada by Routledge 270 Madison Ave, New York, NY 10016 This edition published in the Taylor & Francis e-Library, 2007. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Routledge is an imprint of the Taylor & Francis Group, an informa business © 2008 Brian P. McCall and John J. McCall All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data McCall, Brian Patrick. The economics of search / Brian P. McCall and John J. McCall. p.cm. Includes bibliographical references and index. 1. Economics—Mathematical models. 2. Search theory. I. McCall, John Joseph, 1933– II. Title. HB135.M398 2007 330.01′1—dc22 2007022716 ISBN 0-203-49603-5 Master e-book ISBN ISBN10: 0–415–29992–6 (hbk) ISBN10: 0–203–49603–5 (ebk) ISBN13: 978–0–415–29992–3 (hbk) ISBN13: 978–0–203–49603–9 (ebk)

To Megan and Conor, the next generation

Contents

List of figures List of tables Preface 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6

Preliminaries Contracts, incentives and asymmetric information Search theory Institutional aspects of search economics The explanatory power of economic search Outline of this volume

2 Mathematical methods 2.1 2.2 2.3 2.4 2.5

Introduction Markov chains and related processes An introduction to stochastic dynamic programming Some results in probability theory: a measure theoretic approach The Poisson process

3 The history and evolution of sequential analysis 3.1 3.2 3.3 3.4 3.5 3.6

Introduction Early literary and intuitive contributions to the economics of job search Stigler’s insights Stigler’s model Early mathematical contributions to the economics of job search The sequential job search model

xiii xiv xv 1 1 6 8 9 10 11 19 19 19 34 41 45 49 49 50 52 53 55 57

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Contents 3.7 3.8 3.9 3.10

Sequential analysis: a deeper mathematical perspective Optimal stopping theory Markov decision processes The four stellar examples of MDPs

4 The basic sequential search model and its ramifications 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17

Introduction An important example The fixed-sample size model The basic sequential search model: infinite time horizon, no discounting An alternative view of the reservation wage Infinite time horizon: discounting Infinite time horizon with discounting and a random number of job offers Risky ordering and sequential search The basic sequential model: search for the lowest price Optimal stopping Finite time horizon On-the-job search Search in the presence of experience qualities: the Lippman and McCall belated information model The Lucas equilibrium model for the basic sequential search model Reservation wage property preserved in a continuous search model Optimal replacement policy Martingales and the existence of a reservation wage Appendix 4.I: A naive comparison of FSM and BSM Appendix 4.II: A more sophisticated comparison of FSM and BSM Appendix 4.III: The French version of BSM

5 Estimation methods for duration models 5.1 5.2 5.3 5.4 5.5 5.6

Introduction Hazard functions Counting processes and martingales Parametric methods for continuous-time data with covariates The Cox regression model Discrete-time duration data

58 59 63 63 65 65 69 70 72 74 75 76 77 81 89 90 92 94 97 98 100 101 103 106 110 116 116 117 121 129 136 143

Contents 5.7 5.8 5.9 5.10

Multi-spell discrete-time models Competing risk models General discrete-time life history models Specification tests for duration models

6 Unemployment, unemployment insurance and sequential job search 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Introduction Mismatch unemployment Layoff unemployment with positive probability of recall: the temporary layoff Unemployment insurance and job search decisions (Mortensen 1977) More on the incentive effects of unemployment insurance Efficient unemployment insurance (Acemoglu and Shimer 1999) More on optimal umemployment insurance

7 Job search in a dynamic economy 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Introduction Search in a dynamic economy (Lippman and McCall) Labor market interpretation Variable intensity of search The basic theorem Wealth and search Systematic search Optimal quitting policies Monotone Markov chains and their applications: Conlisk’s research 7.10 Recent research by Muller and Stoyan 8 Expected utility maximizing job search 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Introduction Characterizing risk and risk aversion Comparative statics results Applications of Theorem 8.8 The basic job search model Expected utility maximizing search with recall Expected utility maximizing search without recall Risk aversion and the wage gap Consumption commitments and local risk aversion

ix 149 151 154 159 161 162 163 168 171 177 184 191 198 198 199 202 203 205 206 207 208 208 214 217 217 218 225 226 227 229 241 244 244

x

Contents 9 Multi-armed bandits and their economic applications 9.1 9.2 9.3 9.4 9.5 9.6 9.7

Introduction The multi-armed bandit problem General MAB framework A model of job search with heterogeneous opportunities Miller’s model of job matching and occupational choice Superprocesses: adding decisions to bandits Discounted restless bandits

10 A sample of early responses to Diamond’s paradox and Rothschild’s complaint 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction Price dispersion Matchmaking Search technology Principal–agent problems and durable matches Equilibrium models of price dispersion General structure of equilibrium models

11 Equilibrium search after the Diamond–Mortensen–Pissarides breakthrough 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15

Introduction Two-sided search and wage determination: the Mortensen–Pissarides approach Matching technology Search equilibrium Equilibrium unemployment (Mortensen and Pissarides 1999) Wage differentials, employer size, and unemployment Equilibrium search with offers and counteroffers Competitive search equilibrium Labor market policies in an equilibrium search model (Lucas–Prescott in action) The search contributions of Lars Ljungqvist and Tom Sargent Two-sided search, marriage and matchmakers Bilateral search and vertical heterogeneity Two-sided search with fixed search costs Rocheteau–Wright models of search and money Recent discoveries

246 246 246 256 257 262 265 272

277 277 279 280 280 281 281 300

303 303 305 306 308 309 313 317 324 325 331 345 349 350 353 354

Contents 12 Structural estimation methods 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10

Introduction Early models (Kiefer and Neumann 1979a, 1979b) Illustrations of structural estimation The general model Solution methods Finite horizon (T < ∞) search model Estimation methods Structural estimation of equilibrium job search models Structural estimation and model complexity Dacre, Glazebrook, Nino-Mora (1999)

13 The ubiquity of search 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14 13.15 13.16 13.17 13.18

Historical prelude Introduction The evolution of money Liquidity and search The housing market and liquidity (Krainer and Leroy) The house-selling problem: the view from operations research The natural rate of unemployment A modern approach to the natural rate of unemployment Adam Smith on dynamic stochastic models Coordination and inflation Economic growth and coordination Auctions versus sequential search Auctions and bidding by McAfee and McMillan The analytics of search with posted prices Sale of a deteriorating asset via sequential search Middlemen, market-makers, and electronic search Real options Resource allocation in a random environment

14 Topics for further inquiry 14.1 14.2 14.3 14.4

Exchangeability Polya urns Urn methods The secretary problem

xi 356 356 356 357 365 367 367 368 369 380 383 387 387 389 392 397 407 413 414 416 418 418 419 420 425 434 437 437 446 462 471 471 474 475 483

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Contents 14.5 14.6

The economics of information, behavioral biology, and neuro-economics Equilibrium search models revisited

Notes Bibliography Index

487 490 491 506 540

Figures

4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 6.1 6.2 6.3

Graphical representation of reservation wage determination Change in reservation wage with an increase in the risk of distribution Determination of optimal search strategy in on-the-job search model Determination of optimal search strategy in on-the-job search model Weibull hazard function Log-logistic hazard function Nelson–Aalen cumulative re-employment hazard estimate Smoothed re-employment hazard estimate Weibull model: cumulative hazard estimate Weibull model: survivor function estimate Weibull model: survivor function estimate by unemployment receipt Weibull-Gamma model: cumulative re-employment hazard estimate Weibull-Gamma model: survivor function estimate Weibull-Gamma model: survivor function estimates by unemployment insurance receipt Cox regression model: cumulative hazard function estimate Cox regression model: survivor function estimate Cox regression model: survivor function estimate by unemployment insurance receipt Example of a sample path Graphical depiction of equilibrium condition Changes in preferences Change in unemployment benefits

75 79 93 94 119 120 129 130 134 134 135 135 136 136 141 142 142 157 189 190 190

Tables

4.1 Assumptions of the BSM A4.1 Reservation prices for different costs of search under price distribution F1 and F2 A4.2 Values of the expected fall in minimum price (d(n)) by increasing sample size by one A4.3 Expected total cost comparison of fixed-sample and basic search models 9.1 Dynamic allocation index of occupation (Z(0)(1 − β)) 10.1 Taxonomy of equilibrium dispersion models 10.2 Number of different market equilibria, by search method

68 104 105 105 270 282 299

Preface

Our plan is to review optimal stopping rules from their discovery by A. Wald et al. to their matching generalization by P. Diamond, D. Mortensen, and C. Pissarides. This review is conducted within the boundaries of the economics of search. As far as we know, this is the first book to survey the entire field of search economics including its history, theory, and econometric applications. The economics of search is a prominent component of today’s economic curriculum. It has many practical applications supported by a rich and elegant theory. Our book commences with a historical survey. We then expose the theory of optimal stopping and conclude with several empirical applications. The driving force behind the search enterprise was to comprehend the economics of unemployment. Most of the book is devoted to ramifications of this fundamental problem. After a brief introduction in Chapter 1, we review some of the probabilistic techniques that are used repeatedly in the book in Chapter 2. Readers who are familiar with this material may wish to skim this chapter to become familiar with our notation. The history of sequential analysis is presented in the third chapter. Once again some readers may wish to give this chapter a quick read, while those interested in the evolution of search will want to read some of the elegant references, especially Bellman, Blackwell, Breiman, Karlin, Gittins, Lai, Shapley, Siegmund, Weiss, and MacQueen. Chapter 4 gives the details of the basic search model and its ramifications throughout economics. In Chapter 5 we study duration analysis which is a crucial econometric method for analyzing the sequential data comprising the basic search model. Chapter 6 studies unemployment and unemployment insurance from a search perspective. The first study is the mismatch unemployment that attends youth unemployment. Temporary unemployment is also analyzed. The chapter concludes with a discussion of the moral hazard problem inherent to unemployment insurance. Chapter 7 is an intensive treatment of job search in a dynamic economy. This model has been useful in many operations research studies. As pointed out by John Conlisk, the model has an important monotonicity property. Economic models of search and information proliferated at a rapid rate

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after Stigler’s seminal work in the area. These models were useful in explaining the phenomena of unemployed resources in the economy. While the search literature represents a significant improvement over the earlier deterministic approaches, a standard assumption has been that the objective of the searcher is to maximize expected income. This conflicts with the traditional view that individuals maximize utility functions that are concave in income or wealth. It is the purpose of Chapter 8 to consider the implications of the introduction of a utility-based objective into the search problem. The result is a significant restructuring of the qualitative properties of optimal search. Section 8.5 reviews the basic search model, explicitly delineating its assumptions and implications, and set forth the expected utility maximizing model that will be examined here. Section 8.6 derives the qualitative properties of expected utility maximizing search under conditions of recall. Section 8.7 considers the case of non-recall. Section 8.8 reviews some basic issues in risk and risk aversion. It is based on the survey by Scarsini (1994). The concluding section (Section 8.9) presents analyses of insurance based on the two-state framework introduced by Erhlich and Becker (1972), Hirshleifer (1971), and extended by Rothschild and Stiglitz (1976) and Lippman and McCall (1981). In Chapter 9 we show how the theory of multi-armed bandits applies to many problems of optimal search. The multi-armed bandit (MAB) problem was first posed around the end of World War II and defied solution for three decades until it was finally solved by John Gittins (1979). Gittins showed that the optimal solution to this decision problem involved an index policy in which each bandit is assigned an index and the optimal policy is to sample the bandit that currently has the highest index. These indices are sometimes referred to as Gittins’ indices in honor of Gittins and we will follow this tradition. The next section of the chapter sets out a proof of the optimality of the index policy. The proof is based on that of Whittle (1980) who alternatively used a stochastic dynamic programming approach to prove Gittins’s result. Section 9.3 recasts the result in a more general framework and shows an alternative characterization of the index (which is actually its original characterization). Section 9.4 then describes the generalization of MAB to superprocesses and how an additional sufficient condition is needed to ensure an index solution. We then show how the basic sequential search model (BSM) is simply a special case of the MAB problem and derive its solution using Gittins’s indices. The next several sections present applications of MAB theory to more general problems in search, including the model occupational choice and job matching by Miller (1984) and the model of occupational matching by McCall (1991). Chapter 10 presents some of the early work on equilibrium models with price (wage) dispersion that arose in response to the Diamond paradox. As frequently occurs in science, the idea of a matching equilibrium was sparked

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almost simultaneously in the minds of three prominent economists: Peter Diamond, Dale Mortensen, and Christopher Pissarides. Chapter 11 surveys these insights. These illuminations happened around 1980. In their excellent survey, Mortensen and Pissarides (1999a) note that the ensuing match equilibrium literature is composed of two distinct branches. The objective of the first branch is to provide an explanation of worker and job flows and unemployment in a setting where frictions accompany the stochastic process that matches jobs and workers. This matching approach studies the incentives to invest in search, recruitment, job learning, and all the forms of specific human capital which produce equilibrium levels of employment and unemployment. The second branch investigates the impact of market frictions on wage formation assuming that employers are able to post wages, whereas workers search for the best wage. The search friction is the time that workers spend in their search activity. These two branches have generated an enormous and sophisticated literature which we only skim in this chapter. The companion volume will discuss additional topics in equilibrium search. The first section of Chapter 11 describes two-sided search and wage determination as formulated by Mortensen and Pissarides. The following section exposes the matching technology which is basic to this new approach. As part of the exposition, we present an urn model that exemplifies the properties of this important function. Section 11.4 is a brief description of the search equilibrium. The properties of equilibrium unemployment are then presented for this version of the matching model. The best way to comprehend the Diamond–Mortensen–Pissarides (DMP) innovation is to see how it operates in a relatively simple setting. Hence, Section 11.5 shows a superb use of this innovation including some new ramifications. The model presented in this section has been devised by Mortensen and Pissarides. The Diamond paradox (1971) is resolved by Burdett and Mortensen. In brief, buyers and sellers are identical, respectively, and have the same search cost. In this setting sellers charge the monopoly price. On the other hand, if buyers make take-it-or-leave-it offers and sellers engage in costly search for buyers, then the competitive price clears the market. This perplex has preoccupied search theorists for the last thirty-five years. One way of cutting the Gordian knot is to introduce heterogeneity. Butters did this in decisive fashion in his 1977 article. What is remarkable about Burdett and Mortensen is that the conundrum is solved when there are identical buyers and identical sellers as postulated by Diamond. On-the-job-search is the key to the Burdett–Mortensen resolution. The Burdett–Mortensen model is presented in Section 6. Perhaps the most important feature of the matching equilibrium analysis is its welfare properties. Using a criterion developed by Hosios, Section 11.6 also describes the razor-like condition required for efficiency. Inefficiency is anticipated in DMP models and its source is described in Section 6. Postel-Vinay and Robin expand upon the Burdett–Mortensen model by

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allowing for a series of offers and counteroffers among employers when an employee receives an outside job offer. This model is discussed in Section 11.7. It will be surprising to some to see a directed search model exhibit efficiency. This is Moen’s contribution and it is elaborated in Section 11.8. The following section presents the seminal island model by Lucas and Prescott. Here we have undirected search as in the basic sequential search model. This combination of undirected sequential search and the Lucas– Prescott model was instigated by Alvarez and Veracierto. It is their model which we describe in Section 11.9 together with its empirical implications. The next section shifts to another seminal model originated by Gale and Shapley. Brock has shown that this model and the illustrious human capital marriage model by Becker are similar. The application is by Bloch and Ryder, where basic insights were made by Collins and McNamara. Section 11.11 presents the important work by Rocheteau and Wright which in many ways unifies the previous research and describes a search-theoretic foundation for monetary theory, which was composed by Kiyotaki and Wright. The final section reports recent and important empirical research by Shimer and Hall. Structural econometric methods are the subject of Chapter 12. There have been several excellent surveys on structural estimation in economics, two of which are books, the volume by Devine and Kiefer (1991), which presents commentaries on more than 100 research studies, and the monograph by Wolpin (1995), an illuminating portrait of structural estimation. The survey articles include: a brief discussion of the original model by Eckstein and Wolpin (1989), which we summarize in Section 2,1 and the splendid articles by Rust (1994a, 1994b), which expose the over-ambition in structural estimation. This defect centers on the computational complexity of dynamic programming. The complexity of dynamic programming is mitigated by the LP methods associated with a new approach to the Gittins index. A concise summary of this approach is contained in Section 12.8. The recent survey by Eckstein and Van Den Berg (2007) is exploited in Section 12.10. Search drenches economics and transforms a static set of definitions into a dynamic, inter-related construct. In Chapter 13, we consider several important topics in search economics: exchange, liquidity, economic growth, house selling, the natural rate of unemployment, insurance, real options, the evolution of money, property rights, division of labor, and trust. A proper explanation of each of these topics entails a description of how the particular topic interacts with search. We now present a brief description of this chapter. The first section of this chapter presented some pertinent historical information which may cast some light on the remaining sections. Actually, this historical information is transmitted throughout the entire chapter. The main purpose of Section 13.2 is to alert the reader to the contents of the subsequent sections.

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Section 13.3 traces the evolution of money beginning with a barter economy and progressing to the money economies now operating in almost every country. The presentation leans on the excellent article by Robbie Jones which emphasizes the role of sequential search and the work by Kiyotaki and Wright, which centers on the matching models of Mortensen, Diamond, and Pissarides. Section 13.4 surveys liquidity from a search perspective. The general setting is described first. This is followed by alternative definitions of liquidity and their compatibility with a search definition. Liquidity is then discussed vis-à-vis the thickness of markets, predictability and flexibility. Next, in Section 13.5 we analyze the interplay between the housing market and liquidity. The section is drawn from an article by Kramer and LeRoy. Both the demand and supply of housing are studied. An equilibrium is derived and characterized. Under certain conditions, the market has the martingale property. Section 13.6 alerts the reader to the equivalence between the basic sequential search model and the house-selling problem.2 A brief description of the natural rate of unemployment is contained in Section 13.7. We emphasize its historical development beginning with its almost simultaneous discovery by Ned Phelps and Milton Friedman in 1967. Section 13.8 presents a modern approach to the natural rate of unemployment based on Rogerson’s 1997 article. Rogerson describes the natural rate in terms of the matching methods devised by Diamond, Mortensen, and Pissarides, and presents a critical assessment. After briefly considering Smith’s views on economic dynamics, Section 13.10 discusses the Ball–Romer model that illustrates the role coordination plays in inflation. Section 13.11 describes a search/matching approach to growth in an environment of technological advance. We expose the recent research on this topic by Jones and Newman (1995). Their article models this process of adaptation and re-adaptation to ongoing shocks emanating from continuous technological changes, by taking a search/matching perspective. Their central idea is that each state of technical knowledge reveals a set of feasible activities and also reveals a set of potential matching opportunities which agents can exploit. Beneficial matches improve technology, but are not realized immediately. Their realization requires a learning process in which agents search over and/or experiment with the novel opportunities. Learning in this manner is an investment in adaptive information. A key assumption is that each advance triggered by technical improvement increases potential rewards to feasible matches, but its immediate effect is to lower matching efficiency as it diminishes the usefulness of adaptive information acquired in earlier states. This is Schumpeter’s creative destruction—the enhancement of one aspect of informational capital destroys the usefulness of its adaptive counterpart. The following section examines the close relationship between auctions and sequential search. Arnold and Lippman show that there are at least three connections. The analysis first generalizes BSM to encompass the sale of several units. With respect to the auction regime, they next demonstrate that

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the expected return per unit is increasing when the valuations of bidders are random variables with exponential or uniform distributions. Their third discovery shows that the preferred mechanism depends on the number of units for sale. More precisely, they prove that there is a crossing point n* such that BSM is preferred if n < n* and an auction is more desirable if n ≥ n*. The predictions of Arnold and Lippman are confirmed in a careful empirical study of livestock markets. Section 13.13 is the comprehensive and illuminating study of auctions and bidding by McAfee and McMillan. Their analysis begins with a set of eight assertions based on their own paper together with the insights of Akerlof regarding asymmetric information. McAfee and McMillan believe that asymmetric information lies at the core of auction theory. In the next section we explore search with posted prices in which a seller sets the price of a good and then a buyer arrives and determines whether or not to purchase the item according to whether the valuation exceeds the posted price. Section 13.15 concentrates on middlemen, market makers, and electronic search. Each of these topics deserves a separate book, so our treatment is necessarily incomplete. We begin with Shevchenko (2004) on middlemen, who cites Rubenstein and Wolinsky (1987) as a commencement to his study. Shevchenko’s fundamental idea is that in an economy with many goods and heterogeneous tastes, there is a definite role for agents who hold inventories of a broad spectrum of items.3 This critical tradeoff facing intermediaries is similar to that confronting individual firms when they are deciding on the level of inventory to hold for each good.4 Shevchenko obtains the steady state equilibrium number of middlemen, together with their size, and the distributions of inventories and prices. This section also surveys the article by Rust and Hall (2003). They study two types of competitive intermediaries—middlemen (dealer/broker) and market makers (specialists). They show that with free entry into market making and search and transactions costs going to zero, the bid-ask spreads are driven (competitively) to zero. Thus, a fully efficient Walrasian equilibrium is obtained asymptotically. Their findings imply that middlemen and market makers survive in this competitive environment with each allocated half of the total trade volume. Section 13.16 considers electronic search. Autor (2001) obtains three consequences of the labor market internet. First, in their job search, workers use many sources of job information: friends, co-workers, private employment agencies, newspaper advertisements, etc. Second, the Internet labor markets may facilitate the acquisition of new skills, and third, e-commerce has a potent effect on labor outsourcing and specialization. A brief summary of the recent article by Ellison and Ellison (2005) comprises Section 13.16. They enumerate several important lessons about markets which have been taught via the Internet. They address two basic questions: how has the Internet affected commerce and how can it generate frictionless commerce? They explain how the Internet has revealed new

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insights with respect to the operation of markets with negligible search costs. Empirical analysis can be conducted where only search costs differ. The Internet has also decisively improved the ability to conduct field experiments. The Ellisons’ note that some observations coming from an economic analysis of the Internet challenge the basic economics of search costs and product differentiation. The next topic related to BSM is learning from experience. We present the research of Jones and Newman, Muth, and Rothschild. The recent article by Muller and Ruschendorf (2001) is then covered. They show that the “main tool” for a constructive proof of Strassen’s theorem is the BSM. This is important because this classic theorem links the BSM to the max-flow-min-cut theorem of Ford and Fulkerson and then to a host of graph-theoretic results, including matching and networks that have assumed importance in the economic literature. The final section is devoted to a presentation of the famous assignment theorem by Derman, Lieberman, and Ross (DLR) (1972b). This is also closely related to the BSM and other important “empirical” theorems. Chapter 14 concludes our review and presents a glimpse of Volume II. We note that Volume II contains a review of real options and their nexus with optimal stopping. The ecology of search is also portrayed in our study of the honeybee. We also consider recent Bayesian methods including the landmark innovation: the Markov Chain Monte Carlo (MCMC) technique. Adaptive search is also surveyed along with exchangeability. The secretary problem bears a striking connection with optimal stopping and has generated a large literature, which we review. We also show how neuroeconomics is related to optimal search. Finally, the important papers by Hall, Ljungqvist and Sargent, and Shimer which question the usefulness of Nash matching. Ljungqvist and Sargent construct generalized versions of the basic search model and both Hall and Shimer raise deep objections to Nash bargaining. While a substantial amount of the work done on this book was completed while B. McCall was at the University of Minnesota, some of the work was also completed when B. McCall was on sabbatical at the Center for Labor Economics (CLE) at the University of California, Berkeley. Support from the CLE is gratefully acknowledged. B. McCall is also grateful to his wife, Toni, for her support during the “never ending” process of writing this book. J. McCall thanks his wife, Kathy, for her unlimited encouragement and common sense during this uncertain adventure. Many have contributed to this volume. Unfortunately, the literature is so large, it is impossible to acknowledge all those who have written in the search area. K. Velupillai read and made important critiques of the earliest versions. We owe him a great debt. Others who made substantial suggestions include: Brian Ellickson, Glenn Gotz, Jack Hirshleifer, Robbie Jones, Axel Leijonhufvud, Steve LeRoy, and John Riley. Major revisions were performed in response to the extensive and penetrating remarks by Buz Brock, Bob Lucas,

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The Economics of Search

and Dale Mortensen. Siegfried Berninghaus contributed a draft of Chapter 12. A special debt is owed to Steve Lippman, who not only collaborated with J. McCall on several articles discussed here, but also worked on an earlier version of this book which was terminated by J. McCall’s illness. Finally, we owe much to Lorraine Grams who typed the many versions of this volume with great care and cheer.

1

Introduction

1.1 PRELIMINARIES We live surrounded with the protection of thousands of arts and we have devised schemes of insurance which mitigate and distribute the evil which accrue. Barring the fears which war leaves in its train, it is perhaps a safe speculation that if contemporary western man were completely deprived of all the old beliefs about knowledge and actions he would assume, with a fair degree of confidence, that it lies within this power to achieve a reasonable degree of security in life. . . . primitive men had none of the elaborate arts of protection and use which we now enjoy and no confidence in his own powers when they were reinforced by appliances of art. He lived under conditions in which he was extraordinarily exposed to peril, and at the same time he was without the means of defense which are today matters of course. Most of our simplest tools and utensils did not exist; there was no accurate foresight; men faced the forces of nature in a state of nakedness which was more than physical; save under unusually benign conditions he was beset with dangers that knew no remission. . . . Any scene or object that was implicated in any conspicuous tragedy or triumph, in no matter how accidental a way, got a peculiar significance. It was seized upon as a harbinger of good or as an omen of evil. (Dewey 1929: 9–10) The quest for certainty was the basic force fusing primitive men into tribal groups. The institutions and risk-controlling mechanisms designed in this social setting allowed the protected members to gather special information and concentrate on the corresponding activities. No doubt the “family” was the first institution to be formed, perhaps simultaneously with the realization that hunting groups of size n were much more efficient than n solitary hunters. This insight would lead quickly to the development of clans or tribes. The problem of cooperation and conflict must have been resolved, however crudely, in order that these social groups could reduce the relentless uncertainty of the external environment. One would expect the more hostile

2

The Economics of Search

environments to produce rapid evolution of these primitive social groups with stringent controls on any internal conflict which threatened group stability. The role of information and search in these early times was clear. Failure to recognize an enemy, looking too long in the wrong place, or not following the advice of the experienced hunter could produce deadly consequences. The incentives for correct and quick inference were never greater. It is to the survivors of these harsh natural experiments that modern man owes his inductive prowess. It was in this setting that language evolved. The ability to transmit information was essential for survival. The high value of information was the glue that stabilized groups, promoted group harmony, and group allegiance. Just as internal conflict was controlled by a variety of institutional arrangements, “treaties” among neighboring tribes eventually evolved to limit the risk of warfare. The terrain occupied by these social groups was sufficiently diverse that diets probably were variable. Furthermore, environmental conditions varied, so that some clans were thriving while others were starving. Intermarriage and other “hostage” arrangements were probably used to cement treaties. This in turn led to informational flows among these tribes regarding diet, hunting and gathering technologies, and the material welfare of the tribes comprising this informational network. It is easy to see how trade and insurance accompanied the development of these informational flows. A tribe in one region might produce an excess of some commodity, convey some of the excess across a tribal boundary and “give” it to the resident tribe. These gifts were implicit exchanges, although the temporal separation of the tit from the tat might be rather large. The key point is the fact that these primitive exchanges were founded on information and search. The transition from primitive to civilized life was marked by the classification of risks according to their predictability, controllability and destructive power. This marked the beginning of economics. Individuals and groups could pursue their comparative advantage. This gave rise to trade and, finally, to money as the medium of exchange.1 Hayek (1978a: 59) reminds us that the transition from the hunter-gatherer society to one with trading and explicit experimentation was lengthy: We must not forget that before the last 10,000 years, during which man has developed agriculture, towns and ultimately the “Great Society”, he existed for at least a hundred times as long in small food-sharing hunting bands of 50 or so, with a strict order of dominance within the defended common territory of the band. The needs of this ancient primitive kind of society determined much of the moral feelings which still govern us, and which we approve in others. It was a grouping in which, at least for all makes, the common pursuit of a perceived physical common object under the direction of the alpha male was as much a condition of its continued existence as the assignment of different shares in the prey to

Introduction

3

the different members according to their importance for the survival of the band. It is more than probable that many of the moral feelings then acquired have not merely been culturally transmitted by teaching or imitation, but have become innate or genetically determined. There were two essential prerequisites to these commercial evolutions: protection of private property and the protection of contracts. When society guaranteed these conditions, individuals played the exchange game and converted the heterogeneous into the homogeneous. That is, the boundaries separating societies were pierced. “Outsiders” became buyers and sellers of commodities. These commercial adventures were games and discovery processes. They possessed all the ingredients of modern decision-making under uncertainty. Decision-making under uncertainty is decomposable into stages: (a) experiments are designed and ranked by their anticipated net values of information; (b) the information generated by the preferred experiment until the stopping time2 is reduced to the minimal information required for the decision. The reduced statistics are called minimal sufficient statistics—they compress data without distortion or value-reduction relative to the decision; (c) a decision is made and the reward collected. The choice among experimental designs is based on Rothschild–Stiglitz stochastic dominance, a partial-ordering of informational content.3 In the statistical analysis of these data, von Neumann–Morgenstern determines which decision is best.4 In the earlier example this decision might be: produce x, cross border y and “exchange” in region z. In relating search to economic processes it is difficult to separate the two. Economics is perhaps the most pervasive of the disciplines influencing most individual decisions and almost every human relationship. The experience, discovery, stability and surprise that comprise the essence of life are mirrored by the experiments, information accumulation, ranking of alternatives and decision-making that characterizes economics. Responding to the flux and novelty of life in a rational manner requires constant vigilance. Experience can be divided into those that are repetitive and relatively stable and those that are novel and infrequent. Individuals learn quickly how to accommodate to the former by designing routines. The latter requires continuous alertness that exploits events that seem favorable and deters those that appear detrimental. It is these continuous experiments that generate new information and reordering of alternatives. In short, opportunity cost is a random process driven by these continuous reorderings. Each member of society is involved in a similar type of sequential stochastic control and their interactions produce oscillating prices. Information about price changes is a prerequisite for economical decision-making. Just as it is now impossible to contemplate economic processes without considering search, one quickly realizes that economic search is at the core of information mechanisms. To exemplify these relations, consider the job search process. The process

4

The Economics of Search

commences when the individual develops those talents comprising his/her genetic makeup. Obviously much of this early investment is parental and much of it entails the acquisition of those attributes possessed by all productive citizens—language, “manners,” and civilized behavior. As this learning unfolds and formal education begins, parents discover natural comparative advantages which they foster. The child also recognizes these and, as education proceeds, focuses on these talents. During this crucial educational process, the individual, guided by parents, teachers, and other informational institutions available to him/her, discovers the potential value of the alternative human capital portfolio he/she can acquire. These discoveries influence the educational path and are tested when the marketplace is entered and actual job offers are forthcoming. The individual experiments in the job market, trying out jobs until a stable match is achieved. It is important to note that both employer and employee continue to search after this match. The partition to which the individual has been assigned may be inappropriate. In addition the partition and the individual fluctuate over time. This oscillating heterogeneity often is overlooked. If the worker outgrows the partition the employer may promote or the employee may quit. If the partition becomes too demanding for the worker,5 a dismissal or reassignment may be necessary. It is edifying and awe-inspiring to pause and compare the transformation of today’s “savage” infant into a mature citizen, with the corresponding transformation of primitive man into our “civilized” counterpart. The former occurs in less than twenty years, whereas the latter consumed thousands of years. Reflections like this evoke a reverence for past accomplishments which have given us the institutions to perform this miraculous compression of time. The recent literature on organizational theory is intimately related to the classification problem in search theory. The relation between incentives (moral hazard) and search was stated clearly in the remarkable book review by Cannan (1930). In his appraisal of unemployment insurance, Cannan (1930: 47) observes: “Whenever an evil can be increased by human slackness or carelessness, insurance against that evil tends to increase it.” A key component of search—the transmission of information—is sometimes subverted by one of the most attractive features of the price mechanism. Individuals collect information that is pertinent to their own welfare and unless properly motivated may transmit signals that are detrimental to the organization. As anticipated, organizations have evolved institutions to mitigate this incentive problem. Information economics has grown enormously over the last fifty years. The remarkable research of Akerlof, Spence, and Stiglitz has earned them Nobel prizes. This was anticipated by Stigler (1962: 104): The amounts and kinds of information needed for the efficient allocation of labor, whether judged from the viewpoint of the laborer, the employer, or the community, extend far beyond the determination of wage rates.

Introduction

5

The kinds and amounts of skill men should acquire pose parallel informational problems, and so too do the non-monetary conditions of employment. The traditional literature has not done these problems justice. It is doubtful that justice would be more closely approached by making exaggerated claims of the importance of the problem of information. There is no exaggeration however, in the suggestion that the analysis of the precise problems of information and of the methods an economy uses to deal with them appears to be a highly rewarding area for future research. We conclude this preliminary section with a definition of economic search. The status of citizen and personal expenditures on insurance protect the individual from much of uncertainty’s ferocity. The methods used to adjust the remaining risks that afflict the individual include: continuous (or partial) information acquisition as the flux unfolds; experimentation and learning; the continuous (or partial) re-ranking of alternative actions, stopping some and starting others. All of these activities involve classification. As time passes, some mistakes (misclassifications) are corrected, while the new information produces new errors. It is convenient to consider the flow of information as emanating from a collection of intertwining networks, with each node representing an explicit or implicit contract. These networks are spatial, temporal and hierarchical. They are in constant flux and comprise the essence of economic search. Many of these contracts are with intermediaries who forge connections with large, and otherwise inaccessible, information sources. Economic search finds, makes, and terminates contracts. The principal–agent problem is embedded in economic search. Definition Economic search is the collection, transmission, and implementation of information required for the sequential decisions entailed in expected utility maximization. Our treatment of concepts like information transmission and asymmetric information is necessarily superficial. Even if we restricted our attention to the economic contents of these ideas, each would deserve at least a chapter. As Green and Laffont (1986) have stated, the economics of information and incentives initially concentrated on the transmission of information about the “subsystems” of large organizations in a timely fashion. Marschak’s early research in the 1920s was on this topic. Hayek’s celebrated 1945 article was a penetrating analysis of the price system as a conductor of information about the fluctuating economy to the responsive economic agent. Marschak and Radner developed an economic theory of team communication in a decision-theoretic setting. The more recent research focuses on the incentive problems of organizations. Various schemes have been proposed to elicit true and timely information from agents with goals different from those of the organization.6 An early paper on this incentive approach to information

6

The Economics of Search

transmission is Hurwicz (1969). The importance of incentives in the recent literature on organization theory has tended to eclipse the transmission problem. We now provide the reader with a brief sketch of the role and significance of asymmetric information and game theory. It also indicates where one can obtain thorough discussions of these vital components of the economics of information.

1.2 CONTRACTS, INCENTIVES AND ASYMMETRIC INFORMATION

1.2.1 Introduction The past two decades have witnessed dramatic changes in economic theory. Several years ago most graduate texts in economic theory were founded on price theory. Today, in the major economic departments, economic theory revolves around noncooperative game theory, while price theory has been relegated to the back seat. Before the ascendance of game theory, the emphasis in economics was on the usefulness of economic models in explaining empirical phenomena. While this emphasis persists, it has been overshadowed by the quest to apply strategic thinking to resolve economic problems. The key ingredients of this remarkable transformation are: the Nash equilibrium concept, Harsanyi’s characterization of contracts with asymmetric information as noncooperative games, and Selten’s explicit consideration of time and the elimination of equilibria associated with non-credible threats and promises. Van Damme and Weibull (1995: 16) observe that: “John Harsanyi showed that games with incomplete information could be remodeled as games with complete but imperfect information, thereby enabling analysis of this important class of games and providing a theoretical foundation for ‘the economics of information’.” Signaling, moral hazard, and adverse selection are prominent members of this class. A game is said to be one of complete information if Nature makes the first move and this move is observed by all players. A game has perfect information if each information set is a singleton (see Rasmussen 1989). Harsanyi adopts a Bayesian approach assuming that each player has a type—a type specifies the information a player possesses regarding the game. The resulting refinement of Nash’s equilibrium is called a Bayes–Nash equilibrium. Reinhard Selten was the first to refine the Nash equilibrium for analysis of dynamic strategic interactions. Such refinement is necessary since many equilibria entail non-credible threats and do not make economic sense. Subgame perfect equilibrium, Selten’s formalization of the requirement that only credible threats should be considered, is used widely in the industrial organization

Introduction

7

literature. It has generated significant insights there and in other fields of economics. At first it was not clear how the problems of asymmetric information could be formulated as noncooperative games. Indeed, pioneering research in signaling and insurance was performed in the 1970s by Akerlof, Rothschild, Spence, and Stiglitz with little reference to game theory. While these models generated important economic implications, they were stymied by problems of equilibrium and credibility. This changes suddenly when fundamental papers by Kreps and Wilson (1982) and others showed that this research could use the deep insights of Harsanyi and Selten to illuminate credibility and equilibrium. The ensuing unification of industrial organization was analogous to the epiphany that occurred in search theory when optimal stopping, dynamic programming, and matching were applied to search problems.

1.2.2 Principal–agent models This section is a brief discussion of three of the most important asymmetric information models: moral hazard, adverse selection, and signaling.7 All three belong to the class of principal–agent models. In these models, the principal P designs the contract. The agent A either accepts or rejects the contract depending on the expected utility of the contract vis-à-vis the utility from other alternatives. Finally, the agent performs the task specified by the principal. The two parties are opposed in that the revenue for the agent is a cost for the principal and the effort of the agent is costly to him/her and beneficial to the principal. In determining the optimal contract between principal and agent, this opposition must be resolved. Macho-Stadler and PerezCastrillo (1997: 20) see this as “one of the most important objectives of the economics of information.” There are two parties to a principal–agent (P–A) model: one is informed while the other does not know a crucial piece of information. Consider the insurance P–A model where the insurance company is the principal and the insuree is the agent. It is usually assumed that the agent knows his health status, whereas the principal is uncertain. The P–A model is a bilateral monopoly. Hence, the nature of the bargaining process must be specified. For simplicity, it is assumed that either P or A receives all of the surplus. For example, P does this by stipulating a “take-it-or-leave-it” contract, which the agent either accepts or rejects. Salanie (1997: 5) notes that bargaining under asymmetric information is very complicated: “There is presently no consensus among theorists on what equilibrium concept should be used.” Salanie observes that: “the P-A game is a Stackelberg game in which the leader (who proposes the contract) is called the principal and the follower (the party who just has to accept or reject the contract) is called the agent.”

8

The Economics of Search

1.2.3 Three principal–agent models with asymmetric information Moral hazard occurs when the agent’s actions are not observed by the principal. More specifically, moral hazard is present when the subject’s actions are influenced by (a) the terms of the contract and, (b), in a way that is not fully specified in the contract. These hidden actions are evident in automobile insurance contracts, where the driving behavior of the insuree is not known by the insurance company. The contract usually has a deductible to mitigate the risk to the insurance company. Adverse selection occurs when the agent is privy to pertinent information before the contract is finalized. This information is hidden from the principal. Adverse selection is evident in health insurance contracts where certain characteristics (health status) of the agent are imperfectly known by the insurance company. An insurance contract is designed for each group with a particular health characteristic (sickly or healthy) so that each member of the group has an incentive to buy it. Signaling occurs when one of the parties (the agent) to the contract has pertinent information regarding his/her type. Before entering the contract, the agent can signal the principal that he/she has this type; thus the behavior of the informed party conveys (signals) the type of information to the uninformed party. A key observation is that each of these asymmetric information models can be represented as a noncooperative game. The crucial equilibrium concept is a refinement of the Nash equilibrium. For example, the signaling game has a perfect Bayesian equilibrium. Excellent discussions are given in Gibbons (1992) and Macho-Stadler and Perez-Castrillo (1997). One of the most important topics in the economics of information is the optimal design of contracts under symmetric and asymmetric information. In the symmetric case, the principal designs the contract so that the expected marginal payoff equals the marginal cost of effort. If the principal is risk neutral, then the optimal contract is one in which the principal accepts all the risk and the agent is therefore fully insured, receiving a payoff that is independent of the outcome. The situation with asymmetric information is much more complicated because the optimal contract must balance the conflict between two opposing goals: efficiency in the allocation of risk between principal and agent and maintaining the incentives of the agent.

1.3 SEARCH THEORY In the first stage of search theory the samples sizes were set a priori at some optimally determined fixed number. The second stage of search theory replaced this with a random sample size determined by an optimal stopping rule. The third stage seeks the source of an endogenous distribution of prices (wages) to replace the exogenous distribution in the Stigler and sequential

Introduction

9

models. This large and complex research effort was initiated by Rothschild (1973). The fourth stage replaced the one-sided sequential model with a twosided equilibrium problem, where there is competition and interaction among optimizing agents. This ongoing research is based on the matching and bargaining models devised by Diamond, Jovanovic, Maskin, Mortensen, and Pissarides.8 In the early work on search, many economists placed a much higher value on the verbal insights of common sense, when they sought a solution to a search perplex. Mathematical formulations of these insights were undervalued. Today, the situation is somewhat reversed in that formal analysis is applauded, while common sense is frequently treated with suspicion. Both attitudes diminish economics. The first does not encapsulate insights in a formal model. Hence, testing and extension are difficult if not impossible. The second closes off economics from the imaginative intuitions of common sense. The commencement of any economic analysis is preceded by information accumulation comprising the search process. These information flows nourish the economic process over its lifetime and determine the longevity of the economic activity. For example, consider the amount of information that is accumulated from the time at which a firm is conceived until it dissolves. Each decision made during this time span is based on information flowing from a search process. With this information the firm can coordinate its activities and adapt to a changing environment. While the price mechanism retains its primacy as the coordinator par excellence, information about prices must be acquired via search. For example, the smooth operation of a shopping mall belies its complexity. It is precisely its synchronous and orderly behavior that accounts for this oversight. A moment’s reflection reveals the complexity of this exchange mechanism and the quality and quantity of information required for the coordination of the countless transactions that comprise a single day’s operation.

1.4 INSTITUTIONAL ASPECTS OF SEARCH ECONOMICS Institutional dynamics was perhaps the strongest current flowing into the Stigler dynamo. Stigler’s substantial contributions to this current reflected his propensity for concrete, historical, economic phenomena. Hence, it is unsurprising that institutional dynamics would be the centerpiece of search economics. We have already commented on the institutional arrangements that preceded the growth of economic search. In addition to these general rules that might be called common sense, there is a multitude of special institutions constantly adapting to the changing environment. Sometimes they are quite subtle and easily overlooked. Yet if their presence is ignored, economic analysis is distorted. The significance of institutions in analyzing search economics was studied

10

The Economics of Search

early on by Alchian (1969). Becker, Demsetz, DeVany, and others continue in this tradition. Their major thesis is that market institutions adapt to exchange environments so that the appropriate information is collected and transmitted efficiently. Thus they implicitly connect the formation of institutions with the economics of search. A fluctuating environment induces institutional oscillations that are modulated by adopting alternative strategies for collecting and transmitting information. For example, one trading environment may be most economical under an inflexible price system, while another may perform best when prices vibrate. The central observation is the distinction between the Walrasian market institution adopted by many general equilibrium theorists and the plethora of market institutions that actually drive the economy.9 Each member of society must keep track of those prices affecting his personal welfare. Ostroy (1973) presents a similar argument: interpretations of the standard theory say that the Walrasian auctioneer, after announcing equilibrium prices, expedites demands and supplies. If there is one theme which distinguishes the present treatment from the standard theory, it is that exchange is a do-it-yourself affair. Individuals will not exchange with “the market”; they will exchange with each other. This elementary logistical consideration is the basis upon which we shall construct an argument for monetary exchange.

1.5 THE EXPLANATORY POWER OF ECONOMIC SEARCH A host of activities that were incomprehensible to the classical deterministic theory of exchange suddenly emerge as rational responses to uncertainty in the economic theory of search. Furthermore, there is a set of phenomena which the deterministic model found perplexing. The amount of time spent looking and waiting was inexplicable in the timeless classical model. These temporal activities are now easily understood by search-theoretic economics. The time spent looking for the combination of goods that maximizes expected utility is as much a productive enterprise as is the time and energy consumed in locating the “right” job. The dollar value of the resources invested in searching, and hence waiting, is enormous, but so is the return to these investments. Mistakes are inevitable in a probabilistic economy and there is no question that a prophet could reduce the expenditure on searching and improve the quality of the outcome. We will show that economic search in theory does remarkably well in approximating these perfect foresight allocations. Perhaps the most significant contribution of search to economic theory is the explicit realization that search unemployment, consumer search, and queues, “idle” resources like vacant apartments and unsold commodities, job vacancies, are all productive activities and are essential for the orderly behavior of a free enterprise economy. The formation of contracts is time-consuming and contractual relations require surveillance. Compliance is not automatic.

Introduction

11

Individuals must ascertain when a contract should be terminated. Exogenous shocks may cause the severing of old connections and the formation of new ones. The basic search model would be a fairy tale in the neoclassical clockwork world. The temporal, spatial, and hierarchical networks established by individuals to learn, adapt, exchange, and, in short, to make rational decisions in a fluctuating environment, where information is costly and subject to unreliability and obsolescence, are essential for survival in the real world. Another phenomenon puzzling to the static theory was the prevalence of “sticky prices.” In the search environment with costly information, sellers are reluctant to alter their signals to customers until they are sure the relation between demand and supply of a particular good has had a permanent change. It also is costly to change prices every time the market moves. Finally, costs are imposed on customers who enter stores anticipating a quick purchase of one combination of goods only to discover that this combination breaks their budget and they must spend time shopping for the best feasible combination. When imperfect information and costly search are considered, several basic theorems of neoclassical economics must be modified.10 Competitive economics are Pareto optimal in the classical model. Marginal rates of substitution equal price ratios and there is no difference between the purchase and selling prices. When buyers and sellers must search for one another and prices are disperse, intermediaries usually emerge to facilitate the matching process. Intermediaries perform a valuable function, but they must be rewarded.11 The spread between the buy and sell price covers their opportunity costs. But now marginal rates of substitution are not equal and Pareto optimality is lost. Similarly, information costs invalidate Coase’s theorem. In a frictionless economy any discrepancy between private and social cost would be eliminated by contractual arrangements among the affected parties.

1.6 OUTLINE OF THIS VOLUME Over the last twenty-five years the economics of search has achieved a central position in economics. During this period search has grown rapidly and has merged with the other part of information theory which focuses on auctions, Nash equilibria and bargaining, and other game theoretic aspects. Many of the puzzles that confronted search twenty-five years ago have been either solved or are now much closer to a resolution. Of course, there are also new perplexes that have attracted a whole new class of outstanding economists. Search is remarkable in that the econometrics of search was born shortly after Stigler’s seminal article in 1961, though its rapid growth did wait until search theory was formulated as an optimal stopping problem. This took place in the 1960s and early 1970s with the birth of the basic sequential search model (BSM) and almost simultaneously the conception of a corresponding econometrics program.

12

The Economics of Search

In the BSM individuals are confronted with a distribution of prices (wages) and most determined when to stop sampling from the distribution and buy the good (accept the job). Each draw entails a cost and the environment is stationary in that the individual knows the distribution of prices and this distribution never changes. Moreover, individuals never need worry about time running out, since they live forever. Under these circumstances individuals who want to minimize their overall expected costs of purchase (maximize net expected net wages), including the costs of search, will do best by setting a “reservation” price (wage) so that they buy the good (accept the job) the first time the price (wage) is equal or less than the (greater than or equal to) reservation price (wage). Intuitively it makes sense never to alter one’s strategy over time, since the conditions facing the individual never change. Moreover, it would be a strange individual indeed who would reject a price that was lower than one he/she would accept. After the introduction of the basic search model in 1970 (McCall 1970; Mortensen 1970) much of the latter literature devoted itself to relaxing the assumptions of this canonical model. For example, bargaining and Nash equilibria were adopted by search theorists in the early 1980s in order to relax the assumption of an exogenous price (wage) distribution. This union has proved to be remarkably fertile. There are many important articles which, when combined, establish a new search frontier. Search theory is approximately forty-five years old. Yet there has not been a book dedicated to search that surveys the entire search terrain. However, there have been many comprehensive survey articles, but often these are dated. The closest competitors to this two-volume overview include: Mortensen (1986), Mortensen and Pissarides (1999a, 1999b), Ljungqvist and Sargent (2000, 2004b), Levine and Lippman (1995), Sargent (1987), Stokey and Lucas (1989), McMillan and Rothschild (1993), Devine and Kiefer (1991), Rothschild (1973, 1974), Kohn and Shavell (1974), Ferguson (1989), Diamond (1984), Phelps et al. (1970) and Lippman and McCall (1976a, 1976b, 2002).12 An innovation of our work is the combination of the theory of sequential search with the econometrics of sequential search. There are many outstanding articles and books on search econometrics that include: Heckman and Singer (1984), Kiefer and Neumann (1979a, 1979b), Devine and Kiefer (1991), Lancaster (1990), Eckstein and Wolpin (1989). The absence of any survey of search and the econometrics of search should give us pause. It appears that the rapidity of search and econometric innovations quickly obsoletes any book presenting the current status of these subjects. This observation is supported by the search chapters in the macroeconomic texts by Sargent (1987), and Ljungqvist and Sargent (2000, 2004b). Although the books are aimed at those studying macroeconomics, they are also one of the best sources of the evolving search literature. The macro and search literatures have undergone changes so rapid that new editions of Ljungqvist and Sargent are required every four or five years.

Introduction

13

The rapidity of growth and novelty in the macro and search literatures is nicely portrayed by comparing the classic study by Don Patinkin with the Ljungqvist and Sargent (2004b) masterpiece and the innovative articles by Mortensen and Pissarides. Consider an economist who, having read and understood Patinkin’s classic, fell into a thirty-year slumber (this might be the same as getting tenure for the same period!) On awakening he immediately buys Ljungqvist and Sargent to see what has transpired in macro and search. He quickly perceives that he must essentially learn a new and more difficult subject. This task may appear so arduous that our Rip van Winkel becomes a lawyer. There is also the relatively recent surge in Bayesian econometric articles and books. The development of Markov Chain Monte Carlo (MCMC) methods stimulated this burst of research. Indeed, several economists wrote original articles on MCMC.13 See the seminal articles by Geweke (1989), and Chib (1991). There have been a host of Bayesian articles by economists. See Devine and Kiefer (1991) and Hirano (2002). An excellent introductory Bayesian econometric text has been composed by Lancaster (2004). Of course, Zellner, the “father of Bayesian econometrics,” wrote the classic text on Bayesian econometrics in (1971) and has contributed to the MCMC explosion. See Zellner and Rossi (1997). Let us reflect for a moment on those topics which have or might benefit from applications of search theory. First, observe that when search theory is posed as an optimal stopping rule, it becomes clear that search is a basic response to uncertainty with which all search decision-making must contend. These decisions include job choice, the selection of consumer goods, choice of a marriage mate, deciding where to live, and choosing schools for your children, where to shop, choice of doctors, the composition of investment portfolios, etc. As we will see in Volume II, search theory explains animal and insect behavior. The three basic decisions of animals and insects which can be formulated as stopping rules are: mate choice, food selection, and the choice of nest or cave. Young (1988: 3) also sees search as a vital activity of the brain: Two things in particular are missing from most lay and philosophical treatments of life and mind. First they do not show appreciation of the intense and complex continuous internal activity that directs organisms to search for means of survival. This incessant pursuit of aims is the essence of the maintenance of life. The search is not random but is directed towards the achievement of particular immediate targets that are pre-set for each creature. Secondly, this continuity of life is made possible only by calling from moment to moment upon the stored information derived from past history. As the philosopher Collingwood once put it, “Life is history.” A major deterrent to implementing these Bayesian search models was the

14

The Economics of Search

complicated posterior distribution that required an approximation in many Bayesian applications. Geman and Geman (1984) and others devised Markov Chain Monte Carlo’s procedures for accomplishing this difficult estimation in relatively short order. A large group of statisticians and economists contributed to the growth of MCMC. These many economists included Zellner, Chib, and Geweke. By now it should be clear to the reader that we may have bitten off more than we can chew. We hope to mitigate this problem by dividing our presentation into two volumes. We conclude this Introduction with the topics covered in each chapter of this volume. In Chapter 4, the nine assumptions of the basic sequential search model are enumerated. Two especially nice properties are lodged in the BSM. The first is the reservation wage property (for labor markets), mentioned above, which identifies a reservation or critical wage, ξ such that the job searcher accepts a job offer w if, and only if, w ≥ ξ. The second pleasant property associated with BSM is myopia which means that the reservation wage ξ is calculated by comparing the return from one more search with the return from stopping. There are three challenges to be met as each assumption of the BSM is relaxed: 1 2 3

Does the reservation wage property remain when a particular assumption is relaxed? Is the myopia property invariant to the relaxation of each assumption? Do these nonequilibrium structures hold-up when subjected to econometric and empirical scrutiny?

The responses to these challenges comprise almost the entire Volume I. We now give a summary of the contents of Volume I. Chapter 2 of Volume I follows this Introduction, and presents the basic mathematics used in search theory and search econometrics. It gives an idea of who comprise our audience. As we will see there is a powerful unity between search theory and the econometrics used to measure the strength of alternative search hypotheses. Chapter 3 describes the history and evolution of search. The contributions by Cayley, Wald, Bellman, Shapley and Robbins are some of the most outstanding. The informal analysis of economic search was extremely important. These contributions by Hicks, Alchian, Hutt, etc., defined the search problem and were essential to the startling illumination when these informal notions were joined to the formal theory. From a theoretical and practical viewpoint the existence of an easily calculated reservation wage is the basic property of BSM. In Chapter 4 the nine assumptions underlying the construction of BSM are enumerated. There is a challenge in this enumeration. Show that if an assumption is relaxed, the highly desirable reservation wage is preserved! This systematic one-by-one

Introduction

15

relaxation of the assumptions corresponds almost exactly to the Table of Contents of Volume I. In Chapter 4 the basic sequential search model is presented. As we move through the chapters of the book, it will become clear that the BSM is the foundation for much of the subsequent search innovations. The BSM is a special kind of Markov decision process, that is, it is an optimal stopping rule which informs the searcher as to when he/she should stop. These optimal stopping rules are at the heart of most of search theory. The BSM tries to emulate the search experience of a typical agent. It raises questions that are important to the manner in which search is conducted. Many of these queries are not answered in Chapter 4 as these more complicated questions require a more sophisticated analysis. A key observation is that the structure of the BSM has a potent influence on how these questions are answered. Chapter 5 presents the econometrics that are used to measure the influence of search models. Seminal work on the early versions of these models was conducted by Heckman and Singer and Kiefer and Neumann. They fit into a class of models called duration analysis. Search and unemployment are joined in Chapter 6. The BSM reveals a negative relationship between the cost of search and the length of search (frictional) unemployment. We explore models that investigate the impact of unemployment insurance (UI) on unemployment. Since in most countries UI benefits only last for a specific amount of time, we expand the BSM to incorporate the fact that the cost of search may vary over time. We also review some of the empirical work that uses duration analysis to measure the impact of UI benefits on unemployment durations. The analysis of BSM assumed that the distribution of wages or prices was constant over time. In fact, we expect this distribution to fluctuate over time. “Job search in a dynamic economy” (Chapter 7) was constructed to address situations where wage and price distributions move up (down) as the economy improves (regresses). The stochastic process is a Monotone Markov chain. Chapter 7 considers the wide-ranging implications of Monotone Markov chains based on the insights of Conlisk, Daley, and others. One of the assumptions of BSM is that the searcher’s utility function is a linear meaning that he/she responds to expected values, but not to variability. This assumption is relaxed and we study several cases in Chapter 8. Chapter 9 is an introduction to the Gittins index and we show that the reservation wage is a special case of this index. A more complete (and modern) exposition of the Gittins index is described in Volume II. The next two chapters consider models of equilibrium search. While covering many of the topics in equilibrium search these chapters do not give a comprehensive overview. Thus, we will return once gain to equilibrium search in Volume II. Chapter 10 first surveys the large literature addressing equilibrium search models before the addition of Nash bargaining by Diamond, Mortensen, and

16

The Economics of Search

Pissarides. The basic conundrum addressed by these early equilibrium models was: Does the sale of homogeneous goods (or jobs) at different prices (wages) violate the law of one price? Rothschild (1973) raised the important question of the existence of an equilibrium price dispersion, namely, what gives rise to the nondegenerate price distribution assumed in simple search models. This question provoked an enormous literature on equilibrium search models which sought a positive answer to Rothschild thereby negating Diamond’s (1971) finding that all sellers charge the monopoly price. In Diamond’s model, buyers are identical, sellers are identical, and buyers have the same search cost. If the rules are reversed—buyers instead of sellers make take-it-or-leave-it offers and sellers incur search costs—then markets clear at the competitive price. The favorite way out of this dilemma is to introduce heterogeneity into the model. In McMillan and Rothschild (1993), there are several sources of heterogeneity among the thirty-one papers surveyed. They include differences in sellers’ production costs, buyers’ search costs, buyers’ beliefs about the offer distribution, sellers’ time varying prices, buyers’ inventory holdings, and advertising which yields different buyer information. Many of these early papers are surveyed in Chapter 10. Over the past decade there has been an increasing interest in economic models that study the flow of jobs and workers among diverse states. Most of this research relies on equilibrium models and emphasizes the frictions in both search and recruiting. Using a rich set of data on flows in labor markets, this research studies job creation, job closure, unemployment and labor mobility. Matching and equilibrium are crucial to the theoretical approach. Decisions made by employers on the demand side of the market are highlighted. A comprehensive review of this prodigious literature is Mortensen and Pissarides (1999a). Ljungqvist and Sargent (1998, 2005a, 2005b) have devised sophisticated extensions of the simple sequential search model to study unemployment behavior in European countries. In Sweden, they recommend a smoothing of the distorted incentives in the labor market by adopting less generous unemployment benefits. In European welfare states, their equilibrium search model finds that: when the generosity of unemployment insurance increases, the upward pressure on the unemployment rate eventually dominates the reduction in labor mobility due to high marginal income taxes. The higher unemployment rate is caused by unemployed workers reducing their search intensity and increasing their reservation rates. . . . Higher unemployment pushes taxes upward which, in turn, increases the unemployment rate further, and so on. Alvarez and Veracierto (1999) combines McCall (1970) with Lucas and Prescott (1974) to study the degree to which differences in employment and

Introduction

17

unemployment across countries are attributable to labor market policies. They find: i) that minimum wages have small effects; ii) firing taxes have similar effects to those found in frictionless general equilibrium models; iii) unions have large and negative effects on employment, unemployment and welfare; and iv) unemployment benefits substantially increase unemployment and reduce welfare. (Alvarez and Veracierto 1999: 1) Recent research in this area, together with comparisons between matching models and simple stopping equilibrium models, are considered in Chapter 11. A novel estimation procedure emerged from the empirical demands of the BSM. It is called structural analysis and has been widely used in assessing search hypotheses. There are several excellent books and articles providing an overview of this methodology. Included in this set are Gotz and McCall (1984), Miller (1984), Rust (1987), and Wolpin (1984). We provide an introduction to this topic in Chapter 12. There are many search settings that drive an economy. Unfortunately, we cannot treat each of them in depth. In fact, some will only be listed. Chapter 13 covers some of the important applications of search in an uncertain economic environment. We first show that search and auctions are alternative methods for allocating resources. In fact, there is a critical number such that search is preferred when the number is not exceeded and a switch to auctions occurs when the critical number is surpassed. Arnold and Lippman (1995) spell out the theory so that these critical numbers can be calculated. They also conduct an empirical study of the sale of cattle. They find that the cattle are auctioned or allocated by a search mechanism depending on factors that Arnold and Lippman have hypothesized in their theory. We also present the results contained in McAfee and McMillan (1987a) which describes four basic auctions. They give a running account of the importance of asymmetric information. The authors then derive a thorough and cogent presentation of auction theory. Finally, in Chapter 14 we some give some concluding remarks to Volume I. While this volume is lengthy it is nevertheless incomplete. Our next volume considers many additional topics although the focus is mainly on Bayesian models. We will begin that volume with a survey of exchangeability which is essential for understanding the Bayesian models we analyze. We also will explore “exchangeable search” and provide an in-depth treatment of the Polya urn model and show its intimate connection to exchangeable models. The secretary problem is one of optimal stopping. It has many similarities with optimal stopping, search, and many new implications for resolving practical problems. The move in this direction has been championed by Bruss and Ferguson. In Volume II we will present a rather detailed study of the secretary problem.

18

The Economics of Search

Over the last fifteen years there has been a surging research interest in real options; they are closely related to financial options. Both types have their roots in optimal stopping. Sequential search, the Gittins index, and the real option are close relatives of the BSM. We explore these connections in Volume II. Also in Volume II we will introduce Markov Chain Monte Carlo methods. As we note, estimating the posterior distribution in Bayesian studies has almost always been a difficult or impossible task. By a clever use of Monte Carlo methods and Markov chains, these estimation problems have been virtually eliminated: they no longer impede Bayesian analysis. A powerful method for econometric and statistical analysis is nonparametric Bayesian estimation. This method has its origin in the early research of Ferguson (1974). However, it only became feasible when it was combined with MCMC. Several very important econometric studies can be completed using these Bayes nonparametric methods. The parallel between economic search and the search behavior of animals has been long recognized. Hirshleifer pointed it out to us in 1983 (private communication). We have communicated with the late William Hamilton and he also encouraged us, as well as his fellow biologists, to apply optimal stopping rules to animal behavior. There are essentially three search problems that animals and insects must solve: first find food, second locate a desirable mate, and third discover a livable cave or nest. These problems are also explored in Volume II. There is the remarkable growth in information economics, which encompasses signaling, asymmetric information, moral hazard, adverse selection, Nash bargaining, Nash equilibrium, and auctions. In this volume we show the close connection between the BSM and auctions. There is also some use of Nash equilibria. Volume II makes more use of signaling in animal behavior, where it is related to the “handicap principle” and Nash equilibria attached to game theory are also employed. While we introduce equilibrium models of job search and cover some topics in structural estimation in this volume, given the rapid expansions of the literature in these areas, our coverage in this volume is necessarily incomplete. So, in Volume II we will expand upon both of these topics.

2

Mathematical methods

2.1 INTRODUCTION In this chapter we will review some mathematical concepts and techniques that will be used later in the book. Our goal here is not to give a detailed presentation of the different mathematical topics but to provide a brief overview or refresher. At the end of the chapter we list a set of books that give more thorough treatments of the topics discussed in this chapter. We assume that the reader has some basic familiarity with probability theory at a level that might be covered in an undergraduate statistics or econometrics course. We will first review some basics of Markov chains and stochastic dynamic programming where, throughout, state spaces, and any accompanying random variables, are assumed to be discrete. We then introduce some more advanced probability notions associated with continuous time stochastic processes using measure theoretic concepts.

2.2 MARKOV CHAINS AND RELATED PROCESSES

2.2.1 Preliminaries A discrete-time stochastic process is a sequence of random variables Χ(t), t = 1, 2, 3, . . . A Markov chain is one of the simplest dependent stochastic processes. The fundamental assumption is that the probability of an individual moving from state i in period t to state j in period t + 1 depends only on i, j, and t and is denoted by Pij(t). The future manifestations of a Markov chain are completely determined by the present state of the system and are independent of the past. More precisely, the stochastic process Χ(t) is a Markov chain if Χ(t) assumes only a finite (or countably infinite) number of values as t runs over the positive integers and the following condition (called the Markov property) is satisfied: P[Χ(tn) = xn|Χ(t1) = x1, . . ., Χ(tn − 1) = xn − 1)] = P[Χ(tn) = xn|Χ(tn − 1) = xn − 1]

20

The Economics of Search

The conditional probabilities P[Χ(tn) = j|Χ(tn − 1) = i] = Pij(t) are called the transition probabilities of the Markov chain. If the transition probabilities are independent of t, Pij(t) = Pij, for all t, then the Markov chain is said to be stationary. Many of the properties of Markov chains can be illustrated with a simple two-state example. Suppose an individual may occupy one of two states, unemployment or employment. If an individual is unemployed and searching in period n, then the probability of transiting to employment in period n + 1 is P1. On the other hand, if the individual is employed in period n, the probability is P2 that he/she becomes unemployed and begins job search in period n + 1. The probability of being in the search state at the initiation of the stochastic process (n = 0) is π0. Let Χn be the random variable denoting the state of the system at period n. The state of unemployment (employment) corresponds to Χn = 0(Χn = 1). The transition probabilities of this two-state model are: P(Χn + 1 = 1|Χn = 0) = p1 P(Χn + 1 = 0|Χn = 0) = 1 − p1 = q1

(2.1)

P(Χn + 1 = 1|Χn = 1) = 1 − p2 = q2 P(Χn + 1 = 0|Χn = 1) = p2 The transition probabilities of a Markov chain can be compactly described by the transition probability matrix: P=

1 − p1 p2

冤

p1 . 1 − p2

冥

The probabilities of being unemployed and employed initially are P(Χ0 = 0) = π0(0) and P(Χ0 = 1) = π0(1) = 1 − π0(0), respectively. The initial distribution of the chain can be compactly represented by the row vector π0 = [π0(0), π0(1)]. Given the initial probability distribution π0 and the transition matrix P, the

21

Mathematical methods

probabilities of unemployment and employment can be calculated for any future period n. First note the following relations: P(Χn + 1 = 0) = P(Χn = 0 and Χn + 1 = 0) + P(Χn = 1 and Χn + 1 = 0) = P(Χn = 0)P(Χn + 1 = 0|Χn = 0) + P(Χn = 1)P(Χn + 1 = 0|Χn = 1) = P(Χn = 0)(1 − p1) + P(Χn = 1)p2

(2.2)

= (1 − p1 − p2)P(Χn = 0) + p2 The first equality is based on the addition rule for mutually exclusive events, the second on the definition of conditional probability, the third on the relations in (2.1), and the fourth on the relation P(Χn = 1) = 1 − P(Χn = 0). From (2.2) it is clear that P(Χ1 = 0) = (1 − p1 − p2)π0(0) + p2 since P(Χ0 = 0) = π0(0). Similarly, P(Χ2 = 0) = (1 − p1 − p2)2π0(0) + p2[1 + (− p1 − p2)]. On repetition of this argument, the general expression for P(Χn = 0) is given by n−1

P(Χn = 0) = (1 − p1 − p2) π0(0) + p2 n

冱(1 − p

1

− p2)i.

(2.3)

0=1

Now since n−1

冱(1 − p

1

− p2)i =

i=0

1 − (1 − p1 − p2)n p1 + p2

Equation (2.3) can be rewritten (assuming p1 + p2 > 0 ) as P(Χn = 0) =

p2 p2 + (1 − p1 − p2)n π0(0) − p1 + p2 p1 + p2

冢

冣

(2.4)

and P(Χn = 1) = 1 − P(Χn = 0) =

p1 p1 + (1 − p1 − p2)n π0(1) − . (2.5) p1 + p2 p1 + p2

冢

冣

As n gets large, the contribution of the second term on the right-hand sides of (2.4) and (2.5) diminishes in importance, assuming as we will that

22

The Economics of Search

1 − p1 − p2 < 1. In the limit as n approaches infinity (4) and (5) converge, respectively, to π(0) = lim p(Χn = 0) =

p2 p1 + p2

(2.6)

π(1) = lim P(Χn = 1) =

p1 p1 + p2

(2.7)

n→∞

and

n→∞

both of which are independent of the initial state of the system. Equations (2.6) and (2.7) have the following implications for the job transition example. The probability that an individual is in unemployment (employment) at some future date becomes less and less dependent on the initial state as the future state becomes more distant. In the limit, complete independence of the initial conditions is achieved. In matrix notation, the distribution πn = [πn(0), πn(1)] of the Markov chain at period n is given by πn = πn − 1P which on iteration reduces to πn = π0Pn, where Pn is the matrix of the n step transition probabilities. The (i, j) entry of Pn, say P ijn, is simply pijn = p(Χn = j|Χ0 = i). A Markov chain is said to be regular if some power of the Markov transition matrix is composed of only strictly positive elements. When the Markov chain is regular, an equilibrium vector of the steady-state probability distribution π exists and is the solution to π = πP with 1′π = 1 where 1 is a vector of ones. In the two-state example, this is a system of three equations in two unknowns and under the assumptions made has the following solution: π(0) =

p2 p1 + p2

π(1) =

p1 , p1 + p2

which are the same as the results obtained in (2.6) and (2.7).

Mathematical methods

23

Example 2.1 In a mythical society suppose that movements between unemployment and employment are controlled by the following Markov transition matrix:

冤

9 10 1 4

冥

1 10 3 4

where job search and employment are represented by 0 and 1, respectively. Assume also that the initial distribution of the population in job search and employment is π0(0) = .1 and π0(1) = .9 that is 90 percent of the members of this society are employed at the start of the process. Then from equation (2.4) the probability of being in job search after n periods or, equivalently, after n moves according to transition matrix (2.7) is P(Χn = 0) = 5/7 + (3/20)n(1/10 − 5/7). The corresponding probability for employment is P(Χn = 1) = 2/7 + (13/20)n(9/10 − 2/7). From these relations it is clear that the steady-state distribution of unemployment and employment differs markedly from the initial distribution. In the beginning only one-tenth of the population was unemployed and nine-tenths were employed whereas in the long-run five-sevenths of the population were frictionally unemployed and two-sevenths were employed, respectively. Example 2.2 (Random Walks): Let Χ0, Χ1, . . . be a sequence of integervalued random variables with the same distribution; the partial sum n

Sn =

冱Χ

j

j=0

is a Markov chain with a stationary transition matrix. Observe the intimate relation between the analysis of Markov chains and classical probability theory. That is, the dependencies of Markov analysis are captured by the transition matrix. Example 2.3 (The Polya urn): An urn contains one ball of each of k ≥ 2 different colors. A sequence of balls is drawn randomly from the urn. After

24

The Economics of Search

each observation, the sampled ball is replaced in the urn along with a second ball of the same color. The sequence of random variables Yn, n = 0, 1, 2, . . . having values in the set of k-tuples of positive integers (i.e. the Yn are random vectors) is such that the j th component of Yn equals the number of balls of the j th color in the run after the n th trial, j = 1, 2, . . ., k. This sequence {Yn} is a Markov chain with the typical entry of the transition matrix being P(S, S + e(j)) =

sj , #S

where S is an element on the k-tuples of positive integers and e(j) is a kdimensional vector with all elements equal to 0 except for the j th which equals k

1, sj equals the j element of S and #S = th

冱s . i

i=1

2.2.2 Classification of states Let us restrict attention to Markov chains that are stationary and have positive integers as parameters. The state space may be finite or countably infinite. The initial state of the chain is chosen according to π(x), a probability function defined over the state space of integers. Subsequent movements of the chain are governed by the square probability transition matrix, P, which is defined in terms of the transition probabilities given by:

p00 p01 p02 L  p10 p11 p12  P = p p21 p22  20 M O

(2.8)

The initial distribution π(x) and the transition matrix P completely describe the Markov chain with an integer state space. The probability of moving from state i to state j in exactly n steps is n denoted by P(n) ij . Furthermore, it is easily verified that the i,j entry in P is this n step transition probability, and

P(nij + m) =

冱P

(n) ik

P(m) kj .

k

In matrix terms P n + m = P nP m. State j can be reached from state i if there is some n > 0 such that P(n) ij > 0.

Mathematical methods

25

In the unrestricted symmetric random walk every state can be reached from every other (indeed each occurs infinitely often) whereas an absorbing state only communicates with itself, i.e., if i is absorbing there is no n > 0 such that P(n) ij > 0, i ≠ j. A set C of states is closed if no other state outside C can be reached from a state in C. A Markov chain is irreducible if the only closed set is the set of all states. In such a chain every state can be reached from every other state. The state j has period t > 1 if P(n) jj = 0 except when n = rt is a multiple of t and t is the largest integer possessing this property. If no such t > 1 exists, the state is aperiodic. For example, the unrestricted random walk has a periodicity of 2 while the random walk with a reflecting barrier is aperiodic. Let fij denote the probability that starting from the state i the process ever goes through state j. Clearly ∞

fij =

冱f

(n) ij

n=1

where f (n) ij is the probability of first reaching j from i exactly n trials. The mean recurrence time for visits to state j is given by ∞

uj =

冱nf

(n) jj

.

n=1

State j is said to be persistent (recurrent) if fjj = 1. For example, each of the infinite states in the unrestricted random walk is persistent. State j is said to be transient if fjj < 1. For example, the interior states of a random walk with two absorbing states are transient. A persistent state is called null if uj = ∞. For example, all the states in the unrestricted symmetric random walk are null (null recurrent). Each occurs infinitely often, but the mean time between recurrence is infinite! An aperiodic persistent state is called ergodic if uj < ∞. For example, each state in a random walk with two reflecting barriers is ergodic. Theorem 2.1 For a persistent state j there is a unique irreducible closed set containing j such that for every two states i, k in C fik = fki = 1. Note that if state j is transient. lim P(n) ij = 0,

n→∞

26

The Economics of Search

i.e., the probability that Markov chain is in a transient state is “close” to zero for n sufficiently large. Clearly, then an equilibrium distribution will not involve transient states. Theorem 2.2 The states of a Markov chain can be decomposed uniquely into nonoverlapping sets T, C1, C2, . . . such that T contains all the transient states and Cj, j = 1, 2, . . . are irreducible and contains only persistent states of the same type, i.e., same period and either null recurrent or positive recurrent. Note that if a Markov chain is decomposable or reducible there is no chance of obtaining an equilibrium distribution. For example, let the chain be reducible into two disjoint closed sets C1 and C2. Then if i ∈ C1, j ∈ C2

0

P = P (C1) if i and j ∈ C1 (n) ij

(n) ij

if i and j ∈ C2

P(n) ij (C2)

Hence, the starting state i has a relentless influence on the limit. If the chain is periodic, an equilibrium distribution also does not exist. However, it is easy to convert a periodic chain into an aperiodic chain (see Feller) so periodic chains are a nuisance, but not really embarrassing. It should be clear from the definitions that a finite chain can contain no null state and must include at least one positive recurrent state. For suppose all were transient. Then

0=

冱 lim P j n→∞

(n) ij

= lim

冱P

n→∞ j

(n) ij

=1

Theorem 2.3 (Finite Markov chains) A Markov chain is finite if its values comprise a finite set of n values. The associated Markov transition matrix P is {Pij}i, j = 1, 2 . . . n. Finite Markov chains have the following properties: 1 2 3

4

The number of visits to the set of transient states is finite with probability one. A unique equilibrium distribution exists provided the chain is irreducible. (There can be no null recurrent states.) All states are recurrent and are members of the same class, they are associated with a unit eigenvalue and an accompanying eigenvector composed of nonzero entries. All communicating classes are aperiodic if and only if unity is the only eigenvalue of absolute value 1.

Mathematical methods

27

2.2.3 Fundamental convergence theorem In an irreducible ergodic Markov chain the limits πk = lim P(n) jk n→∞

exist and are independent of the initial state j. Furthermore, the πk constitutes an equilibrium probability distribution over the set of states k = 1, 2, . . . such that πk > 0 ∀k (2.9)

冱π

=1

πj =

冱π P

k

k

and

i

ij

for all j (i.e. π = πP).

(2.10)

If the chain is irreducible and aperiodic and there are numbers πk > 0 satisfying (2.9) and (2.10), then the chain is ergodic and πk is the reciprocal of the mean recurrence time, i.e., πk = uk− 1 Note that if the state space is finite (N states) the equilibrium distribution is the solution to the N linearly independent equations given by (2.9) and (2.10). If the state space is countably infinite and an equilibrium exists, then an approximate solution can be had by solving (2.9) and (2.10) for large N.

2.2.3 Markov chains and random walks Many extremely important economic phenomena display characteristics that are akin to those of the simple random walk. The behavior of stock market prices is the outstanding example, but others are also notable. The wealth position of individuals can be analyzed using the methods of random walk. A bankruptcy model could be used to elucidate the survival behavior of firms. An extraordinary property of random walks is the unexpectedly long duration of positive (negative) accumulated gains. In assessing the relative performance of managers companies are “slow” in their evaluations of success and failure. Salaries and layoffs lag behind the accumulated record. If a manager’s activity is buffeted by random walk-like forces, then firms should be congratulated for pursuing what appears to be a sluggish “wait and see” evaluation (remuneration) policy. The reason for applause is that under

28

The Economics of Search

random walk conditions, the optimal evaluation policy is to wait until a lot of experience is accumulated. The usefulness of the random walk concept in economics is apparent. As we will see, many of the properties of random walks are contrary to intuition, and, indeed, some are truly astonishing. The discussion is drawn from the classic work of Feller (1970). The coin-tossing experiment Recall the simple coin-tossing experiment. The sequence {Χi}, i = 1, 2, . . . are independent identically distributed random variables. Each Χi takes on a value of +1(−1) if a head (tail) occurs. The probabilities of each event are p and 1 − p = q, respectively. Let Sn be defined as Sn = So + Χ1 + Χ2 + . . . + Χn where So can be graphically interpreted as the gambler’s initial wealth with each Χi a unitary depletion or addition to total wealth; total wealth at time n is simply Sn. The sequence (Sn) is a simple random walk. Furthermore, (Sn) has the martingale property when p = q, i.e., E(Χn + 1|Χ1, Χ2, . . ., Χn) = Sn + E(Χn + 1|Χ1, . . ., Χn) = Sn. If p > q (p < q), {Sn} is a submartingale (supermartingale). The sequence (Sn) is also a Markov chain with transition matrix P = O … … … … … … N

M q 0 0 0 0 0 M

M 0 q 0 0 0 0 M

M p 0 q 0 0 0 M

M 0 p 0 q 0 0 M

M 0 0 p 0 q 0 M

M 0 0 0 p 0 q M

N … … … … … … O

Note that some authors define the simple random walk with 1 − p − q > 0, i.e., the particle does not move with positive probability. Properties of the random walk This is a brief summary of the properties of the random walk stochastic process. Many results will be stated without proof. Let P be the number of heads (+1’s) in n trials and Q the number of tails (−1’s). Let So = 0 and note that Sn = P − Q.

Mathematical methods Definition 2.1 with Sn = x.

29

A path (S1, S2, . . ., Sn) from (0, So) to (n,x) is a polygonal line

We will call the length of the path n. Thus, there are 2n paths with n = P + Q and x = P − Q. The number of paths from the origin to (n,x) is given by Nn,x =

冢

P+Q P

P+Q . Q

冣冢

冣

x Nn,x paths from (0,0) to (n,x) such that n S1 > 0, S2 > 0, . . ., Sn > 0, where n and x are positive integers.

Theorem 2.4 (Ballot)

Proof

There exist

(See Feller 1970: 73)

A random walk is said to return to the origin at time k if Sk = 0. The following rather surprising result is proved in Feller (1970: 77, 76). Theorem 2.5 (Return) The probability of no return to the origin (note that returns can only occur for even values of the positive integer n) up to and including 2n is identical to the probability of a return at 2n. The probability is given by: P{S1 ≠ 0, S2 ≠ 0, . . ., S2n ≠ 0} = P{S2n = 0} = U2n

(2.11)

where U2n =

2n

冢n冣2

−2n

Using Sterling’s rule we get the approximation: U2n~

1 √xn

An immediate corollary of this theorem gives the probability distribution of the first return to the origin. To say that no return occurs until 2n means that {S1 ≠ 0, . . ., S2n − 2 ≠ 0 and S2n = 0} From (2.11)

30

The Economics of Search P{S1 ≠ 0, . . ., S2n − 2 ≠ 0} = U2n − 2

And P{S1 ≠ 0, . . ., S2n ≠ 0} = U2n Which implies that, f2n, the probability of the first return at 2n is f2n = U2n − 2 − U2n or from the definition of U2n, 1

冢2n − 1冣U

f2n =

2n

Note that by assumption Uo = 1 and hence ∞

冱f

2n

= 1;

n=1

the likelihood of a return to the origin approaches one as the number of trials increases. A common misinterpretation of the law of large numbers is the expectation that in a fair coin-tossing game each of the two participants will be leading for one-half the duration of the game. In fact, if we let 2k denote the last tie in a game of duration 2n, 1 < k < n, then the probability of k is the same as the probability of n − k. This implies that P{k > n2} = P{k < n2} = 12. The probability of no tie in the first half of the game equals the probability of no tie in the second half of the game equals 12. This is independent of the length of the game. It is also surprising that the most probable values for k are 0 and n. These are consequences of: Theorem 2.6 (First arcsine) Let α2k,2n denote the probability that in a random walk of 2n trials the last return to the origin occurred at 2k then α2k,2n = U2kU2n − 2k, k = 0, 1, . . ., n. Proof The total number of possible paths is 22n. The event whose probability we wish to calculate is {S2k = 0, S2k ± 1 ≠ 0, . . ., S2n ≠ 0}. There are 22k ways of picking the first 2k vertices. But we require that S2k = 0 and the probability of this is U2k. Thus the number of paths satisfying the first condition is 22kU2k. By the Return Theorem, the remaining (2n − 2k) vertices can be

Mathematical methods

31

chosen in 2(2n − 2k)U2n − 2k ways. The total number of paths giving rise to the desired event is 22nU2kU2n − 2k. The desired probability is obtained by dividing by 22n. Feller shows that α2k,2n, can be approximated by 1/n f(xk) where xk =

k n

and f(x) =

1 π√x(1 − x)

, 0 < x < 1.

It follows that for fixed 0 < x < 1 and large n the cumulative distribution can be approximated as follows:

冱α

2k,2n

k < xn

≈

2 arcsin√x. π

Feller also proves (1970: 83) the following: Theorem 2.7 (Second arcsine) During a random walk of 2n trials, the probability that a player is winning for 2k periods and losing for 2n − 2k periods is also given by α2k,2n. Random walk with two absorbing barriers Let qw and pw be probabilities that a particle (gambler) who begins (with wealth w) at point 0 < w < a will be, respectively, absorbed at 0 (ruined) or absorbed at a (win the game). After the first trial he is either at w + 1 or w − 1 with probabilities p and q respectively. This gives the recursive difference equation qw = pqw + 1 + q qw − 1, 0 < w < a − 1 If w = 1 q1 = p q2 + q qo If w = a − 1, qa − 1 = p qa + q qa − 2

(2.12)

32

The Economics of Search

where qo = 1, qa = 0

(2.13)

Suppose p ≠ q. Two solutions to (2.12) are qw = 1 ∀w (qw = 1 = p qw + 1 + q qw − 1 = p + q) and qw = (q/p)w qw = (q/p)w = p(q/p)w + 1 + q(q/p)w − 1 =

qw + 1 qw + pw pw − 1

=

qw(q + p) q = pw p

w

冢冣

qw = A + B(q/p)w is a formal solution to (2.12), where by the boundary conditions (2.13), A and B must satisfy A+B=1 A + B(q/p)a = 0 qo = 1 1 = A + B(q/p)o = A + B qa = 0 0 = A + B(q/p)a This implies: qw =

(q/p)a − (q/p)w (q/p)a − 1

(2.14)

If p = q = 1/2. Equation (2.14) breaks down because qw = 1 and qw = (q/p)w are identical. But when p = q = 1/2, qw = w is a solution since 1

1

1

1

冢w = p(w + 1) + q(w − 1) = 2 w + 2 + 2 w − 2 = w冣 Thus, qw = A + Bw is a formal solution with A = 1, A + Ba = 0 which implies

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33

w qw = 1 − . a The expected gain g to the gambler in this game is g = (a − w)(1 − qw) − wqw

冢

= a(1 − qw) − w = a 1 − 1 −

w − w = 0. a

冣

Martingales: a first look In this section we give a brief introduction to the concept of martingales. Later when we turn to issues in estimation, a more in-depth treatment will be presented. Definition 2.2 Let {Xn; n = 0, 1, . . . } and {Yn; n = 0, 1, . . . } be the stochastic processes. We say that {Xn} is a Martingale w.r.t. {Yn} if, for n = 0, 1, . . . (i)

E [|Χ|] < ∞ and

(ii) E [Χn + 1|Y0, . . .,Yn] = Χn.

(2.15) (2.16)

(Y0, . . .,Yn) can be interpreted as the information or history up to the stage n. Thus, in a gambling context, this history could include more information than the sequence of past fortunes (Χ1, . . . , Χn) as, e.g., the outcomes on plays in which the player did not bet. The history determines Χn in the sense that Χn is a function of Y0, . . .,Yn. From (2.16) note that Χn is the particular function Χn = E [Χn + 1|Y0, . . .,Yn] of Y0, . . .,Yn. From the property of conditional expectation, namely, E [g(Y0, . . .,Yn)|Y0, . . .,Yn] = g(Y0, . . .,Yn), we infer that E [Χn|Y0, . . .,Yn] = Χn and invoking the law of total probability E [Χn + 1] = E{E [Χn + 1|Y0, . . .,Yn]} = E [Χn] so by induction E [Χn] = E [Χ0] for all n.

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The Economics of Search

2.3 AN INTRODUCTION TO STOCHASTIC DYNAMIC PROGRAMMING We now turn to a discussion of stochastic dynamic programming. To keep matters simple, at this point we shall look only at situations where the action space and state space are finite. The following discussion draws extensively from Ross (1983a). Suppose that a stochastic process is observed at time points n = 0, 1, 2, . . . and can be in one of a number of possible states. Denote the state space by S. After observing the state of the process, an action must be chosen, and we let A (assumed finite) denote the set of all possible actions. If the process is in state i at time n and action a is chosen, then independent of the past two things occur 1 2

We receive an expected reward R(i,a) The next state of the system is chosen according to the transition probabilities Pij (a); thus, for a given action, the transition probabilities of the state of the system follow a Markov chain.

If we let Χn denote the state of the process at time n and an the action chosen at that time, then assumption 2 is equivalent to stating: P(Χn + 1 = j|Χ0, a0, Χ1, a1, . . ., Χn = i, an = a) = Pij (a). So, both the rewards and the transition probabilities are functions only of the last state and subsequent action. To choose actions we must follow some policy. We will place no restrictions on the class of allowable policies and therefore define a policy to be any sort of rule for choosing actions. An important subclass of all policies is the class of stationary policies. Definition 2.3 A policy is called stationary if it is nonrandomized and the action it chooses at time t only depends on the state of the process at time t. A stationary policy is a function, f, mapping the state space into the action space. For stationary policies the sequence of states {Χn, n = 0, 1, 2, . . .} forms a Markov chain with transition probabilities Pij = Pij (f(i) ) and it is for this reason that these processes are called Markov decision processes. To determine policies that are optimal in some sense we first need to decide on an optimality criterion. We will use the total expected discounted return as our criterion. The criterion assumes a discount factor, 0 < β < 1, and among all policies π attempts to maximize

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35

∞

Vπ(i) = Eπ

冤冱β R(x , a )| x = i冥. n

n

0

n

n=0

The use of a discount factor is economically motivated by the fact that dollars received in the future are less valuable than dollars received today. If R(x,a) < B for all x and a, then it is straightforward to show that |Vπ(i)| < B/(1 − β). The optimality equation and optimal policy Let V(i) = supπVπ(i). A policy π* is said to be β-optimal if Vπ*(i) = V(i) for all i ∈ S. Hence, a policy is β-optimal if the expected β-discounted return is maximal for every initial state. Theorem 2.8

V(i) satisfies the optimality equation

冱P (a)V(j)],i ≥ 0

V(i) = maxa[R(i,a) + β

(2.17)

ij

j

Proof Let π be any arbitrary policy, and suppose that π chooses action a at time 0 with probability Pa, a ∈A. Then, Vπ(i) =

冱P [R(i,a) + β冱P (a)W (j)], a

ij

a ∈A

π

j

where Wπ represents the expected discounted return from time 1 onward, given that policy π is being used and that the state at time 1 is j. However, if the state at time 1 is j, the situation is the same as if the process had started in state j with the exception that all returns are now multiplied by β. Hence, Wπ(j) = βVπ(j) and thus Vπ(i) ≤

冱P [R(i,a) + β冱P (a)V(j)] ≤ 冱P [max {R(i,a) a

ij

a ∈A

a

a

a ∈A

j

冱P (a)V(j)}] = max [R(i,a) + β冱P (a)V(j)].

+β

ij

a

ij

j

j

Because π is arbitrary, we have

冱P (a)V(j)]

V(i) ≤ maxa[R(i,a) + β

ij

j

To go the other way, let a0 be an action such that

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The Economics of Search

冱P (a )V(j) = max [R(i,a) + β冱P (a)V(j)]

R(i,a0) + β

0

ij

a

ij

j

j

Let π be the policy that chooses a0 at time 0 and, if the next state is j, then views the process as originating in state j, following a πj policy such that Vπ ≥ V(j) − ε. j

Hence,

冱P (a )V (j) ≥ R(i,a ) + β冱P (a )V(j) − βε.

Vπ(i) = R(i,a0) + β

πj

0

ij

0

j

0

ij

j

Because Vπ(i) ≥ Vπ, this implies that

冱P (a )V(j) − βε.

V(i) ≥ R(i,a0) + β

0

ij

j

Hence, we obtain the inequality

冱P (a)V(j)] − βε.

V(i) ≥ maxa[R(i,a) + β

ij

j

The result then follows because ε is arbitrary. We are now ready to prove that the policy determined by the optimality equation is optimal Theorem 2.9 Let f be the stationary policy that when the process is in state i, selects the action (or an action) maximizing the right side of (2.17), that is f(i) is a policy such that R(i, f(i)) + β

冱P ( f(i))V(j) = − max [R(i,a) + β冱P (a)V(j)], i ≥ 0 ij

a

j

ij

j

then Vf (i) = V(i) for all i ∈ S, and hence f is β-optimal. Proof

Because,

冱P (a)V(j)] = R(i, f(i)) + β冱P ( f(i))V(j),

V(i) = maxa[R(i,a) + β

ij

j

ij

j

we see that V is equal to the expected discounted return of a two-stage problem in which we use f for the first stage and then receive a terminal reward

Mathematical methods

37

V(j) if we end in state j. But this terminal reward has the same value as using f for another stage and then receiving terminal reward V. Thus, we see that V is equal to the expected reward of a three-stage problem in which we use f for two stages and then receive the terminal reward V. Continuing the argument in this fashion shows that V(i) = E(n-stage return under f | Χ0 = i) + βnE(V(Χn)|Χ0 = i). Letting n → ∞ and using V(j) < B/(1 − β) and 0 < β < 1, we get V(i) = Vf (i). Theorem 2.10

V is the unique bounded solution of the optimality equation.

Proof Suppose that u(i), i ∈ S, is a bounded function that satisfies the optimality equation

冱P (a)u(j)], i ≥ 0

u(i) = maxa[R(i,a) + β

ij

j

For a fixed i, let a* be such that

冱P (a*)u(j)

u(i) = R(i,a*) + β

ij

j

Because V also satisfies the optimality equation, we have

冱P (a*)u(j) − max [R(i,a) + β冱P (a)V(j)]

u(i) − V(i) = R(i,a*) + β

ij

a

j

ij

j

冱P (a*)[u(j) − V(j)] ≤ β冱P (a*)|u(j) − V(j)|

≤β

ij

ij

j

j

冱P (a*)sup |u(j) − V(j)| = βsup |u(j) − V(j)|.

≤β

ij

j

j

By reversing the roles of u and V we can similarly conclude that V(i) − u(i) ≤ βsupj|V(j) − u(j)| Therefore, |V(i) − u(i)| ≤ βsupj|V(j) − u(j)|, and consequently supi|V(i) − u(i)| ≤ βsupj|V(j) − u(j)|.

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Because β < 1, this implies that supj|V(j) − u(j)| = 0. For any stationary policy g, Vg is the unique solution of

Theorem 2.11

冱P (g(i))V (j)

Vg(i) = R(i,g(i)) + β

ij

(2.18)

g

j

Proof It is immediate that Vg satisfies (2.18) because R(i,g(i)) is the one stage return and

冱P (g(i))V (j)

β

ij

g

j

is the expected additional return obtained by conditioning on the next state visited. That it is the unique solution follows exactly as in the proof of the preceding theorem. Method of successive approximations It follows from the results above that if we could determine the optimal value function V, then we would know the optimal policy: it would be the stationary policy that, when in state i, chooses an action that maximizes

冱P (a)V(j)

R(i,a) + β

ij

j

Of course what is the function V? In this section we show that V can be obtained as a limit of the n-stage optimal return problem. Note that for any policy π and initial state j, ∞

冱 β R(x ,a )|Χ i

|En[return from time (n + 1) onwards|Χ0 = j ]| = |En[

i

i

0

= j ]|

i=n+1

≤

β n + 1B 1−β

The method of successive approximation is as follows: let V0(i) be any arbitrary bounded function, and define V1 by

冱P (a)V (j)]

V1(i) = maxa[R(i,a) + β

ij

j

0

Mathematical methods

39

In general, for n > 1, let

冱P (a)V

Vn(i) = maxa[R(i,a) + β

ij

(j)]

n−1

j

Intuitively, Vn is the maximal expected discounted return of an n-stage problem that confers a terminal reward V0(j) if the process ends in state j. The following proposition shows that Vn converges uniformly to V as n → ∞. Theorem 2.12 1 2

If V0 ≡ 0, then |V(i) − Vn(i)| ≤ β n + 1B/(1 − β). For any bounded V0(i), Vn(i) → V(i) uniformly as n → ∞.

Proof Suppose V0 ≡ 0, so Vn(i) equals the maximal expected reward in an n-stage problem starting in i. Now, for the β-optimal policy f, V(i) = Ef [return during first n-stages] + Ef [additional returns] ≤ Vn(i) + β n + 1B/(1 − β), where the inequality follows from R(i,a) < B and the definition of Vn. To go the other way, note that V must be larger than the expected return of the policy that uses the n-stage optimal policy for the first n-stages and any arbitrary policy for the remaining time. Hence, V(i) ≥ Vn(i) + Ef [additional returns from n + 1 onwards] ≥ Vn(i) − β n + 1B/(1 − β) which together with the proceeding, proves (i). To prove 2 let V 0n denote Vn when V0 ≡ 0. Then for any bounded V0 it can be shown that |Vn(i) − V 0n(i)| ≤ β n supj|V0(j)|. Example 2.4 When should you buy a new car? Suppose that at the beginning of each time period you inspect your car and determine its condition or state. After observing this state, a decision as to whether or not to buy a new car must be made. If your decision is to buy a new car then a cost of R is immediately incurred and the state at the beginning of next time period is 0, which is the state of a new car. If the present state is i and you decide to keep your old car, then the state at the beginning of the next period will be j with probability Pij. In addition, when the car is in state i at the beginning of a time period, an operating cost C(i) is incurred. Let V(i) denote the minimal expected total β-discounted cost, given that the state is i. Then V satisfies the optimality equation

40

The Economics of Search V(i) = C(i) + min[R + βV(0), β

冱P (a)V(j)]. ij

j

Under what conditions on C(i) and the transition probability matrix P = [Pij] will V(i) be increasing in i ? First, it is clear that the operating costs must be increasing, so let us assume the following condition: Condition 1 C(i) is increasing in i. However, condition 1 by itself is insufficient to imply that V(i) is increasing in i. It is possible for states i and k where k > i that k has higher operating costs than i but it might take the process into a better state than i. To ensure that this does not occur, we suppose that, under no replacement, the next state from i is stochastically increasing in i. That is, we have the following condition. ∞

Condition 2

For each k,

冱P

ij

increases in i.

j=k

In other words, if Ti is a random variable representing the next state visited after i (assuming you keep your current car) then P(Ti = j) = Pij, so condition 2 states that Ti + 1 is stochastically larger than Ti or Ti ≤ stTi + 1, i = 0, 1, 2, . . . Because this is equivalent to E(f(Ti)) increasing in i for all increasing functions, f, condition 2 is equivalent to the statement that 冱Pij f(j) increases in i, for all increasing f. We now prove by induction that, under conditions 1 and 2, V(i) increases in i. Theorem 2.13

Under conditions 1 and 2, V(i) increases in i.

Proof Let V1(i) = C(i) and for n > 1 Vn(i) = C(i) + min[R + βVn − 1(0) + β

冱P V ij

n−1

(j)]

j

It follows from condition 1 that V1(i) increases in i. Hence, assume that Vn − 1(j) increases in j, so, from condition 2, 冱jPijVn − 1(j) increases in i for all n, and because V(i) = limnVn(i), the result follows. Theorem 2.14 Under conditions 1 and 2, there exists an v, with v ≤ ∞, such that the β-optimal policy replaces the car when the state is i if i ≥ v and does not replace the car if i < v. Proof It follows from the optimality equation (2.17) that it is optimal to buy a new car when the condition of your current car is i if

Mathematical methods

41

冱P V(j)

R + βV(0) ≤ β

ij

j

冱

Because V(j) increasing in j implies that jPijV(j) increases in i (from condition 2), the result follows with v being given by v = min[i:R + βV(0) ≤ β

冱P V(j)], ij

j

where v is taken to be ∞ if the preceding set is empty. Thus for this example the optimal policy consists of a “reservation state” such that if the state of the car equals or exceeds this state then it is optimal to buy another car.

2.4 SOME RESULTS IN PROBABILITY THEORY: A MEASURE THEORETIC APPROACH In this section we develop some notions in probability theory using measuretheoretic concepts.

2.4.1 Probability spaces We denote the sample space by Ω and a particular element (outcome) by ω. The collection (set) of subsets of elements of Ω will be denoted by F. Definition 2.4 The collection of subsets of F is referred to as a σ-algebra (σ-field) if and only if 1 2 3

Ω, ∅ ∈ F F is closed (stable) under countable unions F is closed (stable) under countable intersections.

Finally, we have a function P that maps elements of F into [0,1]. Definition 2.5 A probability space is the triplet (Ω, F, P). A random variable Χ is a mapping from Ω into another space E (Χ: Ω → E). Thus, the random variable Χ assigns a value e ∈ E to each ω ∈ Ω. If we let E denote the set of subsets of E and we further restrict Χ to be a mapping where the inverse image X −1 is such that Χ −1 (B) ∈ F

for all B ∈ E

then Χ is called a measurable mapping from (Ω,F ) to (E,E) and symbolized by Χ: (Ω,F ) → (E,E ).

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2.4.2 Expectations The expected value of a nonnegative random variable Χ with respect to P is the integral 2n − 1

冮

EP(Χ) = Χ(ω)dP(ω) = lim

n→∞

冱 2 P冢冦ω| 2 i

n

i n

≤ Χ(ω) <

i=0

i+1 2n

冧冣

+ nP({ω|Χ(ω) ≥ n}). Extensions to random variables that are not nonnegative is straightforward. We may simply omit the P subscript in cases where no confusion is possible. If g is a real function defined on (E,E) then

冮

E [g(Χ)] = g(Χ(ω))dP(ω). Definition 2.6 A random variable is integrable with respect to probability measure P (P-integrable) if

冮

E [|Χ|] = |Χ(ω)|dP(ω) < ∞ Two classic inequalities involving expectations are Chebyshev’s inequality and Jensen’s inequality.

Chebyshev’s inequality If Χ is a random variable such that E [|Χ|a] < ∞ for some a > 0 then for any c > 0, P[|Χ| ≥ c] ≤ c−aE [|Χ|a]. Jensen’s inequality Let Χ be a random variable and :R → R a convex function. Further suppose that E [Χ] < ∞ and E [(Χ)] < ∞. Then, (E [Χ]) ≤ E [(Χ)]. Definition 2.7 Let µ1 and µ2 be two measures on the measurable space (Ω,F ). The measure µ2 is said to be absolutely continuous with respect to the measure µ1 if for all A ∈ F such that µ1(A) = 0 ⇒ µ2(A) = 0. The Radon–Nikodym theorem forms the basis for conditional expectations Theorem 2.15 (Radon–Nikodym) Let µ1 and µ2 be two measures on the measurable space (Ω,F ) that are σ-finite. If µ2 is absolutely continuous with respect to µ1, then there exists a nonnegative finite random variable, denoted

Mathematical methods

43

dµ2/dµ1 and called the Radon–Nykodym derivative of µ2 with respect to µ1, such that µ2(A) =

冮

A

dµ2 dµ1 for all A ∈ F. dµ1

A more formal definition of a transition probability is the following: let (E,E) and (Ω,F ) be two measurable spaces, and let (Px, x ∈ E) be a family of probability measures on (Ω,F ) such that for all sets A ∈ F, the function x → Px(A) is measurable with respect to E. Then (Px, x ∈ E) is called a transition probability from (E,E) into (Ω,F ). Theorem 2.16 (Fubini) Consider the product space (U,U) = (E × Ω, E ⊗ F) and let f be a nonnegative random variable defined on this space. Then for a probability measure Q on (E,E) there exists a probability measure S on (U,U) and a transition probability (Px, x ∈ E) such that for U = A × B in U, S(A × B) =

冮 P (B)Q(dx) A

x

and the mapping from x into

冮

U

f(u)S(du) =

冮 冤冮

Ω

E

冮

Ω

f(x,ω)Px(dω) is E-measurable and

冥

f(x,ω)Px(dω) Q(dx)

Definition 2.8 Let Χ be an integrable random variable defined on the probability space (Ω, F, P). Let G be a sub-sigma algebra of F. If there exists a random variable Y such that Y is G-measurable and P-integrable and

冮 Χ(ω)dP(ω) = 冮 Y(ω)dP(ω) for all C ∈ G, C

C

then Y is called (a version of ) the conditional expectation of X. Law of iterated expectations Let Χ be a random variable that is integrable with respect to P. Let G1 and G2 be two sub-sigma fields of F with G1 ⊂ G2. Then E [E [Χ|G2]|G1] = E [Χ|G1] Thus, when G1 is “coarser” than G2, the expected value of Χ given G1 is the same as the expected value of Χ given G2 when the information set is G1.

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2.4.3 Histories Definition 2.9 Let (Ω,F ) be a measurable space. A history (or filtration) (Ft, t ≥ 0), or simply Ft, on (Ω,F) is a family of sub-σ fields of F such that for all 0 ≤ s ≤ t, Fs ⊂ Ft. Definition 2.10

A history is said to be right continuous if

Ft = 傽 Ft + dt = Ft +

dt > 0

Now, consider a family of random variables (Xt, t > 0) defined on (Ω,F). We will refer to this family as a stochastic process. When no confusion arises the stochastic process will simply be denoted by Xt. The internal history of Xt is the filtration defined by F Χt = σ(Χs, s ∈ [0, t]) for t > 0. Any history Ft that contains F Χt (i.e. Ft ⊃ F Χt for t > 0) is called a history of Xt and Xt is said to be adapted to Ft. The path of the stochastic process Xt is the mapping ω → Χt(ω) for a given ω ∈ Ω. A stochastic process is said to be continuous, right continuous, or left continuous if its paths are respectively, continuous, right continuous, or left continuous, P-almost surely. Definition 2.11 A stochastic process Xt is said to be Ft – progressive if and only if for all t ≥ 0 the mapping (t, ω) → Χt(ω) is measurable with respect to B + ⊗ F where B + is the sigma-algebra of Borel sets on the positive real line. Theorem 2.17 If Xt is a stochastic process that is adapted to Ft and is either right-continuous or left-continuous then Xt is called Ft-progressive. Definition 2.12 Let Ft be a history and T be an R +-random variable. Then T is called an Ft-stopping time if and only if {T ≤ t} ⊂ Ft

for t ≥ 0.

So, a random variable T is a stopping time if at any time t it is possible to tell whether the stopping time event has occurred or not. Next, we expand the notion of the Markov chain to continuous time processes. In what follows we assume that the stochastic process Xt is E valued on the measurable space (E,E) but for simplicity we will suppress any notation referring to this fact when no confusion arises. Definition 2.13 Let Xt be a stochastic process defined on (Ω, F, P) and adapted to some history Ft. Then Xt is called a (P, Ft) – Markov process (or

Mathematical methods

45

when no confusion arises simply a Markov process) if and only if for all t ≥ 0, σ(Χs, s ≥ t) and Ft are P-independent given Xt. For Markov processes then “knowing” Xt is as good as knowing the entire history of the process up to t in terms of forecasting the future values of the stochastic process. In other words, if Xt is a Markov process, then for all 0 ≤ s ≤ t and bounded, measurable function f, E [f(Χt)|Fs] = E [f(Χt)|σ(Χs)]. Recall that in discrete-time a Markov chain had an associated transition matrix P. The continuous-time analog of this is the transition function: Definition 2.14 Let (E, E) be a measurable space, and for each 0 ≤ s ≤ t let Ps,t(x, A) with x ∈ E and A ∈ E, be a function from E × E into R+ with the following properties: 1 2 3

A → Ps,t(x, A) is a probability on (E, E) for all x ∈ E. x → Ps,t(x, A) is in B+/E for all A ∈ E. for all 0 ≤ s ≤ u ≤ t, x ∈ E and A ∈ E, Ps,t(x, A) = Ps,u(x, dy)Pu,t(dy, A).

冮

Then the function Ps,t(x, A) is called a Markov transition function. If we further assume that Ps,t(x, A) = Pt − s(x, A) for all t and s, the Markov transition function is said to be homogeneous. Definition 2.15 function. If

Let Xt be a Markov process and let Ps,t(x, A) be a transition

冮

E [ f(Χt)|Fs] = Ps,t(Χs, dy)f(y) for all 0 ≤ s ≤ t and bounded, measurable functions f, then the Markov process Χt is said to admit the Markov transition function Ps,t(x, A).

2.5 THE POISSON PROCESS Before concluding this chapter on mathematical methods, we discuss the properties of a simple counting process the “Poisson process”. In Chapter 5 on empirical methods for duration models we expand this discussion. A stochastic process {N(t), t ≥ 0} is called a counting process if N(t) represents the total number of “events” that have occurred up to time t. A counting process N(t) must satisfy 1 2 3

N(t) ≥ 0 N(t) is integer valued If s < t, then N(s) ≤ N(t)

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4

For s < t, N(t) − N(s) equals the number of events that occurred in the interval (s, t).

A counting process is said to have independent increments if the number of events which occur in disjoint time intervals are independent. For example, the number of events which have occurred by time 10, N(10), must be independent of the number of events occurring between times 10 and 15 (N(15) − N(10)). A counting process is said to have stationary increments if the distribution of the number of events which occur in any interval of time depends only on the length of the interval. Thus, the process has stationary increments if the number of events in the interval (t1 + s, t2 + s)[N(t2 + s) − N(t1 + s)] has the same distribution as the number of events in the interval (t1, t2)[N(t2) = N(t1)] for all t1 < t2 and s > 0. One of the most important counting processes is the Poisson process. Definition 2.16 The counting process {N(t), t ≥ 0} is called a Poisson process with arrival rate λ, λ > 0, if 1 2 3

N(0) = 0 The process has independent increments The number of events in any interval of length t is Poisson distributed with mean λt.

Hence, for all s, t ≥ 0 P{N(t + s) − N(s) = n} = eλt

(λt)n , n = 0,1, . . . n!

From 3 above it follows that a Poisson process has stationary increments and also that E(N(t)) = λt. To decide if an arbitrary counting process is Poisson one must verify conditions 1, 2, and 3. Condition 3 is hard to verify. This suggests a second definition of a Poisson process. Before continuing, however, we need to define some notions of convergence. Definition 2.17 A sequence of (xn) ∈ E is said to converge to a point x0 ∈ E if for every ε > 0 there exists an N such that || xn − x0 || < ε for all n > N where ||⋅|| denotes a measurable distance function on E. When a sequence converges we denote it by lim xn = x0. n→∞

Definition 2.18

A sequence of random variables (Χn) is said to converge to

Mathematical methods

47

a point x0 ∈ E if for every ε > 0 and δ > 0 there exists an N such that P(|| Χn − x0 || > ε) < δ for all n > N. When a sequence of random variables converges we denote it by plim xn = x0 or simply plim xn = x0. n→∞

A function f(x) is o(h) or “little oh of h” if

Definition 2.19 lim h→0

f(h) = 0. h

The function f(h) h2 = lim , h→0 h h→0h

f(x) = x2 is o(h), lim

while the function f(x) = x is not o(h). In order for a function f(·) to be o(h) it is necessary that f(h)/h go to zero as h goes to zero. But if h goes to 0, the only way for f(h)/h to go to zero is for f(h) to go to zero “faster” than h does. Similarly, we have that a random function is op(h) when the function converges to 0 in probability “fast” enough. Definition 2.20

A random function Χ(h) is op(h) if plim

Χ(h) = 0. h

Definition 2.21 The counting process {N(t), t ≥ 0} is a Poisson process with rate λ, λ > 0, if 1 2 3 4

N(0), = 0 The process has stationary and independent increments. P{N(h) = 1} = λh + op(h) P{N(h) ≥ 2} = op(h).

Theorem 2.18

Definition (2.16) ⇔ Definition (2.21).

2.5.1 Interarrival times Consider a Poisson process and denote the time of the first event by T1. For n > 1, let Tn be the elapsed time between the (n − 1)th and nth event. The sequence {Tn, n = 1, 2, . . .} is the sequence of interarrival times. Note that event {T1 > t} takes place iff no events occur in [0, t] and hence P{T1 > t} = P{N(t) = 0} = e − λt. Therefore, T1 has an exponential distribution with mean 1/λ.

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Theorem 2.19 Tn, n = 1, 2, . . . are iid exponential random variables with mean 1/λ. Finally, we have: Theorem 2.20 {Nt − λt, t > 0} is a martingale. A more general form of Theorem 2.20 is discussed in Chapter 5 on estimation for duration models. Intuitively it can be interpreted as the expected number of arrivals in a small interval of time dt which is approximately equal to λdt or E(Nt + dt − Nt) = λdt + op(dt).

3

The history and evolution of sequential analysis

3.1 INTRODUCTION Speculations about economic phenomena almost always precede quantitative studies of these phenomena. The speculations are frequently profound and plausible, but they provide no apparatus for judging their quantitative validity. A stream of rhetoric often accompanies speculative insights and, indeed, may convince most people of their validity. However, a quantitative appraisal is possible only when these conjectures have been translated into a mathematical model, together with a procedure for estimating the model. The development of the mathematical model is relatively free of practical constraints. Yet it is always true that in the beginning, the mathematical edifice was a seed perceived by the senses. When a connection is established between a speculative economic model and a fully developed mathematical model, an epiphany or enlightenment occurs. After such a contact the economic model may be transformed into a scientific edifice and many practical problems are clarified and frequently resolved.1 A process like this is similar to what actually happened in the discovery and growth of the sequential search paradigm, which we briefly describe. We first survey the economic insights obtained via speculative (philosophical) discourse and then turn to the mathematical models that are connected to this discourse. These quantitative models include econometric techniques for estimating the actual behavior of economic agents. Three distinct enlightenments have occurred so far in the development of search. The first was by Stigler who saw the nexus joining early search speculations with a fixedsample size statistical model. Stigler’s insight presented economics with its first quantitative procedure for studying search behavior. The second perceived the connection between the sequential speculations by Hutt, Knight, Simon, Alchian, and others and the mathematical model instigated by Cayley and growing into the enormous literature comprising sequential analysis.2 The third illumination took place in the 1980s when several economists including Diamond, Mortensen,3 Pissarides, Rothschild, and others realized that there are important interactions among agents engaged in sequential search processes and that the most striking interactions can be addressed

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using the equilibrium and bargaining methods discovered by John Nash in the 1950s. Nash’s insights stimulated a large search literature on equilibrium and matching models which continues to flourish. It should be emphasized that, while dazzling, Nash does not replace Wald et al. Sequential decisionmaking remains a foundation of these matching models. Furthermore, there are competitive search models which are almost completely based on the stopping rule paradigm. These include the seminal papers by Jovanovic (1979b), Lucas and Prescott (1974), and the recent research by Ljungqvist and Sargent (see, for example, 2005a).

3.2 EARLY LITERARY AND INTUITIVE CONTRIBUTIONS TO THE ECONOMICS OF JOB SEARCH4 Many illustrious economists attempted to link information gathering with labor markets. Though Stigler’s 1962 paper “Information in the Labor Market” offered very little in the way of new insights, it did succeed in forging a quantitative model of job search with testable implications. This was an enormous contribution. The notion of job search as an explanation for unemployment is present either explicitly or implicitly in the writings of several economists before Stigler’s 1962 article. While suggestive for policy, their verbal models contained no mathematical formula and were neither vulnerable to empirical testing nor sufficiently well developed to converse with the analytical work. Cannan acknowledges the existence of a moral hazard problem associated with unemployment insurance in 1930 when he denies “that unemployment insurance leads to the refusal of available work” (p. 45) and asserts that “the insurance scheme has reduced the economic pressure which used to make persons grab at every chance of employment . . . made them, like the old British army, ‘ready to go anywhere and do anything’ ” (p. 46). In fact, his statement that the job-seeker’s strategy is not to go on the dole for the remainder of his life but rather to “Take what you can get now, or hold out another week, when something better may turn up” (pp. 46–7) appears, with the advantage of hindsight, like a description of the reservation wage policy. Writing about wages in 1932, Hicks (p. 45) recognizes the unanticipated unemployment effects of the search which necessarily emanates from the presence of imperfect information when he asserts that firms’ “knowledge of opportunities is imperfect [whence] it is not surprising that an interval of time elapses between dismissal and reengagement, during which the workman is unemployed.” His statement (p. 45) that: “the unemployment of the man who gives up his job in order to look for a better . . . may believe that he could get higher wages elsewhere” implicitly recognizes that voluntary unemployment joined with search can be a productive activity. The earliest explicit statement of search unemployment as a productive activity was made by Hutt in 1939:

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A worker . . . may refuse immediately available work . . . because he feels that to accept it will prevent him from seeking for better openings. . . . When actively searching for work, . . . he is really investing in himself. . . . He is doing what he would pay an efficient employment agency to do . . . the search for a better opening is worth the risk of immediately foregone income. (1977: 83; original emphasis) In addition to the labor literature anticipating the job search model, other prominent precursors in the economics of search include Knight, Schumpeter, and Alchian who were concerned with the search for natural resources and the search for inventions. Knight (1921) was keenly aware of the significance of the ancient activity of prospecting for natural resources. We find his statement (p. 338), “where the possibility of securing wealth by the discovery of natural resources is known . . . resources will be attracted into the field of searching for them in accordance with men’s estimates of the chances of success in relation to the outlays to be incurred” closer to a Marshallian than a Stiglerian search-theoretic view. While the research and development (R&D) literature had not yet begun, its beginnings are recognizable in the writings of both Knight and Schumpeter. One of the premises of this literature is that innovations resulting in lower costs of production and/or improved product quality must be sought in much the same way that a consumer searches for a lower price. Again, Knight’s stance (1921: 341; original emphasis), “In the case of new knowledge . . . it is clear that in so far as the results can be predicted the investment of resources in the acquisition of new knowledge will be so adjusted as to . . . equate realized value to costs and eliminate profits” is Marshallian. Schumpeter (1942) continued the advance toward Stigler’s introduction of search in his concern for uncertainty and the opportunistic contractual behavior which uncertainty fosters. He not only recognized the need for “safeguarding activities” such as patent protection but also anticipated the tendency to over-search when property rights are assigned imperfectly. Practically any investment entails, as a necessary complement of entrepreneurial action, certain safeguarding activities such as insuring or hedging. Long-range investing under rapidly changing conditions . . . is like shooting at a target that is not only indistinct but moving . . . and moving jerkily at that. Hence it becomes necessary to resort to such protecting devices as patents or temporary secrecy of processes, or, in some cases, long-period contracts in advance. (1942: 88) In Alchian’s (1951) early evolutionary view, refined and extended by Nelson and Winter (1985), innovations may result from bungled imitation as

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well as from conscious efforts to improve. His statement (Alchian 1951: 215), “individual motivated behavior based on the pervasiveness of uncertainty and incomplete information. Adaptive, imitative, and trial-and-error behavior in the pursuit of positive profits,” reveals his view that innovation, like prospecting for natural resources, is a form of search. Simon (1951, 1991) developed a stochastic theory of the employment relation. The earlier paper resembles Wald’s sequential rule and the more recent discussion contains many insights.

3.3 STIGLER’S INSIGHTS In his Nobel laureate address Stigler (1983: 533) claims that “The Central task of an empirical science such as economics is to provide general understanding of events in the real world” and continues with the caveat that “ultimately all of its theories and techniques must be instrumental to that task.” A key point of his address is the importance of the accumulation of economic knowledge, with the inevitable dispute and competition among ideas. One long-accepted idea in the chain of the history of economics is the decisive influence and impact upon market behavior of the economic actors’ information. Both this impact and the actors’ behavior would, of course, vary with the extent and accuracy of the information possessed by the economic actors. Whereas economists have been aware of the pervasiveness and importance of information for eons and renowned economists such as Knight, Hicks, Schumpeter, Simon, and Alchian understood and wrote about the force of incomplete information in economic life, it has only been since the appearance of Stigler’s 1961 article that the economics of information has evolved in any significant fashion. Because search is one of the mechanisms for acquiring price information and because price formation is the central issue of economic inquiry, Stigler implores us to incorporate search into our models and “systematically take account of the cold winds of ignorance” (1961: 225). Referring to the “amounts and kinds of information needed for the efficient allocation of labor” (Stigler 1962: 104) as well as other informational problems such as “the amounts of skill men should acquire,” he asserts that “The traditional literature has not done these problems justice . . . the analysis of the precise problems of information and methods an economy uses to deal with them appears to be a highly rewarding area for future research” (1962: 104). Since then not only has the economics of information been removed from Stigler’s “slum dwelling in the town of economics” (1961: 213), it has become one of the most important and active fields in economics. Marginal utility was the centerpiece of mid-twentieth-century economics, an engine of analysis that was virtually silent about some of the most important manifestations of economic behavior, including the design, formation, and adaptation of institutions; the persistence of profits in the presence of

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mobile resources; the existence and persistence of price variability; and the prominence of unemployed resources. Whereas many economic institutions—particularly hedging, insurance, and a host of other pooling mechanisms including share cropping and corporate share holding—owe their existence and structure to the presence of uncertainty, this clockwork centerpiece ignored the impact of chance on behavior. The gathering, processing, and transmission of information are the distinguishing characteristics of advertising, queuing and inventory accumulation, reputation, matching, goodwill, and intermediaries. Waiting and shopping also can be productive activities in an uncertain environment. Shopping enables the buyer to accumulate information until an item with the desired qualities and price is selected; similarly, sellers wait until a desirable buyer is found. These dynamic phenomena revealed a flaw in the Marshallian apparatus long recognized by institutional, theoretical, and applied economists. Prior to Stigler’s seminal 1961 article, the role of information had not been analyzed; moreover, “the amount of information possessed by individuals in any market was arbitrarily postulated rather than derived from economic principles” (Stigler 1983: 539). With his simple fixed-sample model of search, however, standard economic theory was easily applied to endogenously determine precisely how much information people would acquire. Thus, the profession’s ready acceptance of Stigler’s contribution is explained by the long-perceived need for such a theory, the theory’s use of extant tools, and the fact that the theory extended rather than challenged the existing Marshallian framework. Similarly, the sequential recasting of Stigler’s framework in 1970 was instantly accepted by the profession for the same reasons. In particular, this extension was facilitated by the recent development of optimal stopping in the statistical literature. The two were simply joined in what appears to be, as it were, a marriage made in heaven.

3.4 STIGLER’S MODEL We can’t overemphasize our agreement with Lucas (1987: 56) when he states that a “model’s explicitness invites hard questioning.” Like those that preceded him, Stigler saw that the theory of price, the coordinating mechanism of economic allocation, was seriously handicapped by the absence of a formal theory of information accumulation. Stigler’s genius was not in asserting that information should be incorporated into a model in which analysis could be effected but rather in providing such a model. His contribution consists in explicitly modeling the information acquisition activity—and doing so in a manner which satisfied the practical, theoretical, institutional, and empirical demands for a dynamic economic system while maintaining and strengthening the bond with the Marshallian tradition. In particular, his search model enables the analyst to apply the conventional first-order condition: continue gathering information until the marginal value equals the

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The Economics of Search

marginal cost. Stigler converted advertising, price variability, and unemployed resources into the basic elements of a Marshallian search model. Just like Hayek (1945: 522), who stressed that it is a gross mistake to suppose that all knowledge of “the particular circumstances of time and place . . . should as a matter of course be readily at the command of everybody . . .,”5 Stigler starts from the fact (1961: 213) that “no one will know all the prices which various sellers . . . quote at any given time.” From this it followed that search entailed a buyer who, wishing to “ascertain the most favorable price, must canvass various sellers” (p. 213).6 Taking price dispersion as a manifestation of ignorance in the market (p. 214),7 Stigler posited this canvassing as the buyer’s gathering of n independent offers—say Χ1, . . ., Χn—from some cumulative distribution of function F. With the view that the buyer’s chief cost is time, the buyer’s cost of search is proportional to the number of offers solicited (p. 216). Thus, in a few short paragraphs Stigler cast his principles of search in mathematical terms: select the sample size n so as to minimize Emin{Χ1, . . ., Χn} + nc. A number of important results flow from this one-equation model. First, Stigler observes (1961: 215) that “increased search yields diminishing (marginal) returns,” and he also obtains the familiar first order condition (p. 216): “If the cost of search is equated to its expected marginal return, the optimum amount of search will be found.” Second, because the uncertain outcomes associated with search entail transactions at different prices, quite literally, by the luck of the draw/sample, Stigler’s new search theory explained violations of the law of one price without taking recourse, as did Jevons, to explanations involving differences in preferences. Third, to the extent that the act of gathering information is not timeless, search is dynamic. Consider a model with discounting—namely, minimize β n Emin{Χ1, . . ., Χn} − c(1 − β n)/(1 − β). Due to the delay associated with additional search, the total quantity of search is reduced as the discount factor β falls (so as not to too greatly diminish its usefulness). Fourth, when the agent doing the search is a job-seeker or an individual seeking to sell an asset, we obtain a full-fledged explanation of idle resources: the activity of unemployment (i.e., search) is an investment incurred to improve the future.8 As regards price formation, the search explanation of idle resources enables two Marshallian work horses of economic action, waiting and effort, to be harnessed in tandem. Stigler proceeded to speak of advertising as the obvious modern method of identifying buyers and sellers (1961: 216). Noting that the cost of collecting information is nearly independent of its use, the need for “firms which specialize in collecting and selling information” was clear (p. 220). He also briefly discussed reputation. Finally, with some considerable foresight and concern for the inefficiency generated by price dispersion, he comments (pp. 223–4)

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that manufacturers seek the reduction of uncertainty by setting uniform prices for nationally advertised brands for “if they have eliminated price variation, they have reduced the cost of the commodity (including search) to the buyer.” Prompted by the success of Stigler’s new model, the impact upon econometric methods was large though substantially delayed. Whereas uncertainty had previously been modeled as the error term, the new econometrics inserted uncertainty into the structure of the model. The power of this new theory was most apparent in labor economics where an explanation and understanding of unemployment remains elusive.9 Using the sequential version of the search model, the new econometrics, as exemplified by Heckman and Singer (1984, 1985), Kiefer and Neumann (1989) and Lancaster (1990) succeeded in studying the behavioral effects of unemployment insurance, training programs, and welfare transfers in the labor market. Fully nine years after Stigler “opened the door,” the economics of information was combined in a simple sequential model with the emerging mathematical developments including Wald’s sequential analysis (1947), Bellman’s dynamic programming (1957),10 and Markov decision processes.11 Whereas statisticians had already produced and analyzed systems considerably more advanced than Stigler’s simple model, the orientation and presentation was such that their work had little impact in economics until 1970.

3.5 EARLY MATHEMATICAL CONTRIBUTIONS TO THE ECONOMICS OF JOB SEARCH In the urgent wartime environment,12 Abraham Wald13 improved upon the double sampling inspection method of Dodge and Romig (1959) and developed the theory of sequential analysis.14 While Wald’s work was being further developed by statisticians, probabilists were working on the theory of martingales and stopping times.15 With probabilists located in statistics departments at that time, the timing was ripe for interaction and the development of optimal stopping. During the period of this development, MacQueen and Miller (1960) formulated and solved the house-selling problem. As shown in Example 3 below, it is precisely the sequential search problem. Cayley’s problem and the secretary problem— Examples 1 and 2 below—are its two prominent precursors. See Ferguson (1989) for an insightful history and delineation of the development of all of these problems. Example 3.1 (Cayley’s problem) The first stopping problem was proposed by Arthur Cayley in 1875. His problem is a version of the house-hunting problem. Given n objects with known values V1 < . . . < Vn, the decision-maker seeks to select an object so as to minimize the expected value of the object

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selected. The objects are presented in a random order, and there is no recall of rejected objects. In contrast to the usual house-hunting problem, the distribution of the value of the next object to be considered is not independent of the values of the previously viewed objects. Example 3.2 (The secretary problem) The decision-maker is going to select a secretary from among n candidates. He interviews them one at a time. The candidates are presented in a random order, and there is no recall of rejected candidates. Each candidate’s ability level is observable at the time of interview; however, the actual magnitude of any candidate’s ability level supplies no information whatsoever as regards the ability level of remaining candidates. The payoff function for the secretary problem is 1 if the best secretary is selected; otherwise, it is 0. The optimal policy consists in interviewing and rejecting 1/e of the candidates and then selecting the next candidate with the highest rank observed. The probability of selecting the best secretary is 1/e.16 Example 3.3 (The house-selling problem) Offers Χ1, Χ2, . . . arrive one at a time. The offers are independent and identically distributed with known cumulative distribution F. The value of an offer is observable upon arrival, and each observation costs c dollars. The return Rn from stopping after n observations is Rn = max(Χ1, Χ2, . . ., Χn) − nc. The problem is to design a stopping rule N to maximize ERN. Shortly after the MacQueen and Miller paper appeared, a host of advances were made.17 Applications did not lag. In particular, we mention the optimal control of Brownian motion and, especially, the now-enormous option pricing literature.18 Of course, numerous developments in the theory of dynamic programming occurred during the 1970s accompanied by an even larger set of applications in the economics and operations research literatures.19 Beginning with the efforts of the Anti-Submarine Warfare Operations Research Group in England during World War II, an enormous literature on detection or search for a hidden object was started, and development continued into the 1960s and 1970s (see Stone, 1975, for a comprehensive synthesis). Shortly thereafter, the literature on “Multi-Armed Bandit Problems” (MABP), related both to sequential analysis and stopping rules, began to develop. In the bandit problems each observation of the ith arm induces the decision-maker to Bayesianly update his beliefs regarding the payoff associated with arm i. See Berry and Fristedt (1985), Gittins (1979, 1989), and Whittle (1980) for a discussion of MABP, and Rothschild (1974) for an application to economic search.

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3.6 THE SEQUENTIAL JOB SEARCH MODEL In light of the advances in optimal stopping and the formulation of the house-selling problem, today it seems natural and even obvious that economic job search should be placed in a sequential setting. This was done by J. McCall in his 1970 article, “Economics of Information and Job Search.” He begins (p. 115): In the simplest job search model the searcher is assumed to know both the distribution of wages for his particular skills and the cost of generating a job offer. Job offers are independent random selections from the distribution of wages. These offers occur periodically and are either accepted or rejected. Under these conditions it is easy to show that the optimal policy for the job searcher is to reject all offers below a single critical number. . . . Unemployment terminates whenever an offer exceeds the critical value. Just as in Stigler’s original search model (where F is the offer distribution), the sequential model is encapsulated in one equation.20 The only difference between the two equations is that the fixed-sample size n is now replaced by a stopping time N: select a stopping time N so as to maximize Emax{Χ1, . . ., ΧN} − Nc. The optimal stopping time is simply described: accept the first offer which exceeds the reservation wage ξ.21 The reservation wage ξ solves the first-order condition ∞

冮

c = (y − x)dF(y) = H(x), x

where H(x), the marginal gain from obtaining exactly one more offer, is strictly decreasing (if F has a positive density) and convex. As H ′ < 0, the first-order condition has ξ as its unique solution, and it also has a simple interpretation: equate the marginal cost c of taking an additional offer with the expected increase in the return from taking exactly one more search/offer. Thus, the reservation wage is easily computed from this myopic condition. Moreover, it has been shown (see Lippman and McCall, 1976) that ξ is the expected return to search. The time until an acceptable offer is obtained, called the period of frictional unemployment,22 is a geometric random variable with mean 1/(1 − F(ξ) ). With this model and its analytic first-order condition for determining the reservation wage, a number of insights are easily obtained. The discouraged worker phenomenon is analyzed for the first time by McCall (1970: 117):

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“Consider an individual whose expected returns from remaining unemployed are ε0.” She stops out of the labor market if the return ξ to search is less than ε0. To paraphrase Lucas (1987: 67), focusing upon unemployment as an individual rather than an aggregate problem—so that the worker can be viewed as having elected voluntary unemployment—enabled economists to formulate social policies for dealing with it. The view of unemployment as voluntary has the advantage of inviting an analysis of the forces such as unemployment insurance which impact the choice of unemployment. Two methods for reducing the number of dropouts are suggested: lowering the cost of search and shifting the offer distribution by adding a constant to each offer. Both changes increase the return to search. Lowering the cost c of search, say by additional government benefits available only while actively seeking employment,23 induces the worker to search. If the wage of each employed worker is augmented by a fixed amount δ either by a direct government subsidy to the employer or by introducing a training program which increases the worker’s marginal product, then the effective offer increases by δ and Hδ(x) = H(x + δ): the H functions shifts to the right, and the reservation wage increases by δ. In the first case the period of frictional unemployment does indeed increase as claimed on p. 119 of McCall (1970). In the second case, however, it remains constant. In fact, in a model with discounting, adding a constant δ to each job offer causes the period of frictional unemployment to decrease (and the reservation wage to increase by less than δ) precisely because the return to search increases.24 An analysis of a minimum wage law on discouraged workers proceeds naturally in this framework. If a regime with minimum wage m converts each pre-regime offer less than m to an offer of 0, then the new H function, label it H0, decreases in a particular manner: H0(x) = H(x) for x > m, but H0(x) = H(m) for x < m. Each worker whose reservation wage was less than m has a search cost c > H(m); in the minimum wage regime such a worker is forced to drop out of the labor market as c > H0(x) for all x.25 Finally, a model with discounting and jobs which last a finite (random) amount of time is introduced so the searcher must plan to re-enter the market to seek future jobs. This model anticipates the soon-to-emerge work on quits and layoffs.

3.7 SEQUENTIAL ANALYSIS: A DEEPER MATHEMATICAL PERSPECTIVE In his recent appraisal of Herbert Robbins’s research in sequential analysis, Siegmund (2003) divides the review into four sections: (1) sequential allocation, (2) optimal stopping theory, (3) sequential estimation and (4) sequential hypothesis testing. We summarize his remarks in 3.7 and Section 3.8.

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3.7(a) Sequential allocation Siegmund states that Robbins was the first to conduct a sequential allocation and this occurred in Robbins (1952). The following problem is studied. The agent (experimenter) receives a sequence of rewards Χ1, Χ2, . . . by choosing at each time n = 1, 2, . . . to observe a random variable with distributions function F and mean µ or a random variable with distribution function G with mean γ ≠ µ. The decision to select F or G to generate Χn can be based on Χ1, . . ., Χn − 1, but once a distribution is selected Χn is produced independently of the previous Χi. The goal is to maximize the expected average reward. E [(Χ1 + . . . + Χn)/n]

(3.1)

The most famous version of this setup is the so-called bandit problem. For example, it can be shown that if one chooses two sparse but infinite sequences of integers, makes forced choices of F at time specified by one sequence and chooses G at specified times of the other sequence and at all other times selects the distribution with the better record in the past (larger expected reward), then in the limit as n → ∞, (3.1) converges to max(µ,γ). This illustrates a general conundrum in problems of sequential decision-making under uncertainty. The problem is the conflict between maximizing one’s immediate expected reward and collecting more information to use in the long run. These problems are known as “multi-armed bandit” (MAB) problems. As we will see, it is this ubiquitous problem in discounted decision-making which bestows extraordinary generality on the MAB analysis.26 Gittins and Jones (1974) and Gittins (1979) demonstrated that a discounted version of the allocation problem with product prior distributions was reducible to a parametric family of optimal stopping problems. Surprisingly, these optimal stopping problems do not entail relationships among the different “arms” and therefore are relatively simple to solve. To honor Gittins’s role in discovering this breakthrough, Whittle called the index associated with the optimal solution the Gittins index.27 Briefly put, an index is assigned to each bandit. The optimal policy selects the machine with the largest index, namely, the Gittins index. 3.8 OPTIMAL STOPPING THEORY28 Let us reflect on Wald’s contributions after first presenting a brief biography. Wald was born on October 31, 1902 in Cluj, Hungary. Wald was not permitted to attend school on Saturday, the Jewish Sabbath. As a consequence Wald was educated at home until he entered the local university. In 1927 he transferred to the University of Vienna and earned a PhD in Mathematics in 1931. Between 1931 and 1937 he published twenty-one articles in geometry. However, in spite of his brilliance and productivity he was unable to arrange any academic appointment because as a Jew he was a noncitizen. He did manage

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to support himself by tutoring the Viennese banker and economist, Karl Schlesinger. This experience created an interest in economics and econometrics evidenced by the publication of several articles and a monograph. It was through this contact with econometrics that Wald became interested in statistics. The Cowles Commission invited Wald to become a Fellow and he arrived at the University of Chicago in 1938. Weiss notes that this appointment saved Wald’s life as almost his entire family in Europe were consumed by the Holocaust. Unfortunately, Wald had only twelve years to make seminal contributions to statistics. In 1950, he and his wife were killed in an airplane crash in India. The year 1933 was pivotal for probability theory and statistics. Kolmogorov axiomitized probability and opened the door for subsequent contributions to stochastic processes and limit theorems. The Neyman–Pearson theory was also culminated in 1933 and led to the formalization of hypothesis testing, estimation, and decision theory.29 In “Sequential Tests of Statistical Hypotheses” (1945), Wald unified these contributions to statistics and probability. He presented a host of new results in random walks, martingales, stochastic processes, and limit theorems to probabilists. He also demonstrated to statisticians that statistical inference is not merely the “significance” of a body of data. It also requires that statisticians enter the experimental process and engage in a continuous analysis of the data as it unfolds. This is the essence of the sequential probability ratio test (SPRT). DeFinetti considered this test and all of Wald’s decision theoretic approach to statistical inference as “the most decisive involuntary contribution toward the erosion of the objectivist positions” in his fundamental statement that admissible decisions are Bayesian.30 The genesis of SPRT was described in Wallis (1980) and summarized in Ghosh (1992): in March 1943 Friedman and Wallis discussed the following problem. If x1, . . ., xn are independent observations with distribution f and a test of the hypothesis H0: f = f0 against H1: f = f1 is undertaken, the Neyman– Pearson theory gives a most powerful test when it accepts H0 when the likelihood ratio λN is small, λN ≤ c, or rejects H0 when λN > c. They made the following crucial observation. In some cases, the inequality λN ≤ c reduces ψ(x1) + . . . + ψ(xN) ≤ cN for some nonnegative function ψ. Then, in these circumstances the statistician can stop after observing m < n observations if ψ(x1) + . . . + ψ(xm) > cN and reject H0. This yields a saving equal to the cost of observing (xm + 1, . . ., xn) which in many wartime applications was quite large. Friedman and Wallis conjectured that in many cases it is more economical to observe x1, x2, . . . one by one and accept H0, reject H0, or continue sampling depending on the information in successive observations. The economy of the reduced average sample size relative to the size of the most powerful test would, they conjectured, offset any reduction in power. Wald’s initial reaction to this proposal was negative. However, after two days of contemplation, he confirmed the Friedman–Wallis conjecture and

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described the details of the theorem proving the optimality of the sequential probability ratio test. There are several important aspects of Wald’s contributions. First, Wald himself gives a Bayesian interpretation of his SPRT theorem.31 Second, the sequential probability ratio test is one of the first applications of dynamic programming which had not yet been developed by Bellman. In “Statistical Decision Functions” (1949), Wald unifies almost all of statistics by analyzing statistical problems as special cases of zero-sum, two-person games. Third, Weiss (1992) notes that Wald’s 1949 article converts statistical decision problems into zero-sum two-person in such a way that sequential sampling is allowable. That is, observations can be taken one at a time and after each observation the statistician can decide between stopping and continuing. This stopping rule was viewed by Wald as a game in extensive form. Fourth, Weiss observes: “Remarkably, Wald had constructed an optimal way for the statistician to play such a game with his sequential probability ratio test, without any reference to the theory of games.” Fifth, the key variable in SPRT is Sn − nµ, where n

Sn =

冱x and µ = E(Χ ), i = 1, . . ., n. i

i

i=1

This key variable is a martingale. And sixth, choosing which observations to be used in the selection of a terminal decision was an important feature of Wald’s formulation. Weiss notes that “this certainly gave a strong impetus to the study of optimal experimental design.” These design problems are the hallmark of multi-armed bandit problems. In concluding our summary of Wald’s contribution to search theory we see that he was the first to use dynamic programming. He also introduced optimal stopping rules, the essence of optimal search, into statistics. He realized that sequential decisions like search are fundamentally Bayesian. He also saw that optimal stopping and martingales were sisters. All of this was done in a zero-sum gaming framework, without any reference to von Neumann– Morgenstern game theory. Siegmund (2003) notes that optimal stopping theory is rooted in the study of the optimality properties of the sequential probability ratio test of Wald and Wolfowitz (1948) and Arrow et al. (1949).32 Both papers proceeded by converting the sequential problem into a formal Bayes problem, which we would now regard as an optimal stopping problem. A decision-maker observes an adapted sequence {Rn, Fn, n ≥ 1}, with E|Rn| < ∞ for all n. At each time n, a choice is made, to halt sampling and collect Rn, or continue sampling, hoping to receive a larger reward. An optimal stopping rule N is one that maximizes E(Rn). Siegmund notes that the key to discovering an optimal or near-optimal solution is the family of equations: Zn = max(Rn, E(Zn + 1|Fn) ), n = 1, 2

(3.2)

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Informally, Zn is the most one can expect to win given that stage n has been attained. Equation (3.2) states that these winnings are a maximum of what one can expect by terminating at stage n and what the expected winnings are if one more observation is taken and the procedure continues in optimal fashion. A reasonable candidate for an optimal rule is: stop when N equals min{n: Rn ≥ E [Zn + 1|Fn]}. Equation (3.2) reveals that {Zn, Fn} is a supermartingale, whereas {Zmin(N,n), Fn} is a martingale. When n is bounded, say 1 ≤ n ≤ m for some given value of m, the solution satisfies Zm = Rm. Thus (3.2) can be solved and the optimal stopping rule is obtained by “backward induction,” i.e., dynamic programming. After the analysis of Arrow et al. (1949), Wald’s problem of testing a simple hypothesis against a simple alternative was recognized by Snell (1952) as an abstract optimal stopping problem. In 1957, Bellman devised dynamic programming by applying the principles of (Equ. 3.2) to a wide class of sequential decision problems, stochastic and deterministic. This gave rise to Markov decision processes. In 1961, Chow and Robbins showed that the monotone case defines the only large class of optimal stopping rules that have explicit solutions. Siegmund also discusses the secretary problem as belonging to the class of optimal stopping rules. In his brief discussion of sequential estimation, Siegmund observes that sequential hypothesis testing “purports” to be more efficient than a corresponding fixed sample size. In estimation, sequential methods permit the statistician to obtain estimates of fixed precision. This is usually not possible for fixed-size samples. Optimal stopping in discrete time, as just presented, is, in principle, completely understood. However, it is almost always necessary to use numerical methods to compute an optimal rule. Siegmund notes that Blackwell was studying the foundations of dynamic programming. Blackwell (1962, 1965, 1967) and Dubins and Savage (1965) investigated the basis of gambling, while Chow and Robbins were constructing optimal stopping theory. These two approaches did not imitate optimal stopping theory and began with a fixed probability space with a given stochastic process defined on it. Consequently, an uncountably infinite number of actions may occur at each stage, thereby creating barriers to a rigorous analysis of an appropriately modified version of (Equ. 3.2). This failure has led some to speculate on the proper role of measure theory as the foundation of probability theory. From this brief review it is clear that dynamic programming, optimal stopping, multi-armed bandits, game theory, and Bayes analysis are closely linked. We now turn to a concise description of another important connected discipline: Markov decision processes.

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3.9 MARKOV DECISION PROCESSES Kallenberg (2002) regards Bellman (1957) as the beginning of Markov decision processes (MDP), although he notes that the Shapley (1953) paper on stochastic games contains the value iteration procedure for MDPs. However, this was not realized by its readers until after 1957. In and around 1960 other computational methods (policy iteration and linear programming) were designed by Howard (1960), DeGhellinck (1960), Manne (1960), and Blackwell (1962). Finite state Markov chains (MCs) are widely used probabilistic events. Avrachenkov et al. (2002) note that MCs embrace a “here and now” philosophy by means of the Markov property which stipulates that transitions from the current state to some subsequent state depend only on the current state and time. The MC is not endowed with the ability to model a process where a decision-maker by an adroit selection of actions can influence the motion of the process. Howard (1960) and Blackwell (1962) were the first ones to supply the missing endowment thereby generating MDPs. Since their inception MDPs have evolved rapidly so that today there is an almost complete existence theory and good algorithms for obtaining optimal policies with respect to discounted expected reward and other criteria. Many stochastic optimization problems can be characterized as MDPs. These include: optimal stopping, replacement problems, inventory problems, multi-armed bandit problems, and problems of finance. Kallenberg (2002) restricts his survey to discrete, finite, Markovian decision problems, that is, the state space Χ and the action site A(i), i ∈ Χ are finite and the decision times are equidistant. If at t the system occupies state i and action a ∈ A(i) is taken, the following occurs independently of the process history: 1 2

a reward r(i,a) is earned immediately; the process moves to state j ∈ Χ according to the transition probability p(j|i,a), where p( j|i,a) ≥ 0, and

冱 p( j|i,a) = 1 for all i, j, and a. j

The objective is to design a policy, i.e., a rule at every decision point which optimizes the performance of the system. Usually performance is measured by a utility function. 3.10 THE FOUR STELLAR EXAMPLES OF MDPs 1

Optimal stopping. In a stopping problem there are two actions in each state—the first is to stop while the second is to continue. If we continue in state i, a cost ci is incurred and the system moves to j with probability Pij. If a stopping decision is made in state i, the decision-maker receives ri

64

2 3 4

The Economics of Search and the process halts. An optimal policy chooses the best action for each state so as to maximize the total expected reward. This type of problem usually has its optimal policy characterized by a control limit policy. In a search setting the control limit policy is a reservation wage policy which, as we will show, is the Gittins index for a one-armed bandit. Inventory and replacement problems and maintenance and repair problems can be solved using MDPs Gambling problems are also amenable to this type of analysis. A nice discussion is Ross (1983b). Multi-armed bandit (MAB) problem. The MAB problem models the dynamic allocation of resources to one of n independent projects. At each period the decision-maker may exercise exactly one of the projects. An optimal policy chooses the project with the largest Gittins index. These indices can be calculated for each project separately. Thus, the MAB problem can be solved by a sequence of n-one armed-bandit problems.

Gittins and Jones in their seminal insight realized that a discounted version of the allocation problem with product prior distributions was reducible to a parametric family of optimal stopping problems which entail no relations among different arms and therefore are fairly easy to solve. Thus we see that the MAB which can be viewed as a MDP entails optimal stopping in a Bayesian setting facilitated by dynamic programming. The MAB addresses a fundamental problem in discounted sequential decision-making: the conflict or tension between maximizing one’s immediate expected reward and collecting more information having a long-run benefit. This tension is called a bandit problem, but of course it is a basic economic problem that penetrates all human decisions.

4

The basic sequential search model and its ramifications

Where the sequential method is applicable, it leads to remarkable savings because the test can be stopped as soon as sufficient evidence is available. But this is of no less interest from a theoretical point of view, because the decision whether to stop or proceed and how is based on the changing likelihood function of the experiment in progress. One seems almost compelled to return to the Bayes formulation since so many of its elements have reappeared spontaneously and of necessity. (Bruno deFinetti [on Wald] 1972: 194; emphasis added) The sequential decision process in economics is an idea with a powerful theoretical structure which, at the same time, provides a valid description of empirical decision-making. We believe that it can be the building block for dynamic models that marginalism has been for static models. (Cyert and DeGroot 1987: 188)

4.1 INTRODUCTION Looking back over almost fifty years there seems little doubt that search is the best way of treating the accumulation of information in a dynamic setting. It is also evident that a dynamic environment entails a process with a history where learning via search occurs, a present where actions are taken, and a future where the consequences are endured. This dynamism endows the search process with expediency. The time for action is almost always rapidly approaching and this urgency is reflected in the stopping rule which accompanies sequential search. Search requires an attentive mind adapting to the unfolding environment and always prepared to seize an optimal opportunity which tends to appear suddenly. The appearance of such an opportunity causes the searcher to stop searching and redirects his attention to the selected opportunity. Thus, optimal stopping is analogous to the phase transition of thermodynamics. There is a structural change in behavior when the stopping time is reached—the job-searcher becomes a worker.

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To summarize, dynamic implies that information is accumulated through a sequential search process; the actions must be taken before a certain time means that search must be formulated as a stopping rule, i.e., search is a sequential process which fits almost perfectly in the Wald–Bellman–Shapley framework of optimal stopping. It is almost obvious that the stopping decision should be in accordance with a Bayesian formulation. Indeed, the likelihood function is a key component of Wald’s stopping rule. For example, consider Wald’s sequential probability ratio test (SPRT). Let ᐉi be the likelihood under Hi, (i = 0,1) and πi the prior probability of Hi; the SPRT continues search (sampling) as long as A < ᐉ0/ᐉ1 < B, A and B constants, and stops as soon as the inequality is violated. To see its Bayesian content observe that the SPRT can be rewritten in terms of the posterior odds as A′ < π0ᐉ0/π1ᐉ1 < B′ < = > to stop as soon as your belief in one of the two hypotheses is sufficiently strong. Wald also proved that the only admissible solutions to a decision problem are Bayes and comprise a minimally complete class. The Bayes position with respect to stopping decisions is clearly articulated by deFinetti in the quote that begins this chapter. From our excursion into the history of statistics, we learned that search (information accumulation) in a dynamic economy belongs to the Wald stopping framework and is essentially a Bayesian endeavor. For our purposes, this means that the stopping rule framework is the appropriate setting for studying the manifold search problems that arise in economics and other disciplines. Following an important example by Veinott and a concise statement of Stigler’s model, this chapter presents a detailed study of the basic sequential search model (BSM) with a complete description of its assumptions. The important research on search over the past fifty years has been of two kinds. The first is the relaxation of the assumptions of the BSM so that it is better fit to address real problems. Some of these relaxations are fairly straightforward, while others are quite difficult. Discounting and moving from a discrete to a continuous model represent relatively easy transitions. On the other hand, risk aversion and the design of equilibrium search models have been very difficult. Relaxation of these assumptions provides a richer and more useful model. In most cases, the reservation wage structure is preserved, that is, stopping occurs if and only if the wage offer exceeds the reservation wage. If we think of today’s search model, where most of the assumptions of the simple model have been relaxed, as a “search cathedral” constructed over a fifty-year period, we immediately realize the large number of economists (over 1000) who have contributed to the modern search edifice. We call it a cathedral because it provokes wonder, and indeed astonishment, that such a simple beginning would culminate in such a beautiful structure. Almost every economist who has worked in the search arena has contributed an important idea that is now reflected in today’s cathedral. Alas, most of these contributions are absorbed, but not acknowledged. The range of application of BSM is extraordinary. Steve Lippman (in a private communication) has observed that the BSM

The basic sequential search model and its ramifications

67

is so wonderfully powerful analytically and conceptually flexible that it has been applied in arenas so diverse that use of the search paradigm need not include the search activity in a sampling sense. For example, information gathering prior to production a la Lode Li is a search activity. But the R&D patent race literature as well as Lippman and Rumelt’s uncertain imitability utilizes the paradigm in spite of the fact that there is no “information gathering” as such. Instead, the productive effort to bring about technological change are modeled as if it were search activity. Even introducing stochastic components to the learning curve concept can be viewed as utilizing the search paradigm. We can add several other sequential search applications. The first linkage is with auction theory, which we exposit in Chapter 13.1 A second area that benefits from search is the evolution of money and the recent search-theoretic exchange models by Kiyotaki, Wright, and others. We also discuss this in Chapter 13. The third and perhaps, from a mathematical perspective, the most important tie is the yoking of BSM and multi-armed bandits. This is presented in Chapter 9.2 We will see several of what we consider to be the most prominent manifestations of BSM, while many others will only receive silence. The subdisciplines in economics that have been most influenced by search include: labor economics, macroeconomics, econometrics, urban economics, migration, and the marriage market. These comprise the foundation of the search cathedral. The influence of search has also spread to other disciplines. Some of the most conspicuous are sociology, biology, neuroscience, psychology, and computer science (many simulations entail stopping rules). The search cathedral has accommodated many of these noneconomic applications of search, which are like stained-glass windows. The cathedral is always adapting its structure as new applications appear. Turning to the remaining contents of this chapter, we first study the elementary sequential search model and derive the reservations wage, which has the property that the searcher accepts an offer and terminates his search unemployment, if and only if an offer exceeds the reservation wage. Our simple model can be formulated as a Markov decision process, which means that the versatile dynamic programming can be enlisted. The assumptions of the simple sequential search model are listed in Table 4.1. One by one, we relax these assumptions. In this way, we are able to survey almost the entire thirty-year search effort on versions of the simple search rule policy. Of course, this research output is enormous and we will try to select those contributions which are insightful and innovative. Unfortunately, only some of these articles are covered in this volume.3 We first modify BSM by adding discounting and infinite horizons. The finite-time horizon model is then studied, when offers may be recalled and when such recall is impossible. We then consider the important case of on-the-job search where the worker has three alternatives: full-time work,

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The Economics of Search

Table 4.1 Assumptions of the BSM 1. 2. 3. 4. 5. 6. 7. 8. 9.

Infinite horizon (unlimited number of searches) Discrete time model No discounting The offer distribution F is known, nondegenerate, and invariant over the business cycle F is given exogenously Employment is an absorbing state; no quits or layoffs Utility function is risk neutral Cost of generating an offer c is constant over time Searcher retains highest offer, i.e., there is recall

BASIC INQUIRY: Can each of these assumptions be relaxed without destroying the reservation wage property of the simple model?

on-the-job search, and full-time search. The optimal policy is derived when these three options are present. We then study search in the presence of experience qualities—a version of belated information models, where features of the job are only revealed after employment commences. So far we have been applying a discrete version of BSM. It is easy to convert this model into a continuous time search model with variable intensity of search. We do this and then present a rigorous proof of the existence of a reservation wage using Lippman’s martingale argument. The continuous time search model has an immediate interpretation as a real option. We present a variant of Mortensen’s proof of this important property. The optimal replacement policy for stochastically failing equipment has a structure similar to the search model. This was first noticed by Breiman (1964) and we present an extended version of his observation using a basic theorem by Derman. Finally, the concluding section is we hope a “fair” comparison of BSM and Stigler’s fixed-sample rule. There has been some controversy over the claim that sequential search is preferable to fixed-sample search. In our first discussion of the Stigler model, we present a simple and rather naive comparison showing sequential to be the better of the two. Later in the chapter we present a fairly rigorous examination of this issue based on the seminal research of Manning, Morgan, and Schmitz. The dominance of the sequential approach is less apparent when there are fixed costs of search and we permit the fixed-sample model to take on some sequential features. The founders of sequential models had hoped that they would be applied to medical research. This has not occurred on the scale anticipated. We conclude the chapter with a brief appraisal of why this has been so.

The basic sequential search model and its ramifications

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4.2 AN IMPORTANT EXAMPLE Sequential decision-making is concerned with decision-making under uncertainty regarding the outcomes of available actions. The job-searcher must decide whether to accept or reject a job offer when rejection could lead to a long period of unemployment until another similar offer occurs and acceptance could entail the loss of a much higher job offer if only the searcher had waited a little longer. The dynamics aspect of sequential decision-making requires that actions be taken at different points of time. The specific actions taken can have a powerful impact on the individuals welfare. He/she may become entrenched in an undesirable job when he/she could have perhaps been a little more choosy and much more successful. Thus, regret and blame may accompany sequential decision-making. However, if the individual knows that he/she followed the best strategy, these poor outcomes could not have been avoided by a better choice of actions. There may be regret, but blame is banished and tomorrow is another day for decision-making. Veinott (1974) has an excellent example of a Markov decision process which is an optimal stopping problem. More specifically, it is a sequential Bayesian search decision. Example 4.1: Searching for sunken treasure: an optimal stopping problem Veinott presents an S-state Markov chain having transition probabilities pst, 1 ≤ s,t ≤ S. When the process is in state s two actions may be taken. The action σ stops the process and the decision-maker receives a reward of rs. The other action γ entails continuation of the process. The decision-maker pays an entrance fee cs and in return observes the process during the next period of time. The decision-maker wishes to design his/her stop and continuation actions so that he/she maximizes total expected net reward. As noted by Veinott, the central issue in these stopping problems is whether one should stop in a given state with a sure reward or should an entrance fee be paid to continue with the hope of stopping at a later time with a higher net return. Notationally, the Markov process can be characterized as follows: for each 1 ≤ s,t ≤ S, As = {σ,γ}, r(s,σ) = rs, r(s,γ) = −cs, p(t|s,σ) = 0, q(s,σ) = 1 − p(t|s,σ) = 1, p(t|s,γ) = pst, and q(s,γ) = 1 − p(t|s,γ) = qst. Veinott next spells out the conditions such that a myopic (one-step-lookahead [OSLA]) policy is optimal. If the stopping problem has the following two properties: (i) rs = 0, for all s and (ii) pst = 0 whenever cs ≥ 0 and ct < 0, the myopic policy (continue in states with negative entry fees and stop otherwise) is optimal. This is clear because it always pays to continue in states with negative entry fees which are equivalent to positive rewards, because one can always stop and receive a zero reward in the following period. Moreover, since the stopping reward is always zero, one continues in a state with nonnegative entry fees because there is a prospect of eventually hitting a state with a

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negative entry fee. This possibility has been abolished by (ii), so nothing is lost from stopping. Veinott examines the following stopping problem where (i) and (ii) are satisfied: consider how a salvage firm should search for sunken treasure with value r > 0 dollars. Suppose the firm has prior probability p that the treasure is in a specified region, 0 < p < 1. At a cost of c per dive, the firm can utilize the services of a diver. The conditional probability that a dive is unsuccessful given that the treasure occupies the specified region is 0 < q < 1. When should the firm quit in order to maximize the expected net reward from salvage? Suppose the first s dives are unsuccessful. Then the posterior probability that the treasure is in the region is, by Bayes rule, ps =

pqs , s = 0, 1, . . . pq + 1 − p s

Note that p0 = p, ps is decreasing in s, and ps ↓ 0 as s → ∞. If the (s + 1)th dive is successful, the posterior probability that the treasure is in the region is 1. If the dive is unsuccessful, the posterior probability that it is in the region ps + 1 will drop below ps. If the treasure is(is not) there and all dives are unsuccessful with probability zero(one) then the posterior probability the treasure is(is not) there converges to one as the number of dives goes to ∞. If we limit diving to at most S + 1 dives, the problem is how to optimally terminate an S + 1 state Markov chain with one stopped state. The states are the possible numbers 0,1, . . .,S of unsuccessful dives. Note that for each s,t = 0,1, . . ., S, one has rs = 0, cs = c − rps, ps,s + 1 = qps + 1 − ps, and qs = 1 − ps,s + 1 = (1 − q)ps, s < S, pst = 0 for s = S or t ≠ s + 1 ≤ S, and qs = 1. Moreover ps decreases in s, whereas cs is increasing in s. Hence, there exists a unique integer s*, 0 ≤ s* ≤ S + 1, such that cs < 0 for 0 ≤ s < s* and cs ≥ 0, for s* ≤ s ≤ S. The conditions (i) and (ii) are satisfied. Therefore, a myopic policy is optimal. After s unsuccessful dives, it is optimal to dive again if s < s* and to stop searching altogether if s ≥ s*.4 This example shows the close connections among MDPs and Bayes adaptive policies. It also reveals the special conditions giving a myopic policy and those conditions yielding a threshold (or control limit) policy.

4.3 THE FIXED-SAMPLE SIZE MODEL5 Historically, search theory was first applied to the problem faced by a consumer who tries to find the lowest price among several different sellers of an identical good. In fact, the earliest search theoretic model was created by George Stigler (1961). In his formulation, Stigler considered a consumer searching for the lowest price p for a certain good x. It is assumed that the vendors of good x are geographically dispersed and that the x sold by any one of them is identical to that sold by any other. The prices charged by different

The basic sequential search model and its ramifications

71

vendors for x, however, are not identical and are initially not known to the consumer. From his point of view, the price charged by any one seller is a random variable p with a known distribution F(p). Finding out what price any seller is in fact charging (searching that seller) is therefore equivalent to sampling once from this distribution. The consumer is assumed to face constant cost c per vendor searched. In the Stigler formulation, he must choose n, the number of sellers he will search, in advance of starting the search. His problem, then is to select n so that the expected total costs of purchase, inclusive of search costs, are minimized. Formally, his objective is to minimize ∞

冮

f(n) = [1 − F(p)n] dp + nc

(4.1)

0

where the first term on the right-hand side, which we denote by M(n), is the expected minimum price among the n vendors searched6 and nc is the total costs of searching n different vendors. It is clear that M(n) is a decreasing function of n. Let d(n) = f(n − 1) − f(n). Then ∞

冮

∞

d(n) = f(n − 1) − f(n) = [1 − F(p)

n−1

0

冮

] dp − [1 − F(p)n] dp 0

or ∞

冮

d(n) = [1 − F(p)n − 1]F(p)dp

(4.2)

0

Now d(n) is the expected decrease in the minimum price resulting from the nth search, and thus represents the marginal benefit of search. It is clear from equation (4.2) that d(n) > 0 for all n and that d(n) is decreasing in n and that lim d(n) = 0. Given that the consumer is selecting the sample size in advance, it n→∞

is clearly optimal for him to select n* such that d(n*) ≥ c > d(n* + 1). In other words, it pays to increase the sample size until the marginal benefit, d(n), is no longer at least as great as the constant marginal cost, c. It is clear from equation (4.2) that the optimal sample size, n*, is a function both of c and of F. Thus, if c increases, since d(n), for a nondegenerate F, is a strictly decreasing function, n* decreases. In fact, n* is a decreasing step function of c, holding F constant. While the Stigler model is quite descriptive of those few search processes in which the searcher is constrained to decide in advance the size of the sample to be searched,7 its usefulness in situations in which he/she is not so constrained has been questioned. The main reason for this is that most real-world

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search, including consumer search, is done sequentially. In other words, reports of prices charged by different vendors generally come in one at a time, separated by periods of nontrivial length. Between such reports, there is usually time for the shopper to decide whether to accept the best price discovered so far or to go on searching. A strategy which enabled him to take full advantage, not only of knowledge of F and c, but of the price information obtained in the actual search process as it comes in would likely be superior to one which does not allow such adaptation. Since the Stigler searcher must decide in advance of the beginning of search what his/her search strategy will be, this mode of search does not permit the consumer to adapt optimally to the new information. For example, suppose that on his/her first inquiry a consumer using the fixed-sample size strategy with n* > 1 discovered a seller offering the desired product at the minimum possible price, a value of p for which F(p) = 0. It will clearly not be optimal to search the rest of the predetermined sample, since the total cost of doing so will be at least c and the benefits are zero. Yet the fixed-sample size strategy requires him/her to do just that. Another deficiency of the fixed-sample size model is that it is not at all descriptive of situations in which the opportunity to buy the good at an offered price pi is non-recallable, i.e., when the consumer, after at first declining to buy the good for pi from vendor i, cannot after sampling more vendors be sure of coming back to i and purchasing at pi. Many real-world situations obviously involve non-recallable opportunities. The model, however, assumes the good will still be available at the best of the n* prices obtained when the search is over. As we shall see in the next section, the sequential search model does not suffer from either of these defects. In fact, as the example at the end of that section will illustrate, it may dominate the fixed-sample size model in the sense that the expected net benefits of following the optimal policy are sometimes greater with sequential search than with the Stigler model.

4.4 THE BASIC SEQUENTIAL SEARCH MODEL: INFINITE TIME HORIZON, NO DISCOUNTING8 Each and every day (until he/she accepts a job) the searcher ventures out to find a job, and each day he/she generates exactly one job offer (he/she is not allowed to vary the intensity of his/her search effort). The cost of generating each offer (which includes all out of pocket expenditures such as advertising and transportation that are incurred each time a job offer is obtained) is a constant cost c, and there is no limit to the amount of offers that the searcher can obtain. In this simple model, when an offer is accepted, the searcher transits to the permanent state of employment (quits and layoffs are not permitted). Whereas the searcher’s skills are unvarying, prospective employers do not necessarily evaluate or value them equally. The induced dispersion of offers is incorporated into the model by assuming that there is a cumulative

The basic sequential search model and its ramifications

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distribution function F of wages which governs the offers tendered. The distribution is invariant over time. Thus on any given day the probability that the searcher will receive an offer of w or less is F(w), independent of all past offers and the time the offer is made. In this simple setting the offer can be interpreted as the (discounted present value of the) lifetime earnings from a job. All participants are assumed to be risk neutral and seek to maximize their expected net benefits. The only decision the searcher must make is when to stop searching and accept an offer. Under the foregoing set of assumptions,9 it will be demonstrated that the optimal policy for the searcher is to reject all offers below a single critical number, termed the reservation wage, and to accept any offer above this critical number. To be more precise, a job offer, Χi, is presented each period, where each Χi is a random variable with cumulative distribution function F(·), E(Χi) < ∞, and the Χi’s are mutually independent. The job searcher is assumed to retain the highest job offer so that the return from stopping after the nth search is given by Yn = max(Χ1,Χ2, . . .,Χn) − nc The objective is to find a stopping rule that maximizes E(YN) where N is the random stopping time, i.e., the random number of job offers until one of them is accepted. Let ξ denote the expected return from following the best stopping rule. Now suppose that the searcher receives a wage w. If the searcher rejects the offer and continues then since tomorrow is the same as today except for the already incurred cost c, the searcher following the optimal stopping rule from tomorrow forward will expect a return of ξ. Continuing then yields a return of ξ. Thus, a searcher should stop and take x only if x ≥ ξ. The critical number ξ is called the reservation wage and any search policy with this form is said to possess the reservation wage property. This simple argument shows that the optimal expected return from the optimal stopping rule satisfies: ξ = Emax(ξ,Χ1) − c

(4.3)

Now Emax(ξ,Χ1) can be rewritten as: ξ

∞

冮

冮

Emax(ξ,Χ1) = ξ dF(w) + xdF(x) ξ

0 ξ

∞

冮

冮

∞

冮

∞

冮

= ξ dF(x) + ξ dF(x) + xdF(x) − ξ dF(x)ξ ξ

0

ξ

冮

= ξ + (x − ξ)dF(x). 0

ξ

ξ

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The Economics of Search

So (4.3) can be rewritten as ∞

冮

c = (x − ξ)dF(x) = H(ξ).

(4.4)

ξ

where ∞

冮

H(x) = (y − x)dF(y). x

Note that the function H is convex, nonnegative, strictly decreasing with lim H(x) = 0, x→∞

lim H(x) = E(Χ1).

x → −∞

To show that H is decreasing: ∞

∞

冮

冮

dH(x)/dx = (y − x)dF(y)/dx = −(x − x)dF(x) − dF(y) = −(1 − F(x)).10 x

x

Convexity follows from: d 2H(x)/dx2 = d(−(1 − F(x))/dx = f(x) > 0. Thus we can graph (4.4) as in Figure 4.1. Equation (4.4) has a simple economic interpretation: the critical value ξ associated with the optimal stopping rule is chosen to equate the marginal cost of obtaining one more job offer with H(ξ), the expected marginal return from one more observation. It is useful to observe that the random variable, N, the number of offers needed until ξ is exceeded, has a geometric distribution with parameter λ = 1 − F(ξ) and E(N) = 1/ λ.

(4.5)

4.5 AN ALTERNATIVE VIEW OF THE RESERVATION WAGE Let g be the expected gain from following the optimal policy in (4.3). Then, assuming (1 − F(ξ)) > 0, g satisfies ∞

冮

c g=− + xdF(ξ)/(1 − F(ξ)) 1 − F(ξ) ξ

(4.6)

The first term on the right-hand side (RHS) reflects the fact that the number

The basic sequential search model and its ramifications

75

Figure 4.1 Graphical representation of reservation wage determination

of observations required to find an offer of at least ξ is a geometric random variable with parameter (1 − F(ξ)); the second term on the RHS is the conditional expected value of an offer given that it is at least ξ. Rearranging equation (4.6) yields ∞

冮

∞

冮

c = (x − g)dF(x) = (x − ξ)dF(x). ξ

ξ

But from (4.4), c = H(ξ). Thus it follows that (g − ξ)(1 − F(ξ)) = 0 and g = ξ (assuming (1 − F(ξ)) > 0). 4.6 INFINITE TIME HORIZON: DISCOUNTING We now alter the elementary search model by the explicit introduction of discounting. In this version it is appropriate to interpret the lump-sum earnings of the job as the discounted present value of all future wages. We analyze the case where the search cost is incurred at the beginning of the period. Let β

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denote the discount factor with β ≡ 1/(1 + r) and r is the appropriate rate of interest. In this case the reservation wage satisfies ξ = −c + βEmax(ξ,Χ1) and c = βH(ξ) − ξ(1 − β).

(4.7)

Since the right-hand side is decreasing in ξ, an increase in r that decreases β results in a decrease in the reservation wage. 4.7 INFINITE TIME HORIZON WITH DISCOUNTING AND A RANDOM NUMBER OF JOB OFFERS The structure of the optimal policy remains the same when the number of offers received per period is not always one but rather a random variable. We will consider the case where the number of offers received is at most one. Let q be the probability of receiving an offer in any period, (1 − q) is the probability of receiving no offer in a given period, and assume that the search cost is incurred at the beginning of the period. Invoking the same argument as above the basic functional equation becomes τ

ξ = −E

冢冱cβ 冣 + Eβ max(ξ,Χ ) k−1

τ

1

k=0

where τ is a geometric random variable with parameter q. Now, ∞

τ

E(β ) = β

冱[β(1 − q)]

k−1

q = qβ/ [1 − β(1 − q)] = q/[q + r]

k=1

and E(c + cβ + cβ2 + . . . + cβτ − 1) = c[1 − E(βτ)]/(1 − β) = c(1 + r)/(q + r). So, ξ = [q/(q + r)][ξ + H(ξ)] − [(1 + r)/(q + r)]c or c = [q/(1 + r)]H(ξ) − [r/(1 + r)]ξ.

(4.8)

Since H is a decreasing function, we get the intuitive result that the reservation wage declines as q decreases.

The basic sequential search model and its ramifications

77

4.8 RISKY ORDERING AND SEQUENTIAL SEARCH Let Χ and Y be random variables with c.d.f.s F and G. Theorem 4.1 A necessary and sufficient condition for Χ to be stochastically larger than Y is Eu(Χ) ≥ Eu(Y) for all u ∈ L1, where L1 is the class of nondecreasing functions. This is a partial order: a partial order ≤p on a set is a binary, transitive, reflexive, antisymmetric relation. [If A ≤p B and B ≤p A, A = B by definition of anti-symmetric.] Here the partial order is defined over set of c.d.f.s. Definition We say that the random variable Χ is stochastically larger than the random variable Y, written Χ >1 Y, if and only if G(t) − F(t) ≥ 0 for all t. When this is true we say that Χ dominates Y according to the criterion of the first order stochastic dominance (FSD). Definition We say that the r.v. Χ is less risky than the r.v. Y written Χ >2 Y if and only if t

冮 (G(x) − F(x))ds ≥ 0, for all t.

(4.9)

−∞

When F and G satisfy (4.9) we say that Χ dominates Y (or F dominates G) in the sense of second order stochastic dominance (SSD). If Χ >1 Y, then the integrand in (4.9) is nonnegative when the integral itself is always nonnegative. Thus Χ >1 Y implies Χ >2 Y. Hence SSD is a weaker requirement than FSD. Theorem 4.2 than Y is

(a) A necessary and sufficient condition for Χ to be less risky

Eu(Χ) ≥ Eu(Y), all u ∈ L2 where L2 is the set of nondecreasing concave functions. (b) Moreover, if Χ >2 Y and E(Χ) = E(Y), then Eu(Χ) ≥ Eu(Y) for all concave functions. One consequence of Theorem 4.1 is the fact that the means of Χ and Y have the same ordering as Χ and Y. That is, if Χ >1 Y, then EX ≥ EY. This

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The Economics of Search

follows by considering the nondecreasing function u(t) = t. Thus FSD is so strong a condition as to preclude comparison (ordering) of random variables with the same mean. ∞

冮

Recall that for Χ nonnegative, E(Χ) = (1 − F(t))dt. Thus 0

∞

冮

E(Χ) − E(Y) = (G(t) − F(t))dt. 0

It follows from the definition of SSD that Χ >2 Y also implies E(Χ) ≥ E(Y). If Χ >2 Y and E(Χ) = E(Y) then Var(Χ) ≤ Var(Y). Take u(t) = −t2 so that u is concave. Observing that Eu(Χ) = −E(Χ 2), it follows from EX = EY and Theorem 4.2, that Var(Y) − Var(Χ) = E(Χ 2) − E(Y 2) ≥ 0. Let Χ and Z be nonnegative random variables with cumulative distribution functions F and G, respectively. Suppose Z(G) is riskier than Χ(F) in the sense of second order stochastic dominance and E(Χ) = E(Z). Query

Does Z have a larger reservation wage than Χ ?

Let HF and HG be the H-functions associated with F and G (with ξF and ξG the corresponding reservation wages). Consider the convex increasing function uY(x) defined by uY(x) =

冦0,x − y,

x≤y x > y.

If Χ >2 Z and E(Χ) = E(Z), then Eu(Χ) ≤ Eu(Z), all convex u. Thus, ∞

冮

HG(y) = (z − y)dG(z) = EuY (z) ≥ EuY (x) = HF (y), all y ≥ 0. y

Hence, HG(y) lies above HF (y), as depicted in Figure 4.2, and so ξF < ξG. Ljungqvist and Sargent (2004b: 90) have an interesting option pricing explanation of the preference for the risky alternatives by an unemployed worker. In option pricing theory, the value of an option is an increasing function of the variance in the price

The basic sequential search model and its ramifications

79

Figure 4.2 Change in reservation wage with an increase in the risk of distribution

of the underlying asset. This is so because the option holder receives payoffs only from the tail of the distribution. In our context, the unemployed worker has the option to accept a job with asset value w/(1 − β). Given a mean preserving increase in risk, the higher incidence of very good wage offers increases the value of searching for a job while the higher incidence of very bad wage offers is less detrimental because the option to work will in any case not be exercised at such low wages. Recall ξ satisfies: ∞

c=

冮 (x − ξ)dF(x). ξ

The decreasing nature of HF (when coupled with HF (ξF) = c = HG(ξG) ≥ HF (ξG)) implies ξF ≤ ξG. The “riskier” distribution is preferred!

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The Economics of Search

4.8.1

The price elasticity of search

Information is a purchasable commodity and as such has an associated demand curve. Thus, in any situation in which information is in fact purchased, one can properly inquire whether or not the demand for information is elastic. As we have seen, search activities can be regarded as the purchase of information about the agent’s economic opportunities. In the context of the basic sequential search model without discounting, we therefore inquire in this section whether or not the searcher’s expected total expenditures11 on search increase with the price c of obtaining each observation. Recalling that an elastic demand is characterized by falling total expenditures as the price of the good increases, and that the opposite is true for an inelastic demand, an affirmative answer to this question will show that the demand for information in the context of this model is inelastic, and a negative answer will indicate an elastic demand. From equation (4.5), the expected total cost of search, E(TC) is given by E(TC) =

c . 1 − F(ξ)

(4.10)

Where c is the per period cost of search and ξ is the reservation wage, we know that the reservation wage decreases as c increases. Consequently, it can be shown that the amount of search, T, decreases stochastically with c. Thus E(T), the expected number of observations, decreases as c increases.12 But does c increase faster than E(T) falls? The following theorem reveals that the answer to this question depends on the hazard rate function of the offer distribution F.13 Search is inelastic if F has an increasing hazard rate function (IHR) and elastic if F has a decreasing hazard rate function (DHR). Recall that with an IHR, the hazard function λ(x) is increasing in x, and that the reverse is true for DHR distributions. Theorem 4.3 If F is IFR, then E(TC) is an increasing function of c and the demand for search in the undiscounted basic sequential model is inelastic. If F is DHR, then E(TC) is a decreasing function of c and the demand for search is elastic. Proof Differentiating the first order condition of the model (equation [4.4]) with respect to c yields 1 = −[1 − F(ξ)]dξ/dc. Using this fact and the fact that ∞

冮

¯¯ (x) = [1 − F(y)]dy, H xi

¯¯ (x) is the function defined in equation (4.4), and denoting the hazwhere H ard function at x by λ(x), the density function of F by f, and E(TC) by E(c), we have

The basic sequential search model and its ramifications

81

[1 − F(ξ)]2 E′(c) = [1 − F(ξ) + cf(ξ)dξ/dc] = [1 − F(ξ) − cλ(ξ)] = [1 − F(ξ) − λ(ξ)H(ξ)] using equation (4.4) =

∞

∞

ξ

ξ

冮 f(x)dx − λ(ξ)冮[1 − F(x)]dx ∞

=

ξ

∞

=

1 − F(x) f(x)

冮 冤1 − λ(ξ)冤

冥冥 f(x)dx

λ(ξ)

冮 冤1 − 冤λ(x)冥冥 f(x)dx ξ

which is greater than zero if λ is increasing and less than zero if λ is decreasing. 4.9 THE BASIC SEQUENTIAL MODEL: SEARCH FOR THE LOWEST PRICE At this point in our study of search, it may be useful to consider the searcher’s strategy as she looks for the lowest price of a particular consumption good. Assume there is no recall of past opportunities.

4.9.1

The sequential model

The agent searches by obtaining information on prices charged by different vendors in a one-at-a-time fashion. The probability distribution F of prices is known but initially he/she is ignorant of the price charged by any particular seller. Searching different sellers then, is tantamount to selecting independent draws from F. Each time such a draw is taken, the consumer is assumed to incur a cost of c. This aspect of the model captures the important fact that general knowledge (such as the probability distribution of prices alluded to here) being true for relatively large expanses of time and/or space, is relatively inexpensive to generate and disseminate, because of the economics of scale involved. These economies are usually not present in the case of more particular knowledge (such as the knowledge of the individual prices). The result is that for any given economic agent, general knowledge is often very cheap to acquire relative to particular knowledge. Thus in our basic model we neglect the cost of obtaining the general knowledge which is the distribution of prices. This knowledge can be thought of as corresponding to what in everyday parlance is sometimes referred to as knowing “about what it (the good) costs,” information which is often gleaned from casual conversation with

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The Economics of Search

friends or relatives, watching television, or other similar activities which involve a zero or near zero cost, however measured. Let V(p) be the expected price paid plus the expected total search costs from this point forward when the agent has the chance to buy the good at a price of p and assuming he follows an optimal policy now and in the future. Applying the techniques of dynamic programming,14 Bellman’s equation is ∞

冮

冤

冥

V(p) = min p,c + V(q)dF(q) . 0

(4.11)

Since the graph of the first term in brackets in equation (4.11) is just the 45 degree line in the positive quadrant and the graph of the second is that of a positive constant, the graph of V(p) must have a shape such that there is a ∞

冮

critical number p¯ , where p¯ = c + V(q)dF(q) and 0

V(p) = p for p ≤ p¯ ,

(4.12a)

∞

冮

V(p) = p¯ = c + V(q)dF(q) for p ≥ p¯ .

(4.12b)

0

The optimal strategy is to purchase if and only if search reveals a price less than or equal to p¯ . Search continues so long as a price of p¯ or less has not yet been encountered. Such a strategy is said to have the “reservation price property.” Equation (4.12b) says that the reservation price, p¯ , is equal to the expected total cost of not purchasing the good at the price at which it is currently offered, paying c, and continuing search in an optimal fashion starting with the next vendor. The equation also states that when the currently available price is higher than the expected cost of continuing search, one should continue. Equation (4.12a) says that when the good is currently available at a price less than the expected total future costs of an optimal search strategy from this point on one should purchase now and discontinue search. It is important to note that in these comparisons, only prospective costs (those incurred from the present onward) are considered. Sunk costs, whatever they may be, are irrelevant and may be ignored. Observe that p¯ is also, at any point in time before a price of p¯ or lower has been encountered, the expected total cost of purchasing the good, i.e., the expected minimum price plus expected total search costs. Specifically, p¯ is the expected total cost of purchase before search has started. Rewriting equation (4.12b) we have p¯

冮

∞

冮

p¯ = c + dF(q) + p¯ dF(q) 0

p¯

(4.13a)

The basic sequential search model and its ramifications

83

or ∞

p¯

冮

冮

p¯

冮

∞

冮

p¯ dF(q) + p¯ dF(q) = c + qdF(q) + p¯ dF(q). 0

p¯

0

(4.13b)

0

Rearranging equation (4.13b) we obtain p¯

冮(p¯ − q)dF(q) = c.

(4.14)

0

The left-hand side of equation (4.14) is the expected reduction in the price at which the good will be available if one searches one more vendor, given that it is now available at a price of p¯ . In other words, it is the expected gain of search one more seller when one has an offer of p¯ in hand. The right-hand side of (4.14), of course, is the cost of searching one more seller. Equation (4.14) says that the p¯ is that price at which the expected gain of taking one more draw from F is just equal to the cost of doing so. Since the function x

冮

h(x) = (x − y)dF(y)

(4.15)

0

is increasing in x, it is evident that equation (4.15) implies that for any price lower than p¯ , the expected benefits of searching another vendor are less than c, whereas the opposite is true for any price in excess of p¯ . One way of interpreting (4.15) is that when an optimal search strategy is followed, the agent always chooses the same action as he/she would if he/she were only allowed to search one more vendor. In other words, the strategy which is best in the very short run is also the best long-run strategy. Thus the optimal search strategy is often called “myopic,” since in order to pursue it one only needs to compare paying the currently offered price with the expected cost of searching one more seller and paying whatever he asks. Once again it is important to distinguish between the reservation price property and the myopic property: the former tells us which price offers are acceptable, specifically those less than or equal to p¯ , whereas the latter provides us with a simple method for calculating p¯ . Now if we use the integration by parts formula, 冮udv = uv − 冮vdu, to integrate the left-hand side of equation (4.14) with u = (p¯ − q) and dv = dF(q), we obtain p¯

冮F(q)dq = c. 0

(4.16)

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The Economics of Search

If we define the function g(p) as p

冮

g(p) = F(q)dq

(4.17)

0

we see that g(p) is the expected gain from searching one more seller when one has an offer of p in hand. It is obvious from inspection that g(0) = 0, and that for all p ≥ 0: g(p) ≥ 0, g′(p) = f(p) > 0, and g″(p) ≥ 0. Hence g(p) is a nonnegative, increasing, convex function of p. Consider next the case where there is recall of past opportunities. In other words, at any point the agent is free to continue searching or pay the lowest offer received so far. Let Vr(p) be the expected price paid plus the total expected search costs when the lowest offer received so far is p and an optimal search strategy is followed form this point on. Since no matter how many more vendors are searched, the agent will always have the option to go back to the current offer and pay p, once an offer of p has been received; at any point thereafter the agent will never face expected total future costs higher than Vr(p). Thus the Bellman equation is ∞

冮

冤

p

冮

冥

Vr(p) = min p,c + Vr(p) dF(q) + Vr(q)dF(q) . p

0

(4.18)

Since in the no-recall case, the agent’s optimal strategy, taking the first offer less than or equal to p¯ , was such that in pursuing it, he/she seemed not at all constrained by his/her inability to recall past offers, we initially try to solve equation (4.18) by conjecturing15 that its solution has the same form as that of equation (4.11). Thus our candidate solution to (4.18) is Vr(p) = p for p ≤ p¯ ,

(4.19a)

Vr(p) = p¯ for p ≥ p¯

(4.19b)

where ∞

冮

p¯

冮

p¯ = c + p¯ dF(q) + qdF(q). p¯

(4.19c)

0

We shall now show that Vr(p), as given in equation (4.19) is the unique16 solution of equation (4.18). First, we verify that it is in fact a solution. To do this we simply insert Vr(p) as defined by equations (4.19a) and (4.19b) in equation (4.18) and see the latter equation is in fact satisfied. For p ≥ p¯ , substituting equation (4.19b) in (4.18), we have

The basic sequential search model and its ramifications ∞

冤

85

p¯

冮

冮

冥

Vr(p) = min p,c + p¯ dF(q) + qdF(q) 0

p¯

Substituting from equation (4.19b) for the second term in the brackets we obtain Vr(p) = min[p, p¯ ] = p¯ which verifies our conjecture for values of p ≥ p ¯ . Now for p ≤ p¯ , substituting equation (4.19a) in equation (4.18) we obtain ∞

冤

p

冮

冮

冥

V(p) = min p,c + p dF(q) + qdF(q) . 0

p

(4.20)

If we define ∞

冮

p

冮

k(p) = c + p dF(q) + qdF(q), p

0

we notice that k(p) is the second term in brackets in equation (4.20). Also observe that k(p¯ ) = p¯ . Thus if we can prove that k(p) > p for p ≤ p¯ , we will have Vr(p) = p for p ≤ p¯ , and thus the Vr(p) defined by equations (4.19a) and (4.19b) satisfies (4.18) for all values of p. Now ∞

冮

k′(p) = dF(q) < 1. p

This observation, along with the fact that k(p¯ ) = p¯ , proves that for all values of p ≤ p¯ we have k(p) > p. Thus, our guess is a solution of equation (4.18). It is well known that functional equations of the form of (4.18) have unique solutions. Thus we have proved that the optimal strategy, in the case of consumer search with recall has the reservation price property. In this simple case, the recall option is worthless. Furthermore, because the recall option is never utilized the agent can pursue that strategy corresponding to the no recall problem. Since every strategy available in the no-recall case is also available in the recall case, the optimal strategies must be identical in the two cases. Thus the two reservation prices must be the same.

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The Economics of Search

4.9.2 Comparative statics for the basic lowest price search model Since the comparative statics results are the same in the no-recall and recall versions of the basic sequential model, we consider only the no-recall case. It is evident from equations (4.13a) and (4.15) that the reservation price p¯ depends on c and F. Since the function defined in equation (4.15) is increasing in x, it is clear from equation (4.14) that as c increases, p¯ also increases. The economic intuition is that as the cost of sampling from F goes up, we expect that the agent will change his strategy such that the expected number of vendors searched goes down. Since N, the number of vendors sampled, is a geometric random variable with parameter F(p¯ ), we have E(N) = 1/F(p¯ ), an increase in p¯ reduces E(N). Let us examine the effect on p¯ of changes in F. Suppose F and G are two distributions such that F is stochastically larger than G. We denote this fact by F ≥ G. st

Let p¯ G be the reservation price if the price distribution is G and let p¯ F be the reservation price if the distribution is F. We assume c, the cost of searching a single seller, is the same in both cases. Now by the fact that F is stochastically larger than G, if f is a decreasing function of x we have EF[ f(x)] = 冮 f(x)dF(x) ≤ 冮 f(x)dG(x) = EG[f(x)]. But equation (4.14) can be rewritten ∞

冮h(q)dF(q) = E [h(q)] = c F

(4.21)

0

where h(q) = p¯ − q for q ≤ p¯ h(q) = 0 for q > p¯ . It is obvious that h is a decreasing function of q. Thus we have p¯ F

冮(p¯ 0

p¯ F F

冮

− q)dF(q) ≤ (p¯ F − q)dG(q).

(4.22)

0

Using this, equation (4.14), and the fact that the function h(x) defined in equation (4.15) is increasing in x and that c is the same in both cases, we

The basic sequential search model and its ramifications

87

conclude that p¯ F ≥ p¯ G. Thus when the new distribution stochastically dominates the old, the reservation price increases, other things being equal. Next we consider the effect of a mean preserving spread in F. Let F¯¯ be a mean preserving spread of F. Thus for any price p p

p

冮F¯¯(q)dq ≥ 冮F(q)dq. 0

(4.23)

0

Let p¯ F¯¯ be the reservation price when the distribution is F¯¯ and p¯ F be the reservation price when it is F. Again, c is assumed to be the same in both cases. Equation (4.15) then implies that p¯ F¯¯ ≤ pF. Thus a mean preserving spread in the price distribution, other things being equal, lowers the reservation price. The economic intuition behind this result is as follows. A mean preserving spread can be thought of as pushing probability toward the tails of the distribution while keeping the mean constant. In this sense, the distribution is made more “risky.” Since there is now more likelihood of finding exceptionally low prices at some locations, it pays the shopper to hold out for these bargains. Thus, he/she lowers his/her reservation price. Notice that the shopper is not harmed by the “upper tail” since he/she never buys when the price is in that region.

4.10 Optimal stopping We present the optimal stopping problem as a special case of a Markov decision process. We follow the classic description by Ross (1983a).17 The decision-maker begins in state i and may choose to stop in which case he receives R(i) or he may continue by making a payment of C(i). If he continues the next state is j with probability Pij. Ross assumes that both C(i) and R(i) are nonnegative. Let us consider a stopping decision that takes the process to state ∞. Then we have a two-action decision process with action 1 representing the stopping action and with C(i,1) = −R(i),

Pij(2) = Pij

Pi,∞(1) = 1,

P∞,∞(a) = 1

C(i,2) = C(i),

C(∞,a) = 0.

The following two assumptions ensure that the process belongs to the class of negative dynamic programming: 1

inf C(i) > 0

2

sup R(i) < ∞.

i i

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The Economics of Search

Let R = sup R(i) and ponder the related problem in which, when in state i the i

decision-maker must stop and receive a final reward R(i) − R or pay a cost C(i) which permits the process to continue. In the latter circumstance, the process transits from i to j with probability Pij, j ≥ 0. ¯¯π represent the expected costs for Let π represent any policy. Then Vπ and V the original and related problems, respectively. It follows that for a policy stopping with probability 1, ¯¯π(i) = Vπ(i) + R, i = 0, 1, 2, . . . V

(4.24)

By assumption (1), any policy that doesn’t stop with probability 1 has ¯¯π(i) = ∞. Thus, any optimal policy for the original process is also Vπ(i) = V optimal for the related process, and vice versa. Now the related process is a Markov decision process with nonnegative ¯¯ satisfies costs. Hence, the optimal cost function V ¯¯(i) = min{R − R(i),C(i) + ΣPijV ¯¯( j)}, i ≥ 0. V Similarly, the policy choosing the minimizing actions is optimal. This ¯¯(i) − R so that follows by V(i) = V V(i) = min{−R(i),C(i) + ΣPijV( j)}, and such a policy is optimal. Let Vo(i) = − R(i), and for n > 0 let Vn(i) = min{−R(i),C(i) + ΣjPijVn − 1(u)}. The minimal expected total cost incurred if we begin in state i and are permitted a total of n stages before stopping is represented by Vn(i). Thus, Vn(i) ≥ Vn + 1 (i) ≥ V(i) and, consequently, lim Vn(i) ≥ V(i).

n→∞

The process is called stable if lim Vn(i) = V(i).

n→∞

Ross states the following proposition showing that conditions (1) and (2) insure stability. Proposition Under conditions (1) and (2) with R = sup R(i) and C = inf C(i), then i

i

The basic sequential search model and its ramifications Vn(i) − V(i) ≤ Proof

89

(R − C)[R − R(i)] ∀n and ∀i. (n + 1)C

See Ross (1983a: 53).

4.10(a) Not all stopping problems are stable Let the integers comprise the state space and let C(i) ≡ 0 R(i) = i Pi,i + 1 = 1/2 = Pi,i − 1. Then Vo(i) = −i and by induction Vn(i) = −i. Until stopping occurs the state moves according to a symmetric random walk. Such a Markov chain is null recurrent. Hence with probability 1 any state N is eventually hit. Thus V(i) = ∞ and neither (1) or (2) is satisfied. This stopping problem is not stable. Ross now shows the conditions under which the one-stage-look-ahead (or myopic) policy is optimal. Let ∞

B = {i: −R(i) ≤ C(i) −

冱P

ij

R( j)},

j=0

∞

= {i: R(i) ≥

冱P

ij

R( j) − C(i)}.

j=0

The set B represents those states such that stopping is at least as good as continuing for exactly one more period and then stopping. A policy that terminates upon entry into B is called a myopic policy. If B is a closed set of states and the process is stable, the myopic policy is optimal. For proof see Ross (1983a: 54). From the discussion thus far, it is clear that dynamic programming and Markov decision processes are inextricably joined. In their introduction to the Handbook of Markov Decision Processes, Feinberg and Shwartz (2002) begin their overview by noting that: The theory of Markov Decision Processes—also known under several other names including sequential stochastic optimization, discrete-time stochastic control, and stochastic dynamic programming—studies sequential optimization of discrete time stochastic systems. The basic object is a discrete-time stochastic system whose transition mechanism can be controlled over time. . . . The goal is to select a “good” control policy.

90

The Economics of Search In real life, decisions that humans and computers make on all levels usually have two types of impacts: (i) they cost or save time, money, or other resources or they bring revenues, as well as (ii) they have an impact on the future, by influencing the dynamics. In many situations, decisions with the largest immediate profits may not be as good in view of future events. MDPs model this paradigm and provide results on the structure and existence of good policies and on methods for their calculation.

Kallenberg (2002) begins his articles with the following statement: “Bellman’s book (Dynamic Programming), can be considered as the starting point of Markov decision processes.” From the comments of four major contributors to MDPs, Ross, Feinberg, Shwartz, and Kallenberg it is clear that dynamic programming has been crucial to the development of Markov decision processes. Furthermore, a premier application of MDPs is to optimal stopping problems; and as we saw in Ross’s outline of optimal stopping, a myopic policy is a special case of MDPs with great practical significance. 4.11 FINITE TIME HORIZON In the infinite time horizon models considered so far, the reservation wage was invariant over time, so that if an offer is refused once, we will continue to refuse it for all time. Thus, the search with recall and search without recall cases are identical since the recall option is never exercised. One reason why the reservation wage may fluctuate over time is that the time horizon is finite. Suppose that the searcher envisages n periods during which he/she might search so n opportunities unfold sequentially. The process stops when one of the opportunities is chosen. We wish to choose the opportunity yielding the highest net benefit. That is, we want a stopping rule such that the expected gain from following it is a maximum. In calculating the optimal stopping rule, let Vn(x) be the maximal net benefit attainable when sampling without recall (i.e., offers not immediately accepted are lost), n periods remain (i.e., we have yet to observe n offers) and we currently have an offer x available. Then V0(x) ≡ x and for n > 0: ∞

冦

冮

冧

Vn(x) = max x,−c + Vn − 1(y)dF(y) . 0

(4.25)

When sampling with recall define x as the maximum offer received thus far and define V*n(x) as the maximal return function with n periods remaining. Then V*0(x) = x and for n > 0: ∞

* n

冦

V (x) = max x,−c + V

* n−1

冮

冧

(x)F(x) + Vn −* 1(y)dF(y) . x

The basic sequential search model and its ramifications

91

4.11.1 The no recall case First, consider the case of sampling without recall. We might expect there to be a minimally acceptable offer (reservation wage) when n periods remain; label this quantity ξn. Thus, with n job offers remaining to be received we stop if the current offer is at least ξn and continue otherwise. From (4.25) it is clear that ∞

冮

ξn = −c + Vn − 1(y)dF(y) 0

for n > 0 and ξ0 = 0. An induction argument establishes that for each n, ξn ≥ ξn − 1. Assuming that −c + E(Χ0) > 0 then ξ1 ≥ ξ0 = 0. Assume that ξn − 1 ≥ ξn − 2. Then, ξn = −c + E(max(x,ξn − 1)) ≥ −c + E(max(x,ξn − 2)) = ξn − 1. Q.E.D.

4.11.2 The recall case The possibility of selecting a previously passed over offer adds considerable complexity to the problem. Define ∞

zn(x) = −c + V

* n−1

冮

(x)F(x) + V *n − 1(y)dF(y). x

Thus, V *n(x) = max{x,zn(x)}. Whereas with no recall it is fairly obvious that the reservation wage was a nondecreasing function of time (i.e. time remaining), with recall it is not only not clear whether the reservation wage is a nondecreasing function of time, we are not even certain if there is a reservation wage. To begin the analysis, an easy induction argument shows that for each n and x, V *n(x) ≥ V *n − 1(x) and V *n(y) ≥ V *n(x) if y ≥ x so that zn(x) is nondecreasing in n and x. Define: ξ*n = min{x : x ≥ zn(x)} with ξ*0 = 0. We hope to verify that ξ*n is the reservation wage when n periods remain. That is, x ≥ ξ*n if and only if x ≥ zn(x). To verify this we need to show that x − zn(x) is nondecreasing in x. First note that

92

The Economics of Search ∞

冮

冤

冥

x − z1(x) = x − −c + xF(x) + ydF(y) x

∞

冮

= c − (y − x)dF(y) = c − H(x). x

So the statement holds for n = 1. Assuming that it holds for n − 1, we have x − V *n − 1(x) = min{0,x − zn − 1(x)} is nondecreasing in x so that ∞

x − zn(x) = c + x − V

(x) −

* n−1

冮冤V

冥

(y) − V *n − 1(x) dF(y)

* n−1

x

is also nondecreasing in x as desired, completing the induction argument and verifying the existence of the reservation wage ξ*n. It is immediate from the fact that zn(x) is a nondecreasing function of n that ξ*n ≥ ξn −* 1. 4.12 On-the-job search Next, consider the problem of on-the-job search in which the worker can reduce his/her cost of search by remaining employed while searching. At each point in time, regardless of past decisions, the worker has three alternatives: work full time, on-the-job search, and search full time. We consider here the infinite time horizon, discounted model in which the offer is simply one period’s wage. Let cE denote the cost of on-the-job search and assume that cE > c. Moreover, we assume that the worker engaged in on-the-job search incurs no loss in wages. Let V(x) be the return from following an optimal search strategy when the current wage rate is x. Then V(x) is the unique solution to ∞

冮

冦

V(x) = max x + βV(x),x − cE + βF(x)V(x) + β V(y)dF(y),−c

(4.26)

x

∞

冮

冧

+β V(y)dF(y) 0

where the first option within the brackets is the no search option, the second option is the employed search option and the third option is the unemployed search option.

The basic sequential search model and its ramifications

93

Moreover, a search strategy which selects an action maximizing the righthand side of (4.26) for each x ≥ 0 is optimal. Define: ∞

冮

h(x) = [V(y) − V(x)]dF(y) x

and ∞

冤

冮

冥

g(x) = cE − c − x + β (V(x) − V(y))dF(y) . 0

Then we can rewrite (4.26) as V(x) = x + βV(x) + max{0, −cE + βh(x), −cE + βh(x) + g(x)} Notice that g(0) = cE − c > 0 and that both h and g are nonincreasing functions. Define ξE, ξ*, and η to be the unique solutions to −cE + βh(x) = 0, −cE + βh(x) + g(x) = 0 and g(x) = 0, respectively. Then, if (a) η < ξE and the current wage rate is x, then search full time if x ≤ ξ*, use on-the-job search if ξ* < x < ξE and work full time if x ≥ ξE. (b) η ≥ ξ, then search full time if x ≤ ξ* and work full time if x > ξ*.

Figure 4.3 Determination of optimal search strategy in on-the-job search model: Case A

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The Economics of Search

Figure 4.4 Determination of optimal search strategy in on-the-job search model: Case B

4.13 SEARCH IN THE PRESENCE OF EXPERIENCE QUALITIES: THE LIPPMAN AND McCALL BELATED INFORMATION MODEL Let Z be a random variable that equals +1 or −1 with equal probability. We assume that there is some belated information or experience information that arrives after a worker begins his/her job and that it can be described as affecting his/her per period return to work by Y = αZ where α measures the amount of uncertainty. This information is assumed to arrive with probability p each period the job is worked. Thus if the initial wage offer is w then before belated information arrives the per period return to working equals w while after belated information arrives the per period return to working is either w + α or w − α according to whether positive (Z = +1) or negative information (Z = −1) is received. If τ is the arrival time of the information, then τ has a geometric distribution with parameter p. P(τ = n) = (1 − p)n − 1p, n = 1, 2, . . . Assume that the initial wage is drawn from the distribution F, that the cost of search is c and that individuals maximize expected discounted returns with

The basic sequential search model and its ramifications

95

discount factor β. Moreover, once Y is learned workers are allowed to quit there current job and continue search. Let Vα(x) be the maximal expected discounted returns attainable over an infinite horizon when the current available wage is w. Similarly, let Cα be the optimal return when the currently available wage offer is rejected. Note ∞

τ

Eβ =

冱β p(1 − p)

n−1

n

=

n=1

βp [1 − β(1 − p)]

and E(1 + β + . . . + β τ − 1) =

(1 − Eβ τ) 1 = (1 − β) [1 − β(1 − p)]

it follows that

冦

1 − Eβ τ w + αZ w + Eβ τE max Cα; 1−β 1−β

冦

w βp w−α + 1/2 max Cα; 1 − β(1 − p) 1 − β(1 − p) 1−β

Vα(w) = max Cα; = max Cα;

冢

冤

+ max Cα;

冥冧

冤

w + α w + αZ 1−β 1−β

冣

冢

冣

冥冧 ≡ max{C ;S (w)}. α

α

where ∞

冮

Cα = −c + β Vα(w)dF(w). 0

Clearly the value of continued search is Cα whereas Sα(w) is the return to accepting the offer and then deciding (in an optimal manner), on the basis of revealed information whether to quit or to remain permanently on the job. As Sα(w) is continuous and strictly increasing in w, there is a unique solution of Sα(w) = Cα; label it xα. If (w + α)/(1 − β) ≤ Cα (which indicates that the searcher would quit no matter what the belated information turns out to be), then it is clear that the job offer would not be considered in the first place so that Vα(w) = Cα. On the other hand, if (w − α)/(1 − β) ≥ Cα which indicates that the searcher would never quit his job, then Sα(w) =

1 w 1 α ≥ Cα + > Cα 1 − β(1 − p) 1 − β 1 − β(1 − p) 1−β

冢

冣

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The Economics of Search

which means that the offer will in fact be accepted so that Vα(w) = Sα(w). Thus it must be that (1 − β)Cα − α < xα < (1 − β)Cα + α and Sα(w) =

1

w + 1/2βp冤C 1 − β(1 − p) 冢

α

+

w+α 1−β

冥冣.

Setting Sα(w) = Cα and solving then yields xα = (1 − β)Cα −

βp α. 2 − 2β + βp

(4.27)

Note that xα > (1 − β)Cα − α. Defining yα ≡ (1 − β)Cα + α, we have w≤x

C

α  α Vα(w) = w+α −1  冤 w + 1/2βp冢Cα 1 − β 冣冥 [1 − β(1 − p)] xα < w < yα   w w > yα  1−β

It is appropriate to refer to xα as the reservation wage rate while remembering that the searcher will quit unless w + αZ > (1 − β)Cα. Inspection of (4.27) reveals that xα is decreasing in p. The longer it takes to discover what the belated information is (the smaller p) the more cautious the searcher. Now, Cα = −c + βF(xα)Cα −

1 β2p α [F(yα) − F(xα)] Cα + 2 1 − β(1 − p) 1−β

冢

冣

(4.28)

∞

yα

1 β 1 − 2β + βp β + wdF(w) + wdF(w). 2 1 − β(1 − p) 1−β x 1 − βy

冮

冮

α

α

Differentiating (4.28) with respect to α and utilizing equation (4.27) to manipulate the resulting expression we have β 2pF(yα) − F(xα) 1 − 2β − β(1 − p) Ca′ = β 2p 2 − 2βF(xα) − [F(yα) − F(xα)] 1 − β(1 − p)

冦

冧

The basic sequential search model and its ramifications

97

Since 2 − 2βF(xα) > 2β[F(yα) − F(xα)] and βp/[1 − β(1 − p) < β for all 0 < p < 1, Ca′ > 0 for α > 0. Thus the value of search increases with α but xα is decreasing in α.

4.14 THE LUCAS EQUILIBRIUM MODEL FOR THE BASIC SEQUENTIAL SEARCH MODEL Consider a world in which all consumption consists of apples, grown on trees of various heights. Suppose there are many trees relative to the number of people—so many in fact that their price is zero. In order to produce, a worker must be matched to a tree with one worker per tree. Short workers work best on short trees whereas tall workers prefer tall trees. Let w be the number of apples a given worker can pick per period on a particular tree. Let G be the distribution of this random productivity. Lastly, suppose that any tree can become barren between one period and the next with probability θ. What Lucas has done is to construct a technology so that the Bellman equation of the classic model holds for each agent with w representing an idiosyncratic productivity rather than a market wage. Lucas completes his general equilibrium system by giving the worker preferences over sequences {ct} of consumption in the form: ∞

冦冱β U(c )冧. t

E

t

t=0

Thus if each worker consumes his own earnings wt each period, the Bellman equation becomes: v(w) = max[U(w) + β(1 − θ)v(w) + βθv(0), U(0) + β冮v(w′)dG(w′)]. The problem has the same structure as before. But the solution function v to the Bellman equation and the reservation wage ξ will be different as they depend on the shape of U. Lucas thus has discovered a plausible answer to the question: why do workers face a distribution of wage offers? This resolution is not in violation of the “law of one price.” It flows from the capture in a tractable manner of the idea that different people are good at different tasks, and it takes time to make productive matches. This general equilibrium is one of autarchy: capital is so plentiful that it does not pay to create property rights in it. There is also no trade. Changes do take place as the model is made more complex. No one has a motive for spot trading, but there is good reason for workers to be

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The Economics of Search

interested in contingent claims on future goods. Each worker in autarchic equilibrium runs the risk of his job vanishing and that any search may be long-lived. Suppose there is a continuum of these workers and their circumstances are public knowledge. Then workers will engage in pooling arrangements that smooth the consumption of each perfectly requiring each member of the arrangement to take his/her work/search decisions to maximize the value of his/her earnings. This initiates a return to the original Bellman equation not for reasons of risk neutrality, but because none has any risk to bear.

4.15 RESERVATION WAGE PROPERTY PRESERVED IN A CONTINUOUS SEARCH MODEL Suppose instead of paying a fixed cost c to draw from F the searcher looks for opportunities with an intensity he/she controls λ. Offers arrive as a Poisson process with an arrival rate λ. Someone who searches with an intensity λ for a length of time T incurs search costs of c(λ)T. The Poisson assumption, which preserves stationarity in this continuous time setup, means that in a short interval of time ∆ the probability that exactly one offer arrives if λ∆ + o(∆). When the searcher controls both the intensity with which he/she searches and the reservation value, the probability of accepting an offer in the interval ∆ is approximately λ∆(1 − F(x)) or the probability that he/she gets an offer times the probability of acceptance. Hence, if the searcher follows a policy with values λ and x his/her expected gain is V(λ,x), where V(λ,x) satisfies ∞

冦[1 − λ∆(1 − F(x))]V(λ,x) + λ∆冮ydF(y)冧 − c(λ)∆

−r∆

V(λ,x) ⬵ e

(4.29)

x

with probability (λ∆(1 − F(x)) the searcher accepts an offer with expected value ∞

冮ydF(y)/(1 − F(x)); x

with probability [1 − λ∆(1 − F(x))] he/she rejects the offer and searches again. The searcher incurs a cost of c(λ)∆ whether or not he/she continues search. Approximate e−r∆ by 1 − r∆, discard all terms involving ∆2 and obtain

The basic sequential search model and its ramifications

99

∞

冮

λ ydF(y) − c(λ) V(λ,x) =

x

r + λ(1 − F(x))

.

(4.30)

If x* is chosen to maximize V(λ,x), then V(λ,x*) = x*. Apply Leibnitz rule when differentiating the integral in (4.30) ∞

冢

冮

d λ ydF(y) = − xλf(x) yields dx x

冣

dV(λx) = −λxf(x) + V(λx)(λf(x)) = 0 ⇒ x* = V(λ,x*)] dx ∞

冮

λ ydF(y) − c(λ) V(λ,x*) =

x*

. r + λ(1 − F(x*))

Now choose λ* to maximize: V(λ,x*) ∞

冮

ydF(y) − c′(λ) − V(λ,x*)(1 − F(x*)) dV(λ,x*) x* = =0 dλ r + λ(1 − F(x*)) ∞

c′(λ) =

冮 (y − x*)dF(y)

(4.31)

x*

Search intensity is set so that the marginal cost of increasing search intensity equals the expected improvement from another search. Finally, equation (4.30) can be rewritten as ∞

冮

rV(λ,x*) = rx* = λ (y − x*)dF(y) − c(λ) x*

or c(λ) = λH(x*) − rX* or x* =

λH(x*) c(λ) 18 − . r r

(4.32)

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The Economics of Search

4.16 OPTIMAL REPLACEMENT POLICY19 A reservation wage type property is not unique to replacement policies where stochastically failing equipment is more costly to replace after a failure occurs than before failure. Breiman (1964) was the first to point out the similarity between the search model and the replacement model. We suppose that the equipment fails according to a stochastically monotone Markov chain. Definition A Markov chain {ΧT,T ≥ 0} is called stochastically monotone if ΧT + 1 given ΧT = i, is stochastically larger than ΧT + 1 given that ΧT = j, for all T ≥ 0 and i > j.20 Stochastic monotonicity is the key to obtaining optimal control limit rules, i.e., reservation age-like policies, for stochastically failing equipment. Derman (1963) was the first to consider control limit rules. He assumed that the equipment is inspected at regular time intervals and after inspection is classified into one of (m + 1) states denoted by 0,1, . . ., m. A control limit rule ᐉ requires that the equipment be replaced if the observed state is one of the states k, k + 1, . . ., m, for some specified state k. The state k is called the control limit of ᐉ. Derman assumed that {Χn, n ≥ 0} is a stationary Markov chain. The cost incurred when the equipment is in state j is denoted by c(j). For ᐉ ∈ L, the asymptotic expected average cost is given by n

A(ᐉ) = lim n−1 n→∞

冱c(Χ ). k

k=1

Derman (1963) proved the following. Theorem 4.4 Let the Markov chain {Χn, n ≥ 0} be stochastically monotone. Then there is a control limit rule ᐉ* such that A(ᐉ*) = min A(ᐉ). ᐉ∈L

Many Markov processes are stochastically monotone. Markov diffusions are a conspicuous example. In general, the class of totally positive Markov processes is a proper subset of the class of stochastically monotone Markov processes. Stochastically monotone Markov chains with partially ordered state spaces were presented in Kamae et al. (1977). Theorem 4.5 characterizes the class of increasing failure rate (hazard rate) or IFRA distributions with stochastically monotone Markov chains. Theorem 4.5

Let S be a partially ordered countable set. Let {Χn, n ≥ 0} be a

The basic sequential search model and its ramifications

101

stochastically monotone Markov chain with monotone paths and state space S. Let C be an increasing subset of S, with finite complement. Then the first passage time from state i to set C is IFRA.21 4.17 MARTINGALES AND THE EXISTENCE OF A RESERVATION WAGE To conclude this chapter we derive the existence of a reservation wage using results from the theory of martingales. Recall from Chapter 2 that that {Χn} is a Martingale w.r.t. {Yn} if, for n = 1, 2, . . . E [|Χ |] < ∞

(4.33)

E [Xn + 1 |Y1, . . .,Yn] = Χn

(4.34)

and

where (Y1, . . .,Yn) can be interpreted as the information or history up to the stage n and also that E [Χn] = E [Χ1] for all n. We show that there is an optimal policy and that it has the form accept a job offer x iff x ≥ ξ, the reservation wage.

(4.35)

In obtaining these results we use several concepts from the theory of martingales and optimal stopping.22 Consider an arbitrary sequence Χ1,Χ2, . . . of r.v.s. and for each n let Yn be an r.v. whose value is determined by the first n observations Χ1,Χ2, . . .,Χn. The sequence Y1,Y2, . . . is said to be a supermartingale w.r.t. the sequence Χ1,Χ2, . . . if for each n, E(Yn) exists and (with probability 1) E(Yn + 1 | Χ1, . . .,Χn) ≤ Yn.

(4.36)

Moreover, if E(YN) ≤ E(Y1) for every stopping rule N for which E(YN) exists, then the supermartingale is said to be regular. Applied to job search, take Χi to be the ith offer and Yi = max[Χ1, . . .,Χi] − ic.23 The objective is to choose a stopping rule N, called optimal, so as to make E(YN) as large as possible. This raises the question whether there exists an optimal stopping rule. Regardless of whether the sequence Y1,Y2, . . . forms a supermartingale, it can be shown (DeGroot 1970: 347) that among the class of stopping rules that actually stop w.p.1 there is, in fact, an optimal stopping rule if the following conditions hold: lim Yn = −∞

n→∞

and

(4.37)

102

The Economics of Search E(| Z |) < ∞, where Z ≡ supn Yn.

(4.38)

Z can be thought of as the payoff one would receive if one had perfect foresight (w.r.t the Χi’s). Assuming E(Χ1) < ∞ in the job search model, a direct application of the Borel–Cantelli lemma shows that (4.37) and (4.38) hold, so that there is an optimal stopping rule for job search models. In addition, the assumption E(Χ 21) < ∞ applied to the job search model implies E(Χ 21) ≤ M < ∞, for all n

(4.39)

so that the sequence Y1,Y2, . . . is uniformly integral (u.i.).24 Interest in (4.39) flows from the fact that a u.i. supermartingale is regular. The fruit of these technicalities is the following important result referred to as the monotone case by Chow, Robbins, and Siegmund (1971): consider a stopping problem in which an optimal stopping rule exists. Suppose that for any set of observed values Χ1 = x1, Χ2 = x2, . . .,Χn = xn which satisfies E(Yn + 1 | x1, . . .,xn) ≤ yn

(4.40)

the sequence Yn + 1, Yn + 2, . . . is a regular supermartingale w.r.t the sequence Χn + 1, Χn + 2, . . . . Then there is an optimal stopping rule which stops if (4.40) holds and continues otherwise. In other words, if the hypotheses of this theorem hold, then the myopic (OSLA) stopping rule is optimal. Of course, the myopic stopping rule is the rule which stops iff the current return from stopping (i.e., the right-hand side of (4.40)) exceeds the expected value of stopping after taking exactly one more observation (the left-hand side [LHS] of (4.40)). In order to apply this theorem to the job search model, we need to verify all of its hypotheses. Assuming that E(Χ21) < ∞, it remains to verify that Yn + 1,Yn + 2, . . . is a supermartingale whenever (4.40) holds. To achieve this, define ξ to be the unique solution of H(y) − c = 0 so that y ≥ ξ implies H(y) − c ≤ 0. Suppose (4.40) holds for Χ1 = x1, . . ., Χn = xn and define Zn = Yn + nc = max(Χ1, . . ., Χn). Then ∞

冮

E(Yn + 1 | Χ1 = x1, . . ., Χn = xn) = −(n + 1)c + max[Zn,x]dF(x)

(4.41)

0

= −(n + 1)c + Zn + H(Zn) = Yn + [H(Zn) − c], so that H(Zn) − c ≤ 0 since (4.40) holds. Since H is strictly decreasing and Zn increases in n, the desired inequalities (4.36) hold. Moreover, (4.41) reveals that (4.40) holds iff Zn ≥ ξ.

The basic sequential search model and its ramifications

103

APPENDIX 4.I A NAIVE COMPARISON OF FSM AND BSM In this appendix we compare the fixed sample strategy with the reservation strategy. Suppose prices for the good are uniformly distributed over the interval [m,M]. Then F has the density f(x) = 1/(M − n) for m ≤ x ≤ M f(x) = 0 otherwise. Then by equation (4.4) ξ

冮

c = (ξ − x)/(M − m)dx

(A4.1)

0

Performing the integration and simplifying, we obtain ξ2 − 2p¯ m + m2 − 2(M − m)c = 0 Applying the quadratic formula we have ξ = m ± √2(M − m)c since the reservation price cannot be below m, we eliminate the second root from consideration and thus ξ = m + √2(M − m)c

(A4.2)

Suppose we consider the two uniform distributions F1 and F2, where for F1, M = 24 and m = 16, whereas for F2, M = 30 and m = 10. Then F2 is clearly a mean preserving spread of F1. If, for each distribution, we allow c to vary in jumps of .1 from a low of .1 to a high of 2 and use equation (A4.2) to calculate ξ for each value of c for each of the two distributions, we obtain Table A4.1. The following are clear from the table: (1) As expected, a rise in c increases the reservation price. (2) Also as expected, the mean preserving spread for any level of c, lowers the reservation price. (3) The relative advantage of the “riskier” distribution is inversely related to c. This is due to the fact that a lower cost of search increases E(N), the expected number of vendors searched,25 and this increases the likelihood that an exceptionally low value of p will be encountered with the more “spread out” distribution. Let us compute the total expected cost of purchasing the good using a Stiglerian optimal search strategy. We concentrate on F1. By equation (4.2),

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The Economics of Search Table A4.1 c

ξ under F1

ξ under F2

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

17.27 17.79 18.19 18.53 18.83 19.10 19.35 19.58 19.80 20.00 20.20 20.38 20.56 20.73 20.89 21.06 21.22 21.36 21.51 21.67

12.00 12.83 13.46 14.00 14.47 14.90 15.29 15.67 16.00 16.33 16.63 16.93 17.21 17.48 17.74 18.00 18.25 18.49 18.72 18.94

d(n), the expected fall in the minimum price resulting from increasing the sample size from (n − 1) to n, is given by 24

d(n) =

24 − x 8

冮冤

16

(n − 1)

冥

x − 16 dx. 8

冤

冥

(A4.3)

Table A4.2 gives the value of d(n) as computed by equation (A4.3) for n between 1 and 10, inclusive. For any given value of c, the optimal sample size, n¯ , is that value of n such that d(n¯ ) ≥ c > d(n¯ + 1). Then E(TC), the expected total cost of purchase under a Stiglerian optimal strategy, is, 24

E(TC) = 16 +

24 − x n¯ dx + n¯ c. 8

冮冤

16

冥

(A4.4)

Table A4.3 gives the values of n¯ and E(TC) for the same values of c used in Table A4.2, and compares the latter with the corresponding results for the sequential optimal strategy (recall that the expected total cost of purchase under that strategy is p¯ ). The table reveals: (1) as c increases, the optimal sample size under the Stiglerian strategy decreases; (2) the expected total cost

The basic sequential search model and its ramifications

105

Table A4.2 n

d(n)

1 2 3 4 5 6 7 8 9 10

4.00 1.33 .67 .40 .27 .19 .14 .11 .09 .07

of search, for any given level of c, is, as expected, lower for sequential than for Stiglerian search.26 There have been several comparisons of the classic sequential model with Stigler’s search model. Lai et al. (1980) evaluated three stopping rules based on regret, R(δ,T) where R(δ,T) is defined to be the expected total difference in response between the ideal procedure, which assigns all N patients to the superior treatment, and the procedure determined by the stopping rule T. The difference in the effects of the two treatments is measured by a normal random variable z with mean δ and variance σ2. Table A4.3 c

n¯

E(TC)

p¯ under F1

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

8 5 4 3 3 3 2 2 2 2 2 2 2 1 1 1 1 1 1 1

17.69 18.33 18.80 19.20 19.50 19.80 20.07 20.27 20.47 20.67 20.87 21.07 21.27 21.40 21.50 21.60 21.70 21.80 21.90 22.00

17.27 17.79 18.19 18.53 18.83 19.10 19.35 19.58 19.80 20.00 20.20 20.38 20.56 20.73 20.90 21.06 21.22 21.37 21.51 21.66

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The three rules, a Bayes rule, an ad hoc rule, and a second ad hoc rule were studied by Monte Carlo simulation. Their general conclusion was that all three were about equally good from both a frequentist and Bayesian perspective. All three are markedly superior to any fixed-sample size procedure. A more recent study is that by Yee (1994). He compared the classic sequential search model with Stigler’s fixed-sample size and found the former to be superior. Consumers do better by using the information from search as it occurs. These are standard arguments favoring the sequential model over Stigler’s rule. We show that these arguments may be naive and present a more complete comparison.

APPENDIX 4.II A MORE SOPHISTICATED COMPARISON OF FSM AND BSM The favorable (for sequential sampling) comparisons made earlier were recognized as naive. Over a twenty-year period, Morgan and Manning have demonstrated the fallacies latent in these facile comparisons. Two fallacies are exposed in their 1985 article. The first is that an expected utility maximizing sequential strategy is always preferred to a fixed-sample size model (FSM) strategy. See Examples A4.1 to A4.4. The second fallacy is that an optimal search strategy is sequential. In our earlier comparisons an FSM searcher is viewed as choosing the total number of observations and then inspecting each of them even when an exceptional offer is presented before the total number is inspected. Morgan and Manning give a different interpretation of the FSM strategy: they consider an FSM strategy in which one sample is drawn and all the observations are witnessed simultaneously. The advantage of this FSM strategy is the speedy accumulation of information. The searcher solicits offers from all n candidates rather than proceeding one by one till the reservation wage is exceeded. Nevertheless, the FSM policy may over-invest in information relative to a sequential policy. Morgan and Manning conclude that “optimal search will encompass both as special cases”. The remainder of their article addresses this issue. Schmitz (1993) considers some of the points made by Manning and Morgan and presents an illuminating analysis. Example A4.1 (Schmitz 1993) Consider the following acceptance sampling result. A batch of items from a producer are subjected to an acceptance sample procedure defined by: the batch is rejected if a sample of 20 units contains more than 2 defects; otherwise the batch is accepted. This is called a (20,2)-sampling plan. Suppose that inspection of the 20 items produces the following sequence of G (good) and D (defective): G G D G G D G G D G G G G G D G G G G G. Note that with the occurrence of a defect at the ninth observation, we can reject the batch. The (20,2) sampling plan can be modi-

The basic sequential search model and its ramifications

107

fied by stopping the inspection as soon as 3 defective units are discovered or 18 good units are observed. Schmitz calls this a curtailed inspection plan. That is, for an (n,a) sampling plan (n is the sample size and a the acceptance number) it is modified so that items are inspected until either the number of defective items is a + 1 or the number of good items is n − 1. Note that in the curtailed plan the sample size is a random variable N instead of a fixed number n and N ≤ n. The average sample number (ASN) is given by E(N) < n, where the expectation is with respect to one of the possible sample distributions. This simple example reveals a general feature of sequential statistical procedures: The sample size N is a random variable. The data are said to be very informative if EN << n. If EN is close to n, the data contain little information. Example A4.2 (Schmitz 1993) dom variables such that Χi =

Let Χ1, . . ., Χn be independent binomial ran-

ith item is defective . 冦10 ifif the the ith item is not defective

Let k

Sk =

冱 Χ, i

i=1

the number of defectives revealed by the first k observations. The curtailed sampling plan is defined by stopping at N = min{m < n : Sm = a + 1 or Sm = m + a − n} leading to decisions Acceptance if SN = N + a − n Rejection if SN + a + 1 Example A4.3 (Schmitz): sequential probability ratio test (SPRT) for independent experiments Let Χ1,Χ2, . . . be i.i.d. random variables and let Hj: P* = Qj, j = 1, 2 be two simple hypotheses regarding the distribution Qj of Χ1. In the SPRT one continues sampling until one of the two hypotheses is accepted, i.e., let fj be densities of Qj w.r.t. a dominating measure µ, i.e.,

108

The Economics of Search fj =

dQj , j = 1, 2. dµ

Let q be the likelihood ratio of f1 and f2,

 f2(x)/f1(x),  q(x) =  ∞,  1,

f1(x) ≥ 0,

f2(x) ≥ 0

f1(x) = 0,

f2(x) > 0

f1(x) = f2(x)

and let n

qn(x1, . . ., xn) =

冲q(x ) n ∈ N i

i=1

and k1, k2 ∈ ℜ such that 0 < k1 ≤ 1 ≤ k2 < ∞, k1 ≠ k2. The decision procedure δk ,k is to continue sampling until qn exits the interval (k1,k2), that is, until N = inf{n ∈ N: qn(Χ1, . . ., Χn) ∉ (k1,k2)}. One decides that H1 is true if qN(Χ1, . . ., ΧN) ≤ k1 and decides that H2 is true if qN(Χ1, . . ., ΧN) ≥ k2. The process is called SPRT with stopping bounds k1 and k2. 1

2

Objections to sequential statistical procedures If the cost function is given by C(ᐉ,i) = Co + C · i where Co = c, the expected costs for (20,2) then it can be shown that for small p the curtailed procedure is much more expensive than the classical (20,2) procedure. Schmitz notes that in agricultural studies it takes one period of growth to produce a single observation. He then enquires whether one can have one’s cake and eat it too, i.e., design a procedure with the advantages of both procedures without the disadvantages. Example A4.4 given by

Consider the quality control example with inspection costs

Co + C · i, Co, C > 0. The following perplex occurs:

The basic sequential search model and its ramifications

109

(a) On the one hand information acquired during the inspection should be used to “save” costs by early decisions. (b) On the other hand one-at-a-time sampling yields very high costs for small values of p. Consider a three-stage plan. 1

First, select a sample of size 10 and inspect; if (i) (ii) (iii) (iv)

2

3

S10 ≥ 3 S10 = 2 S10 = 1 S10 = 0

reject draw a second sample of size

 5 10  8

If in 1(ii) S15 ≥ 3, reject; otherwise take another sample of size 5. If in 1(iv) S18 ≥ 3, reject S18 = 0, accept elsewhere take a third sample of size 2. If S20 ≤ 2 accept whereas if S20 ≥ 3 reject.

It has the same rejection probabilities as (20,2). The curtailed inspection associated with small p and producing high costs is avoided. Yet it permits early detection of large P. Schmitz observes that Wald himself was alert to the fact that sequential procedures could cause very high costs and that group-sequential methods could be more advantageous. “For practical reasons it may sometimes be preferable to take the observations in groups, rather than singly” (Wald 1947: 101). Since Wald may have suggested using variable group sizes, Schmitz (1993) is a detailed study of these methods, which up until 1993 had not been performed. Schmitz refers to these modified group-sequential procedures as sequentially planned procedures. The specification and existence of sequentially planned procedures rely heavily on measure theoretic arguments. For our purposes, we simply state Schmitz’s assertion that applications of sequentially planned statistical decision procedures are “quite easy.” First, draw a sample of size a, prescribed by τ. Based on observed data decide according to the sampling plan τ

• •

whether to stop sampling or take a second sample of size a2, an r.v. determined by (x1, . . ., xa1).

In the first case a terminal decision is made. In the second case a sample of size a2 is taken and the decision prescribed by τ and based on x1, . . ., xa1, . . . xa1 + a2 is to stop or continue sampling. The key chapter in Schmitz’s monograph is chapter 4, Bayes-Optimal Sequentially Planned Decision Procedures. The methods developed by

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Schmitz are similar to the group procedures advocated by Manning and Morgan. Sequential procedures encounter additional “ethical” obstacles when they are applied to clinical trials. Lai (2001: 301) observes: While sequential analysis had an immediate impact on weapons testing when it was introduced during WWII to reduce the sample sizes of such tests, its refinements for testing new drugs and treatments received little attention from the biomedical community until the Beta-Blocker Heart Attack Trial (BHAT . . .) Lai (2001: 312; original emphasis) notes that the past two decades (1982– 2002) following the “success story of BHAT saw increased use of group sequential designs in clinical trials . . . and major advances in the development of group sequential methods.”27 It should be emphasized that Wald’s two best-known contributions are his invention of the sequential, probability-ratio test (SPRT) (Wald 1947) and its optimality property (Wald and Wolfowitz 1948). The second contribution shows that the only admissible solutions to a decision problem is basically Bayesian. These Bayes solutions comprise a minimally complete class (Wald 1950). Lindley (1990: 45) observes that “both of these results are Bayesian.” Lindley finds it astounding that in the U.S., Wald’s basic research led to the construction of a “school of statistics that is resolutely anti-Bayesian.” Chernoff, who made fundamental contributions to continuous stopping problems and much else, makes the following remarks in his comments on Lai (2001): Also, I feel that a statistical problem is not well understood unless one can describe it from the point of view of a Bayesian decision problem. The very formulation of the relevant possible alternatives requires some subjective consideration based on previous experience. Having understood the problem it is often not necessary to analyze it from a Bayesian or decision-theoretic point of view. (Chernoff 2001: 351; emphasis added)

APPENDIX 4.III THE FRENCH VERSION OF BSM A central concept of the general theory of stochastic processes is the compensator which is also known as the dual predictable projection. This theory was developed primarily by French probabilists, two of the most prominent being Dellacherie and Meyer. The compensator is, for many of us, so abstract it is difficult to use in applications that would benefit from this elegant theory. In an important paper, He (1989) solves several optimal stopping problems using dual predictable projections. In this Poisson setting, the compensator

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111

analysis is clear and rigorous. In particular, he solves the house-selling problem which as we saw is mathematically equivalent to BSM. The following is taken from He’s article.

The Poisson random measure Let (Ω,F,P) be a probability space. Assume that N = (Nt)t ≥ 0 is a point process, that is, ∞

Nt =

冱1

Tn

≤ t,

n=1

when Tn, n ≥ 1, is a sequence of positive random variables such that (i) To = 0, (ii) Tn ≤ Tn + 1 for all n ≥ 0, and if Tn < ∞, Tn < Tn + 1, (iii) Tn ↑ ∞. Let (E,βE) be a measurable space designated as a mark space. Assume, for simplicity, that the mark space is Rd and its Borel σ-algebra. Let (Χn)n ≥ 1 be a sequence of random elements taking values in E. Then (Tn,Χn)n ≥ 1 is called a marked point process, with N its counting process. An alternative representation of the marked point process employs an integer-values random measure: ∞

µ(dt,dx) =

冱E(T ,Χ )(dt,dx,)1 n

n

Tn

< ∞,

(A4.5)

n=1

where E(a) is the Dirac measure concentrated on a. For each B ∈ βE define N Bt = µ([0,t] × B) = 冱1T

n

, t ≥ 0,

≤ t,Χn ∈ B

and Ft = σ{N Bs, s ≤ t, B ∈ βE}, t ≥ 0. The knowledge accumulated up till time t is represented by Ft. Clearly, (Ft )t ≥ 0 satisfies (i) Fs ⊂ Ft, 0 ≤ s ≤ t, (ii) Ft = 傽 Fs, t ≥ 0. s>t

(Ft)t ≥ 0 is called the natural filtration of the marked point process. If Tn < ∞ is equivalent to Χn ≠ 0 for all n ≥ 1, a jump process Zt can be built as follows:

112

The Economics of Search Nt

Zt =

冱Χ , t ≥ 0 n

n=0

and Ft = σ[Zs,s ≤ t], t ≥ 0. Thus, (Ft)t ≥ 0 is also the natural filtration of Z. A stopping time T is a mapping from Ω to [0,∞] such that {T ≤ t} ∈ Ft for all t ≥ 0. A process Y = (Yt) t ≥ 0 is adapted to (Ft) if Yt is Ft-measurable for all t ≥ 0. The predictable σ-field ℘ is the σ-field on Ω × [0,∞] generated by all ¯ is the σ-field on adapted left-continuous processes. The predictable σ-field ℘ ¯ = ℘ × βE. Ω × [0,∞] × E: ℘ A process defined on Ω × [0,∞](Ω × [0,∞] × E) is called predictable if it is ¯) measurable. A crucial example is the following: If t is a stopping time, ℘(℘ then Z(w,t) =

冦0,1,

if 0 ≤ t ≤ T(w) otherwise,

is a predictable process. Given a random measure µ, there is a random measure v, called the compensator or dual predictable projection of µ, such that for all nonnegative predictable processes t

冮冮W(s,x)v(ds,dx)

W = (W(t,x))t ≥ 0,x ∈ E,

0E

is a predictable process and

E

∞

∞

0E

0E

冦冮冮W(t,x)µ(dt,dx)冧 = E冦冮冮W(t,x)v(dt,dx)冧.

Now set G0 = {,Ω}, Gn = σ{T1, . . .,Tn, Χ1, . . ., Χn}, n ≥ 1. Let Fn(dt,dx) be the regular conditional distribution of (Tn,Χn), given Gn − 1, n ≥ 1, it follows that ∞

v(dt,dx) =

冱 F ([t,∞]×E),1 Fn(dt,dx)

Tn −1

n=1

n

< t ≤ Tn.

(A4.6)

The basic sequential search model and its ramifications

113

We now specialize to the Poisson case. If (i) N = (Nt)t ≥ 0 is a Poisson process with parameter λ, (ii) (Χn) are i.i.d. random elements with distribution F, (iii) N and (Χn)n ≥ 1 are independent. For the Poisson case, the random measure µ given in (A4.5) is called a Poisson random measure. Note that (Tn − Tn − 1)n ≥ 1 are i.i.d. random variables with an exponential distribution with parameter λ. This observation gives Fn(dt,dx) = λ1T

e−λ(t − T )dtF(dx).


n−1

n−1

An application of (A4.6) yields the compensator of µ: v(dt,dx) = λdtF(dx).

(A4.7)

Aven and Gaarder (1987) also use the general theory of stochastic processes to obtain optimal replacement policies for equipment subject to Poisson shocks. The theory is drawn from Bremaud (1981) who deftly employs compensators to study the optimal behavior of shock models. Meyer’s (1966) optional sampling theorem also plays a key role. There are two reasons for presenting these two examples of stochastic calculus. The first is to introduce this complex theory in a relatively simple manner. From this introduction the reader should be able to penetrate the large literature on option pricing which uses these French stochastic methods. The second reason is to demonstrate to those readers who are familiar with this advanced probability that it is useful even when studying simple processes like search and replacement. We now apply these results to the classic house-selling problem.

House-selling problem (MacQueen and Miller 1960) A particular house is for sale and the arrival time of the nth prospective buyer is Tn, n ≥ 1. Let Χn be the price-offer of the nth buyer. At t, the seller can choose the maximum offer up to time t and incurs a maintenance expense of ct, where c is a constant. The problem is to determine when to accept an offer so that the seller’s income is maximized. Let µ be a Poisson random measure and ∞

冮

0 < c < λ xF(dx)<∞. 0

Set

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The Economics of Search

max{Χn,1 ≤ n ≤ Nt}, if Nt ≥ 1

Zt = 

0,



if Nt = 0 for t ≥ 0.

Now the problem is to find the optimal stopping time T which maximizes E{ZT − cT}. Observe that for Tn ≤ Tn + 1, n

Zt = ZT = n

n

冱

(ZT − ZT ) = i−1

i

i=1

冱[max(Χ ,Z i

) − ZT ].

Ti − 1

i−1

i=1

Now Z can be described by a stochastic integral, t∞

Zt =

冮冮[max(x,Z

) − Zs−]µ(ds,dx),

s−

00

where Zs− is the left limit of Z at s. For any stopping time T, by applying (A4.5) and (A4.7) we obtain T∞

E[ZT − cT] = E

冦冮冮[max(x,Z ) − Z ]µ(dt,dx) − cT冧 t−

t−

00

T

∞

冮冦 冮

冧

= E λ [max(x,Zt−) − Zt−]F(dx) − c dt 0

0

T

∞

冮冦 冮

冧

= E λ (x − Zt−)F(dx) − c dt 0

Zt−

T

∞

冮冦 冮

冧

= E λ (x − Zt)F(dx) − c dt, 0

Zt

(A4.8)

since Zt− = (Zt−) t ≥ 0 is a predictable process and the number of t’s, such that Zt ≠ Zt− is denumerable. Let ∞

冮

h(y) = λ (x − y)F(dx). y

Clearly, h(y) is a monotone decreasing and right-continuous on [0,∞]. Furthermore,

The basic sequential search model and its ramifications

115

∞

冮

h(0) = λ xdF(dx) > c, and 0

h(y) → 0, as y → ∞. But, note that Zt is monotone increasing and right-continuous. From (A4.8) we finally obtain the familiar optimal solution: T = inf{t:h(Zt) ≤ c} = inf{t:Zt ≥ m} Tn if ZT <m
n

where m = inf{y − h(y) ≤ c}.

5

Estimation methods for duration models

5.1 INTRODUCTION Many empirical tests of search theory employ duration data (see Devine and Kiefer, 1991). For example, the BSM implies that unemployment durations have an exponential distribution. In this chapter we develop some statistical tools used to analyze duration data (for more thorough treatment see Lancaster 1990, or Klein and Moeschberger 1997). Duration analysis is also referred to as survivor analysis, where the duration of interest is the survival time of a subject (e.g. person or machine). Much of the recent statistical analyses of duration data focus on the hazard function. The hazard function is related to the probability of exiting the initial state within a short interval, conditional on having survived up to the starting time of the interval. In many applications hazard functions are conditional on a set of covariates. An important feature of the hazard function is that it can be made to depend on covariates that change over time. In Section 5.2 we review the basic definition of a hazard function and its relation to the probability density and cumulative distribution function. Section 5.3 then gives a brief description of counting process theory and martingales. This framework is useful for analyzing duration data, including multiple spell duration data (see Andersen and Borgan 1985, Arjas 1989, Fleming and Harrington 1991, and Anderson et al. 1993 for more thorough discussions). Parametric estimation methods for continuous time duration models with covariates are presented in Section 5.4 while the semi-parametric Cox regression model is discussed in Section 5.5. Section 5.6 presents estimation techniques for grouped or discrete-time duration data. In many situations we are interested in studying an individual’s movement through several labor market states over time. After extending discrete-time methods to a multi-spell framework in Section 5.7, and competing risks models in Section 5.8, Section 5.9 presents the general estimation methods for discrete-time life history data. The chapter concludes with a brief discussion of some specification diagnostic methods that can be derived from the counting process approach.

Estimation methods for duration models 117 5.2 HAZARD FUNCTIONS This section presents a brief overview of hazard functions. Initially we will focus on models without covariates. Later sections of the chapter will then introduce both time-constant and time-varying covariates into the hazard framework. Let T ≥ 0 represent a positive random duration variable, which has some probability distribution in the population; t denotes a particular value of T. In survival analysis, T is the length of time that an individual lives. In many economic applications T is the duration of an unemployment spell or the duration of job tenure. The cumulative distribution function (c.d.f.) of T is defined as F(t) = P(T ≤ t),

t ≥ 0.

The survivor function is defined as S(t) ≡ 1 − F(t) = P(T > t). Thus, S(t) represents the probability that an event has not occurred by time t or that the individual has “survived past” t. Throughout this section we assume that T is continuous and denote the probability density function dF (p.d.f.) of T by f(t) = (t). For ∆t > 0, P(t ≤ T < t + ∆t|T ≥ t) is the probdt ability of leaving the initial state in the interval (t, t + ∆t) given survival until time t. The hazard function for T is defined as λ(t) = lim ∆t↓0

P(t ≤ T < t + ∆t|T ≥ t) ∆t

(5.1)

Thus, the hazard function is the instantaneous rate of leaving per unit time (the “escape” rate). From equation (5.1) it follows that, for “small” ∆t, P(t ≤ T < t + ∆t|T ≥ t) ≈ λ(t)∆t. The hazard can then be used to approximate a conditional probability. We can express the hazard function in terms of the density and c.d.f. very simply. First, write P(t ≤ T < t + ∆t|T ≥ t) =

P(t ≤ T < t + ∆t) F(t + ∆t) − F(t) = . P(T ≥ t) 1 − F(t)

When the c.d.f. is differentiable, we can take the limit of the right-hand side, divided by ∆t, as ∆t approaches zero from above:

118

The Economics of Search λ(t) = lim h↓0

F(t + ∆t) − F(t) 1 f(t) f(t) · = = ∆t 1 − F(t) 1 − F(t) S(t)

Because the derivative of S(t) is −f(t), we have λ(t) =

d log S(t) dt

(5.2)

and, using F(0) = 0, we can integrate (5.2) to get t

冤 冮

冥

F(t) = 1 − exp − λ(s)ds , 0

t≥0

(5.3)

Straight forward differentiation of (5.3) gives the p.d.f. of T in terms of the hazard function: t

冤 冮

冥

f(t) = λ(t)exp − λ(s)ds . 0

Therefore, all probabilities can be computed using the hazard function. For example, for all points a1 < a2, a2

1 − F(a2) = exp − λ(s)ds P(T ≥ a2|T ≥ a1) = 1 − F(a1) 1

冤 冮

冥

and a2

冤 冮

冥

P(a1 ≤ T ≤ a2|T ≥ a1) = 1 − exp − λ(s)ds . 1

In many empirical applications the shape of the hazard function is of primary interest. In the simplest case, the hazard function is constant: λ(t) = λ, for all t ≥ 0. In this case the exit process is memoryless: the probability of exit in the next interval of time does not depend on how much time has been spent in the current state. The standard continuous-time model of stationary job search with a constant offer arrival rate ρ and wage distribution F implies a constant re-employment hazard rate λ(t) = λ = ρ(1 − G(ξ)) where ξ denotes the reservation wage. For a constant hazard function, equation (5.3) implies that F(t) = 1 − exp(λt) which is the c.d.f. of the exponential distribution.

Estimation methods for duration models 119 When the hazard function is not constant we say that it exhibits duration dependence. Assuming that λ(·) is differentiable, the hazard exhibits positive duration dependence at time t if dλ(t)/dt > 0 and negative duration dependence at time t if dλ(t)/dt < 0. If dλ(t)/dt > 0 for all t we say the process exhibits positive duration dependence. With positive duration dependence, the probability of exiting the initial state increases the longer one is in the initial state. Example 5.1: Weibull distribution A popular parametric distribution used in empirical analysis is the Weibull distribution. The random variable T is said to have a Weibull distribution if its c.d.f. is given by F(t) = 1 − exp(−γtα) where γ and α are nonnegative parameters. The p.d.f. is given by λ(t) =

f(t) = γαtα − 1 S(t)

and the hazard function is f(t) = αγtα − 1 exp(−γtα). When α = 1, the Weibull distribution reduces to the exponential with λ = γ. If α > 1, the hazard is monotonically increasing, so the hazard everywhere exhibits positive duration dependence; for α < 1, the hazard is monotonically decreasing. Graphs of the Weibull hazard function for different values of α are presented in Figure 5.1. Example 5.2: Log-logistic distribution The random variable T has a loglogistic distribution if its c.d.f. is given by F(t) = 1 −

1 1 + tα

Figure 5.1 Weibull hazard function

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The Economics of Search

and its hazard function is given by λ(t) =

f(t) αtα − 1 = S(t) 1 + tα

(5.4)

for α > 0 and  > 0. To examine whether the hazard function exhibits positive or negative duration dependence in some ranges we differentiate (5.4) with respect to t: λ′(t) = = =

α(α − 1)tα − 2(1 + tα) − α22t2α − 2 (1 + tα)2 α2tα − 2(1 + tα) − αtα − 2(1 + tα) − α22t2α − 2 (1 + tα)2 α(α − 1)tα − 2 − α2t2α − 2 αtα − 2[(α − 1) − tα] = (1 + tα)2 (1 + tα)2

For α ≤ 1, λ′(t) < 0 for all t and for α > 1, λ′(t) > 0 for t <

冢

α−1 

冣

1/α

and λ′(t) < 0

α − 1 1/α . Graphs of the log-logistic hazard function for different values  of α are presented in Figure 5.2.

if t >

冢

冣

Figure 5.2 Log-logistic hazard function

Estimation methods for duration models 121 5.3 COUNTING PROCESSES AND MARTINGALES The theory of counting processes and their accompanying martingales is useful in developing estimation procedures for duration data. Here we will be content with giving a cursory overview of the counting process approach. More detailed discussions can be found in Bremaud (1980), Fleming and Harrington (1991), and Andersen et al. (1993). Recall that a counting process is a process that counts the number of events that occur at random times. Let Tn, n = 0,1, 2, . . . be a sequence of positive random variables such that 1 2 3

T0 = 0 Tn < Tn + 1 P a.s. lim Tn = ∞ P a.s. n→∞

Condition (3) is needed to insure that the counting process does not blow up to infinity in finite time. The counting process N(t) is then defined by ∞

N(t) =

冱I{T

n

≤ t}.

n=1

Thus, N(0) = 0, N(t) < ∞ almost surely, and the sample paths of N(t) are piecewise constant, right continuous and nondecreasing with jumps of size 1. Recall from Chapter 2 that a filtration or history denoted by {Ft,t ≥ 0} is a sequence of sigma algebras indexed by t that measure the accumulated information up to time t. As time progresses information increases and so Fs ⊂ Ft for s < t. That is A ∈ Fs → A ∈ Ft. The history “just before” time t is denoted by Ft and is the sigma algebra generated by all sets in Fs for all s < t. −

Definition 5.1 for all t.

A process Χ(t) is adapted to {Ft,t ≥ 0} if Χ(t) is Ft measurable

Before continuing we shall review some results from martingale theory. Definition 5.2 A right-continuous stochastic process Χ(t) with left-hand limits is said to be a martingale with respect to the history Ft if 1 2 3

Χ(t) is adapted to Ft Χ(t) is integrable (E(|Χ(t)|) < ∞) for all t For all 0 ≤ s ≤ t, E(Χ(t)|Fs) = Χ(s) P − a.s.

Χ(t) is called a submartingale if we replace (3) by 3(a) For all 0 ≤ s ≤ t, E(Χ(t)|Fs) ≥ Χ(s) P − a.s. and Χ(t) is called a supermartingale if we replace (3) by

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3(b) For all 0 ≤ s ≤ t, E(Χ(t)|Fs) ≤ Χ(s) P − a.s. The next theorem is important for deriving martingales associated with stochastic processes. Theorem 5.1 Doob–Meyer decomposition Let Χ(t) be a right-continuous nonnegative submartingale with respect to history Ft. Then there exists a right-continuous martingale M(t) and an increasing right-continuous predictable process A(t) such that E(A(t)) < ∞ and Χ(t) = M(t) + A(t) P a.s. Corollary Let N(t) be a counting process with associated “intensity process” λ(t). Then t

冮

M(t) = N(t) − λ(t)ds 0

is an Ft − martingale. Definition 5.3 A process Χ(t) is said to be predictable with respect to {Ft,t ≥ 0} if Χ(t) is Ft − measurable for all t. −

Another useful theorem is the following: Theorem 5.2

If V(t) is a predictable process such that

t

E

冤冮|V(s)|λ(s)ds冥 < ∞ P a.s. then 0

t

t

t

冮V(s)dM(s) = 冮V(s)dN(s) − 冮V(s)λ(s)ds 0

0

(5.5)

0

is an Ft − martingale. Let C be a censoring time and define Y(t) to be the stochastic process Y(t) = I{C ≥ t}. Thus, Y(t) equals one up until and including the time at which an observation is censored and equals zero, thereafter. We assume that this stochastic process is measurable with respect to Ft . Further define Z(t) to be the stochastic process Z(t) = I{T1 > t}. Using this theorem and defining V(t) = Z(t)Y(t) it can then be shown that −

t

t

冮Y(s)Z(s)dN(s) − 冮Y(s)Z(s)λ(s)dt 0

0

is an Ft − martingale. Thus,

Estimation methods for duration models 123 t + ∆t

E

t + ∆t

冢 冮 Y(s)Z(s)dN(s)| F 冣 = E 冢 冮 Y(s)λ(s)ds| F 冣 t−

t−

t

(5.6)

t

for all ∆t > 0. This result enables us to derive an estimator for the integrated hazard function Λ(t): t

冮

Λ(t) ≡ λ(s)ds. 0

From (5.6) we have that E [Y(t)(Z(t)dN(t)| Ft ] ≈ Y(t)λ(t)E [Z(t)| Ft ]dt. −

(5.7)

−

Suppose we have a random sample of size N and let Yi(t), Zi(t), and Ni(t) denote the sample paths of Y(t), Z(t), and N(t) for the ith individual, i = 1, . . ., N. Then appealing to the law of large numbers gives: N

冱Y (t)Z (t)dN (t) i

i

N

N

冱Y (t)Z (t)

λ(t)dt

i

i=1

i

i

i=1

≈

.

N

Hence, N

N

冱

冱Y (t)Z (t)dN (t)

Yi(t)Zi(t)dNi(t)

λ(t)dt =

i=1

i

=

N

i

R(t)

冱Y (t)Z (t) i

i

i=1

i

i=1

where R(t) is the number “at risk” set at time t and includes all those who have not been censored or have failed by t: N

R(t) =

冱Y (t)Z (t). i

i

i=1

Thus, we have t N

Λˆ(t) =

冮冱

t

冮

Yi(s)Zi(s) J(s) dNi(s) = dN*(s) R(s) R(s) 0i=1 0

(5.8)

124

The Economics of Search

where N

dN *(s) ≡

冱Z (s)Y (s)dN (s) i

i

i

i=1

and J(s) = 1 if R(s) > 0 and J(s) = 0 if R(s) = 0 with the convention that 0/0 = 1. The estimator Λˆ(t) in (5.8) is referred to as the Nelson–Aalen estimator of the integrated hazard function and is a step function that is constant at all times except failure times and jumps up by 1/R(t) at time t when a failure occurs at time t. The integrated hazard function of the single duration variable T1 equals t

t

冮

冮

Λ*(t) = λ*(s)ds = Z(s)λ(s)ds. 0

0

Note that this integrated hazard function is stochastic because of Z(t). Thus, t N

t N

冮冱J(s)Y (s)Z (s)dN (s) − 冮冱J(s)Y (s)Z (s)dΛ(s) i

i

i

0i=1

t N

=

i

i

0i=1

t N

冮冱J(s)dN *(s) − 冮冱J(s)Y (s)dΛ*(s) i

i

0i=1

(5.9)

i

0i=1

t N

=

冮冱J(s)Y (s)dM *(s). i

i

0i=1

So, t

N

Λˆ(t) −

冱冮 i=1 t

=

J(s)Yi(s) dΛi*(s) = R(s) 0

t N

冮冱冤 0i=1

Yi(s)J(s)Zi(s) d(Ni(s) − Λ(s)) R(s)

冥

N

冮冱 0

Yi(s)J(s) dM i*(s). R(s) i=1

(5.10)

t

冮

Which shows that for all t, Λˆ(t) is an unbiased estimator of π(s)dΛ(s) where π(s) = P(J(s) = 1), since

0

Estimation methods for duration models 125 t

N

t

冱冮 i=1

J(s)Yi(s) dΛi*(s) = R(s) 0

t

N

冮冤 冥冤冱Y (s)Z (s)冥 dΛ(s) = 冮冤R(s)冥[R(s)]dΛ(s) 0

J(s) R(s)

J(s)

i

i

i=1

0

t

冮

= J(s)dΛ(s). 0

Furthermore, as N → ∞, P(J(s) = 1) → 1 a.s. and, hence, Λˆ(t) is a consistent estimator of Λ(t). The next two corollaries are useful applications of the Doob–Meyer decomposition theorem: Corollary 1 Let M(t) be a right-continuous martingale with respect to Ft and assume that E [M 2(t)] < ∞ for all t. Then there exists a unique rightcontinuous predictable process called the predictable quadratic variation of M(t) and denoted by 〈M,M 〉 (t) such that 〈M,M 〉 (0) = 0, E〈M,M 〉(t) < ∞ for all t and M 2(t) − 〈M,M 〉 (t) is a right-continuous martingale. Corollary 2 Let Mi(t) be a right-continuous martingale with respect to Ft and assume that E [M2i (t)] < ∞ for all t, i = 1,2. Then there exists a unique right-continuous predictable process called the predictable covariation process of M1(t) and M2(t) and denoted by 〈M1,M2〉(t) such that 〈M1,M2〉(0) = 0, E〈M1,M2〉(t) < ∞ for all t and M1(t)M2(t) − 〈M1,M2〉(t) is a rightcontinuous martingale and 〈M1,M2〉(t) is the difference of two increasing right-continuous predictable processes. Finally, we have: Theorem 5.3 Let V1(t) and V2(t) be bounded predictable processes and M1(t) and M2(t) martingales with respect to Ft such that M 2i (t) < ∞, i = 1,2. Then 冮V1(t)dM1(t)冮V2(t)dM2(t) − 冮V1(t)V2(t)d 〈M1,M2〉(t)

(5.11)

is a martingale. Finally we present a theorem that relates the compensator of M2(t) to the compensator of M(t). Theorem 5.4 Let N(t) be a counting process with compensator A(t). Assume that almost all sample paths of A(t) are continuous and that E [M 2(t)] < ∞. Then 〈M,M〉 (t) = A(t). Or in other words, M2(t) − A(t) is a martingale. Sketch of proof

Integration by parts shows that

t

冮

M (t) = 2 M(s−)dM(s) + 2

0

冱{∆M(s)} . 2

s≤t

126

The Economics of Search

Now, since M(t) = N(t) − A(t) we have ∆M(s) = ∆N(s) − ∆A(s). Substituting into above yields t

冮

M (t) = 2 M(s−)dM(s) + 2

冱{∆N(s) − ∆A(s)}

2

s≤t

0 t

t

冮

= 2 M(s−)dM(s) +

冱{∆N(s)}

2

s≤t

0

冮

= 2 M(s−)dM(s) + N(t) 0

where the second equality follows from the assumption that A(t) has no jumps, P − a.s., and the third equality follows from the fact that N(t) is a counting process and so N(t) =

冱∆N(s). s≤t

Since M(t) = N(t) − A(t) we then have t

冮

M (t) − A(t) = 2 M(s−)dM(s) + M(t). 2

0

Now M(s−) is a predictable process, so both terms on the right-hand side are martingales which shows that A(t) is the compensator of M 2(t). Q.E.D. Next, we have: Theorem 5.5

If Ni(t), i = 1, . . ., N are independent counting processes with t

冮

compensators, Ai(t), defined by λi(s)ds and Hi(t) are Ft – predictable functions 0

then N

M(t) ≡

t

冱 冮H (s)d(N (t) − A (t)) i

i

i

i=1 0

is an Ft − martingale with 1

E(M(t) = 0 for all t N

2

var(M(t)) =

t

冱 冮E [H (s)]λ (s)ds. 2 i

i

i=1 0

Applying this theorem to the Nelson–Aalen estimator we have

Estimation methods for duration models 127 t

t

冮

冢

冣

Var Λˆ(t) − J(s)dΛ*(s) = Var 0

t

=E

冢冮R(s) dM *(s)冣 J(s)

0

J(s) 2 d 〈M *,M *〉 = E R(s)

冤 冮冢 冣 0

冥

t

2

冤 冮冢R(s)冣 R(s)dΛ(s)冥 J(s)

0

t

=E

冤 冮冢R(s)冣(s)dΛ(s)冥 J(s)

0

this can be estimated by t

冮冢R(s) 冣dN *(s). J(s)

2

0

For large n, t

冮

冢

t

2

冣

lim nE Λˆ(t) − J(s)dΛ*(s) = lim E

n→∞

0

n→∞

t

冤冮冢 0

冥 冮冢π(s)冣dΛ*(s)

nJ(s) dΛ*(s) = R(s)

冣

1

0

Finally, note that t

t

冮

冢

冣

√n Λˆ(t) − J(s)dΛ*(s) = √n 0

t

1 = √n

冢冮

n

冱

冢冮R(s) dM *(s)冣 J(s)

0

nJ(s) 1 d M i*(s) = R(s) i = 1 √n 0

冣

n

t

冢冱 冮 R(s) dM *(s)冣 nJ(s)

i

i=1 0

which from the martingale central limit theorem leads to the result that the Nelson–Aalen estimator is asymptotically normally distributed. Summarizing our results for the Nelson–Aalen estimator t

冮

J(s) Λˆ(t) = dN *(s) R(s) 0 we have: Theorem 5.6 t

1

冮

Λˆ(t) is an unbiased estimator of J(s)dΛ(s). 0

128 2 3

The Economics of Search Λˆ(t) is a consistent estimator of Λ(t). √n(Λˆ(t) − Λ*(t)) is asymptotically normally distributed with mean 0 t

and variance

冮冢π(s)冣dΛ*(s). 1

0

5.3.1 Kaplan–Meier estimator of the survivor function The Nelson–Aalen estimator can be used to derive an estimator for the survivor function S(t). Note that since dΛ*(t) =

dF(t) 1 − F(t−)

we have t

t

冮

冮

S(t) = 1− (1 − F(s−))dΛ*(s) = 1 − S(s−)dΛ*(s) 0

0

So, we can think of an estimator of S(t) as being defined recursively using t

冮

Sˆ (t) = 1 − Sˆ (s−)dΛˆ(s)

(5.12)

0

where Λˆ(t) is the Nelson–Aalen estimator of the integrated hazard function. Substituting the definition of the Nelson–Aalen estimator into (5.12) yields: 1 , if dN *(t) = 1 Sˆ (t−) J(t)dN*(t) R(t) dSˆ (t) = Sˆ (t−) = R(t) 0 , if dN *(t) = 0

冢

冢 冣

冣

Since Sˆ (0) = 1 we then have Sˆ (t) =

1

dN*(s)

冲冤1 − 冢 R(s) 冣冥 = 冲冤1 − 冢R(t )冣冥. s≤t

ti ≤ t

(5.13)

i

Before turning to models with covariates, we present an example using joblessness spell data from the February 1996 Current Population Survey’s Displaced Worker Supplement (CPS-DWS). In the CPS-DWS workers who have been displaced from a job in the previous three years are asked how

Estimation methods for duration models 129 many weeks it took before they were re-employed. Joblessness duration data are right censored if the spell was ongoing at the time of the survey. For convenience we also censor all spells after 100 weeks. The Nelson–Aalen estimate of the integrated or cumulative hazard function is presented in Figure 5.3. Since the integrated hazard is discontinuous it is not possible to directly estimate the baseline hazard. However, applying kernel smoothing techniques an estimate can be derived. This is presented in Figure 5.4.

5.4 PARAMETRIC METHODS FOR CONTINUOUS-TIME DATA WITH COVARIATES

5.4.1 Time-constant covariates An especially important class of models with time-constant regressors consists of the proportional hazards model. A proportional hazards model can be written as λ(t;x) = lim ∆t↓0

P(t ≤ T < t + ∆t|T ≥ t,x) ∆t

(5.14)

where x is a vector of explanatory variables. All the formulas introduced in Section 5.2 above continue to hold provided the c.d.f. and density are defined conditional on x. Often we are interested in the partial effects of the xj on λ(t;x), which are defined as the partial derivatives for continuous xj and differences for discrete xj. While the durations defined by (5.14) refer to some “internal” time until the occurrence of an event, the impact of calendar time can be modeled by incorporating suitable covariates into x where ρ(·) > 0 is a

Figure 5.3 Nelson–Aalen cumulative re-employment hazard estimate

130

The Economics of Search

Figure 5.4 Smoothed re-employment hazard estimate

nonnegative function of x and λ0(t) > 0 is called the baseline hazard. The baseline hazard is common to all individuals in the population; individual hazards differ proportionately based on a function ρ(x) of observed covariates. Typically, ρ(·) is parameterized as ρ(x) = exp(x′β) where β is a vector of parameters.

5.4.2 Time-varying covariates Studying hazard functions is more complicated when we wish to model the effects of time-varying covariates on the hazard function. For one thing it makes no sense to specify the distribution of the duration T conditional on the covariates at only one time period. Nevertheless, we can still define the appropriate conditional probabilities that lead to a conditional hazard function. Let x(t) denote the vector of regressors at time t. For t ≥ 0, let X(t), t ≥ 0 denote the covariate path through time t: X(t) = {x(s): 0 ≤ s ≤ t}. We define the conditional hazard function at time t by λ(t;Χ(t)) = lim ∆t↓0

P(t ≤ T < t + ∆t|T ≥ t,Χ(t + ∆t)) . ∆t

(5.15)

The proportional hazard form is commonly used when covariates are timevarying: λ(t:Χ(t)) = ρ(x(t))λ0(t). Usually ρ(t) = exp[x(t)′β]. Below, we shall focus on techniques primarily for what is termed “flow sampling.” With flow sampling, the sample consists of individuals who enter the state at some point during the interval [0,t0] and we record the length of time each individual is in the initial state. Flow data are usually subject to

Estimation methods for duration models 131 right censoring. That is, after a certain amount of time (t0), we stop following the individuals in the sample, which we must do in order to analyze the data. For individuals who have completed their spells in their initial state we observe the exact duration. But for those still in the initial state, we only know that the duration lasted at least as long as the tracking period.

5.4.3 Maximum likelihood estimation with censored data1 For a random draw i from the population, let ei ∈ [0, t0] denote the time at which individuals i enters the initial state (the starting time), let t *i denote the length of time in the initial state (the duration), and let xi denote the vector of observed covariates. We assume that t *i has a continuous conditional density f(t|xi;θ), t ≥ 0, where θ is the vector of unknown parameters. Without right censoring we would observe (ei, xi) and estimation would proceed by conditional maximum likelihood estimation. To account for right censoring we assume that the observed duration ti is obtained as ti = min(t *,c i i) where ci is the censoring time for individual i. In some cases, ci is constant across i. For example if you were to track all individuals whose duration starts at the same calendar time and track them for up to two years then the common censoring time would be 104 weeks. We assume that, conditional on the covariates, the true duration distribution is independent of the starting point ei and the censoring time ci. H(t *|x i i,ei,ci) = H(t *|x i i)

(5.16)

where H(·|·) denotes the conditional distribution. Under assumption (5.16), the distribution of t *i given (xi,ei,ci) does not depend on (ei,ci). Therefore, if the duration is not censored, the density of ti = given (xi,ei,ci) is simply f(t|xi;θ). The probability that ti is censored is P(t *i ≥ ci|xi) = 1 − F(ci|xi;θ) Let di be a complete spell indicator (di = 1 of uncensored, di = 0 if censored), the conditional likelihood for observation i can be written as f(ti|xi;θ)d [1 − F(ti|xi;θ)](1 − d ). i

i

For a random sample of size N the maximum likelihood estimator of θ is obtained by maximizing N

冱{d log[f(t |x ;θ)] + (1 − d )log[1 − F(t |x ;θ)]}. i

i

i

i

i

i

i=1

For example, the Weibull distribution with covariates has conditional density

132 The Economics of Search f(ti|xi;θ) = exp(x′i β)αtα − 1exp[ − exp(x′i β)tα] where xi contains unity as its first element for all i.

5.4.4 Unobserved heterogeneity One way to obtain more general duration models is to introduce unobserved heterogeneity into fairly simple duration models. In addition, we sometimes want to test for duration dependence conditional on observed covariates and unobserved heterogeneity. The key assumptions used to incorporate unobserved heterogeneity are that: 1 2 3

Unobserved heterogeneity is independent of the observed covariates. Unobserved heterogeneity distribution is known up to a finite number of parameters. Unobserved heterogeneity enters the hazard function in a multiplicative fashion.

Before moving to a more general framework we will consider the model by Lancaster (1979). For a random draw i from the population it is assumed that the hazard function has the Weibull form conditional on the observed covariates xi and unobserved heterogeneity vi: λ(t;xi,vi) = vi exp(x′i β)αtα − 1

(5.17)

where xi1 ≡ 1 and vi > 0. To identify the parameters α and β we need to normalize the distribution of vi so that E(vi) = 1. In Lancaster (1979), it was assumed that the distribution of vi = Gamma(δ,δ) so that E(vi) = 1 and Var(vi) = 1/δ. In the general case where the c.d.f. of given (xi,vi) is F(t|xi,vi,θ) we obtain the distribution of t *i given xi by integrating out the unobserved effect. Because vi and xi are independent, the c.d.f. of t *i given xi is ∞

冮

G(t|xi;θ,ρ) = F(t|xi,vi;θ)k(v;ρ)dv 0

where it is assumed that the density of vi, k(·;ρ) is assumed to be continuous and depend on the unknown vector of parameters ρ. With censoring and flow data we should assume H(t *|x i i,vi,ei,ci) = H(t *|x i i,vi) and K(vi|xi,ei,ci) = K(vi). Suppose that the unobserved heterogeneity distribution has a gamma distribution and λ(t;xi,vi) = viλ0(t)exp(x′i β).

Estimation methods for duration models 133 Then, t

冮

冤

冥

F(t|xi,vi) = 1 − exp −viexp(x′i β) λ0(s)ds = 1 − exp[ −viexp(x′i β)Λ(t)] 0

t

冮

where Λ(t) ≡ λ0(s)ds. 0

Now the density of vi is k(v) = δδv δ − 1exp(− δv)/Γ(v) where Var(vi) = 1/δ and Γ(·) is the Gamma Function. Thus, ∞

冮

G(t|xi;θ,ρ) = 1 − exp( − viexp(x′i β)Λ(t))δδv δ − 1exp( − δv)/Γ(δ)dv 0

冢

=1− 1+

exp(x′i β)Λ(t) δ

冣

−δ

.

Why would we introduce heterogeneity when the heterogeneity is assumed to be independent of the observed covariates? In many instances in economics, such as job search theory, we are interested in testing for duration dependence conditional on the observed and unobserved heterogeneity, where the unobserved heterogeneity enters the hazard multiplicatively. As shown by Lancaster (1979), ignoring multiplicative heterogeneity in the Weibull model results in asymptotically underestimating α. Therefore, we could very well conclude that there is negative duration dependence conditional on x, whereas there is no duration dependence conditional on x and v. Returning to our example using the CPS-DWS we estimate both Weibull and Weibull-Gamma models controlling for a number of covariates including the weekly benefit amount an individual is qualified to receive (WBA) and an indicator for UI receipt (UI) and the interaction of the two (UI × WBA).2 Figures 5.5 and 5.6 display the estimated cumulative hazard and survivor function, respectively, when the covariates are set to their sample means. To investigate the impact of UI receipt, Figure 5.7 portrays the difference in the estimated survivor function for a UI recipient and UI non-recipient who both qualify for weekly benefits of $200 per week, and whose remaining covariates are fixed at their sample means. As can be seen from the figure, the survivor function of the non-recipient decreases much more rapidly indicating that they find jobs more quickly than UI recipients. Figures 5.8, 5.9, and 5.10 present graphs for the Weibull-Gamma model.

134

The Economics of Search

Figure 5.5 Weibull model: cumulative hazard estimate

Figure 5.6 Weibull model: survivor function estimate

Estimation methods for duration models 135

Figure 5.7 Weibull model: survivor function estimate by unemployment receipt

Figure 5.8 Weibull-Gamma model: cumulative re-employment hazard estimate

136

The Economics of Search

Figure 5.9 Weibull-Gamma model: survivor function estimate

Figure 5.10 Weibull-Gamma model: survivor function estimates by unemployment insurance receipt

5.5 THE COX REGRESSION MODEL

5.5.1 Data with no ties The models presented above impose parametric assumptions on the baseline hazard function. In many instances economic theory provides little help in identifying a particular parametric class. However, if the true baseline hazard function does not belong to the assumed parametric class of functions, estimates will generally be biased. Cox (1972, 1975) developed a technique for obtaining estimates of the β without imposing any parametric form on the

Estimation methods for duration models 137 baseline hazard. This technique is referred to as Cox regression. The model was developed with continuous time duration data in which the probability of two durations ending at the same time equals zero. In most applications however, duration data is grouped to some extent and the model has been modified to accommodate “ties” in the data. First, however, we will look at the case of “no ties.” The Cox regression model assumes that the conditional hazard function follows a proportional hazards model: λ(t;xi) = λ0(t)exp(x′i β). The benefit of the Cox regression method is that it makes no assumptions about the form of the baseline hazard function λ0(t). In fact the estimation method “partials out” the baseline hazard so that it does not appear in the maximand; the only parameters that appear are the regression coefficients. Thus, Cox regression is a semi-parametric estimation method. Cox regression estimation relies on forming a risk pool or risk set at each failure time in the data. The risk set at failure time t′ includes all individuals, i, with ci greater than or equal to t′. Thus at t′ an individual is in the risk set if the event has not occurred before that time or they have not been censored. The partial likelihood function is then constructed by considering the conditional probabilities of failure at each failure time. For example suppose that there are five observations in the data such that:

Obs.

ti

di

xi

1 2 3 4 5

3 5 6 9 11

1 0 1 1 1

2 2 1 0 1

Let R(tj) be the risk set at the (ordered) failure time tj. Thus, R(3) = {1,2,3,4,5}, R(6) = {3,4,5}, R(9) = {4,5} and R(11) = {5}. At each event time tj, we consider the conditional probability that the event occurs for the particular observation among those observations remaining in the risk set “just before” tj, conditional on one event occurring at tj . For any observation, i, in the risk set at time tj the probability of failure at tj approximately equals λ(tj;xi). Thus, if observation j fails at time tj, then the conditional probability of observing j equals exp(x′j βj) exp(x′j βj) or simply where Rj ≡ R(tj). 冱 exp(x′i βi) 冱 exp(x′i βi)

i ∈ R(tj)

i ∈ Rj

138

The Economics of Search

An alternative way to think about the construction of the partial likelihood is to think of drawing balls from an urn at each failure time. A ball is included in the urn for individual i at time tj only if individual i has not been censored or has not failed before time tj. Instead of equal probabilities the relative probability of drawing the ball associated with individual i equals exp(x′i β). exp(x′j βj) Thus probability of drawing individual j equals . 冱 exp(x′i βi) i ∈ R(tj)

The partial likelihood is formed by the product of these conditional probabilities over all failure times: K

exp(x′j β)

冲冢 冱 exp(x′β)冣

PL(β) =

j=1

i

i ∈ Rj

or K

冱x′β − log冢 冱 exp(x′β)冣

log PL(β) =

j

(5.18)

i

i ∈ Rj

j=1

Estimates are obtained by maximizing (5.18) with respect to β. Let βˆ denote the value of β that maximizes (5.18). Then, the first order conditions for a maximum state that the vector βˆ must satisfy: K

s(βˆ) =

冱冢x − j

j=1

冱 xiexp(x′i βˆ)

i ∈ Rj

冱 exp(x′i βˆ)

冣 = 0.

i ∈ Rj

The vector function s is usually referred to as the score function. From our discussion of counting processes it is clear that t

t

冮

冮

Mi(t) = Y(s)Z(s)dNi(s) − Y(s)Z(s)λ0(s)exp(xi β)ds 0

0

is a martingale. Defining the predictable function Hj(t,β,x1,x2, . . .,xn) as 冱 xiexp(x′i β)

Hj(t,β,x1,x2, . . .,xn) = xj −

i ∈ Rj

冱 exp(x′i β)

i ∈ Rj

we have n ∞

sj(β) =

冱冮H (s,β,x ,x , . . .,x )Y (s)Z (s)dN (s) j

j=10

1

2

n

j

j

j

Estimation methods for duration models 139 n ∞

冱冮H (s,β,x ,x , . . .,x )Y (s)Z (s){dM (s) − Y (s)Z (s)λ (s)exp(x′β)}

=

1

j

2

n

j

j

j

j

0

j

i

j=10

n ∞

冱冮H (s,β,x ,x , . . .,x )Y(s)Z(s)dM (s) −

=

1

j

2

n

(5.19)

j

j=10

n ∞

冱冮H (t,β,x ,x , . . .,x )Y (s)Z (s)λ (t)exp(x′β) 1

j

2

n

j

0

j

i

j=10

n ∞

=

冱冮H (s,β,x ,x , . . .,x )Y(s)Z(s)dM (s) 1

j

2

n

j

j=10

since ∞

n

冱 冮H (s,β,x ,x , . . .,x )Y (s)Z (s)λ (s)exp(x′β)ds 1

j

2

n

j

0

j

j

j=1 0

∞

n

=

冱 冮冢x − j

j=1 0

∞

=

冱 xi exp(x′i β)

i ∈ Rj

冱 exp(x′i β)

冣Y (s)Z (s)λ (s)exp(x′β)ds j

n

冮冢冱Y (s)Z (s)x exp(x′β) − j

j

j

冣λ (s)ds

i ∈ Rj

j

0

冱 exp(x′i β)

i ∈ Rj

exp(x′j β) 冱 xi exp(x′i β)

∞

冮冢 冱 x exp(x′β) − 0

j

exp(x′j β) 冱 xi exp(x′i β)

0 j=1

=

0

j

i ∈ Rj

i ∈ Rj

j

i ∈ Rj

冱

j

冱 exp(x′i β)

i ∈ Rj

冣λ (s)ds = 0. 0

i ∈ Rj

Using (5.19) and appealing to results from laws of large numbers and martingale central limit theory, it can be shown that the estimates are both consistent and √n − asymptotically normal with the variance-covariance matrix equal to

冤−E 冢−

∂2 log(PL(β)) ∂β∂β′

冣冥

−1

which can be consistently estimated by k

冤冱 j=1

−

∂2 log(PLj(βˆ)) ∂β∂β′

冥

−1

k

or

∂ log(PLj(βˆ)) ∂ log(PLj(βˆ)) ∂β ∂β′ j=1

冤冱

−1

冥

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The Economics of Search where the vector βˆ denotes the vector of Cox regression estimates and exp(x′j βˆ) . PLj(βˆ) = 冱 exp(x′i β) i ∈ Rj

5.5.2 Data with ties With ties in the data the exact partial likelihood becomes more complex although there are some approximation methods that reduce the complexity and perform well as long as the number of ties are “small”. Suppose at each event time j, dj events occur and let Dj be the set of individuals for which the event occurs at time j. Returning to the urn analogy, then, instead of drawing one ball from the urn at time j, we draw dj balls without replacement. For a particular sequence s = {j(1), . . ., j(dj)} of draws the probability of observing that sequence equals dj

exp(x′j(q)βj(q)) . 冱 exp(x′i βi)

冲

q=1

(5.20)

i ∈ R(tj) − {j(1), . . ., j(q − 1)}

Thus, the probability equals dj

j

exp(x′j(q)βj(q)) 冱 exp(x′i βi)

冲

1 冱 dj! s ∈ P(D ) q = 1

i ∈ R(tj) − {j(1), . . ., j(q − 1)}

where P(Dj) represents the set of permutations of the indices of the Dj individuals who fail at time tj. Since the construction of the exact partial likelihood with ties can be quite complex, several approximations have been suggested. Perhaps the most well know is that by Breslow (1974) who essentially substitutes sampling with replacement for sampling without replacement in the urn analogy. Define sj = 冱 xk. k ∈ Dk

Then the Breslow approximation to the partial likelihood equals: K

PLB(β) =



冲  j=1

exp(s′j β) dj

冱 exp(x′β)冥 冤 i ∈ Rj

i

 .  

Estimation methods for duration models 141 An alternative approximation by Efron (1977) adjusts the denominator of the Breslow approximation to the partial likelihood by subtracting a term for the number of balls drawn from the urn. But instead of deducting the probability weights for the actual balls drawn and then averaging over all permutations, Efron (1977) deducts the average probability weight where the average is over all dj in Dj. The Efron (1977) approximation to the partial likelihood equals: K

PLE(β) =

冲 

 . j=1 k − 1  冲冤 冱 exp(x′j β) − 冱 exp(x′j β)冥  di i ∈ D k=1 i∈R  exp(s′j β)

di

j

j

Since the Cox regression partial likelihood eliminates the baseline hazard function it does not produce estimates of the baseline hazard function. The estimates of β, however, can be used to estimate the cumulative baseline hazard, Λ0(t), using an estimator that weights the data using exp(x′i βˆ): t

Λˆ0(t) =

冮 0

J(s)

dN*(s).

n

冱Y (s)Z (s)exp(x′βˆ) i

i

i

i=1

An estimate of the baseline survivor function, S0(t), then equals:

冢

冣

Sˆ 0(t) = exp −Λˆ0(t) . Returning to the joblessness example presented above, Figures 5.11 and 5.12

Figure 5.11 Cox regression model: cumulative hazard function estimate

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Figure 5.12 Cox regression model: survivor function estimate

Figure 5.13 Cox regression model: survivor function estimate by unemployment insurance receipt

present estimates of the cumulative hazard and survivor functions when covariates are fixed at the sample mean while Figure 5.13 presents estimates of the survivor function for a UI recipient and nonrecipient who qualify for $200 per week in benefits and whose other covariates are fixed at the sample mean.

5.5.3 Stratified Cox regression In some circumstances the proportional hazards assumption may be inappropriate. If the suspect variable is a categorical variable, then one can relax the proportional hazards assumption for that variable by estimating a

Estimation methods for duration models 143 stratified model. Suppose the variable w has H categories and that, a priori, you suspect that the hazards are non-proportional across the H categories. Then the stratified Cox regression involves maximizing the partial likelihood function: H

SPL(β) =

Kh

exp(x′j β)

冲 冲 冢 冱 exp(x′β)冣 h

h = 1 jh = 1

i ∈ R jh

i

where x includes all other variables except w.

5.6 DISCRETE-TIME DURATION DATA

5.6.1 Time-constant covariates Most duration data available in economics is grouped. That is, durations are only known to fall into certain time intervals, such as weeks, months, or even years. For example, unemployment duration data are typically grouped into weeks. One econometric approach that is taken when analyzing grouped duration data summarizes the information on staying in the initial state or exiting that state in each time interval using a sequence of binary responses. As noted by many (e.g. Wooldridge 2003) essentially we have a panel data set where the duration of an individual determines a vector of binary responses. These in conjunction with the covariates can be thought of as creating an unbalanced panel where the number of observations per individual equals Ki = min(Di, Ci) where Di equals the number of periods until the event occurs and Ci equals the number of periods until the observation is (right) censored. In addition to allowing us to treat grouped durations, the panel data approach has at least two advantages. First, in a proportional hazard specification it leads to simple methods for estimating flexible hazard functions. Second, because of the sequential nature of the data, time varying covariates are easily introduced. Throughout this discussion we will assume flow sampling. We divide the time line into M + 1 intervals, (0, k1), (k1, k2), . . ., (kM−1, kM), (kM, ∞), where km are known constants. For example, we might have k1 = 1, k2 = 2, k3 = 3, and so on, but unequally spaced intervals are also feasible. The last interval is chosen so that any duration falling into it is censored at kM: no observed durations are greater than kM. For a random draw from the population, Let ym represent a binary indicator equal one if the event occurs in the mth interval and zero otherwise. For each person i, we observe (yi1, . . .,yiK ) which is an unbalanced panel data set of length Ki. Note that the string of binary indicators is either a sequence of all zeros or a series of zeros ending with a one where the former sequence is i

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observed when the observation is censored and the latter sequence is observed when the series of zeros ends because an event occurred. Let δik = I(Ci = k), ki = 1, . . . Ki. With time invariant covariates, each draw from the population is, {(yi1,δi1), . . ., (yik ,δik ),x}. We assume that a parametric hazard function is specified as λ(t; x,θ), where θ is a vector of unknown parameters. Let T denote the time until exit from the initial state. While we do not fully observe T, either we know which interval it falls into, or we know whether it was censored in a particular interval. Thus we can compute i

i

p(ym = 0|ym − 1 = 0, δm − 1 = 0,x,), p(ym = 1|ym − 1 = 0, δm − 1 = 0,x,), m = 1, . . . M. To compute these probabilities in terms of T, we assume that the duration is conditionally independent of censoring: T is independent of C given C. Thus, P(ym = 1|ym − 1 = 0, δm − 1 = 0,x) = P(km − 1 ≤ T < km|T ≥ km − 1) km

冤 冮 λ(s;x,θ)ds冥

= 1 − exp −

km − 1

km

冤 冮 λ(s;x,θ)ds冥.

≡ 1 − αm(x,θ), m = 1, . . ., M where αm(x,θ) ≡ exp −

km − 1

Therefore, P(ym = 1|ym − 1 = 0, δm − 1 = 0,x,) = αm(x,θ). We can use these probabilities to construct the likelihood function. If, for observation i, uncensored exit occurs in interval mi, the likelihood function is Ki − 1

冤 冲 α (x ,θ)冥[1 − α (x ,θ)]. m

i

ki

(5.21)

i

m=1

The first term represents the probability of remaining in the initial state for the first ki − 1 intervals, and the second term is the (conditional) probability that T falls into interval ki. If the duration is censored in interval ki, we know only that exit did not occur in the first ki − 1 intervals, and the likelihood consists of only the first term in expression (5.21). If di is a censoring indicator equal to one if duration i is uncensored, the log likelihood for observation i can be written as ki − 1

log(Li) =

冱 log[α (x ,θ)] + d log[1 − α (x ,θ)] m

m=1

i

i

ki

i

(5.22)

Estimation methods for duration models 145 Thus, for a sample of size N, the log likelihood function is N

log(L) =

N

ki − 1

冱log(L ) = 冱 冱 log[α (x ,θ)] + d log[1 − α (x ,θ)] i

m

i

i

ki

i

(5.23)

i=1 m=1

i=1

To implement conditional maximum likelihood estimation (MLE), we must specify a hazard function. One hazard function that is popular is a piecewiseconstant proportional hazard: for m = 1, . . ., M, λ(t;x,θ) = γ(x,β)λm,

km − 1 ≤ t ≤ km.

With a piecewise constant hazard and γ(x,β) = exp(x′β) for m = 1, . . ., M we have αm(x,θ) ≡ exp[−exp(x′β)λm(km − km − 1)] where km are known constants (often km = m). Alternatively one could assume an underlying continuous baseline hazard and define km

α0m =

冮 λ (s)ds, 0

m = 1, . . ., M

km − 1

and αm(x,θ) ≡ exp[−exp(x′β)α0m].

(5.24)

Without covariates, maximum likelihood estimation of the α0m in (5.24) leads to a well-known estimator of the survivor function. We can motivate the estimator from the representation of the survivor function as a product of conditional probabilities. For m = 1, . . ., M, the survivor function at time km can be written as m

S(km) = P(T > km) =

冲P(T > k |T > k r

)

r−1

r=1

Now, for each r = 1, 2, . . ., M let Nr denote the number of people in the risk set for interval r: Nr is the number of people who have neither left the initial state nor been censored by kr − 1. Therefore, N1 is the number of individuals in the initial random sample; N2 is the number of people who did not exit the initial state in the first interval, less the number of individuals censored in the first interval, and so on. Let Er be the number of people observed to leave in the rth interval. A consistent estimator of

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The Economics of Search P(T > kr| T > kr − 1) is

(Nr − Er) , r = 1, 2, . . ., M. Nr

(5.25)

It follows from equation (5.25) that a consistent estimator of the survivor function at time kn is m

Sˆ (km) =

(Nr − Er)

冲冤

Nr

r=1

冥,

m = 1, 2, . . ., M.

This is the discrete-time Kaplan–Meier estimator of the survivor function (at points k1, k2, . . ., km). We can derive this Kaplan–Meier estimator by maximizing the likelihood function N

L=

ki − 1

冲 冤 冲 α 冥(1 − α ) m

i=1

ki

di

(5.26)

m=1

with respect to αm, m = 1, . . . M where di is an indicator variable that equals one if the individual spell is not censored. Taking the log of (5.26) gives

ln(L) =

N

ki − 1

i=1

m=1

冱 冤 冱 ln(α )冥 + d ln(1 − α m

i

ki

)

and then rearranging terms yields M

ln(L) =

冱冤

m=1

冱 ln(1 − αm) + 冱 ln(αm)

i ∈ S(m)

i ∈ D(m)

冥

where S(m) denotes those individuals who survivor past m and D(m) denotes the set of individuals for which the event occurs during m. This reduces to M

ln(L) =

冱 冤S (m)ln(α ) + D (m)ln(1 − α )冥 #

m

#

m

m=1

where S #(m) (D#(m) denotes the number of individuals in S(m) (D(m)). Differentiating the log likelihood and setting it to zero yields S #(m) or

1 1 = D#(m) αˆ m 1 − αˆ m

Estimation methods for duration models 147 (S#(m) + D#(m))αˆ m = S#(m). Solving for αˆ m then yields αˆ m =

S #(m) Nm − Em = . S (m) + D#(m) Nm #

Now m

Sˆ (am) =

m

Nr − Er Nr

冲αˆ = 冲冢 r

r=1

r=1

冣

which was the Kaplan–Meier estimator above.

5.6.2 Time-varying covariates For the population, let x1, x2, . . ., xM denote the outcomes of the covariates in each of the M time intervals and let X = (x1, x2, . . ., xM), where we assume that the covariates are constant within the interval. In general we will let Xr = (x1, x2, . . ., xr). We assume that the hazard at time t conditional on the covariates up through time t depends only on the covariates at time t. If past values of the covariates matter, they can simply be included in the covariates at time t. The conditional independence assumption on the censoring indicators is now stated as D(T = km|T ≥ km − 1,xm,δm) = D(T = km|T ≥ km − 1,xm),

m = 1, 2, . . ., M

This assumption allows the censoring decision to depend on the covariates during the time interval (as well as past covariates, provided they are either included in xm or do not affect the distribution of T given xm). Under this assumption, the probability of exiting (without censoring) is P(ym = 1|ym − 1 = 0,xm,δm = 0) = P(km − 1 ≤ T < km|T ≥ km − 1,xm)

(5.27)

km

冤 冮 λ(s;x ,θ)ds冥 ≡ 1 − α (x ,θ).

= 1 − exp

m

m

m

km − 1

We can use equation (5.27) along with P(ym = 0|ym − 1 = 0,xm,δm = 0) = αm(xm,θ) to construct the log likelihood for person i as in (5.22) and the sample log likelihood (5.23).

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5.6.3 Unobserved heterogeneity Unobserved heterogeneity can also be added to hazard models for grouped data. For example, adding unmeasured heterogeneity to (5.24) and letting ξi = exp(vi) gives αm(xm,ξi,θ) ≡ exp[−ξi exp(x′m β)α0m]

(5.28)

Now the survivor function associated with (5.28) equals m

S(m;Χm,ξ,θ) =

m

冤冲α (x ,ξ,θ)冥 = 冤冲exp[−ξ exp(x′ β)α ]冥 r

r

0r

r

r=1

r=1

m

冤 冱exp(x′ β)α 冥.

= exp −ξ

(5.29)

0r

r

r=1

Let ξ have c.d.f. G(ξ). Then m

冤 冱exp(x′ β)α 冥dG(ξ).

S(m;Χm,θ) = 冮exp −ξ

r

0r

r=1

When G is a gamma distribution with E(ξ) = 1 and Var(ξ) = σ2 then (5.29) becomes m

冢

S(m;Χm,θ) = 1 + σ

− σ1

冱exp(x′ β)α 冣

2

r

2

0r

.

(5.30)

r=1

We can then use (5.30) to form the log likelihood function by noting that the probability of the event ending in period m equals S(m − 1;Xm − 1, θ) − S(m;Xm, θ). Letting di equal 1 if event occurs and 0 otherwise (censored) we have the log-likelihood function N

ln(L) =

冱(1 − d )ln[S(k − 1;Χ i

i

,θ)] + di ln[S(ki − 1;Χk − 1,θ) − S(ki;Χk,θ)]

k−1

i=1

where θ = (α′,β′,σ2)′ with α′ = (α01,α02, . . ., α0M). Other unobserved heterogeneity distributions are: m

1 2

冢σ 冤1 − (1 + 2σ 冱exp(x′ β)α )冥 冣

Inverse Gaussian: S(m;Χm,θ) = exp

1

2

r

2

r=1

Stable distribution (Hougaard,1986): m

冱

1 1 − (1 + cσ2 exp(x′r β)α0r) σ2 r=1

冢 冤

S(m;Χm,θ) = exp

1 c

冥 冣

0r

Estimation methods for duration models 149 J

Mass-point distribution: S(m;Χm,θ) =

m

冱p exp(−ξ 冱exp(x′ β)α 冣 j

j

j=1

r

0r

r=1

J

where J equals the number of mass points and

冱p = 1. Rather than fixing j

j=1

the mean of the mixing distribution to 1 for this distribution, empirical implementation is easier by instead fixing α01 = 1.3 5.7 MULTI-SPELL DISCRETE-TIME MODELS Suppose that instead of a single duration we have multiple durations. For example, we may be interested in examining consecutive unemployment durations. The survivor function for the gth spell satisfies m

g

g

S (km) = P(T > km) =

冲P(T

g

> kr|T g > kr − 1)

r=1

We shall assume that P(T g > kmg |T g > kgm − 1) = αmg (xmg ,θ) ≡ exp[−exp(xmg′ βg)αgm0] or P(T g = k gm|T g > k gm − 1) = 1 − αmg (xmg ,θ) ≡ 1 − exp[−exp(xmg′ βg)α gm0]. Thus, m

g

g m

g m

S (k ) = P(T > k ) =

冲exp[−exp(x

g′ r

βg)α0rg ]

r=1

We will have data for up to G spells for an individual. If an individual completes the gth spell with Kg = kg that spell contributes kig−1

P(Kg = kgi) =

冦冲exp冤−exp(x β )α 冥 冧冢exp冤−exp(x β )α 冥冣 g′ g r

g 0r

g′ g kig

g 0kig

r=1

to the likelihood function. If they are censored in the gth spell at time kgi then that spell contributes kig

g

g i

P(K = k ) =

冦冲exp[−exp(x β )α ]冧 g′ g r

r=1

g 0r

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The Economics of Search

to the likelihood function. Let the spell indicator variables vgi = 1 if an individual enters the gth spell and zero otherwise, g = 1, . . .,G and define the censor variables d gi = 1 if the individual completes the gth spell and zero otherwise, g = 1, . . .,G. Then each individual contributes kig − 1

G

Li =

冲冢冦 冲

g=1

冧

exp[−exp(xg′r βg)α0rg ] (1 − exp[−exp(xkg′ βg)αg0k ])d

r=1

g i

g i

冣

vgi

g i

to the likelihood function which is N

L=

N

冲

Li =

i=1

G

kig − 1

冲 冲冤冦 冲 i=1 g=1

冧

exp[−exp(xg′r βg)]α0gr (1 − exp[−exp(xkg′ βg)αg0k ])d

r=1

g i

ν ig

冥

g i

g i

and the log-likelihood function equals N

ln(L) =

kig − 1

G

vgi

冱 冱冢 冱 ln{exp[ − exp(x β )α ]}(1 − exp[−exp(x β )α g′ g r

g 0r

g′ g kig

g 0kig

])

冣

dgi

i=1 g=1 r=1 N

=

kig − 1

G

冱 冱v 冤冦− 冱 exp(x β )α 冧 + d g i

i=1 g=1

g′ g k

g 0r

g i

ln(1 − exp[−exp(xkg′ βg)αg0k ]) g i

g i

r=1

冥

Under the assumption that spell durations are independent, the likelihood decomposes and one can obtain estimates for this model by estimating G single spell models where all individuals with vgi = 1 are included in the gth estimation. If we instead assume that βg = β and αg0m = α0m for all g then we can “stack” observations for each spell within each individual and estimate a single spell duration model with log likelihood N

ln(L) =

kig − 1

G

冱 冱v 冤冦− 冱 exp(x β)α 冧 + d g i

i=1 g=1

g′ r

0r

g i

冥

ln(1 − exp[−exp(xkg′ β)α0k ]) . g i

g i

r=1

An intermediate case would be a model that restricts βg = β for all g but allows the baseline hazard parameters to be spell dependent. N

ln(L) =

G

kig − 1

冱 冱 冱 {−v

g i

exp(xg′r β)αg0r + d gi ln(1 − exp[−exp(xgk ′β)αg0k ])}. g i

g i

i=1 g=1 r=1

This case is similar to the continuous duration Cox regression model that stratifies the stacked data by spell. One could employ single spell discrete duration methods to estimate such a model by stacking the data and incorporating (G−1) × M time-varying covariates that are of the form xgm = I(K = k)×I(S = g) where S is a variable denoting the particular spell.

Estimation methods for duration models 151 Estimation becomes more complicated if we assume that for each duration P(T g > kmg |T g > kmg − 1) has the form g P(T g > kmg |T g > kmg − 1) = αmg (xmg′,θg) ≡ exp[−ξg exp(xmg′βg)α0m ]

where ξg are unobserved random variables which are assumed independent of the covariate processes xmg , g = 1, . . .,G. In general, the ξg may be correlated with each other. Denote this joint distribution of the G × 1 vector ξ by G(ξ ;δ) where we have assumed that the distribution can be parameterized by the Q × 1 vector δ. The unconditional log likelihood function is obtained by integrating out ξ. Thus we have N

ln(L) =

G

kig − 1

冱冮ln冢冲冤冦 冲 (exp(−ξ

g

i=1

g=1

冧

exp(xg′r βg)α0rg )

r=1

(5.31)

g i

v

冥 冣dG(ξ ; δ)

× (1 −exp [−ξg exp (xgk ′βg)α0kg ])d g i

g i

g i

Estimates are obtained by maximizing (5.31) with respect to the βg, α0mg, and δ. Maximum likelihood estimation may prove computationally intensive since the integral in (5.31) is typically multivariate. One may assume a masspoint specification for G where there are M types of individuals in the population and each type as a unique G × 1 vector θ of location parameters. Let pq denote the proportion of the qth type in the population, q = 1, . . .,Q. Then the log-likelihood (5.31) becomes N

ln(L) =

Q

kig − 1

G

冱 ln冢冱 p 冲 冤冦 冲 (exp(−ξ

g q

q

i=1

q=1

g=1

冧

exp(xgr′βg)α0rg )

r=1

vig

× (1 − exp[−ξ exp(x β )α ]) g q

g′ g kig

g 0kig

冥冣

d ig

This likelihood is then maximized with respect to βg, αg0m, g = 1, . . .,G and ξq, Q

and pq, q = 1, . . .,Q where

冱p

q

= 1.

q=1

5.8 COMPETING RISK MODELS In many cases spells end for different reasons. Individuals may quit a job or be laid off, a person may die because of cancer or a heart attack, reemployment may occur into a job that is part-time or full-time. In such cases

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the explanatory variables may have differing effects on the relative probabilities of spells ending for particular reasons. The competing risks framework is meant to allow for this possibility. While without regressors it is not possible in general to distinguish models with correlated risks from those with independent risk, Heckman and Honoré (1989) have derived sufficient conditions on the regressors which enable such identification. We shall assume that such regressors exist. Also, for simplicity, in the discussion below we focus exclusively on the case of two risks. Extension to cases where the number of risks exceeds two is straightforward. Competing risk models assume that there are two latent duration variables, T1 and T2, which represent the time until the occurrence of the type 1 and type 2 events, respectively. What is observed, however, is only T = min(T1, T2) and an indicator I{T = T1}. This is referred to as the “identified minimum.” Thus, we know not only how long it took before at least one of the two types of events occurred but also which one it was. For example, if you were laid off from your job after T months of tenure, we know that you had not quit and were not laid off before this time. We also know the reason for job spell ending (i.e. layoff). What we don’t know is when (if ever) you would have quit your job had you not been laid off first. As in McCall (1996, 1997) we assume that duration data are discrete and proceed by specifying the joint survivor function for the two latent durations T1 and T2 which is denoted by S (T1, T2 ).4 In particular we assume that P(T 1 > k1r |T 1 > k1r − 1) = α1r (x′r,ξ1) ≡ exp[− ξ1 exp(x′r β1)α10r]

(5.32)

P(T 2 > k2r |T 2 > k2r − 1) = α2r (x′r,ξ2) ≡ exp[− ξ2 exp(x′r β2)α20r]

(5.33)

and

where we assume that the variables ξ1 and ξ2 are unobserved and independent of the observed explanatory variables. Correlated risks arise in this model to the extent that ξ1 and ξ2 are correlated. From (5.32) and (5.33) the latent survivor function satisfies t1

1

2

S(t1,t2|Χ,ξ ,ξ ) =

冲exp[ − ξ

t2

1

t=1

exp(x′t β )α ] 1

1 0t

冲exp[− ξ

冤 冱

t2

冱exp(x′β )α 冥

exp(x′t β1)α10t − ξ2

t

exp(x′t β2)α20t]

t=1

t1

= exp − ξ1

2

2

t

2 0t

t

where Χ = {x1,x2, . . .,xmax(t ,t )}. Let G be the distribution function for the unobservables ξ1 and ξ2. Then the unconditional survivor function satisfies 1

2

Estimation methods for duration models 153 k1

冮 冤 冱exp(x′β )α

S(k ,k |Χ) = exp −ξ 1

2

1

1

t

1 0t

k2

−ξ

冱exp(x′β )α 冥dG(ξ ,ξ ).

2

t

2

t

2 0t

1

2

t

(5.34)

To construct the likelihood function in this case suppose the ith individual fails at time k due to cause 1. Then, P(k = min(k1,k2),I(k = k1) = 1|Χ,ξ1,ξ2) = S(k − 1,k − 1|Χ,ξ1,ξ2)[P({k − 1 < T 1 ≤ k} ∩ {T 1 < T 2}|{T 1 > k − 1} ∩ {T 2 > k − 1},Χ,ξ1,ξ2)] = S(k − 1,k − 1|Χ,ξ1,ξ2) + [P(k − 1 < T 1 ≤ k|{T 1 > k − 1} ∩ {T 2 > k},Χ,ξ1,ξ2) + P({k − 1 < T 1 ≤ k} ∩ {T 1 < T 2}|{T 1 > k − 1} ∩ {k − 1 < T 2 ≤ k},Χ,ξ1,ξ2) ] = (S(k − 1,k − 1|Χ,ξ1,ξ2) − S(k − 1,k|Χ,ξ1,ξ2)) + 1/2[S(k − 1,k − 1|Χ,ξ1,ξ2) + S(k,k|Χ,ξ1,ξ2) − S(k − 1,k|Χ,ξ1,ξ2) − S(k,k − 1|Χ,ξ1,ξ2) ] = (S(k − 1,k − 1|Χ,ξ1,ξ2) − S(k,k − 1|Χ,ξ1,ξ2)) + A(k|Χ,ξ1,ξ2) where A(k|Χ,ξ1,ξ2) ≡ 1/2[S(k − 1,k − 1|Χ,ξ1,ξ2) + S(k,k|Χ,ξ1,ξ2) − S(k − 1,k|Χ,ξ1,ξ2) − S(k,k − 1|Χ,ξ1,ξ2) ] assuming that the chances of T1 < T2 given that T1 and T2 both fall in the interval (k − 1, k) = 1/2. In a similar manner we have P(k = min(k1,k2),I(k = k2) = 1|Χ,ξ1,ξ2) = (S(k − 1,k − 1|Χ,ξ1,ξ2) − S(k,k − 1|Χ,ξ1,ξ2)) + A(k|Χ,ξ1,ξ2) Let c1i ( c2i ) be an indicator variable that equals 1 if the ith person spell ends for reason 1 (2) and let c3i be an indicator variable that equals one if the spell is right censored. Then the log likelihood function for the competing risks model Log(L) satisfies: N

ln(L) =

冱c

1 i

ln( 冮{S(k − 1,k − 1|Χi,ξ1,ξ2) − S(k − 1,k|Χi,ξ1,ξ2)

i=1

− A(k|Χ,ξ1,ξ2)}dG(ξ1,ξ2))+ c2i ln(冮{S(k − 1,k − 1|Χi,ξ1,ξ2) − S(k,k − 1|Χi,ξ1,ξ2) − A(k|Χ,ξ1,ξ2)}dG(ξ1,ξ2)) + c3i ln(冮S(k − 1,k − 1|Χi,ξ1,ξ2)dG(ξ1,ξ2))

(5.35)

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The Economics of Search

5.9 GENERAL DISCRETE-TIME LIFE HISTORY MODELS More generally, we consider a discrete-time life history process that is characterized by a discrete-valued state space S and the following conditional transition probabilities Pr(Fk − 1) for r ∈ S which represents the conditional probability that the process is in state r at time k given the history of the process (information) up to (discrete) time k − 1. The history would include not only the past history of transitions of the process but also the history of, possibly time-varying, covariates. If a more coarse history is observed, Gk, with Gk ⊂ Fk we will instead have EG (Pr(Fk − 1)). For example, a particular covariate that affects the transition probabilities may not be observed. A specific form of this probability may be k−1

g (k − 1) Pr(Fk − 1) = 1 − exp(−ξrgs(k(k−−1)1) exp(x′k βgs(k(k −− 1)1))α0m (k − 1)) s

s

s

s

where s(k − 1) is the state occupied at time k − 1, gs(k − 1) represents how many visits to state s have occurred by k − 1 with r

k=

g(s)

冱 冱k

g s

s=1 g=1

and kgs represents the time spent in state s during its gth visit. Note that the s, g(s) and kgs are random variables at time 0. Now, S

冱

S

Pr(Fk − 1) =

s=1

冱1 − exp(− ξ

rg(s) s

g(s) exp(xk β g(s) s )α s0k ) = 1 g s

s=1

and by convention we fix the probability of remaining in the state at t − 1 to S

s(k − 1)

P

(Fk − 1) = 1−

冱

g( j ) [1 − exp(− ξrg(j) exp(x′k β g(j) j j )α j0k )]. g j

j ≠ s(t − 1)

Suppose that we observe only Gk − 1, then g(s) Pr(Gk − 1) = 冮{1 − exp(− ξrg(s) exp(x′k β g(s) s s )α s0k )}dPG . g s

k−1

In particular we assume that durations in each state are observed as well as all x’s. Only the variables ξrg(s) are unobserved and the ξrg(s) are F0 measurable and s s distributed independently of the x’s. Then Pr(Gk − 1) = 冮(Pr(Fk − 1)L(Fk − 1| F0))dB(ξ) where L(Fk − 1| F0) represents the probability (“likelihood”) of observing a particular history Fk − 1 given F0 and B(ξ) is the distribution of the vector ξ.

Estimation methods for duration models 155 Essentially we “integrate out” the vector of variables ξ. Suppose for individual i we observe the sequence of times spent in states and the x’s. Further suppose that the distribution B can be characterized by the parameter δ. Since we have k−1

L(Fk − 1| F0) =

冲 L(F | F j

)=

j−1

j=1

Then the individual contribution to the likelihood function equals ki

Li =

冮冢冲L(F | F )冣dB(ξ,δ) j−1

j

j=1 ki

=

冮冢冲P

r(j)

冣

(Fj − 1) dB(ξ,δ)

j=1

ki

=

冮冲[1 − exp(− ξ

rg(s(j − 1)) s(j − 1)

− 1)) g(s(j − 1)) I(s(j) ≠ s(j − 1) exp(xj′βg(s(j s(j − 1) )α s(j − 1)0k )] s j−1

j=1

S

× [1 −

冱

− 1)) − 1)) [1 − exp( − ξrg(s(j exp(xj′β g(s(j s(j − 1) s(j − 1) )

r ≠ s(j − 1)

×α

g(s(j − 1)) s(j − 1)0ksj − 1

)]]I(s(j) = s(j − 1)dB(ξ,δ)

The log-likelihood function is then N

ln(L) =

冱ln(L ) i

i=1

which is maximized with respect to the parameters βgs, α gs0m, and δ. One example which applies these discrete-time life history methods would be the analysis of employment to unemployment and unemployment to employment transitions when workers may have access to unemployment insurance. In the United States and Canada, workers can still receive benefits while working for jobs with low earnings. In particular, in Canada workers are allowed to earn the maximum of $50 or 25 percent of their weekly benefit amount with no reduction in unemployment insurance benefits. For any additional earnings above this amount, weekly benefits are reduced dollar for dollar. Given the possibility that benefits may be received while employed or unemployed, it may be desirable to specify an econometric model with four distinct states: employed-benefits (EB), employed-no benefits (ENB), unemployed-benefits (UB), unemployed-no benefits (UNB).

156

The Economics of Search

The transitions UNB → ENB, UB → ENB, ENB → UNB, and UNB → UB would commonly be observed where the latter transition occurs when an already unemployed individual files for a claim or when a newly unemployed individual satisfies a waiting period that is required by law (e.g. in Canada there is a two week waiting period before benefits can be received). Other transitions that are possible but perhaps less common are the UB → UNB transition that occurs when an individual exhausts unemployment insurance benefits or is disqualified from receiving benefits for some reason (e.g. inadequate job search). The UB → EB occurs when an individual receiving unemployment insurance benefits begins a low-earnings job that allows them to continue receiving benefits. The transitions EB → UB and ENB → UB occur when an individual who is in the benefit period from a past job loss and has not exhausted all benefits loses their current job. The EB → ENB may occur either when a person who is working and receiving benefits exhausts their benefits or when a person who is working and receiving benefits experiences a sufficiently large increase in earnings that disqualifies them from receiving further benefits while working. Suppose we have panel data that follows a sample of individuals from the time they lose a job forward. Thus, all individuals begin in state UNB. Here, our goal is simply to demonstrate how a likelihood function would be constructed from the individual life histories. So, consider a particular individual (i) who files for and receives benefits after satisfying the two week waiting period, receives benefits without working for the five weeks before working part time. The individual still receives benefits while working and this job lasts four weeks. The individual then is unemployed (and receiving UI benefits) for eight weeks before finding another part-time job. The individual still receives benefits while working at this new part-time job and this job lasts eight weeks before the individual is made a (permanent) full-time job offer which precludes receiving further benefits. This particular history is portrayed in Figure 5.14. The components of the contribution to the likelihood for individual i would be S

P

UNB UNB

(1) = 1 −

冱

r1 r1 [1 − exp(− ξ r1 UNB exp(x′ 1 β UNB)α UNB01)]

r ≠ {UNB} UB UNB

P

UB1 UB1 (2) = 1 − exp(− ξUNB exp(x′2 β UB1 UNB)α UNB02)

for the unemployment period until benefits are received, S

UB UB

P

(3) = 1 −

冱 [1 − exp(− ξ

r ≠ {UB}

M

r1 UB

r1 exp(x′3 β r1 UB)α UB01)]

Estimation methods for duration models 157

Figure 5.14 Example of a sample path S

冱 [1 − exp(− ξ

PUB UB (5) = 1 −

r1 UB

r1 exp(x′5 βUB )α r1 UB03)]

r ≠ {UB}

EB EB1 EB1 PUB (6) = 1 − exp( − ξEB1 UB exp(x′ 6 β UB )α UB04)

for the benefit receipt period until the first part-time job is received, S

P

EB EB

冱 [1 − exp(− ξ

(7) = 1 −

r1 EB

r1 exp(x′7 βEB )α r1 EB01)]

r ≠ {EB}

M S

PEB EB (9) = 1 −

冱 [1 − exp(− ξ

r1 EB

r1 exp(x′9 βEB )α r1 EB03)]

r ≠ {EB}

UB1 UB1 UB1 PUB EB (10) = 1 − exp(− ξEB exp(x′ 10 βEB )αEB04)

for the first period of working on claim, S

UB UB

P

(11) = 1 −

冱 [1 − exp(− ξ

r ≠ {UB}

M

r2 UB

r2 exp(x′11 βUB )α r2 UB01)]

158

The Economics of Search S

冱 [1 − exp(− ξ

(17) = 1 −

UB UB

P

r2 UB

r2 exp(x′17 βUB )α r2 UB07)]

r ≠ {UB}

EB EB2 EB2 PUB (18) = 1 − exp(− ξEB2 UB exp(x′ 18 βUB )αUB08)

for the second period of unemployed benefit receipt, S

P

EB EB

冱 [1 − exp(− ξ

(19) = 1 −

r2 EB

r2 exp(x′19 βEB )α r2 EB01)]

r ≠ {EB}

M S

冱 [1 − exp(− ξ

PEB EB (21) = 1 −

r2 EB

r2 exp(x′21 βEB )α r2 EB03)]

r ≠ {EB}

P

EN2 (22) = 1 − exp(− ξENB2 exp(x′22 βENB2 EB EB )α EB04)

ENB EB

for the second period of working on claim, and S

(23) = 1 −

ENB ENB

P

冱

r1 r1 [1 − exp(− θ r1 ENB exp(x(22)′β ENB)α ENB01)]

r ≠ {ENB}

M S

冱

(23 + K) = 1 −

ENB ENB

P

r1 r1 [1 − exp(− θ r1 ENB exp(x(22 + K)′β ENB)α ENB0K)]

r ≠ {ENB}

for the final period of working “off claim” where we assume that the individual is right censored after K periods on the job. Thus, the contribution to the likelihood equals of this individual

Li =

5

冮冢

冦冲

UB PUNB UNB (1)PUNB (2)

冦 冲P

UB UB

k = 11

冧

21

(k) P

EB UB

冦 冲P

EB EB

(18)

k = 19

冦 冲P

冧

k=3

17

9

EB PUB UB(k) PUB (6)

冧

冧

EB EB

(k) PUB EB (10)

k=7

22 + k

(k) P

ENP EB

冦 冲P

ENB ENB

(22)

k = 23

冧冣

(k) dB(ξ,δ).

In this setup we have allowed for the baseline hazards parameters and coefficients associated with the explanatory variables to depend on the total number of previous visits to that state. More general forms of state dependence could also be incorporated into the model. For example, the transition probabilities may depend not only on the number of times the state was previously visited but also on the time spent in the state on each previous visit.

Estimation methods for duration models 159 5.10 SPECIFICATION TESTS FOR DURATION MODELS To conclude this chapter we briefly consider specification testing for duration models. While there are several different methods to test the validity of a model specification in hazard models, one particularly useful test is based on the notion that if the model is correctly specified then t

t

冮

冮

Mi(t) = Yi(s)Zi(s)dNi(s) − Yi(s)Zi(s)λ0(s)exp(xi′β)ds 0

0

is a martingale.5 Thus, t

t

冮

冮

ˆ i(t) = Yi(s)Zi(s)dNi(s) − Yi(s)Zi(s)λˆ0(s)exp(xi′βˆ)ds M 0

0

which is based on the sample estimates of λ0(s) and β should be approximately a martingale if the model is correctly specified. Using the fact that the estimates are consistent and appealing to the martingale central limit theorem, one can construct Chi-square tests of model specification. Moreover, graphical assessment is feasible since under the null hypothesis that the model ˆ i(t) is “approximately” a martingale for all i and t and, is correctly specified M ˆ i(t) should appear as “white-noise” or hence, plots of weighted sums of M patternless (see Arjas, 1989). The remainder of this section follows McCall (1994b) and focuses on discrete-time duration models. In the discrete case we work with martingale difference sequences. Let Ni(k) = I{Ki = k, Ci > k} where Ci denotes a censoring-time variable. Then, xi(k) = Ni(k) − I{K > k − 1}I{C > k}[1 − αk(xk,θ)] forms a martingale difference sequence. These martingale differences can be standardized by the stochastic variance process vi(k) where vi(k) = I{K > k − 1}I{C > k}αk(xk,θ)[1 − αk(xk,θ)]. Suppose the data is observed over M periods and let α denote the vector of αk(xk,θ) for k = 1, . . ., M. If the model is estimated by maximum likelihood then under suitable conditions (see McCall, 1994b) the test statistic χ2 = N−1xˆ′Σ −0 xˆ is asymptotically chi-square distributed with rank (Σ −0 ) degrees of freedom where

160

The Economics of Search N

xˆ =

冱xˆ and xˆ are M vectors of x i

i

, k = 1, . . ., M evaluated at θˆ and

i(k)

i=1

Σ0 = V0 − EP(∂p0/∂θ′)EP(I0)−1EP(s0x′) − {EP(∂p0/∂θ′)EP(I0)−1EP(s0x′)}′ + EP(∂p0/∂θ′)EP(I0)−1EP(∂p0/∂θ′)′ with s0 the score vector of the log-likelihood function, I0 the Hessian matrix of second derivatives of the log-likelihood function and V0 equal to the M × M diagonal matrix with (m,m)th element EPvi(m) = EP(I{K > m − 1}I{C > m}αm(xm,θ)[1 − αm(xm,θ)]).

6

Unemployment, unemployment insurance and sequential job search

As a matter of social sciences, the issue of whether to focus theoretically on unemployment or to focus on other features of business cycles and hope to learn something about unemployment as a by-product is one of research strategy, neither point of view being usefully enough developed at this point to have proved the other inferior . . . there are clearly other questions, the determination of the best provision of unemployment insurance for example, for which unemployment is inescapably the central issue and from which one cannot abstract if any progress is to be made at all. For some purposes, then, a theory of unemployment is essential. Think, to begin with, about the Walrasian market for a vector of commodities, including as one component “hours of labor services”, that must be at the center of the competitive equilibrium model. In this scenario, households and firms submit supply and demand orders for labor services and other goods at various auctioneer-determined price vectors and, when a market-clearing price vector is found, tracking is consummated at those prices . . . There is no sense in which anyone in this scenario can be said to “have a job” or to lose, seek, or find a job. It seems clear enough that a model in which wages and employment are set in this way . . . can tell us nothing about the list of labor market phenomena that have to do with sustained employer–employee relationships: their formation, their nature, their dissolution . . . such a model will not provide a useful account of observations on quits, fires, layoffs and other phenomena that explicitly refer to aspects of the employer–employee relationships . . . If we are serious about obtaining a theory of unemployment, we want a theory about unemployed people who look for jobs, hold them, lose them, people with all the attendant feelings that go along with these events. Walras’s powerfully simple scenario . . . cannot give us this, with cleared markets or without them. (Lucas 1987: 49–53; emphasis added) Initiative and responsibility, to feel one is useful, and even indispensable, are vital needs of the soul. Complete privation from this point of view is

162

The Economics of Search the case of the unemployed person,* even if he receives assistance to the extent of being able to feed, clothe, and house himself. For he represents nothing at all in the economic life of his country, and the voting paper which represents his share in its political life, doesn’t hold any meaning for him. (Weil 1952: 15; emphasis added)

* It seems clear that Weil is referring to people who have withdrawn from active search because of repeated failures.

6.1 INTRODUCTION One of the advantages of the classic search model is the ease with which one transits from the relatively sophisticated mathematics of Wald, Bellman, and Shapley to the real and pressing problems of unemployment. Lucas’s (1987: 49–53; original emphasis) remarks are pertinent. I have centered this discussion of unemployment on McCall’s original model of the decision problem facing a single unemployed worker. As soon as this simple problem is stated, it leads to a host of questions about this worker’s objectives and his market opportunities, which in turn leads directly to some of the central questions of the theory of unemployment: “What are the arrangements that market economies have for allocating individual workers to specific tasks?”; “What arrangements are available to allocate earnings and employment risks?”; and “What are the possibilities for improving on these arrangements through social insurance and other policies?” . . . In my view, focusing on unemployment as an individual problem, identical in character in business cycle peaks and troughs (though more people have this problem in troughs) is the key step in designing social policies to deal with it. We try to meet Lucas’s standards by presenting several studies of unemployment insurance and job search that focus on the individual problem. These investigations address the two major information problems associated with insurance policies: moral hazard and adverse selection. Unemployment is a fertile field for empirical study of both social insurance policies and sequential search. We first present the search aspect of unemployment assuming risk neutrality. A search model ready made for this exploration is Lippman and McCall (1981), which is a variation of the BSM and called the economics of belated information. At the microeconomic level, it is the intersection of imperfect information and the dynamics of supply and demand, with institutions like unemployment insurance, unions, welfare, and minimum wage laws that gives rise to the phenomenon of unemployment.1 Of course, it is individuals who experience unemployment and their demographic characteristics combine

Unemployment, unemployment insurance and sequential job search

163

with the institutional and informational milieu to determine the incidence and persistence of unemployment. Employment policies must be evaluated within this subtle causal web. The choice among competing employment policies can be accomplished only after this web has been untangled. A major purpose of this chapter is to extract several important forms of unemployment from this complicated web. We begin by studying the mismatch unemployment that characterizes youth unemployment. In mismatch unemployment imperfect information and learning play a prominent role. The employer is not completely informed about the employee’s productivity when he is hired. This information is conveyed only after employment commences. When this revelation occurs, the worker is either fired or retained. Just as imperfect information about the employee gives rise to fires, imperfect information about the employer generates quits. When the worker accepts a job he is not fully informed about its attributes. When this information arrives it may cause the worker to quit. In this way both employer and employee engage in tentative decision-making until perfect information arrives. We are interested in studying the amount of unemployment in this regime of joint decision-making. We do this by analyzing the waiting time—composed of unemployment and temporary employment—until permanent employment. Temporary layoffs are then considered. We begin by considering the choice between welfare and unemployment insurance for the low income worker. We then present optimal search strategies for alternative stochastic specifications with respect to the time till recall. We mention in passing the complications that arise when the laid off worker is risk averse. Finally the moral hazard problem of unemployment insurance is discussed in detail: we summarize several important papers on this topic.

6.2 MISMATCH UNEMPLOYMENT

6.2.1 Introduction An important form of information unemployment is due to mismatches between employer and employee.2 Such mismatches occur because neither employer nor employee has complete information regarding one another’s characteristics when job tenure begins. The employee knows the wage rate before accepting a job, but is unaware of nonpecuniary aspects of the job like the personality of his immediate supervisor. Similarly when the job commences the employer has incomplete information about the worker’s productivity. Mismatches occur when one or both parties to the labor contract are disappointed after they acquire complete information. It is these disappointments that give rise to the mismatch unemployment created by the induced quits and firings. Mismatch unemployment comprises a significant component of youth unemployment. Because of their inexperience much pertinent information arrives after the job is accepted. This belated information

164

The Economics of Search

frequently results in temporary employment terminated by quits and renewed unemployment. It is also quite difficult for employers to measure the productivity of young workers before they are hired. Thus young workers are fired more frequently than their older counterparts. This causes temporary employment that is terminated by a firing which in turn is followed by another spell of search unemployment. We analyze the time till a young worker becomes permanently attached to a job. This time, the length of mismatch unemployment is composed of periods of search unemployment punctuated by periods of temporary employment. Previous research has studied the duration of unemployment when the worker receives belated information; similarly the time until a firm hires an acceptable worker has been studied when the employer receives belated information about the worker’s productivity.3 Here we initiate analysis of the search unemployment induced by the firm’s response to belated information. In this model only the firm is the recipient of belated information. Consequently, there are firings but no quits. In the next subsection the formal model is presented. The reservation wage is calculated and is shown to be a decreasing function of the layoff probability. Finally, our study of the length of mismatch unemployment reveals that an increase in the layoff probability does not necessarily lead to an increase in the period of mismatch unemployment.

6.2.2 Analysis of mismatch unemployment Consider a searcher operating in an environment with infinite horizon, offer distribution F, search cost c, and discount factor β < 1. If there were no belated information, then an offer of x that is accepted by the searcher would garner a present value of x/[1 − β]. However, in contrast to much of the literature we presume the employer receives additional information with regard to the searcher’s suitability for the particular job into which he has been placed. This information arrives one period subsequent to the hiring decision. As a result of this belated information the worker is either fired, after having been employed for one period, or is permanently employed. We assume that the probability p of being fired does not vary with the wage offer. Thus, an accepted offer is worth x/[1 − β] with probability 1 − p and x + V with probability p, where v is the value of continued search. To begin the analysis, let V(x) be the maximal net benefit attainable when there is no recall and the searcher currently has an offer of x available. Then

冦

冤

V(x) = max V : x + β (1 − p) × max {V;x/[1 −β]}, where

x + pV 1−β

冥冧 = pβV + (1 − pβ) (6.1)

Unemployment, unemployment insurance and sequential job search 165 ∞

冮

V = − c + β V(x)dF(x).

(6.2)

0

It is evident from (6.1) that an optimal policy is as follows: accept a job offer of x if and only if x ≥ (1 − β)V ≡ ξ.

(6.3)

Next we solve for the reservation wage ξ. To do so define the usual H function by ∞

冮

H(y) = (x − y)dF(x),

(6.4)

0

then substituting via (6.1) and (6.3) in (6.2) and recalling that ξ = (1 − β)V yields ξ

冮

∞

冮

V = − c + β VdF(x) + β [x(1 − β)/[1 − β] + βpV]dF(x) 0

ξ

∞

冮

= − c + βVF(ξ) + β 2pV[1 − F(ξ)] + β(1 − βp)/[1 − β] (x − ξ)dF(x) ξ

+ β(1 − βp)V[1 − F(ξ)] = − c + βV + β(1 − βp)H(ξ)/[1 − β] so that ξ = − c + β(1 − βp)H(ξ)/[1 − β].

(6.5)

To ensure the existence of a strictly positive solution to (6.5) it is enough to assume that − c + βE(Χ1) > 0

(6.6a)

F(·) is strictly increase on (0, ∞)

(6.6b)

and

so that search is profitable for all values of p, as − c + β(1 − βp)H(0)/ [1 − β] ≥ − c + βH(0) = − c + βE(Χ1) > 0 and by (6.6b) H is continuous and strictly decreasing so that the right-hand side of (6.5) is also continuous and strictly decreasing. (Notice that (6.6b) ensures uniqueness of the solution.)

166

The Economics of Search

As the parameter p is allowed to vary the reservation wage ξ also varies; to be explicit, we shall write ξp in place of ξ to exhibit this dependence. From (6.5) we easily obtain ξ p′ = −

β2 H(ξp)/{(1 + β(1 − βp)[1 − F(ξp)]/[1 − β])}, 1−β

(6.7)

from whence it is clear that the reservation wage ξp declines as the probability p of being fired increases. In particular, if F has a continuous density f, then ξ p′ < 0 and ξ0 > 0 imply that sup{f(ξp): 0 ≤ p ≤ 1} is finite, a fact we shall use later. In his/her quest for a permanent job, the searcher will repeatedly cycle through a spell of unemployment followed by employment (for one period) in a temporary job from which he/she is fired until finally a permanent job is found. Let Np be the time at which a permanent job is found. It is our intention to study Np; specifically, does ENp increase with p? Let θi be the length of the ith spell of unemployment, let M be the number of spells of unemployment, and notice that M−1

Np =

冱 (θ + 1) + θ

M

i

M

=M−1+

i=1

冱 θ.

(6.8)

i

i=1

In addition, all of these random variables are independent, M has a geometric distribution with parameter 1 − p, and θi has a geometric distribution with parameter qp = 1 − F(ξp), the probability of encountering an acceptable offer. The generating function r of a geometric random variable Χ with parameter r is easily seen to be r(θ) = EsΧ = sr/{1 − s(1 − r)}. Utilizing this fact, conditional expectations, the independence of θi and M, and (6.8), we have

冦 冤

M

M−1

Eβ = E{E [β |M]} = E E β N

N

冦

M

θi

i=1

冲 E [ β |M]冧 = β E{[β (β)]

= E βM − 1 =

冲 β |M 冥冧

θi

1

}

M

q

i=1

1 1 − p(βq(β)) β

= βq(1 − p)/{1 − β(1 − q) − β 2qp}. Recalling that the derivative of the generating function evaluated at 1 is the expected value (i.e., ENp = (d/dβ)EβN| β = 1) we have4

Unemployment, unemployment insurance and sequential job search 167 ENp =

1 + pqp . (1 − p)qp

(6.9)

[Notice that EN0 = 1/q0 as per the standard model with no firing.] As p increases qp increases so that each spell of unemployment is (stochastically) shorter; but also the number of M of spells of unemployment (stochastically) increase with p. Thus, these two forces work in opposite directions in regard to their impact on the expected time till permanent employment. We can, however, assert that ENp is increasing for p near 1. To see this note from (6.9) that sign

d ENp = − (1 − p)q p′ + qp + q2p. dp

(6.10)

Now q p′ = − ξ p′ f(ξp) > 0 and, as noted earlier, is bounded on [0,1]. Moreover qp > 0 is increasing. The increasing nature of ENp for p near 1 follows by coupling these two facts with (6.10) and the continuity of d ENp.5 dp The amount Up of search unemployment is closely related to the time Np at which a permanent job is found. In particular, it is easily seen6 that 1 (1 − p)qp

(6.11)

ENp = EUp + p/(1 − p).

(6.12)

EUp = so that

Consequently, ENp will be an increasing function of p whenever EUp is. In fact, both ENp and EUp were monotone increasing functions for a large number of values of c, β, λ and α in the parameter space defined by the inequality (see equation 6.6a) − c + βα/λ ≥ 0 when F is gamma with parameters λ and α, i.e., f(t) = λαtα − 1e−λt/Γ(α), t > 0. However, EUp and ENp were not monotone for α = 0.5, c = 1, λ = 1, and β = 0.7; EUp increased on [0, 0.1] and decreased on [0.1, 1) whereas ENp increased on [0, 0.185). We now supply necessary and sufficient conditions for ENp and EUp to be monotone increasing. In particular, both are monotone increasing when the offer distribution is exponential. Theorem 6.1 Let r be the hazard function associated with F, i.e., r(t) = f(t)/ [1 − F(t)]. The necessary and sufficient conditions for EUp and ENp to be nondecreasing functions of p are, respectively

168

The Economics of Search ξ p′ ≥ −

1 ,0
(6.13)

ξ p′ ≥ −

2 − F(ξp) , 0 < p < 1. (1 − p)r(ξp)

(6.14)

and

Furthermore, (6.13) and (6.14) are satisfied when F is exponential. Proof

Because r is nonnegative, the condition [r and F are evaluated at ξp]

EN p′ =

1 1 1−F + rξ p′ + ≥0 (1 − p)(1 − F) 1 − p 1−p

冦

冧

(6.15)

can be rearranged to yield ξ p′ ≥ −

2−F , (1 − p)r

as desired. Recalling (6.12) enables us to obtain (6.13) by deleting the last term in (6.15). If F is exponential, i.e., f(t) = λe−λt, t > 0, then β = 1, r ≡ λ, and (6.7) can be employed to yield ξ p′ = − β2/{λ(1 − β)eλξ + β(1 − βp)λ} > p

− 1/r[(1 − β)eλξ + β(1 − βp)] > −1/(1 − p)r. p

Q.E.D. 6.3 LAYOFF UNEMPLOYMENT WITH POSITIVE PROBABILITY OF RECALL: THE TEMPORARY LAYOFF

6.3.1 Introduction We now investigate the temporary layoff.7 As is customary we assume that the temporary layoff is triggered by a transitory decline in the demand for the firm’s product. The time till recall is determined by the stochastic properties of the firm’s demand curve, coupled with the loss imposed on the firm when the worker accepts employment elsewhere. For simplicity we assume that the time till recall is known for sure as is the recall wage. This is the situation that seems to underlie much of the analysis of temporary layoffs. We show that in this regime the optimal policy is to search for the first K periods of unemployment and then merely wait for recall during the last N − K periods,

Unemployment, unemployment insurance and sequential job search

169

where N is the total length of the temporary layoff, i.e., N is the time till recall. As we will see there are many circumstances that lead to an optimal value of K equal to zero. This is, of course, the value that has been observed for most temporary layoffs.

6.3.2 Optimal search when the time till recall is a known constant Both the time till recall N and the real wage are known for sure. The amount of UI paid each period is denoted by u. If there were no recall then the reservation wage λ would be the solution to ξ=u−c+

β 1−β

H(ξ).

Thus in order for recall to be attractive we assume w > ξ. Let V(x,n) be the value of the optimal policy when x is the current offer and the worker has been unemployed (due to layoff) for n periods. Then V(x,N) =

1 max{x,w} 1−β

(6.16)

and ∞

冮

x V(x,n) = ; u + βV(0,n + 1); u − c + β V(y,n + 1)dF(y) , 1−β 0 n = 0,1, . . ., N − 1,

冦

冧

(6.17)

where the first term in the brace is the return from accepting x, the second term is the return from not searching, and the third term is the return from search.8 Theorem 6.2 There is a K ≤ N such that the worker engages in search if and only if his length of unemployment is less than K. Moreover, ξ0 ≤ ξ1 ≤ . . . ≤ ξN − 1 ≤ w, where ξn is the reservation wage for a worker who has been unemployed (i.e., laid off) for n periods. Proof Clearly V(x,n) ≤ V(x,n + 1). [Just consider the policy π defined by π(x,t) = π*(x,t − 1) for t < N, π(x,N) = π*(x,N). Then V(x,n + 1) ≥ Vπ(x,n + 1) ≥ V(x,n). [This last inequality requires w > ξ.)] If ξn > ξn + 1, then V(ξn + 1,n) > Thus ξn ≤ ξn + 1.

ξn + 1 = V(ξn + 1,n + 1) ≥ V(ξn + 1,n). 1−β

170 The Economics of Search Now ∞

∆n = u − c + β

冮 V(y,n + 1)dF(y) − {u + βV(0,n + 1)} 0

∞

=−c+β

冮 [V(y,n + 1) − V(0,n + 1)]dF(y) 0

∞

=−c+β

冮 冤1 − β − V(0,n + 1)冥 dF(y) ≥ y

ξn

∞

−c+β

冮 冤1 − β − V(0,n + 1)冥 dF(y) ≥ y

ξn + 1 ∞

−c+β

冮 冤1 − β − V(0,n + 2)冥 dF(y) = ∆ y

.

n+1

ξn + 1

The first inequality follows from the facts that (ξn,∞) ⊃ (ξn + 1,∞) and y/(1 − β) − V(0,n + 1) ≥ 0, whereas the second inequality follows from V(0,n + 1) ≤ V(0,n + 2). Thus, ∆n ≥ ∆n + 1. Finally, ∆N < 0 since xi > w. Q.E.D. Note in passing that risk neutral models of temporary layoffs are compatible with most of the previous literature in that the role of unemployment insurance is emphasized and the decision-maker is assumed to be risk neutral. Thus at each stage of the decision process, the choice between search and waiting for recall is made by comparing the expected discounted returns associated with each. But if risk neutrality were appropriate, unemployment insurance could make little sense, for the worker takes a lower wage while employed in order to finance his unemployment insurance. But with risk neutrality this cannot be optimal due to the deadweight administrative costs. To escape this inconsistency, we must reformulate the temporary layoff models in accord with expected utility maximization. We do not attempt this difficult task here. Instead we proceed by analogy with Danforth’s insights.9 In our simple model the reservation wage declines with time. Following Danforth, this model can be reformulated so that the job searcher maximizes the expected utility of lifetime income. Assuming decreasing absolute risk aversion, he demonstrates that the reservation wage declines as wealth diminishes. Now presumably wealth will decline during a temporary layoff. This runs counter to the forces causing an increasing reservation wage in Theorem 6.2. Clearly, the dynamic properties of the reservation wage—whether it increases, decreases, or fluctuates—depend on the behavior of these opposing forces. Having said all of this we note that some empirical evidence10 suggests that

Unemployment, unemployment insurance and sequential job search

171

the decline in wealth during temporary layoff is small enough to justify linear approximations to the concave utility function, i.e., Theorem 6.2 applies. We now review several articles on unemployment insurance and sequential job search. In his superb review Karni (1999) observes that most research on UI has concentrated on the effect of UI transfers on the duration of search. The fact that the effort associated with the job search activity is known only to the searcher gives rise to the moral hazard problem which is central to most of the research on UI. Karni notes that the UI literature has relied on the seminal research of Mortensen (1977) and has focused on two characteristics of the moral hazard conundrum, namely, the vigor with which searchers act as they seek employment and their propensity to accept job offers. Mortensen demonstrates that when UI is available for a fixed length of time and the benefit rate is invariant over this period then: (1) the transition rate to employment increases with the length of search unemployment; (2) the transition rate of those who are recently unemployed decreases with the length of the benefit premium and (3) the transition rate of the unemployed not receiving UI rises with the benefit rate and the length of the benefit period. Karni refers to (3) as the “entitlement effect” whose cause is that an enhanced UI package marks an increase in future unemployment status. This provides a strong incentive to become employed as soon as possible. Because of its significance, we next present a fairly detailed description of Mortensen’s 1977 article.

6.4 UNEMPLOYMENT INSURANCE AND JOB SEARCH DECISIONS (MORTENSEN 1977) The Mortensen model embodies the following features: the worker is allowed to search while employed; the intensity of search is a choice variable; and the cost of search is viewed as the value of foregone leisure. Two institutional features of the UI program in most states are also explicitly incorporated: benefits are paid only for a specified duration rather than in every period of an unemployment spell, and workers who quit are not qualified for benefits. Moreover a layoff rate is incorporated so that workers realize that future jobs are not permanent and that an involuntary unemployment spell in the future awaits. Thus an increase in benefits makes employment more attractive since it lowers the cost of involuntary unemployment in the future. In particular, those who do not qualify for benefits in the current unemployment spell (quit, new entrant, or exhaustee) will lower their reservation wage in response to this benefit increase. For those receiving UI benefits, there is an additional effect on job search behavior due to the reduction in the net costs of search. We assume that the number of hours worked in jobs is fixed and the same in all jobs. Also assume that all distributions of relevant random variables are known and constant over time.

172

The Economics of Search

Consider the small interval (j, j + dt). If utility is intertemporally separable the utility function is of the form V(j) =

1 [u(yj,ᐉj)dt + V(j + dt)] 1 + rdt

where u(⭈) is the utility flow generated by (yj, lj) and r is the sum of the subjective rate of time preference and the probability of retiring per period when u(0,1) = 0. We denote the entitlement period as T and the benefit level as b. Only qualified individuals (involuntary layoffs) are assumed to receive benefits. The probability that the worker will be laid off in an interval of length dt is assumed to be constant and independent of the worker’s action. Denote it by δdt. The wage distribution is denoted by F(w). The probability that an employed worker who puts forth effort s receives an offer is assumed to be αs. Let V e denote the value function of the worker when employed and u V (t,T) denote the utility obtained when unemployed with t periods of entitlement remaining and the entitlement period equals T. When employed V e(w,T) =

1 Max [u(w,l0 − s)dt + δdtV u(T,T) 1 + rdt 0 ≤ s ≤ l 0

+ αstdtPr{x ≥ w}E{V e(x,T |x ≥ w)} + (1 − δdt − αstdt Pr{x ≥ w}) V e(w,T)] where x is an offer randomly drawn from the distribution characterized by F(x). Since ¯¯ w

Pr{x ≥ w} =

冮 dF(x) = 1 − F(w) w

and ¯¯ w

冮

Pr{x ≥ w}E{V e(x,T)|x ≥ w} = V e(x,T)dF(x) w

an equivalent representation is V e(w,T) =

1 Max [u(w,l0 − s)dt + V e(x,T) + δdt[V u(T,T) 1 + rdt 0 ≤ s ≤ l 0

¯¯ w

冮

− V (x,T)] + αstdt [V e(x,T) − V e(w,T)]dF(x)]. e

w

(6.18)

Unemployment, unemployment insurance and sequential job search

173

The last two terms on the right-hand side of equation (6.18) are respectively the expected loss in the future discounted stream attributable to being laid off and the expected gain in the present value of future utility attributable to finding a higher paying job during the current interval. An analogous approach yields a characterization of the same worker’s value function when unemployed, denoted as V u(t,T). During the current time interval of length dt, the worker enjoys the utility flow u(b,1 − st)dt if qualified for benefits. Here st denotes the proportion of the current interval devoted to search given that the remaining future benefit period is t. An offer x is generated by search with probability αstdt and is accepted if and only if it is at least as large as the worker’s current reservation wage. An unemployed worker chooses both st and ξt to maximize the expected future discounted utility flow, given t. If an acceptable offer is found, the worker is employed at the end of the current interval and can expect to enjoy the future discounted utility stream V e(x,T) given that the optimal search strategy is followed once employed. Because the remaining benefit period is of length t-dt at the end of the current interval, V u(t−dt,T) is the value function associated with being unemployed at the end given optimal search behavior as an unemployed worker. V(t,T) =

1 Max [u(b,w − st)dt + (1 − αstdtPr{x ≥ ξt})V(t − dt,T) 1 + rdt 0 ≤ s ≤ 1,ξ > 0 t

+ αstdtPr{x ≥ ξt}E{V e(x,T)|x ≥ ξt}] for every 0 < t ≤ T. This can be rewritten as V u(t,T) =

1 Max [u(b,1 − st)dt + V u(t − dt,T) 1 + rdt 0 ≤ s ≤ 1,ξ ≥ 0 t

(6.19)

t

¯¯ w

+ αstdt

冮 [V (x,T) − V (t − dt,T)]dF(x)] e

u

ξt

for every 0 < t ≤ T. Because an exhaustee and a new entrant receive no compensation, it follows immediately that the problem becomes V u(0,T) =

1 Max [u(0,1 − st)dt + V u(0,T) 1 + rdt 0 ≤ s ≤ 1,ξ ≥ 0 t

¯¯ w

+ αstdt

冮 [V (x,T) − V (0,T)]dF(x)] e

u

ξ

Given random search, the probability that an unemployed worker finds an acceptable job in a time interval of length dt is equal to the product of the

174

The Economics of Search

probability that an offer will arrive during the interval, αstdt, and the probability that such an offer is acceptable, Pr{x > ξt} = 1 − F(ξt). In other words the probability that an unemployed worker makes a transition from unemployment to employment in the interval is qdt where q = αst[1 − F(ξt)] is the escape rate or hazard rate, the expected frequency with which acceptable offers are found. An increase in the reservation wage reduces the escape rate because the probability that a random offer will be acceptable declines with every increase in the minimum acceptable wage. If the optimal search strategy of an unemployed worker is to search (s > 0), demand leisure (s < 1) and require a positive wage (w > 0) then equation (6.19) implies that an optimal reservation wage and search intensity combination (w, s) satisfies V e(w,T) = V u(t,T)

(6.20)

and ¯¯ w

u2(b,1 − s) = α

冮 [V (x,T) − V (t,T)]dF(x) e

u

(6.21)

ξt

where u2(·) is the partial derivative of u(·) with respect to leisure. Assume that u22(·) < 0. Then, the optimal s decreases with an increase in t. Why? We need to see how the right side of (6.21) changes with t. Even though w changes with t, by (6.20) the derivative of the right side of (6.21) with respect to w is zero. Now, ¯¯ w

冮

u22(b,1 − s)ds = a

ξt

∂V u(t,T) dF(x)dt ∂t

or ¯¯ w

a ds = dt

冮 w

∂V(t, b, UT) ∂t

u22(b,1 − s)

∂V u(t,T) > 0 and u22(·) < 0, ds/dt < 0 or in other words qualified work∂t ers search more intensely as their benefits run out. Moreover since the right Since

Unemployment, unemployment insurance and sequential job search

175

side of equation (6.20) increases with t, the implication is that a qualified worker’s reservation wage falls with realized unemployment duration until the current benefit period is exhausted. Proposition 6.1 In the case of a qualified worker who has not yet exhausted his/her unemployment benefits, the escape rate increases with realized unemployment duration. The opportunity cost of search is higher when payments are received if and only if income and leisure are strict complements in household production; i.e., for the same s, u2(b,1 − s) > u2(0,1 − s) if u12 > 0 and u2(b,1 − s) < u2(0,1 − s) if u12 < 0. As a consequence it can be shown that the time allocated to search jumps up in a discontinuous manner at the moment benefits are exhausted if income and leisure are strict complements. Proposition 6.2 At the moment benefits are exhausted, the escape rate jumps up (down) if income and leisure are strict complements (substitutes) in household production. The effect of a change in the benefit rate on a worker’s optimal action depends upon his or her current status. Suppose a worker had just been laid off (t = T ). Then we have V e(w,T ) = V u(T,T )

(6.22)

and ¯¯ w

u2(b,1 − s) = α

冮 [V (w,T ) − V (T,T )] dF(x). e

u

(6.23)

ξt

∂ V e(w,T) < 1 ∂V (T,T) because from (6.18) the possibility of being laid off is an event that can occur only in the future and an increase in the value V u(T,T) does not increase the indirect utility of being employed by an equal amount due to time preference. Equation (6.22) implies that the reservation wage of a worker just laid off unambiguously increases with either T or b. Also, an increase in either of the benefit parameters reduces the marginal return to search time on the right side of equation (6.23). If an increase in the benefit rate does not reduce the cost of search (u12(·) ≥ 0), then the time allocated to search falls with both parameters. An increase in b or T increases V u(T,T). However,

u

176

The Economics of Search

Proposition 6.3 In the case of a newly laid off worker, the escape rate decreases with the maximum benefit period. If goods and leisure are complements in household production, then an increase in the benefit rate also decreases the escape rate. Consider a worker who is not currently receiving benefits. In this case, equations (6.20) and (6.21) can be rewritten as V e(w,T) = V u(0, T)

(6.24)

and ¯¯ w

u2(0,1 − s) = a

冮 [V (w,T) − V (0,T)]dF(x). e

u

(6.25)

ξ

Now an increase in V u(T,T) increases the value of being unemployed (righthand side of [6.24]) by less than it increases the utility of being employed since an unemployed worker must first find a job before being laid off. From this we have Proposition 6.4 In the case of an unemployed worker not currently receiving benefit payments, an increase in either benefit parameters increases the escape probability. The implications of these propositions is that an increase in either benefit parameter will decrease the escape rate for workers who have recently been laid off and increase the escape of workers nearing the end of their benefit period. One empirical implication of the Mortensen model is that increases in entitlement increase unemployment durations. In the United States, however, most changes in entitlement arise from the triggering or congressional enactment of extended UI benefits which typically lengthens UI entitlement from 26 to 39 weeks. Unfortunately, these extended benefit programs are enacted only in times of poor economic conditions and so disentangling the incentive effect from economic conditions may be difficult. Recently, however, Card and Levine (2000) examined an extended benefit program in New Jersey where extended UI benefits where instituted due to mainly exogenous factors. Analyzing administrative UI records from before, during and after the extended benefits program, Card and Levine estimated that an increase of entitlement of 13 weeks increased unemployment durations by approximately one week. Van Den Berg (1990) considers many aspects of nonstationarity in job search models but simplifies the discussion by not considering, as in Mortensen (1977), future unemployment spells. In particular, Van Den Berg allows offer arrival rates λ(t), benefits b(t), and wage distributions F(w,t) to vary over an unemployment spell.

Unemployment, unemployment insurance and sequential job search

177

While many theorems are proved by Van Den Berg we will just consider the simple situation where there is only one point in time, T, where the exogenous variables are allowed to change. Let the subscript 1 denote the value of a variable before T and the subscript 2 denote the value of the variable after T. Assume, 1. 2. 3. 4.

b1 > b2 λ1 > λ2 F1 first order stochastically dominates F2. F1 is a mean preserving spread of F2.

Let V1 and V2 denote the stationary value functions in situation where parameters do not change. Let V(t) be the value function for the nonstationary problem. Theorem 6.3 V1 < V2 and for every t ∈ (0,T), V2 < V(t) < V1, V′(t) < 0, V″(t) < 0 and V(T) = V2. Thus as T is approached in each case, reservation wages will fall.

6.5 MORE ON THE INCENTIVE EFFECTS OF UNEMPLOYMENT INSURANCE In this section we first develop a simple continuous-time job search model in which an unemployed individual receives UI benefits that are exhausted after a given time and arrival rates of job offers are constant and exogenous. After analyzing the solution to this simple model we allow for the possibility that individuals may search for different types of jobs (full time and part time) and may be able to continue receiving at least some benefits while working. Finally, we will develop a job search model with recall unemployment where the recall date is unknown with certainty. Let b(t) : t = (0,∞) or simply b(0,∞) denote the function describing the time path of UI benefits. We assume this function is known by an unemployed person at the start of the unemployment spell. Further, we assume an individual’s utility depends on consumption and leisure and denote the utility function by u(c,l). The cumulative distribution function for wages is represented by G(w) and the instantaneous discount rate is denoted by r. Job offers which are a random draw from G are assumed to arrive according to a Poisson process with parameter λ. To derive the form of the value function for this problem we begin by taking a discrete-time approximation to the continuous time problem and take the limit as the length of the time intervals go to zero. It can be shown that the discrete-time approximation to this problem yields the following recursive relation

178

The Economics of Search V u(t − dt) =

U(b(t),l u)dt λdt max E{V u(t),V e(w ˜ ,t)} + 1 + rdt 1 + rdt +

(1 − λfdt)V u(t) 1 + rdt

(6.26)

+ oP(t)

where V e(w,t − dt) =

U(w,l e)dt V e(w,t) + + oP(dt) 1 + rdt 1 + rdt

(6.27)

Multiplying both sides of (6.27) by 1 + rdt, dividing by dt and taking the limit as dt goes to zero yields V e(w,t − dt) − V e(w,t) rV e(w,t − dt)dt + lim dt dt dt → 0 dt → 0 lim

U(w,l e)dt oP(dt) + lim dt dt dt → 0 dt → 0

= lim or

V et(w,t) + rV e(w,t) = U(w,l e)

(6.28)

where V et(w,t) denotes the derivative of V e(w,t) with respect to t. Since once an individual chooses employment the problem becomes stationary, V et(w,t) = 0 and (3) reduces to V e(w,t) =

U(w,l e) r

(6.29)

Substituting (6.29) into (6.26) and taking limits gives V u(b(t − dt),t − dt) − V u(b(t),t) + V u(b(t),t)(r + λ[(1 − G(ξ))]) dt dt → 0 lim

∞

= U(b(t),l ) + λ u

U(w,l e) dG(w) r ξ(t)

冮

(6.30)

or V ut(b(t),t) + V u(b(t),t)(r + λ[(1 − G(ξ))]) ∞

U(w,l e) = U(b(t),l u) + λ dG(w) r ξ(t)

冮

where Vut(b(t),t) denotes the derivative of V u(b(t),t) with respect to t.

(6.31)

Unemployment, unemployment insurance and sequential job search

179

By the definition of the reservation wage, V u(b(t),t) = V e(ξ(t)).

(6.32)

Consider a situation where b(t) = b for t < t*, b(t) = 0 for t ≥ t*. Then for all t < t*, by induction V u(b,t − dt) ≥ V u(b,t) so by (6.32) we have ξ(t − dt) ≥ ξ(t). Meyer’s (1990) influential paper on the impact of UI on unemployment spell investigated how re-employment rates depended on the weeks remaining in the UI spell. Using the discrete hazard of the form of (5.24) and administrative UI data from the continuous wage and benefit history (CWBH), Meyer isolates the impact of remaining entitlement by noting that at any given point in the duration the amount of benefits remaining differs across individuals (due to, for example, to state differences in UI law regarding UI benefit durations). Meyer includes a series of variables that allow him to estimate a spline in time until benefit exhaustion. Meyer finds that moving from six to one week of exhaustion more than triples the hazard rate of leaving the UI system which is consistent with individuals lowering the reservation wage as benefit exhaustion approaches. Using Canadian data Ham and Rea (1987) also look at the impact of remaining entitlement on durations.11 Using a discrete-time hazard approach with a logit specification, αm(x,ξi) ≡

1 1 + exp[−(ξi + x′m β + dom)]

Ham and Rea find that an increase in initial entitlement increases the expected duration of unemployment by approximately 1/3 of a week. Most state UI programs in the United States not only have a limited duration for which an individual can receive UI benefits but also allow in some circumstances an individual to continue receiving benefits while unemployed. In particular, many states set a disregard level where full benefits are continued as long as weekly earnings are below the disregard with benefits reduced on a dollar for dollar basis for earnings above the disregard. The impact of these rules on search behavior was explored by McCall (1996). In a fashion similar to McCall (1996), we will construct a continuous time search model in which an individual searches for both fulltime and part-time jobs. Moreover, we will assume that if an acceptable part-time job has been found search continues on-the-job for a full-time job. McCall, assumed that individuals vary their intensity of search over the unemployment spell as benefits are exhausted. McCall assumed no wage dispersion within full-time jobs or part-time jobs although the wages between full-time and part-time jobs could differ. Alternatively, we assume that the arrival of full-time and part-time job offers are constant over an unemployment spell but that part-time and full-time job wage offers are characterized by some wage dispersion. Denote the full-time (part-time) wage

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The Economics of Search

distribution by G f (G p) and the rate of receiving a full-time (part-time) job offer by λ f (λp). We assume while working at a part-time job full-time job offers arrive at the rate λfe. For simplicity we ignore on-the-job search while working full-time. Denoting the value functions while employed full-time (part-time) by Vf (Vp) we have U(w p,l p)dt λfedt E max{V p(w p,t),V f(w ˜ f,t)} + 1 + rdt 1 + rdt

V p(wp,t − dt) =

+

(6.33)

(1 − λ dt)V (w ,t) + oP(t) 1 + rdt fe

p

p

U(w f,l f)dt V f(w f,t) + + oP(t). 1 + rdt 1 + rdt

V f(w f,t−dt) =

From above we have V e(w,t) =

U(w,l e) . Now (6.33) can be rearranged to give r

V p(w p,t − dt)(1 + rdt) − (1 − λpdt)V p(w p,t) − λpdtG f(ξp(t))V p(w p,t) ∞

= U(w p,l p)dt + λfedt

冮 V (w,t)dt f

ξ (t) fe

Taking limits yields V p(wp,t − dt) − V p(w p,t) + (r + λ fe[1 − G f(ξ fe(t)])V p(w p,t) dt dt → 0 lim

∞

= U(w ,l ) + λ p

p

fe

冮 V (w ,t)dG (w) + f

f

f

ξfe(t)

op(dt) dt

or ∞

V (w ,t) + (r + λ [1 − G (ξ (t))])V (w ,t) = U(w ,l ) + λ p t

p

fe

f

fe

p

p

p

p

fe

冮 V (w,t)dG (w) f

f

ξ (t) fe

where V pt(w p,t) denotes the derivative of V p(w p,t) with respect to t. Suppose that benefits can vary over time and denote the entire time path of benefits by b(0,∞). Letting V u represent the value function associated with unemployment gives:

Unemployment, unemployment insurance and sequential job search V u(b(t) − dt,t − dt) =

181

U(b(t),l f )dt λ fdt E max{V u(b(t),t),Vf(w ˜ f,t)} + 1 + rdt 1 + rdt +

λpdt E max{V u(b(t),t),V p(w ˜ p,t)} 1 + rdt (1 − λ dt − λ dt)V u(b(t),t) + oP(t) 1 + rdt f

+

p

V u(b(t − dt),t − dt) − V u(b(t),t) dt dt → 0 lim

+ V u(b(t),t)(r + λ f[(1 − G f(ξ f(t)))] + λp[(1 − G f(ξp(t)))]) ∞

∞

冮 V (w,t)dG (w) + λ 冮 V (w,t)dG (w)

= U(b(t),l ) + λ u

f

f

f

p

ξf(t)

p

p

ξp(t)

V ut(b(t),t) + V u(b(t),t)(r + λ f[(1 − G f(ξ f(t)))] + λp[(1 − G f(ξp(t)))]) ∞

∞

冮 V (w,t)dG (w) + λ 冮 V (w,t)dG (w)

= U(b(t),l u) + λ f

f

f

p

ξ (t)

p

p

ξ (t)

f

p

By the definition of a reservation wage V u(b(t),t) = Vp(ξp(t)) and V u(b(t),t) = V f(ξ f(t)). Suppose b(t) = b up to t* and b(t) = 0 thereafter. Then for t < t*, since V u(b,t + dt) ≤ V u(b,t) we have ξ f(t + dt) ≤ ξ f(t) and ξp(t + dt) ≤ ξp(t). So, we see that as the benefit exhaustion point approached the reservation wages for both part-time and full-time jobs decrease. In many states individuals can keep some benefits while working. Most states have a disregard so that for weekly earnings less than the disregard full UI benefits are received and UI benefits are reduced on a one for one basis for earnings above the disregard. Assume for all full-time jobs that earnings are such that UI benefits would be 0. To keep matters simple suppose that benefits are never exhausted (i.e. b(t) = b for all t). Let d represent the disregard level. Earnings in part-time jobs are then w + b for w < d, w + b − (w − d) for w > d and w < b + d and w for w > b + d . Thus, ∞

冮

∞

冮

U(b,l ) + λ V (w)dG (w) + λ V p(max{min(b + d,w + b),w})dG p(w) f

V u(b,d) =

f

f

f

p

ξf

ξp

(r + λ [(1 − G (ξ f ))] + λp[(1 − G p(ξ p))]) f

f

.

Now an increase in the disregard amounts to a first order stochastic shift in the wage distribution for part-time work. It is clear that for d′ > d,

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The Economics of Search

V u(b,d ′ ) ≥ V u(b,d). Since V u(b,d) = V f(ξf) it follows that ξf(d′) ≥ ξf(d) and so the rate of finding full-time work would decrease. For part-time jobs the reservation wage falls. To see the impact of an increase in d on the reservation wage for part-time jobs first note that ∂V u(b,ξf(d),ξp(d),d) ∂Vξu ∂ξf ∂Vξu ∂ξp ∂V u ∂V u = f × + × + = ≥ 0. ∂d ∂ξ ∂d ∂ξp ∂d ∂d ∂d f

p

So, ∂Vu(b,d) ∂Vp(ξ p,d) × ∂ξp ∂d ∂d =− ≤ 0. ∂d ∂Vp(ξ p,d) ∂ξp Hence, an increase in d lowers the reservation wage for part-time work and increases the reservation wage for full-time work. Although the analysis is more complicated, it is clear that when benefits are exhausted after some point the subsidy to part-time work ends. Thus, the impact of the subsidy should be greater earlier in the unemployment spell. Conditions under which this result holds were derived in McCall (1996). One complication in empirically studying the impact of the disregard on re-employment behavior is that, in the United States, many unemployed individuals who qualify for UI benefits do not file a claim. McCall’s (1996) analysis focused only on displaced workers who qualified for benefits. To allow for the possibility that changes in the parameters of the UI system will affect the choice to file a UI claim, McCall (1996) modeled the claim filing choice and allowed for the possibility that unobservable determinants of that choice may be correlated with unobservable determinants (i.e. ξ1 and ξ2 in the latent survivor function (5.34)) of the re-employment rates into part-time and full-time jobs. McCall modeled UI receipt by the dichotomous variable UI which equals 1 if an individual files a claim and 0 otherwise where Pr(UI = 1) has the functional form P(UI = 1) = 1 − exp(− ξu exp[z′δ]) where z is a vector of explanatory variables, δ is a vector of parameters and ξu is an unmeasured variable that is uncorrelated with X and z. However, ξu may be correlated with ξ1 and ξ2 in (5.34) and ui (along with its interaction with some variables in X) are added as explanatory variables in (5.34). The likelihood function for this model selectivity-corrected competing risks model is

Unemployment, unemployment insurance and sequential job search

183

N

冱c ln( 冮P(ui = 1|z ,ξ ) 1 i

ln(L) =

i

i

P(uii = 0|zi,ξu)1 − ui

u uii

i

i=1

× {S(k − 1,k − 1|Χi,uii,ξ1,ξ2) − S(k − 1,k|Χi,uii,ξ1,ξ2) − A(k|Χ,uii,ξ1,ξ2)}dG(ξ1,ξ2,ξu)) + c2i ln(冮P(uii = 1|zi,ξu)ui

i

× P(uii = 0|zi,ξu)1 − ui {S(k − 1,k − 1|Χi,uii,ξ1,ξ2) i

− S(k,k − 1|Χi,uii,ξ1,ξ2) − A(k|Χ,uii,ξ1,ξ2)} × dG(ξ1,ξ2,ξu)) + c3i ln(冮P(uii = 1|zi,ξu)ui P(uii = 0|zi,ξu)1 − ui i

i

× S(k − 1,k − 1|Χi,uii,ξ1,ξ2)dG(ξ1,ξ2,ξu)). Using this model McCall [1996] found evidence that increasing the disregard increases the re-employment rate into part-time jobs, at least early on in the unemployment spell. In the United States, where many UI recipients are recalled to their former jobs, recall expectations can impact search behavior. Suppose that the rate of recall equals λr and that the rate of new job offers equals λ. Let the future layoff rate from the job equal u and have wage wm. Assume that new jobs have layoff rate equal to zero. To simplify matters we will again consider only the case where benefits do not expire. Then, ∞

冮

V (r + λ + λ [(1 − G(ξ))]) = U(b,l ) + λ V + λ VdG(w) u

m

p

f

m

m

ξ

where Vm =

wm + uV u u+r

and ∞

b+λ V +λ m

冮 VdG(w) ξ

u

V =

m

r − (1 − λm − λ) + λG(ξ)

.

So, ∂V u Vm{r − (1 − λm − λ) + λG(ξ)} − λmVm = ∂λm {r − (1 − λm − λ) + λG(ξ)}2 =

Vm{r − (1 − λ) + λG(ξ)} >0 {r − (1 − λm − λ) + λG(ξ)}2

and the probability of re-employment into a new job decreases as the probability of recall increases.

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The Economics of Search

Using data from the CWBH, Katz and Meyer (1990) looked at recall expectations impact on re-employment into a new job.12 Using a discrete-time hazard approach model they estimate both a recall-hazard equation and new job-hazard equation. In the former, observations involving a new job are treated as censored observations and in the latter observations involving recalls are treated as censored observations. Thus, they are estimating a discrete-time competing risks model where risks are approximately independent. Katz and Meyer find considerable differences in the recall and new job hazards. Moreover, they find that those who expect recall have lower new job hazards than those who don’t expect recall.

6.6 EFFICIENT UNEMPLOYMENT INSURANCE (ACEMOGLU AND SHIMER 1999) We now explore an equilibrium model of unemployment with the provision of unemployment insurance of Acemoglu and Shimer (1999) with risk-averse workers. Acemoglu and Shimer found that an increase in risk-aversion reduces wages, unemployment, and investment. Unemployment insurance, however, was found to have the opposite effect: insured workers seek highwage jobs with high unemployment risk. An economy with risk-neutral workers achieves maximal output without any unemployment insurance, but an economy with risk-averse workers required a positive level of unemployment insurance to maximize output. Thus, moderate unemployment insurance not only improves risk sharing but also increases output. In the Acemoglu and Shimer model, firms make irreversible investments and post wages. Workers optimally search among posted wages. Risk-averse workers wish to avoid unemployment, and in response, the labor market offers its own version of insurance: an equilibrium with lower unemployment and wages. Because lower wages raises the vacancy risk for firms, it reduces the utilization of, and the returns to, ex ante investments, leading to lower capital/labor ratios and a poorer quality of jobs. Unemployment insurance encourages workers to apply to high-wage jobs with unemployment risk. The impact of unemployment insurance on worker and firm behavior is driven by a form of moral hazard. Because insurers cannot directly control workers’ actions, the increased utility of unemployment induces them to search for higher wages. Firms respond by creating high-wage, high-quality jobs, with greater unemployment risk. When agents are risk-averse, the equilibrium without unemployment insurance fails to maximize output because capital/labor ratios are too low. Next we outline the Acemoglu–Shimer model of job search with risk-averse agents. For simplicity we focus on the case where workers are homogenous.

Unemployment, unemployment insurance and sequential job search 185

6.6.1 A model of risk-averse job search: preferences and technology Assume that there is a continuum of identical workers, each with utility function u(c) over final consumption; u is twice continuously differentiable, strictly increasing and weakly concave. All workers are endowed with initial wealth A0. They may invest or store wealth where the return from storage, R, equals 1. To pay for unemployment insurance a government must tax workers. Worker i’s consumption is therefore equal to his assets, A0, minus lump-sum taxes τ, plus net income from wages or unemployment benefits yi, so utility equals u(A + yi) where A ≡ A0 − τ. Also assume that there is a continuum of potential firms, each with access to the production technology f: (0,∞) → (0,∞) that requires one worker and k > 0 units of capital to produce f(k) units of the consumption good. The function f is assumed to be continuously differentiable, strictly increasing and strictly concave and also satisfies conditions for an interior solution. The price of capital is normalized to R. Since workers own a diversified portfolio, firms maximize expected profit. The large number of potential firms insures free entry, so aggregate profits are zero in equilibrium. Workers and firms come together via search. At the beginning of the period, each firm j decides whether to buy capital kj > 0. If it does, it is active and posts a wage wj. In the next stage, each worker observes all wage offers and decides where to apply. That is, worker i seeks a job with a wage wi ∈ W, where W = {wj, for all active j} is the set of wage offers. If he/she is hired, he/she earns yi = wi. Otherwise s/he is unemployed and obtains unemployment insurance yi = b. A firm that hires a worker produces f(k); an unfilled vacancy produces nothing and its capital remains idle. The Acemoglu–Shimer model is a job search with friction model: depending on workers’ application decisions, there may be more competition for some jobs than for others. To capture this let qj ∈ [0,∞] be the ratio of workers who apply for jobs offering wage wj to the number of firms posting that wage. This will be referred to as the job’s expected queue length, an endogenous measure of the extent of competition for jobs offering wj. Assume that a worker applying to a job with wage wj is hired with probability µ(qj) where µ: [0,∞] → [0,1] is continuously differentiable and decreasing. If many workers apply for one type of job, each has a low probability of employment. On the firm side, the probability that a firm that offers a wage wj hires a worker is η(qj) where η: [0,∞] → [0,1] is continuously differentiable and increasing. One can think of firms opening jobs in different geographic regions or industries and workers directing their search toward one of these labor markets. In labor market j, all firms offer a common wage wj and the ratio of workers to firms, qj, determines the matching probabilities. Standard matching frictions ensure that, within an individual labor market, unemployment and vacancies coexist.

186

The Economics of Search

6.6.2 Equilibrium An allocation is defined as a tuple {K,W,Q,U}, where K ⊂ ⺢ + is a set of capital investment levels, W: ⺢ + → ⺢ + is the set of wages offered by firms making particular capital investments (i.e., W(k)) Q: ⺢ + → ⺢ + is the queue length associated with each wage and U ∈ ⺢ + is workers’ utility level. Definition 6.1. 1

An equilibrium is an allocation {K*,W*,Q*,U*} such that

Profit maximization: for all w, k h(Q*(w))[f(k) − w] − k ≥ 0 with equality if k ∈ K* and w ∈ W*.

Profit maximization ensures that, given the queue length associated with each wage, firms choose wages and capital investments to maximize profits. Free entry drives the maximized value of profits to zero. ¯¯ * = {w|w ∈ W(k),k ∈ K}. Define the set of offered wages as W 2

Optimal application: for all w U* ≥ µ(Q*(w))u(A + w) + [1 − µ(Q*(w)]u(A + b) and Q*(w) ≥ 0, with complementary slackness, where U* = sup µ(Q*(w))u(A + w) + [1 − µ(Q*(w))]u(A + b) w ∈ W′

or U* = u(A + b) ¯¯ * is empty. if W Optimal allocation ensures that workers make their application decisions to maximize utility where queues adjust to make workers earn the maximal level of utility U* at any wage, including wages not offered along the equilibrium path. Note that the queue length function also contains two other pieces of information. First, if w* is the unique equilibrium wage, the number of active firms is 1/Q*(w*). Second, the unemployment rate of workers applying to a wage w′ is ν(w′) = 1 − µ(Q*(w′)). Clearly, ν(w′) is increasing in Q*(w′) since workers who apply to jobs with longer queues suffer higher probability of unemployment. Acemoglu and Shimer establish the existence of an equilibrium allocation:

Unemployment, unemployment insurance and sequential job search

187

Proposition 6.5. An equilibrium always exists. If {K,W,Q,U} is an equilibrium, then any k* ∈ K, w* ∈ W(k*), and q* = Q(w*) solve U = max µ(q)u(A + w) + [1 − µ(q)]u(A + b) w,q,k

subject to η(q)[ f (k) − w] − k = 0

(6.34)

w ≥ b.

(6.35)

and

Conversely, if some {k*,w*,q*} solves this program, then there exists an equilibrium {K,W,Q,U} such that k* ∈ K, w* ∈ W(k*) and q* ∈ Q(w*). Proof: Step 1 Let {K,W,Q,U} be an equilibrium allocation with k* ∈ K, w* ∈ W(k*), and q* = Q(w). It must be shown that {k*,w*,q*} solves the constrained optimization problem. First, profit maximization ensure that {k*,w*,q*} satisfies the constraint (6.34). Next, optimal application implies U = µ(q*)u(A + w*) + [1 − µ(q*)]u(A + b) ≥ u(A + b) so, w* ≥ b satisfying constraint (6.35). Suppose not and that another triple {k,w,q} satisfies (6.35) and achieves a higher objective: u(q)u(A + w) + [1 − µ(q)]u(A + b) > U then it must violate the zero profit constraint (6.34). Since {K,W,Q,U} is an equilibrium, optimal application implies u(Q(w))u(A + w) + [1 − µ(Q(w))]u(A + b) ≤ U. Since by (6.35) w ≥ b, these inequalities imply that µ(q) > µ(Q(w)) ⇒ q < Q(w). Thus, h(q)[ f (k) − w] − k < h(Q(w))[ f (k) − w] − k ≤ 0 where the weak inequality exploits profit maximization. Therefore {k,w,q} violates (6.34) and is not in the constraint set.

188

The Economics of Search

Proof: Step 2 Next it is shown that for any solution {k*,w*,q*} to the constrained maximization problem, there is an equilibrium E = {K,W,Q,U} with K = {k*}, W(k*) = {w*}, and Q(w*) = q*. Set U = µ(q*)u(A + w*) + [1 − µ(q*)]u(A + b) And let Q(w) satisfy U = µ(Q(w))u(A + w) + [1 − µ(Q(w))]u(A + b) Or Q(w) = 0 if there is no solution to the equation, in particular if w < b. It is immediate that E satisfies optimal application. We now show that it also solves profit maximization. Suppose to the contrary that some triple {k′,w′,Q(w′)} violates profit maximization, so h(Q(w′))[ f (k′)− w′ ] −k′ > 0. One implication of this is that Q(w′) > 0 which from above implies that w ≥ b. Another implication is that we can choose q′ < Q(w′) so that h(q′)[ f (k′) − w′] − k′ = 0. Now by the definition of Q(w), q′ < Q(w′), and w ≥ b we have U < µ(q′)u(A + w′) + [1 − µ(q′)]u(A + b). Thus, {k′,w′,q′} would satisfy both constraints and yield a higher value of the objective function than {k*,w*,q*}, a contradiction. Finally, existence can be shown by noting in the more interesting case b < b¯ , the constraint set is nonempty and compact whereas the objective function is continuous implying that a maximum exists and by step (2) also an equilibrium. Since k affects only (6.34), an equilibrium k* must satisfy the first-order condition for capital choice h(q*) f ′(k*) = 1. Combining this with zero profits gives w* = f(k*) − k*f ′(k*).

(6.36)

Thus, despite search frictions, capital earns its marginal product and labor keeps the residual. Next, by reinterpreting the zero profit constraint as a function of w, via (6.28), we can graph the equilibrium in the (q,w) plane. See Figure 6.1.

Unemployment, unemployment insurance and sequential job search 189

Figure 6.1 Graphical depiction of equilibrium condition

Proposition 6.6 1 2

3

Let {ki,wi,qi} be an equilibrium when the utility function is ui. If u1 is a strictly concave transformation of u2, then k1 < k2, w1 < w2, and q1 < q2. Let {ki,wi,qi) be an equilibrium when the initial asset level is Ai. If A1 < A2 and the utility function has decreasing absolute risk aversion (DARA), then k1 < k2, w1 < w2 and q1 < q2. Let {ki,wi,qi} be an equilibrium when the unemployment benefit is bi. If b1 < b2, then k1 < k2, w1 < w2, and q1 < q2.

Proposition 6.6 is portrayed in Figures 6.2 and 6.3. Essentially as the indifference curves become more concave the tangency moves downward and to the left resulting in a lower wages w and output q. So when individuals are more risk averse, equilibrium wages, output and capital are lower but employment rates are higher. As unemployment benefits increase so do the equilibrium wage, capital and output levels but the unemployment rate increases. Acemoglu and Shimer go on to show that if individuals are risk neutral output is maximized at b = 0, but that if individuals are risk-averse output is not maximized at b = 0. Thus in this latter case some level of unemployment benefits yields a Pareto improvement.

190

The Economics of Search

Figure 6.2 Changes in preferences

Figure 6.3 Change in unemployment benefits

Unemployment, unemployment insurance and sequential job search 191 6.7 MORE ON OPTIMAL UNEMPLOYMENT INSURANCE

6.7.1 The Shavell–Weiss analysis This summary is based on the excellent survey article by Karni (1999). The private information on search effort and reservation wages enables the unemployed searchers to search more leisurely and wait until a wage offer is substantially higher than the optimal reservation wage. It is moral hazard that promotes this non-optimal behavior by insulating these searchers from the full force of this behavior. The Shavell–Weiss model assumes a sequential search model, where the relevant wage distribution flows from the efforts of the unemployed searcher. The probability of accepting a job offer in the tth period is given by ∞

p(w*t,et) =

冮 f(w |e )dw , t

t

t

w *t

where f(·|et) is the density function of wt given the level of search effort, et. Shavell and Weiss assume the searcher plans for the future by selecting a sequence of levels of effort and reservation wages, {et,w*t}∞ to maximize discounted expected utility. This behavior underlies the employment probability and, hence, the expected cost of the UI program. Karni notes that “the optimal choice of effort and reservation wage depends on UI benefits,” and concludes that “the main insight provided by the analysis is that if the unemployed agent has no wealth to begin with and has no access to borrowing, then the optimal UI benefits should decline over time, approaching zero in the limit” (Karni 1999: 450; original emphasis). This analysis implies that the best UI benefits schedule increases the level of effort and the reservation wage falls as unemployment persists. These effects show that the departure rate from unemployment rises as the unemployment spell continues. Karni states that the main insight of this analysis is that “the problem is best dealt with if the UI benefits decline monotonically with the duration of the unemployment spell.” This conclusion, while convincing, remains silent regarding the design of UI benefits. The efficiency effects of this UI program are that a long search is often accompanied by an excellent match between employee and employer and hence a long period of employment. Phelps (1972) has also argued that such a policy gives little protection to those who are unlucky in their job search. Let us recast the Shavell–Weiss analysis and show that this new interpretation also supports their position. Let un ≥ 0 be the UI benefit associated with the nth week of unemployment, and assume that this benefit is received without regard to the worker’s search efforts. Then, as per equation (6.17) we have

192

The Economics of Search ∞

冮

x V(x,n) = max ; u + βV(0,n + 1); un − c + β V(y,n + 1)dF(y) , 1−β 0 n = 0,1, . . . (6.37)

冦

冧

The method employed in the proof of Theorem 6.3 easily yields: Theorem 6.4 Let ξn be the reservation wage for a worker in his nth week of unemployment. If u1 ≥ u2 ≥ ..., then ξ1 ≥ ξ2 ≥ .... Moreover, the difference between the return to search and the return to “leisure” increases with the number of weeks of unemployment. Thus, the incentives associated with a decreasing sequence of UI benefits cause pn to increase with n, where pn is the conditional probability that a worker returns to employment in the n + 1st week given that he has been unemployed for a week. In comparing Shavell–Weiss with Mortensen, Karni makes a very important point. Mortensen’s definition of job effort is in terms of search time, whereas Shavell–Weiss view this effort as a source of disutility, producing a stochastically dominant shift in the offer distribution. In Mortensen’s model effort is empirically observable, while Shavell–Weiss do not indicate how this feature of their model can be observed. Karni’s final observation is that it is not obvious “how the optimal UI schedule is to be implemented, since the parameters include the unobservable utility of the unemployed—the assumption that all unemployed are identical seems unacceptable.”

6.7.2 Hopenhayn and Nicolini (1997) Hopenhayn and Nicolini extend the Shavell–Weiss analysis and consider the design of an optional UI program where the insurer has the power to tax or supplement the insured’s income after re-employment. These features are absent from all extant UI programs but Karni notes that “if used correctly, the increased flexibility afforded by the extended set of rewards and punishments have positive welfare implications.” That is, improved design may produce some utility for participants at lower cost. Hopenhayn and Nicolini use a repeated principal–agent relationship with moral hazard between the unemployment insurer and insurees. Karni (1991: 451) identifies the key feature of their model is that the likelihood of finding new employment depends on the search effort of the unemployed which is private information. They differ from Shavell and Weiss in that both job effort and the received job offers by the unemployed are not observed. Hopenhayn and Nicolini assume that only search effort is unobserved. Karni sees this as an unrealistic assumption. Hopenhayn and Nicolini also

Unemployment, unemployment insurance and sequential job search 193 assume that all jobs are identical and pay the same wage. Once employed a worker never is separated. This means that UI insurance is sold only to unemployed individuals. Karni observes that banishing future unemployment eliminates Mortensen’s (1977) “entitlement effect”. This analysis of optimal UI assumes that the cost of UI is convex in the reservation utility level of the unemployed. There are two major conclusions from this analysis: 1 2

UI benefits must decline with the duration of the unemployment spell, and The tax imposed on insurees after they find employment depends on the duration of the unemployment spell prior to finding a job.

The tax increases in the duration of the unemployment spell. Hopenhayn and Nicolini used empirical results (U.S. data) to calibrate their model. They also estimated the gains from their plan relative to a plan where employed are not taxed. The estimated gain was from 15 to 35 percent. There are problems with this comparison, but Karni (1991: 453) notes that their main insight is preserved, namely, “that individuals buying insurance should be taxed according to their unemployment histories.”

6.7.3 Hansen and Imrohoroglu (1992) Hansen and Imrohoroglu (1992) have exposed the dangers of not approaching UI in a general equilibrium construct. Only with such machinery can the distortions of the taxes essential to the design and sustenance of a UI program be properly evaluated using a welfare maximization criterion instead of one that minimizes program cost. In their model, individuals face idiosyncratic employment shocks and are unable to borrow or insure themselves through private markets. They simultaneously study the effects of an unemployment insurance program on equilibrium allocations when these programs can help agents overcome liquidity constraints and allow them to smooth consumption, can subsidize leisure so that in the presence of moral hazard an agent has a reduced incentive to work, and are financed by taxes which can distort allocations. Hansen and Imrohoroglu (1992) operationalize moral hazard by considering the likelihood that UI benefits continue when a job offer is rejected. In their model all job offers yield the same wage so there is no value to society in terms of improved match quality of rejecting a job offer. Results from a calibrated model show that the optimal replacement rate for UI benefits can be quite sensitive to the amount of moral hazard for reasonable amounts of risk aversion.

194 The Economics of Search

6.7.4 Fredriksson and Holmlund (2001) Fredriksson and Holmlund (2001) study the incentive effects of UI on job search behavior using a Pissarides type matching model. They consider random motion among three labor market states: employment, insured unemployment, and uninsured unemployment. They assume that workers suffer job losses and entry into insured unemployment according to an exogenous rate . Those workers who are insured receive UI benefits at a constant replacement rate, b, per period, uninsured workers receive social payments, z, per period of unemployment. Unemployed workers move from insured to noninsured states at rate λ. Unemployed workers search for employment. The state from which they search, I is insured unemployment and N denotes noninsured unemployment, t affects the job search effort, ej, j = {I,N}, which is the fraction of time devoted to search. The arrival rate of employment opportunities depends on the search effort coupled with labor market tightness, given by the ratio, ρ, of job vacancies to the number of active job searchers. That is, the net hiring, H, in a given period is characterized by a job matching, constant return to scale, function H = H(S,v), where S is the effective employment seekers (S ≡ eIuI + eNuN, where uI and uN are the number of insured unemployed and uninsured unemployed, respectively) and v denotes job vacancies. Thus, the transition rate from unemployment to employment is αj = ejH(1,ρ), j = {I,N}. The model results in an “entitlement effect” where an increase in benefits for the insured, b, increases the search intensity of the uninsured. Similarly, an increase in the amount of social assistance to the uninsured increases the search intensity of the insured. This occurs because the marginal benefit of increased search intensity depends on the matching function H. Wage bargaining takes place between employers and employees and yields the Nash bargaining solution. The threat point is easily identified, since the fallback position of employees is the state of insured unemployment. Thus the threat point is assumed to be the utility of insured employment. Fredriksson and Holmlund assume that workers do not save and that the unemployed are unable to smooth consumption by borrowing. The intensity of search is selected to maximize the discounted sum of expected utilities. Both UI benefits and social assistance are financed by taxing wages. Given a UI policy specifying a replacement rate, a social assistance program, and a tax rate τ, an equilibrium in such a model is composed of search intensities, eI, and eN and a wage rate, w, such that: (1) eI and eN maximize individual utilities; (2) the wage rate is set by the Nash bargaining solution; (3) the labor market clearing conditions require that the flow from the employment state to the insured unemployment state equals the flow from the two unemployment states to the employment state and that the flow from

Unemployment, unemployment insurance and sequential job search 195 uninsured unemployment equals the flow into uninsured unemployment; and (4) the balanced government budget constraint τ(1 − uI − uN) = buI + zuN is satisfied. Finally, the social welfare criterion is the weighted sum of individual utilities in the different labor market states, weighted by the proportion of the population in each state. As Karni (1999: 455) noted regarding the optimality of the equilibrium. “The answer is obvious once we realize that individual search efforts have positive externalities.” Increased job search effort tends to reduce the equilibrium rates of unemployment and, therefore, unemployment insurance expenditures and taxes decline. But individuals do not capture the entire gain from their search effort which yields an equilibrium where search intensity is suboptimal. One major conclusion of Fredriksson and Holmlund (2001) is that welfare increases when one moves from a situation where the level of UI benefits and social assistance are equal to one where the former exceeds the latter. More generally, with no discounting, optimal unemployment insurance should involve the amount of unemployment insurance benefits declining over the unemployment spell. Karni (1999: 455) observes in his review article that certain aspects of their model create questions concerning the robustness of the main result. “First, as the authors recognize the assumption that employees do not save is unrealistic. Second, the assumed exogeneity of the rate of transition from the employment state to the insured unemployment state begs the potentially important issue of the effect of the UI program on this rate.” Karni (1999: 456) goes on to note that, if the probability of transition from the employment state to the state of insured unemployment is affected by the employee’s job performance then this performance is suboptimal for the same reasons that the search effort is suboptimal and, other things equal, a higher level of benefits in the insured state tends to reduce social welfare. Finally, the assumption that the transition from the insured to the uninsured is a random process at the individual level introduces another unrealistic feature; it seems obvious that in a society of risk averse individuals, this artificially imposed idiosyncratic risk entails unnecessary loss of welfare. Karni’s review (1999: 463) has an excellent section on “What Have We Learned” which, in concluding this chapter, we quote in its entirety: The variety of approaches to modeling the labor market makes some of the results model specific and thus difficult to compare. Nevertheless,

196

The Economics of Search some general observations seem warranted. First, regarding the formulation of the problem of optimal UI 1

2

The main objective of UI is to enable households to smooth their consumption spending in the face of unemployment risk. The risk involved is the loss of earnings in the event of unanticipated unemployment. Unemployment insurance, being a contingent claim, dominates saving as a mean of consumption smoothing. The modeling of the environment in which the unemployed operate depends on the viewpoint taken. Partial equilibrium analysis tends to favor search models in which the durations of unemployment spells depend on the intensity of the job search activities of the unemployed and their search strategies. General equilibrium analysis emphasizes job matching technologies with the total unemployment and job vacancies as argument. Despite the important role of search intensity the exact interpretation of what this intensity means in practice is not always spelled out. Implicitly it implies spending time and money acquiring information about job vacancies, filling out job applications, and showing up for interviews. Thus a formal treatment of search effort requires explicit attention to the interaction of consumption and leisure. Unfortunately, this is not easy to do; as a result, the issue has not been addressed by the literature.

Second, some conclusions emerge that seem compelling. 1

Under the optimal UI scheme benefits decline with the duration of the unemployment spell. The intensity of search and the reservation wage are private information giving rise to a moral hazard problem. This problem requires foregoing some of the benefits from consumption smoothing in order to induce the appropriate level of search effort and to lower the reservation wage. The work so far suggests that if the only source of moral hazard is search behavior and the unemployed wealth is relatively small, then to produce the right incentives, UI benefits must decrease as a function of the duration of the unemployment spell.

Two related results are also noted: First, the duration of the benefits period should be unlimited even though this entails reduced replacement ratio. Second, the UI premium paid by employees following an unemployment spell should increase as a function of the duration of the spell. The latter result should be interpreted with some caution since it was established in a context in which the only source of moral hazard is the search behavior. If job performance itself is subject to manipulation by employees, then high UI premium may have adverse effects on the motivation to work and increase unemployment.

Unemployment, unemployment insurance and sequential job search 197 2

Under optimal UI scheme, premiums should be based on experience rating. Firms and individuals have hidden characteristics that, in the absence of perfect experience rating, interact with their actions to produce a phenomenon called endogenous adverse selection. The presence of endogenous adverse selection makes first-best allocation of unemployment risk bearing unattainable. Experience rating is costly and perfect experience rating is likely to be infeasible and nonoptimal. However, to minimize the impact of endogenous adverse selection experience rating should be incorporated in optimal UI schemes.

7

Job search in a dynamic economy

7.1 INTRODUCTION This chapter generalizes the basic search model by permitting the wage offer distribution to fluctuate with the business cycle. Ad hoc search arguments have been used extensively to speculate about individual behavior over the business cycle. An indisputable fact emerging from these speculations is that the state of the economy is in constant flux and may have a significant influence on search behavior. In the next subsection a search policy is designed that explicitly considers business cycle effects. A policy that is optimal in a depressed economy is unlikely to be best in a buoyant economy. In particular, the reservation wage should be lower in the former situation. It is precisely behavioral responses like this that generate macroeconomic phenomena. Indeed, most macroeconomic phenomena for which search models were designed manifest themselves only in a dynamic economy. The dynamic model envisages the economy changing according to a known Markov chain in which the probability of reaching a higher state in one transition is a nondecreasing function of the current state. Each state of the economy is characterized by a known distribution of wages. Assuming that the wage distribution is stochastically increasing in the state of the economy, the analysis demonstrates that for each state there is a reservation wage and this wage increases with the state of the economy. At this point it should be emphasized that this is not an equilibrium model of search. Only the employee side of the search market is studied. Each searcher has his/her own reservation wage, and the heterogeneity of labor implies that there is no single positive wage below which firms cannot hire. Section 7.3 presents the implications of the dynamic model for labor market behavior. These behavioral implications are contrasted with those of the static model. Having introduced the business cycle it is obvious that the searcher may wish to vary his intensity of search as the economy changes. The optimal policy is established in the concluding subsection. In contrast to the monotonicity of reservation wages with the state of the economy, it is somewhat

Job search in a dynamic economy

199

surprising that when the individual can vary the rate of search, the optimal search intensity need not increase as the economy improves.

7.2 SEARCH IN A DYNAMIC ECONOMY (LIPPMAN AND McCALL)1 Let {1,2, . . .,K}, K ≤ + ∞, be the set of states of the economy, and denote the distribution function of wages associated with state i by Fi. The discrete time discount factor is β ≤ 1, and the search cost is c > 0. Naturally, we also assume that sup µi < ∞, where µi is the mean of Fi.2 The economy changes according to a discrete time Markov chain, with one-step transition matrix P = (Pij), which is independent of the sequence of offers drawn from {Fi}.3 The job offer available is the value of the random draw from last period’s distribution function. If the searcher accepts an offer of x, the process terminates and the searcher is absorbed into the employment state for the n periods remaining in his working life. In this context, x can be regarded roughly as the discounted value of all future wages. If the searcher rejects x, he must pay a fixed price c for a draw from the distribution Fi. The economy then moves to a new state j according to Pij. After this transition to state j, the searcher must decide whether to accept or reject the new offer y. The searcher is not allowed to retain rejected offers. The goal is to find a stopping rule which maximizes the β-discounted expected net benefits. In order to ensure that the larger numbered states are preferable, we make two fundamental monotonicity assumptions. Assumption 7.1 ing, that is

The distribution functions {Fi} are stochastically increas-

F1(t) ≥ F2(t) ≥ . . . ≥ Fk(t), all t ≥ 0. Assumption 7.1 implies that µ1 ≤ µ2 ≤ . . . ≤ µk where µi is the mean of Fi, but, of course, not conversely. Moreover, it implies that ∞

∞

冮

冮

Hi + 1(ξ) ≡ (x − ξ)dFi + 1(ξ) ≥ (x − ξ)dFi(ξ) ≡ Hi(ξ) ξ

ξ

for each ξ, so that if each state were absorbing and β = 1, the reservation wage associated with state i + 1 would be at least as large as that associated with state i. k

Assumption 7.2

For any k,

冱P j=i

ij

is nondecreasing in i.

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The Economics of Search

Assumption 7.2 rules out situations in which a high number state is inferior to a low number state. This situation could arise if the transition probability from a high state to a lower state was sufficiently large. Furthermore, define k

µ-- (n) i =

冱P

µ.

(n) ij j

j=1

Assumptions 7.1 and 7.2 together imply not only that µi, the expected value of the immediately forthcoming offer, increases in i, but also that µ--(n) i , the expected value of the n + 1st forthcoming offer, increases in i. In fact, the sequence <Χin>n ∞= 1 of offers received when starting from state i is stochastically increasing (term by term) in i. That is, Xi + 1,n > Xi,n, for all n. Thus, in this sense higher states are again preferable to lower states. The higher the state the more “buoyant” the economy. The importance of Assumptions 1 and 2 stems from the following monotonicity factor: if f is a nondecreasing function, then ∞

冮 f(y)dF

is nondecreasing in i

(7.1)

f(j) is nondecreasing in i.

(7.2)

i(y)

0

and k

冱P

ij

j=1

We shall say we are in state (i,x) if the economy is in state i and the currently available job offer is x. Let Vn(i,x) denote the maximal β-discounted expected return attainable when n periods remain and we are in state (i,x). Then Vn(i,x) satisfies the recursive equation (V0 ≡ 0)

冦

∞

k

冱P 冮V (j,y)dF (y)冧

Vn + 1(i,x) = max x, − c + β

ij

j=1

n

(7.3)

i

0

≡ max {x,ξn(i)}. Clearly, ξn(i) is the reservation wage when n + 1 periods remain and the economy is in state i. If we wish to interpret x as the wage rate rather than the discounted present n

value of all future wages, then x should be replaced by x

冱β j=0

associated with stopping; thus equation (7.3) becomes

j

as the reward

Job search in a dynamic economy

201

n

冦 冱 β ,ξ (i)冧. j

Vn + 1(i,x) = max x

(7.3′)

n

j=0

Consequently, we can interpret

ξn(i)

as the reservation wage rate whereas ξn(i)

n

冱β

j

j=0

is merely the minimally acceptable discounted present value of all wages when n + 1 periods remain and the state of the economy is i. Theorem 7.1 The optimal return Vn(i,x) is nondecreasing in n, i, and x, so that the reservation wages ξn(i) satisfy ξn(1) ≤ ξn(2) ≤ . . . ≤ ξn(k),

(7.4)

ξn + 1(i) ≥ ξn(i).

(7.5)

and

If ξn(i) satisfies (7.3′), then ξn + 1(i)/αn + 1 ≥ ξn(i)/αn,

(7.5′)

n

where αn ≡

冱β. j

j=0

Proof Then

Note that V1(i,x) ≡ x, and assume Vn(i,x) increases in both i and in x. ∞

冮

∞

冮

f (j + 1) ≡ Vn(j + 1,y)dFi + 1(y) > Vn(j,y)dFi + 1(y), 0

0

∞

冮

≥ Vn(j,y)dFi(y) ≡ f(j), 0

by hypothesis and (7.1), respectively. Applying (7.1) and (7.2) to the above implies that ξn(i + 1) ≥ ξn(i), so that Vn + 1(i,x) increases in both i and in x, establishing (7.4). Equation (7.5) is immediate from Vn(i,x) nondecreasing in n. Suppose (7.3′) holds. Clearly (7.5) holds. Now V2(i,x)/α1 ≥ xα1/α1 = V1(i,x)/ α0. Assume Vn(i,x)/αn − 1 ≥ Vn − 1(i,x)/αn − 2 for all i and x, then since (7.5) holds and − c/αn ≥ − c/αn − 1, we obtain ξn(i)/αn ≥ ξn − 1(i)/αn − 1 for all i. Therefore

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The Economics of Search Vn + 1(i,x) = αn max {x,ξn(i)/αn} ≥ αn max {x,ξn − 1(i)/αn − 1} = (αn/αn − 1) max {xαn − 1,ξn − 1(i)} = (αn/αn − 1Vn(i,x)).

This completes the induction argument. Q.E.D. Observe that ξn(i) is also a nondecreasing function of β(0 ≤ β ≤ 1). 7.3 LABOR MARKET INTERPRETATION Since the reservation wage increases as the economy improves, there will be fewer dropouts in a buoyant economy. For suppose the economy is in state i and the tth individual has reservation wage ξt(i). Suppose further that the return from not participating in the labor market exceeds ξt(i). Then the optimal policy for the tth individual is to drop out. When the economy moves to a higher state j, this decision must be reconsidered, and since ξt(j) ≥ ξt(i), the tth individual may decide to search for employment. Thus an improving economy will lead to an increase in the labor force as more individuals begin looking for work. It has been noted that the cost of search declines in a growing economy.4 Notice that when the cost of search is state-dependent, with ci denoting the cost of search when the economy is in state i, Theorem 7.1 remains true provided c1 ≥ c2 ≥ . . . ≥ cK. This, of course, strengthens the negative relationship between economic growth and number of dropouts. One would expect the length of search, the period of frictional unemployment, to decline as the economy improves. In general, this is not true. However, a simple but rather severe restriction on the Markov transition matrix, namely, pij = pj, does produce the anticipated result. More specifically, we wish to study the behavior of pi, where pi is the probability that a search from state i results in a job offer that will be accepted. To begin, define qij = 1 − Fi(ξ(j)), where ξ(j) is the reservation wage (in the infinite horizon problem) for state j. Then qij is the conditional probability that a job offer resulting from a search from state i will be accepted given that the economy has moved to state j at the time the offer arrives. Hence, pi, satisfies k

pi =

冱p q . ij ij

j=1

If pij = pj, then pi ≤ pi + 1. This follows since qij ≤ qi − 1, j as Fi + 1 is stochastically

Job search in a dynamic economy

203

larger than Fi. This result states roughly that frictional unemployment decreases as the economy improves. More precisely, let Yi be the number of offers tendered (until one is finally accepted) starting from state i, so that the random variable Yi is precisely the period of frictional unemployment. Defining p = Σj pj pj(= Σj pij pj), we have P(Yi = 1) = pi, P(Yi = k) = p(1 − p)k − 2 (1 − pi), k = 2,3, . . . which, coupled with pi ≤ pi + 1, implies5 that Yi + 1 is stochastically smaller than Yi. In other words, the period of frictional unemployment decreases stochastically in the state of the economy. Recall that in the standard search model the job searcher is assumed to be unaware of the increase (decrease) in the offer distribution and consequently frictional unemployment declines (rises) as the economy improves. Here no such assumption is made about being fooled; the reservation wage increases (decreases) in a precise manner, and yet the behavior of frictional unemployment is, in general, ambiguous. If the standard model is to be consistent, the discouraged workers also will be unaware of changes in the wage distribution, and labor force participation will not vary as the economy changes. Throughout this analysis we assume that the job searcher knows the wage distributions, the state of the economy, and the Markov chain. We do not address the important adaptive problems that arise when this information is imperfect. It is clear, however, that when the economy is in state j and the searcher believes it to be in i with i < j, the reservation wage will be set too low. This will lead to a reduction in frictional unemployment. The converse is true when j < i. Underestimation (overestimation) of the number of periods till retirement also reduces (increases) the reservation wage and, hence, frictional unemployment. We will consider this Bayesian updating in Volume II.

7.4 VARIABLE INTENSITY OF SEARCH Typically the intensity of search is a variable that can be controlled by the job searcher. In general, it will depend on the searcher’s preference for leisure, his wealth, and unemployment compensation. We do not incorporate these explicitly into our model, but instead place rather mild restrictions on the search cost function. The model was constructed as follows. Associated with search intensity µ, 0 ≤ µ ≤ µ--, is (i) the search cost c(µ) and (ii) the probability of receiving a job offer p(µ), both of which are independent of the state of the economy; furthermore, given that an offer is received, it is independent of the search intensity. Naturally, both c(µ) and p(µ) are assumed to be nondecreasing. In addition, we assume that c(0) = 0 = p(0) and c(·) is right continuous. For ease in presentation, we give p(·) the explicit representation p(µ) = µ/(µ + λ), where λ > 0. Since the searcher can have no impact on

204

The Economics of Search

the economy, we assume that the Markov chain is independent of the search intensity chosen by the searcher. It is now clear upon reflection that the analog of (3) is given by (V0 ≡ 0)

冦

K

冢 冱P V (j,0)

冤

Vn + 1(i,x) = max x, max − c(µ) + (β/(µ + λ)) λ 0 ≤ µ ≤ µ--

n

∞

K

+µ

ij

j=1

冱F 冮V (j,y)dF (y)冣冥冧 = max {x,ξ (i)}. ij

j=1

n

i

(7.6)

n

0

We are interested in the behavior of not only the reservation wage ξn(i) but also µn(i), where µn(i) maximizes ξn(i), that is, where µn(i) is the optimal search intensity when n + 1 periods remain and the state of the economy is i. Theorem 7.2 With variable search intensity, Vn(i,x), as given in (7.6) is nondecreasing in n, i, and x, and the reservation wages satisfy (7.4) and (7.5). Proof To begin, observe that [ξn(i + 1) − ξn(i)](µ + λ)/β ≥ [− c(µn(i)) + c(µn(i + 1))] k

冱(P

+λ

i + 1, j

− Pi, j)Vn − 1(j,0)

j=1

∞

k

冤冱P 冮V

+ µn(i)

( j,y)dFi + 1(y) −

n−1

i + 1, j

j=1

0

∞

k

冱P 冮V 0

冥

(j, y)dFi(y) ≥ 0.

n−1

i, j

j=1

The second term on the right-hand side is greater than or equal to zero by (7.2) and the induction hypothesis that Vn(i,x) is nondecreasing in both i and x while the third term exceeds zero by (7.1), (7.2) and the induction hypothesis. The first inequality follows from µn(i) being suboptimal from state i + 1. Q.E.D. In light of the fact that the reservation wage ξn(i) increases with the state of the economy, it would come as no surprise if the optimal search intensity µn(i) increased with i. In general, this result is not true. In fact, all possible orderings of {µn(i)}, with the proviso that µn(K) ≥ µn(1), are possible. Moreover, it is not even necessary that µn(i) be monotone in n. (The added assumption of convexity of c(·) is not relevant in eliminating this seeming pathology.)

Job search in a dynamic economy

205

7.5 THE BASIC THEOREM Theorem 7.3

If Pij = Pj, for all i, then

µn(1) ≤ µn(2) ≤ . . . ≤ µn(K),

(7.7)

µ1(i) ≥ µ2(i) ≥ . . . µn(i) ≥ µn + 1(i) ≥ . . . ≥ µ(i),

(7.8)

and

where µ(i) is the optimal search intensity for the infinite horizon problem. Proof We can reformulate (7.6) as follows: (V0 ≡ 0 and Vn(i) is to be interpreted as the maximal β-discounted expected return attainable when n periods remain, the state of the economy is i, and no offer is currently available): K

冱P V (j)

冦

Vn + 1(i) = max − c(µ) + (λ/(µ + λ))β 0 ≤ µ ≤ µ--

ij

n

j=1

冮

冧

+ (λ/(µ + λ)) max [βy,βΣPijVn(j)]dFi(y) . 0

(7.9)

K

冱P V (j),

In this case, if Pij = Pj for all i, we have, setting Vn = β

ij

n

j=1

冦

Vn + 1(i) = max − c(µ) + (λ/(µ + λ))Vn + (µ/(µ + λ)) 0 ≤ µ ≤ µ--

∞

× [Vn +

冮 (βy − V )dF (y)] 冧 n

(7.10)

i

Vn/β

∞

冦

= max − c(µ) + (µ/(µ + λ)) 0 ≤ µ ≤ µ--

冮 (βy − V )dF (y)冧 + V . n

i

n

Vn/β

It now follows from (7.1) with f(y) = 0 for y ≤ Vn/β and f(y) = βy − Vn for y > Vn/β, that µn(i + 1) ≥ µn(i), for i = 1,2, . . .,K − 1 and all n ≥ 1. ∞

冮

Finally, note that since (y − x)dF(y) decreases as x increases, Vn nonx

decreasing in n would imply that

206

The Economics of Search ∞

cn ≡

冮 (βy − V )dF

∞

n

Vn/β

i(y)

≥

冮

(βy − Vn + 1)dFi(y) ≡ cn + 1.

(7.11)

Vn + 1/β

7.6 WEALTH AND SEARCH The wealth position of the job-searcher also may influence his search behavior. As his assets decline the searcher may become more willing to accept employment. This willingness would be reflected in a declining reservation wage. Three different analyses of this bankruptcy problem have been presented. (1) Danforth (1974a) has investigated the effect of asset holdings on the strategy of a job-searcher who is maximizing expected utility where the utility function is characterized by decreasing absolute risk aversion. This is done for both finite and infinite horizon models. His main result is that the reservation wage decreases (increases) as assets decline (grow). It is of course not surprising that the reservation wage declines in the finite horizon model. By analyzing the infinite horizon model in a stationary setting (only wealth changes and prices are constant over time), Danforth (1974a) is able to isolate the wealth effect: “The individual’s readiness to accept lower income jobs increases in this model solely in response to a falling bank balance.” In his empirical study, Kasper (1967) rejects the hypothesis that the rate of decline of the reservation wage is positively related to age. Even though there are fewer work periods remaining, the searcher does not drop this reservation wage. The influence of wealth as portrayed by Danforth is a reasonable explanation of this phenomenon. Using Dutch data from the Socio-Economic Panel, Bloemen and Stancenelli (2001) find some evidence that the reservation wage is positively related to wealth, although the overall impact of wealth on the probability of employment is small. (2) The model of job search in a dynamic economy can also be applied to the bankruptcy problem. Let i be the assets at the job-searcher’s disposal, and take Χ to be a nonnegative random variable which represents the total return per dollar invested. Thus, if the value of the searcher’s assets is i and he continues his search, he spends c on search and invests i − c, so that the value of his assets after search is given by j = (i − c)Χ. Now, if his assets fall below c he is not permitted any further search (for he is not permitted any reinvestment—this would be palatable if we interpret the search cost c as daily living expenses), and, hence, he will accept the last offer tendered. Assuming that the sampled wage distribution does not depend upon his assets—except to account for the fact that he must permanently drop out of the labor market if his assets should fall below c—the model obtains with {Fi} and {Pij}, given by Fk = Fk − 1 = . . . = Fc,Fc − 1 = Fc − 2 = . . . = F1 = F0 = point mass at zero and

Job search in a dynamic economy Pij =

冦Pr ob[Χ =1, j/(i − c)], ifif

j=0

207

i>c . and i ≤ c

Clearly, the assumptions are satisfied; hence the reservation wage increases with the searcher’s wealth. (3) A martingale argument has also been used to analyze the bankruptcy problem.6 Unanticipated events may cause the searcher to accept the last offer and begin work immediately rather than continuing search. If the assets of the job-searcher fall below some critical level, then search is terminated and the last job offer accepted. The random behavior of the searcher’s assets can be analyzed using the classical ruin model. Let p be the probability that his assets are increased by one and q the probability that they are reduced by one, q > p. Specifically, let Χi be the change in the searcher’s assets on the ith day where the Χi are independent with P(Xi = 1) = p = 1 − P(Χi = −1). If the individual’s assets are unexpectedly increased above a certain level, say a1 + a2, he remains solvent forever whereas he accepts his last offer if his assets drop to 0. Let a1 be the magnitude of his assets when search begins. Letting Sn be his assets at time n, i.e., n

Sn =

冱Χ +a, i

1

i=1

the random variables Y1,Y2, . . ., defined by Yn = (q/p)S

n

form a martingale7 with respect to the Χi’s, and E(Yn) = (q/p)a . Since the expected value of this martingale process is always the same, even if stopped at a random time, the probability π that bankruptcy occurs is the solution to 1

E(YN) = (1 − π) (q/p)a

1

+ a2

+ π(q/p)o = (q/p)a,

where N is the first time his assets reach either 0 or a1 + a2 = a. For convenience assume that the probability of ruin is constant during the period of search (specifically, it depends neither upon his stopping rule nor his current asset level). Furthermore, assume that the searcher accepts the last offer if ruin occurs. Once again, we find that the reservation wage ξ can be computed myopically as follows: ξ = πE(Χ1) + (1 − π)E(max(Χ1,ξ)) − c. 7.7 SYSTEMATIC SEARCH Salop (1973) has discussed yet another reason for the reservation wage to decline as search proceeds.8 The elementary search model assumed that

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individuals sampled randomly without recall from the wage distribution. All firms in which their skills were used as factors of production were treated equally. In fact, job-searchers usually have prior information about job opportunities in these firms. Using this information to rank firms, they then search systematically starting with those firms with the “best” opportunities, rather than randomly. As the searcher marches down his list he recalculated his reservation wage at each step. The reservation wage appropriate for a highly ranked firm will be too large for a lower-ranked firm so that searching according to this systematic process gives rise to a declining reservation wage as search continued.

7.8 OPTIMAL QUITTING POLICIES A question of great interest is when will it benefit a worker to quit his current job and begin to search. Let c denote the out-of-pocket costs of search, β the discount factor, F the distribution of wage rates so that x/(1 − β) is the discounted present value of working forever at x dollars per period, and assume that dollar payments and costs take place at the beginning of the period whereas offers are received at the end of the period. Then during search, the reservation wage rate ξ satisfies ξ/(1 − β) = − c + [β/(1 − β)]E max (ξ,Χ1). Consequently, it is clear that the worker should quit if and only if w/(1 − β) < ξ/(1 − β) − K, where w is his current wage rate and K ≥ 0 is the out-of-pocket cost associated with quitting (e.g., loss of pension plan benefits). Of course, if K > 0, then the worker may not quit even though his current wage rate is below the reservation wage of a worker who has quit.

7.9 MONOTONE MARKOV CHAINS AND THEIR APPLICATIONS: CONLISK’S RESEARCH John Conlisk (1985, 1990; Conlisk et al. 2000) was the first to discover the power of monotone Markov chains in economic models. This section is based on his research. We begin with a brief historical note.

7.9.1 Historical background Daley was the first to formalize monotone Markov chains and exemplify its applications in the noneconomic arena. Simply put, a real-valued discrete

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time Markov chain {Χn} is said to be stochastically monotone when its onestep transition probability function P{Χn + 1 ≤ y | Χn = x} is nonincreasing in x for every fixed y. Daley (1968) notes that a particular stochastically monotone Markov chain can be easily bounded by another chain, with different transition probabilities and it need not be stochastically monotone. Furthermore, a finite subset Χ1, . . ., Χk of a stochastically monotone Markov chain (Χn) satisfies Esary’s et al. (1967) definition of associated random variables: that is, for every pair of functions f (Χ1, . . ., Χk) and g(Χ1, . . ., Χk), monotone decreasing in each argument, cov[ f (Χ1, . . ., Χk), g(Χ1, . . ., Χk)] ≥ 0, when the covariance exists. Daley describes several important applications of stochastically monotone Markov chains including research on random walks, epidemic processes, genetics, Galton Watson branching processes, and queuing processes. Derman (1963, 1970) shows the connection between monotone Markov chains and control-limit policies (economists call these reservation policies). Derman first shows that two conditions A and B are equivalent in characterizing stochastically monotone Markov chains. He then assumes condition A and obtains the reservation property for a replacement policy. That is, Condition A The transition probabilities {Pij} are such that for every nondecreasing function f (j), j = 0,1, . . ., L, the function L

g(i) =

冱P

ij

f (j), i = 0, . . ., L − 1

j=0

is also nondecreasing. Condition B The transition probabilities {Pij} are such that for each k = 0,1, . . ., L, the function L

rk(i) =

冱P , i = 0,1, . . ., L − 1 ij

j=k

is nondecreasing. (Condition B is a version of Assumption 2 in the dynamic economy model.) Lemma 1 Condition A < = > Condition B. Assume a system is inspected at regular intervals and then classified into one of (m + 1) states, 0,1, . . ., m. A control limit rule ᐉ states: replace system if the observed state is in the set k, k + 1, . . ., m, for some predetermined state k. The state k is called the control limit of the rule ᐉ. Assume Χn is the observed state of system at time n ≥ 0 and {Χn, n ≥ 0} is a stationary Markov chain. Let c(j) be the cost incurred when system occupies

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state j. L is the class of all control limit rules. For ᐉ ∈ L, the long run average cost is given by n

冱

1 A(ᐉ) = lim c(Χk). n→∞ n k=1 Theorem 7.4 (Derman 1963, 1970) Let the Markov chain {Χn, n ≥ 0}, be stochastically monotone. Then there is a control limit rule ᐉ* such that A(ᐉ*) = min A(ᐉ). ᐉ∈L

Let us turn to the major contributions of Conlisk. Conlisk (1985)9 observes that monotone chains form a small and specially well-behaved subset of all Markov chains. Yet the subset includes most chains found in economics. Thus, monotonicity is an ideal type of restriction— “mathematically strong and economically weak.” Definition 7.1 Suppose a Markov chain has n states and n × n transition matrix P = [Pij], i,j = 1,2, . . ., n. The entry Pij is the probability of moving to j from i. Each row of P can be considered as a density over (1,2, . . ., n). P is monotone if each row is stochastically dominated by the row below it, i.e., if the cumulative densities for row i exceed those for row i + 1. For n = 3, replacing each row of P by the corresponding cumulative density gives the matrix

P11 P = P21 P31

P11 + P12 P21 + P22 P31 + P32

P11 + P12 + P13 P21 + P22 + P23. P31 + P32 + P33

If each of the first two rows of this matrix is greater than or equal to the row below it, P is said to be monotone. Note that the sum j

冱(P

i,k

− Pi + 1,k)

(7.12)

k=1

is the difference between the j th cumulative density for row i and the j th cumulative density for row i + 1. Row i + 1 dominates row i provided (7.12) is nonnegative for j = 1, . . ., n − 1. If j = n, the sum is zero. Monotonicity of P requires that (7.12) be nonnegative for i = 1, . . ., n − 1. Let (7.12) be the (i, j )th element of (n − 1) × (n − 1) matrix. Definition 7.2

Monotonicity. Let D(P) be the (n − 1) × (n − 1) matrix with

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typical (i, j) element given by (7.12). The transition matrix P is monotone if and only if D(P) ≥ 0. Many economic applications have a structure exemplified by

Example (i):

.6 P = .3 .2

Example (ii)

P=

Example (iii):

1 P = .5 0

.3 .4 .3

.1 .3 .3, D(P) = .1 .5

冤

冥

.2 .2

1

冦n冧, D(P) = 0 0 0 0

0 .5 .5, D(P) = .5 1

冤

冥

.5 .5

Conlisk also identifies several prominent economic applications of Markov monotonicity: 1 2 3

Intergenerational mobility Evolution of asset prices Size distributions of firms.

Roughly speaking, monotonicity means: 1 2 3

The larger a parent’s income, the larger the child’s expected income The higher a share price this period, the greater its expected price next period The larger the firm size this period, the larger its expected size next period.

General application Many P matrices have the G property.

Gibrat’s law of proportionate effect: G = The frequency function(density) of proportional income changes is the same at all income levels. Property G was applied to income mobility by Champernowne (1953). He assumed

α − 1 + α0, α , P = −1 0, 0,

α1, α0, α−1, 0,

α2, α1, α0, α−1,

0, α2, α1, α0,

0, 0, α2, α1,

where the α’s are positive and sum to one.

... 0, 0, α2,

... ... 0

0

   . . .

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Each row of P is the preceding row shifted one step to the right. This implies monotonicity. In general, property G implies monotonicity. Dynamic economy revisited Economy moves among K business cycle states according to a Markov chain with transition matrix P. When economy is in state i, offers are drawn from density given by the ith row of another matrix Q. If there are n states of the economy and m wage-levels, P is n × n and Q is n × m. In this setting, a worker faces a higher dimensional Markov chain. His state in a period is given by a pair of indices (i,h). Index i is the state of the economy and index h is the current wage offer. The worker’s Markov chain has nm states. The probability of transiting from state (i,h) to state (i,k) is pijqhk. The worker partitions nm states into accept or reject states. Note that P is monotone and Q, which is not necessarily a transition matrix since it need not be square, is also monotone in that each row is stochastically dominated by the row below it. These monotonicity assumptions imply that the nm state chain confronting the searcher has a transition matrix which is a special case of a monotone transition matrix. Monotone Markov chains have a host of connections with discrete and continuous stochastic processes and ordering relations.10 For example, birth and death processes and random walks have this monotonicity property. For now we return to Conlisk for his discussion of search and other economic applications.

7.9.2 Austrian assets An Austrian asset earns its entire return in a single period, the “harvest date” chosen by the owner.11 The classic examples are trees and wine, in which cases the questions are when to cut the trees and when to drink the wine. Other timing questions, such as when to exercise a stock option, when to sell a work of art, when to build on idle land, and when to declare bankruptcy can be construed as Austrian capital theory questions. Formally, assume: the asset transits among n states according to P; the agent can harvest the asset only once; the payoff is ri if the asset is harvested while in state i; the discount factor is constant at β (with 0 ≤ β < 1); and the agent wishes to choose the harvest date so as to maximize the expected present value of the payoff. These real asset models belong to the class of real option models.12 See Dixit and Pindyck (1994). Given stationarity, the optimal strategy is to partition the state into a set of wait-states and a set of harvest-states, and to harvest the asset the first time it reaches a harvest-state. The question is how best to choose the partition. The answer revolves around a second question. What are the expected present values by state, assuming an optimal partition is used? Kemeny and Snell (1958) have answered these questions for a model nearly identical to the

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model here. The paragraphs below show how the answers can be sharpened when P is monotone. Though economists often study timing questions under uncertainty, the literature on Austrian capital theory is almost exclusively nonstochastic. Two exceptions are the Lippman and McCall (1976c) search model, which can be viewed as an Austrian capital theory model, and the Brock, Rothschild, and Stiglitz articles.13 By treating states and time continuously, Brock et al. handle more challenging mathematics than the mathematics of a discrete Markov chain by assuming away instantaneous jumps in the value of the asset. Roughly speaking the case of monotone P considered here is somewhere between the case of unrestricted P considered by Kemeny and Snell and the case of tridiagonal P that would be an analog to the continuous case considered by Brock et al. Turning to the analysis, consider vi, the expected present value of the asset when it is in state i. If i is a harvest-state, then vi equals the harvest-value ri. If i is a wait-state, then vi equals the discounted expectation of the asset’s value next period, namely, β (Pv)i. Optimality requires that vi equal the maximum of these two amounts. Thus, for each i, the equation vi = max[ri, β (Pv)i] is a necessary condition for optimality. In vector form, v = max(r, β Pv)

(7.13)

is necessary, where the max-function is understood to apply element by element. It turns out that (7.13) has exactly one solution for v. Given the solution v, the optimal partition of states into harvest-states and wait-states is immediate. If ri > β (Pv)i, then state i is a harvest-state. If the opposite inequality holds, then state i is a wait-state. If ri = β (Pv)i, then state i can be put in either category (hence there may be more than one optimal partition even though v is unique). Solution of the Austrian capital theory problem on a Markov chain can thus be simply stated: solve (7.13) for v and partition states accordingly.14

7.9.3 Interacting Markov chains In Conlisk (1976, 1992) and Conlisk et al. (2000), interactive Markov chains are studied. Their monotonicity and stability are demonstrated and their application to a host of economic phenomena are displayed. This research stimulated several formal treatment of interactive chains.15 Conlisk’s research is similar to the classic segregation model by Schelling (1971), which has been formalized in an article by Pollicott and Weiss (2001).

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7.10 RECENT RESEARCH BY MULLER AND STOYAN Our next application is a significant extension based on three articles by Muller. In Muller (1997a), Markov decision processes (MDP) are assumed to differ only in their transition probabilities. Muller shows that the optimal value function of an MDP is monotone with respect to properly defined stochastic order relations. He also finds conditions for continuity with respect to appropriate probability metrics. He applies his result to the inventory control model by Scarf (1960) and the optimal stopping model by Lippman and McCall (1976c). Muller (1997a) begins by noting that only a small number of Markov decision processes yield explicit solutions. Models that accommodate the complexity of real-world situations are almost always too complicated to be computationally tractable. Hence, there is a continued interest in discovering good approximations. The basic challenge is to find approximate models with solutions that differ slightly from the solution of the problem of interest. Muller addresses this challenge by evaluating the difference between solutions of two MDPs. Muller also studies the effect of replacing the transition probability distribution by a different distribution. He considers how this alters the optimal value of the MDP. Muller does not study the effects of different transition probabilities on the optimal policy. Instead, he concentrates on the influence of the expected value. This is justified by his belief that the expected value is more important in applications which include job search, i.e., optimal stopping, where no reasonable measure of distance between different actions is available. These effects on the optimal value of the MDP are especially interesting for several reasons. First, the distributions are often unknown and must be estimated.16 In addition, the distribution is selected from a parametric family which has an explicit solution. Finally, the use of computers for numerical approximations entails discretizations. A huge literature exists on approximations by discretization of state and action spaces. Quantitative studies usually entail the distance between the transition probabilities in total variation norm and assume the boundedness of the value functions. Muller uses the theory of integral probability metrics,17 which has total variation as a special case. He uses this metric to generate integral stochastic orderings and shows that the value function of a MDP depends “monotonically” on the transition probabilities. In Section 5B, Muller presents the optimal stopping application. Let Χ1, . . .,Χn be a sequence of i.i.d. random variables with distribution P. These random variables may be observed by paying c per observation. If the decision-maker halts after the kth observation, he receives reward max{Χ1, . . .,Χk}. We wish to maximize this reward, i.e., we seek an optimal stopping rule (with recall). From earlier we know that the solution is given by the following value iteration:

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Vn + 1(s) = max {s, − c + β 冮P(dx)Vn max {s,x})}. In Muller and Stoyan (2002), the “job search” example is continued and order relations are derived. Consider the following N period Markov decision process with discount rate β. Denote the action space by A. If the process is in state s ∈ S then the set of feasible actions are D(s). Let D be the restricted set of actions and states defined by D = {(s,a):a ∈ D(s)}. Now if action a is chosen and the process is in state s, the state transits according to the transition kernel Q(s, a, ds′) and reward r(s, a, s′) is earned. Thus the Bellman equation associated with the problem is Vn(s) = max {冮r(s,a,s′) + βVn + 1(s)Q(s,a,ds′)} for n = 0,. . .,N − 1. a ∈ D(s)

First the following theorem can be shown using an induction proof: Theorem 7.5 1 2 3 4

Let the state space S be ordered and assume that

D(·) is increasing in the sense that for s ≤ t we have D(s) ⊆ D(t). For each fixed a ∈ A and s ≤ s′ the transition kernel Q(a,s′,·) first-order stochastically dominates Q(a,s,·). r(·,a,s′) is increasing for all s′, and VN is increasing.

Then Vn(s) is increasing in s for all n = 0, 1, 2,. . ., N. Definition 7.3 The set F is called a generator of an integral stochastic order if for probability measures P and Q, P ≤ Q if 冮fdP ≤ 冮fdQ for all f ∈ F. Let MDP(Q′) and MDP(Q″) be two Markov decision processes differing only with respect to their transition kernels, which are denoted by Q′(s,a,·) and Q″(s,a,·). The value functions for these models are denoted by VQ′ n and VQ″ n , respectively. Then more generally we have from Muller and Stoyan (who reproduced Muller’s 1997a result) the following: Theorem 7.6 1 2 3

If there is a generator such that for all (s,a) ∈ D

Q′(s,a,·) < F Q″(s,a,·) for all (s,a) ∈ D; r(s,a,·) ∈ F for all (s,a) ∈ D; Q″ VQ′ n ,Vn ∈ F for all n.

Q″ Then VQ′ n (s) ≤ Vn (s) for all n and all s. Furthermore, if π* is an optimal policy for MDP(Q′), then Q″ V0π* (s) ≥ VQ′ 0 (s) for all s.

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Muller and Stoyan note that there are many applications where the assumptions of Theorem 7.6 are satisfied with ≤ F being one of the orders: ≤ st, ≤ cx, ≤ icx, and ≤ icv, where st, cx, icx and icv are shorthand for stochastic, convex, increasing convex, and increasing concave, respectively. In the job search problem (with recall), the reward function is r(s,0,s′) = −c and r(s,1,s′) = s. The transition kernels are Q(s,1,·) = δ∞ (where ∞ denotes the absorbing state and δx denotes a one-point distribution in x) and Q(s,0,·) = 冮δmax{s,x}(·)P(dx). They show that Vn is increasing and convex for all n, and, therefore, by Theorem 7.4 P′′ P′ ≤ icx P″ implies VP′ n (s) ≤ Vn (s), for all n and s.

In two papers, Muller (2001a) and Muller and Ruschendorf (2001) use stochastic orders to obtain bounds for the value of the optimal stopping of a dependent sequence Χ1, . . .,ΧN. It is important to note that the article by Stidham and Weber (1993) demonstrates that many optimal control problems for queuing networks can be modeled as MDPs. We believe that many economic problems in search theory and elsewhere can be formulated as networks of queues, in which case, we can apply MDPs to solve them. Then it is straightforward to derive their equilibrium distributions.

8

Expected utility maximizing job search1

8.1 INTRODUCTION Economic models of search and information proliferated at a rapid rate after Stigler’s seminal work in the area. These models were useful in explaining the phenomena of unemployed resources in the economy, most notably in providing a rationale for the Phillips curve relation. While the early search literature represented a significant improvement over the earlier deterministic approaches, a standard assumption that had been maintained, as we saw in Chapter 4, was that the objective of the searcher is to maximize expected income. This conflicts with the traditional view that individuals maximize utility functions that are concave in income or wealth. It is the purpose of this chapter to consider the implications of the introduction of a utility-based objective into the search problem. The result is a significant restructuring of the qualitative properties of optimal search. Section 8.2 reviews some basic issues in risk and risk aversion. It is based on the survey by Scarsini (1994). The next section presents analyses of insurance based on the two-state framework introduced by Ehrlich and Becker (1972), Hirshleifer (1971), and extended by Rothschild and Stiglitz (1976) and Lippman and McCall (1981) with some applications discussed in Section 8.4. Section 8.5 reviews the basic search model from Chapter 4, explicitly delineating its assumptions and implications, and sets forth the expected utility maximizing model that will be examined here. Section 8.6 then derives the qualitative properties of expected utility maximizing search under conditions of recall. Section 8.7 considers the case of nonrecall. Next, Section 8.8 briefly discusses the work of Vesterlund (1997) who studied the impact of risk aversion on the wage distribution and on the equilibrium labor force participation and equilibrium employment in a simple matching model. The final section then discusses work by Chetty (2004) who studied how consumption commitments may lead to differences between local and global risk aversion.

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8.2 CHARACTERIZING RISK AND RISK AVERSION2

8.2.1 Introduction The literature on probabilistic economics is enormous and burgeoning.3 In this section we identify some areas where probabilistic analysis has enriched our understanding of economic behavior and has generated fundamental advances in economic theory. Few would doubt that uncertainty has a decisive influence on economic behavior. Almost every phase of consumption and production is affected by uncertainty. Perhaps the most significant uncertainty is length of life. Also, individuals are uncertain about their incomes and producers are unsure of their sales and costs. The number, size of purchase, and intervals between arrivals of customers at a store are all stochastic, as are the number of employees who arrive at work on a given day. Inventory depletions, equipment breakdowns, wars, depressions, and inflations all occur unpredictably, and the results of research and technological processes are probabilistic. In short, economic agents operate in an environment permeated by stochastic phenomena, and their basic economic decisions are modified accordingly. Consequently, the underlying determinants of supply and demand have stochastic components, and relative prices are random variables. At first blush, risk would appear to have no influence on economic behavior unless people were averse to risk. But, though in many circumstances risk aversion is a fact and is essential to understanding economic behavior, much economic behavior is a direct consequence of uncertainty and is independent of risk aversion. Human behavior adapts to uncertainty and risk aversion in a variety of ways. Insurance, futures markets, contingency clauses in contracts, and use of stock markets are among the most important institutions that facilitate adaptation to risk aversion. On the other hand, methods like inventory control, preventive maintenance, and annual physical examinations are also used by individuals and firms to cope with uncertainty, independent of risk aversion. For example, Jones and Ostroy (1984) have shown that risk aversion is not essential to the increased “flexibility” that is achieved by building plants with flat average cost curves (Stigler 1939), engaging in parallel research and development programs (Nelson 1961), and holding substantial quantities of liquid assets (Makower and Marschak 1938). A position is defined to be “more flexible than another if the range of alternative future positions attainable from it at any given level of cost[subsumes] . . . that of the other.” These adaptations to uncertainty are important manifestations of rational behavior. A deterministic economic theory does not provide an adequate explanation of these responses to a stochastic environment. Similarly, the information accumulation that characterizes the decision processes of people is inexplicable by a purely deterministic model. On these grounds, it is easy to explain and to applaud the development of probabilistic economics. It is not

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easy, however, to explain the long time during which deterministic models have dominated economic theory. In 1958 Arrow remarked that: uncertainty has been long discussed in economic literature, but usually only in a marginal way. Discussion rarely advanced beyond the point of the famous article by Daniel Bernoulli who argued that the individual acts in such a way as to maximize the mathematical expectation of his utility. Despite the elegance and simplicity of this theory, little real use was made of it in explaining the facts of the business world beyond the existence of insurance, at least until the work of Frank Knight in 1921, which was continued somewhat belatedly by J.R. Hicks and Albert G. Hart who made the first fruitful applications of the theory of uncertainty to the behavior of business firms, particularly in regard to such questions as liquidity, the holding of inventories and flexibility of production processes. (Arrow 1958: 4) In spite of these insights by Knight (1921), Hicks (1931), and Hart (1942), for the most part the economic profession continued to propagate the classical economic theory in which costlessly acquired perfect information accompanies all economic activity. Where confronted with stark empirical realities, how could such a theory survive? It was useful in explaining several phenomena: tax effects, demand and supply as resource allocators, explanations of relative prices, specialization in production, foreign exchange rates, etc. But an enormous set of economic activity could not be explained, and, worse yet, came to be regarded as inefficient, undesirable, or irrational. Undoubtedly another part of the answer lies in the manner in which economic theory is tested. Econometricians always include stochastic terms in their econometric models. Hard empirical facts necessitated these additions. However, they were seldom grounded in economic theory and, when present, were merely appended to economic models that had been constructed in a certainty milieu. Furthermore, the stochastic component of econometric models frequently was viewed as an inconvenience which a properly formulated deterministic model would eventually eliminate. Another probable reason for the paucity of probabilistic analysis is that economic theory rarely attempted to explain the behavior of individual firms or individual consumers. Rather, the focus has been on the behavior of the representative firm and the average consumer. Where there are so many firms and consumers, a law of large numbers was implicitly invoked to reduce uncertainty essentially to zero. While invoking the law of large numbers is appropriate for some applications, it is misleading for others. Consumers search for low prices, purchase insurance, place a positive value on information, and diversify holding of risky assets. The firm searches for productive

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employees, insures against fire and other “acts of God,” and purchases many kinds of information. None of these actions is consistent with the certainty model. The success of probabilistic models in other sciences like physics and genetics raises the question: is there some reason to believe that economic behavior is less susceptible to probabilistic formulation than, say, genetic behavior? We think not, believing that economics is the cloudiest of the sciences and thereby agreeing with deFinetti (1974: 197) that: in the field of economics, the importance of probability is, in certain respects, greater than in any other field. Not only is uncertainty a dominant feature, but the course of events is itself largely dependent on people’s behavior. . . . It is, therefore, probability theory, in the broadest and most natural sense, that best aids understanding in this area.

8.2.2 Measures of risk and risk aversion During the course of our analysis we shall assume that the economic agent encounters a stochastic environment and acts so as to maximize his expected utility of the random outcome. The agent’s utility function u is assumed to be nondecreasing and concave, whence ru > 0, where the Arrow–Pratt measure of absolute risk aversions, ru, is defined by ru(t) = −u″(t)/u′(t). Of course, ru ≡ 0 if u is linear. The appropriateness/usefulness of this measure of risk aversion will be revealed in our analysis. Finally, we say that an agent with utility function u is more risk averse than an agent with utility function v if ru > rv. On several occasions (e.g., see Section 8.5) we shall also employ the Arrow–Pratt measure of relative risk aversion, Ru, defined by Ru(t) = tru(t). Having specified a measure of the agent’s aversion to risk, we proceed by supplying two measures of risk, each of which induces a partial order on the random variables in the agent’s environment. To be of use, of course, there must necessarily be a close connection (in fact, an equivalence) between the ordering of the set of random variables and the associated ordering induced by their expected utilities. Let Χ and Y be two random variables with cumulative distribution functions F and G, respectively. Recall that we say that the random variable Χ is stochastically larger than the random variable Y, written Χ >1 Y, if and only if G(t) − F(t) ≥ 0, for all t.

(8.1)

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When F and G satisfy (8.1), we say that Χ dominates Y—or F dominates G according to the criterion of first-order stochastic dominance. Because 0

E(Χ ) =

∞

冮 F(t)dt + 冮[1 − F(t)]dt,

−∞

(8.2)

0

for any random variable Χ [with the proviso that at least one of the two integrals in (8.2) is finite], it is clear that E(Χ ) > E(Y ) whenever Χ >1 Y. The link between first-order stochastic dominance and expected utility is provided in the next theorem. Theorem 8.1 A necessary and sufficient condition for Χ to be stochastically larger than Y is Eh(Χ ) ≥ Eh(Y ), all h ∈ £1.

(8.3)

where £1 is the set of nondecreasing functions. As revealed by (8.2), Χ >1 Y and F ≠ G imply that F(Χ ) > F(Y ). Consequently, we seek a weaker condition, one that will enable us to distinguish between random variables with equal means. Also remember that a random variable Χ is less risky than the random variable Y, written Χ >2 Y, if and only if t

冮 [G(s) − F(s)]ds > 0, for all t.

(8.4)

−∞

When F and G satisfy (8.4), we say that Χ dominates Y in the sense of secondorder stochastic dominance. Again, (8.2) reveals that F(Χ ) > F(Y ) whenever Χ >2 Y, F ≠ G, and E(Χ ) = E(Y ). Additionally, the so-called “mean preserving spread” is simply a special case, namely, the one in which F − G changes sign exactly once and E(Χ ) = E(Y ). The connection between second-order stochastic dominance and expected utility is contained in the next theorem. Of particular significance is (8.6), because it is often applied to u′, the derivative of the agent’s utility function: unlike u, u′ is a decreasing function. Theorem 8.2 Let £2 be the set of nondecreasing concave functions. A necessary and sufficient condition for Χ to be less risky than Y is Eh(Χ ) > Eh(Y ), all h ∈ £2.

(8.5)

Moreover, if Χ >2 Y and E(Χ ) = E(Y ), then Eh(Χ ) > Eh(Y ), all concave functions h.

(8.6)

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8.2.3 Alternative derivations of absolute and relative risk aversion The agent maximizes expected utility, where U is defined on the space of pecuniary outcomes with x denoting initial wealth, a constant. To satisfy the rationality assumption, u is increasing. There is a premium π such that the agent is willing to convert a random amount of money Z into its expected value E[Z }. The premium π depends on x and on the distribution of Z, that is π = π(x,FZ ). A more formal definition requires π to be the unique solution of: u(x + E[Z ] − π(x,FZ )) = E[u(Χ + Z )].

(8.7)

Suppose E[Z ] = 0 and Var(Z ) = σZ2 . By Taylor’s expansion we get u(x − π) = u(x) − πu′(x) + o(π2) and 1 E[u(x + Z )] = E[u(x) + Zu′(x) + Z2u″(x) + o(Z 3)] 2 1 = u(x) + σZ2 u″(x) + o(σZ2 ). 2 An application of (8.7) yields 1 u(π,FZ ) = σZ2 A(x) + o(σZ2 ), 2 where A(x) =

d u″(x) = − log u′(x). u′(x) dx

The function A is the absolute local measure of risk aversion. It can be interpreted as the local tendency to insure at x given u. The risk premium can be stated in relative terms with respect to x. In particular, π*(x,FZ ) is the proportional risk premium satisfying E[u(xZ )] = u(E(xZ ) − xπ*(x,FZ )) and yields 1 π*(x,FZ ) = π(x,FxZ ). 2

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Let E(Z ) = 0 and we obtain 1 π*(x,FZ ) = σZ2 R(x) + o(σZ2 ), 2 where R(x) = xA(x) is the relative local risk aversion measures. Note that these derivations are local, that is, they describe the agent’s response when he/she insures against infinitesimal risks. The following is the celebrated theorem by Pratt (1964) and Arrow (1974). The following conditions are equivalent:

Theorem 8.3

(i) A1(x) ≥ A2(x), for all x. (ii) π1(x,FZ ) ≥ π2(x,FZ ), for all x and all Z. (iii) u1 (u−1 2 (t)) is concave in t. If any one of these equivalent definitions is satisfied, u1 is more risk averse than u2. Since (i) <=> (ii), the absolute local measures of risk aversion and the corresponding risk premia have a similar comparative behavior. Note that a risk-neutral agent who displays indifference between Z and E[Z ] has a zero risk premium and a linear utility function. The risk averse(preferer) has a concave(convex) utility function. Pratt also proved the following two theorems: Theorem 8.4 A(x) decreasing in x <=> π(x,FZ ) decreasing in x, for all Z. (u displays increasing absolute risk aversion.) Theorem 8.5 R(x) decreasing in x <=> π*(x,FZ ) decreasing in x for all Z. (u is said to have decreasing relative risk aversion.)

8.2.4 A reformulation of the Rothschild–Stiglitz notion of increasing risk Scarsini considers increasing risk and notes an error in the famous Rothschild–Stiglitz (RS) formulation. He begins with four definitions: Definition 8.1

Given a distribution function H, define

t

H(t) =

冮 H(s)ds

−∞

H satisfies the following:

224

The Economics of Search lim H(t) = 0, lim H(t) = ∞

t → −∞

t→∞

H is increasing, convex such that if ∞

冮

| xdH(x)| < ∞, −∞

then ∞

lim H(t) − t +

t→∞

冮 xdH(x) = 0.

−∞

Definition 8.2 A distribution function G differs from a distribution function F by a spread [G is a spread of F ] if there is an interval I such that G assigns no greater probability to any open subinterval of I and G assigns at least as much probability as F to any open interval to the right or left of I. Definition 8.3

G is a mean preserving spread of F if

∞

∞

−∞

−∞

冮 xdF(x) = 冮 xdG(x)

and G is a spread of F. Definition 8.4 G is a mean preserving increase in risk with respect to F if there is a sequence Fn of distribution functions such that F = F1, lim Fn = G, n→∞

and Fi is a mean preserving spread of Fi − 1. Rothschild and Stiglitz (1970) analyzed the following conditions: RS1: G is a mean preserving increase in risk with respect to F. RS2: There are random variables W,Z on a same probability space such Χ and Z = Y. that E[Z|W] = W a.s. and W = st st RS3: Χ ≤cx Y or Y ≤cv Χ. RS4: G(t) − F(t) ≥ 0, ∀t ∈ R and lim G(t) − F(t) = 0. t→∞

Theorem 8.6 (Scarsini 1994) Conditions RS1–RS4 are equivalent. Scarsini notes that RS1 meant that a mean preserving spread shifts mass from the center to the tail of the distribution. This is an incorrect interpretation as proven by Landsberger and Meilijson (1990). Scarsini observes that the major defect of Rothschild–Stiglitz’s idea is that to generate a mean preserving increase in risk, a sequence of spreads is performed, taking out probability mass from intervals which need not have a

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nonempty intersection. Landsberger and Meilijson (LM) devise a mean preserving increase in risk at v, which is more restrictive.4 Their comparative concept of risk is in accord with the intuition that moving probability from the center to the tails is what increases risk. Their four conditions are stronger than those of RS. It is the element v which produces this strength. Condition LM1 states that all intervals I of mean preserving spreads contains v. LM2 not only requires that W and Z be a martingale, but also that v lies in the central range of the conditional probability of Z|W. The LM3 condition requires that the inequality of expected utilities holds for a class of functions that contains concave and all functions that are antistarshaped and supported at v.5 Scarsini shows the analogy of LM4 and RS4 by invoking Fubini’s theorem to obtain: x

H(x) =

x

冮 h(x)dt = 冮 (x − t)dH(t).

−∞

−∞

8.3 COMPARATIVE STATICS RESULTS The problem is E[v(Χ,α)], max α where Χ is a random variable that has economic importance for the agent and α* is a control parameter. Let α* solve this problem. Scarsini studies the dependence of α* on changes in risk or in the agents risk aversion. First consider the effects of changes in risk aversion on α*. Theorem 8.7 (Diamond and Stiglitz 1974) Let v1(x,α), v2(x,α) be two smooth functions increasing in x and convex in α. Let α1 = arg max E[v1(Χ,α)] α

α2 = arg max E[v2(Χ,α)]. α Suppose v1 is more risk averse than v2. If there is an x* such that for i = 1 and 2 for all α, ∂ vi(x,α) ≥ 0, x < x* ∂α ∂ vi(x,α) ≤ 0, x > x* ∂α then α1 > α2.

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The Economics of Search

Scarsini observes that many economic problems can be described by the following function: v(x,α) = u(αx − C(α)), with u: R → R increasing and concave (with bounded continuous derivative) and C: (0,∞) → R increasing, convex, continuously differentiable, and such that c(0) ≤ 0. E[v(Χ,α)] < ∞ be achieved at αF and sup E[v(Y,α)] < ∞ be attained Let sup α α at αG. Theorem 8.8 1 2

Under the above conditions for v,u,C:

(Rothschild and Stiglitz, 1971): if G is a mean preserving increase in risk of F, xu′(x) is concave, and u′(x) is convex, then αG ≤ αF. (Landsberger and Meilijson 1993): if G is a mean preserving increase in risk about the mean of F and xu′(x) is concave, then αG ≤ αF.

Sufficient conditions for xu′(x) concave can be stated in terms of Arrow– Pratt measures of risk aversion. Since R(Χ ) = −x

u″(x) , u′(x)

1 (u″′(x)x + u″(x)(1 + R(x))). Thus, if u has decreasing u′(x) relative risk aversion and R ≥ 1, then u″′(x)x + 2 u″(x) > 0, that is, xu′(x) is concave. when R′(x) = −

8.4 APPLICATIONS OF THEOREM 8.8

8.4.1 Rothschild and Stiglitz (1971) An agent has initial wealth w to be allocated between consumption today and consumption tomorrow. The amount not consumed today is invested and tomorrow yields a random return Χ per dollar invested. The utility function has the form u[(1 − α)w + (1 − δ)αwX ] where α is the savings rate and δ denotes the rate of time discount. The agent is risk averse, so u is concave. Conditions of Theorem 8.8 (2) are satisfied so the effect of an increase in risk on saving can be calculated.

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8.4.2 Competitive firm under price uncertainty (McCall 1967; Sandmo 1971; Lippman and McCall 1981) A firm owned by an individual maximizes the expected utility of profits assuming a random price Χ for the output. The output level α is the firm’s control parameter. The profit π is given by π = Χα − C(α) − B, where B are fixed costs and C, the variable costs, are increasing and convex. Risk aversion is assumed so u is increasing and concave. The problem is max E[u(π(x,α))]. α

Theorem 8.8 can be used to discover the changes in the distribution of price which cause an increase or a decrease in firm output. 8.5 THE BASIC JOB SEARCH MODEL Phrased in the paradigm of job search, the basic problem can be summarized as follows: an individual is seeking employment in a market in which jobs are homogeneous in all nonwage attributes. Since he/she does not know which wage is being offered by which prospective employer, but only the distribution function F of wages, he/she proceeds by randomly applying for positions, one at a time. For each application submitted there is a cost c incurred. The problem, then, is that of determining when to terminate the search process and accept a job offer in order to maximize the expected wage net of search costs. It should be noted that the term “wage” as it is used here may be thought of as the total lifetime earnings from the job (i.e., the wage rate times the duration of employment where the duration of employment is assumed the same for all jobs). Recall from Chapter 4 that the basic search model can be summarized by the following assumptions: Assumption 8.1 A job offer Χi is presented in period i, where each xi is a nonnegative random variable with cumulative distribution function F(·), E(Χi) < ∞, and the Χi’s are mutually independent. Furthermore, F has associated density function f. Assumption 8.2

The distribution F of wages is known to the searcher.

Assumption 8.3 received.

The searcher pays a constant amount c for each offer

Assumption 8.4

The searcher may receive an unlimited number of offers.

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The Economics of Search

Assumption 8.5 The searcher may, at any time, accept any offer received to date, so that recall is permitted. Assumption 8.6 search costs.

The searcher seeks to maximize the expected wage net of

The optimal strategy for this problem is the basic search paradigm which has a myopic sequential structure: after each offer is received, the decision is made whether to accept one of the offers received so far, or to search at least one more time. This decision is made by comparing the expected marginal return from soliciting one more offer with the marginal cost of receiving that offer. More formally, if w is the best offer received to date, then H(w), the expected gain from searching one more time, is given by: ∞

冮

H(w) = (x − w)f(x)dx. w

The optimal rule is to stop searching and accept employment whenever w satisfies H(w) ≤ c. Furthermore, the optimal rule is to accept any offer w ≥ ξ where ξ satisfies H(ξ) = c. The quantity ξ is referred to as the reservation wage, and it completely characterizes the optimal strategy. The optimal strategy for the basic search problem has the following important properties: Property 8.1 The optimal strategy is determined by a myopic policy; that is, the decision whether to accept a job is determined at each stage of the search process by considering the expected increase in income from observing one, and only one, more offer. The optimal strategy is then to search if and only if this quantity is positive. Property 8.2 There exists a unique reservation wage ξ such that the optimal strategy is to accept a wage offer w if and only if w ≥ ξ. Property 8.3 The level of the reservation wage ξ depends only on the level of the search costs c and the distribution F of wages. Property 8.4

The optimal strategy is the same under recall and nonrecall.

Property 8.5 The reservation wage increases with a mean preserving increase in the riskiness of the distribution of wages. Property 8.6

The reservation wage decreases as c increases.

While this particular model and concomitant results are neat and elegant, search theorists have considered various relaxations of the assumptions of

Expected utility maximizing job search

229

the basic model so as to enhance the connection between the model and “reality”.6 The theory of expected utility maximizing search have been considered by many researchers with E. Karni and D.C. Nachman the most conspicuous of those not represented in Lippman and McCall (1979).7 Nachman (1972, 1975) and Danforth (1974a) appear to be the first to study successfully the implications of utility maximization in the context of the basic search model. The resulting model is easy to analyze and serves to highlight the importance of the specific assumptions concerning the objective. The model that will be used in this section is the one framed in Assumption 8.1 through Assumption 8.6 with the important exception that Assumption 8.6 is altered as shown by Assumption 8.6′ Let U be a twice continuously differentiable nondecreasing concave (utility) function (defined on the real numbers).8 Denote the searcher’s initial wealth by M. The searcher seeks to maximize the expected utility of his wage plus initial wealth minus search costs. We demonstrate in Section 8.6 that Property 1 through Property 5 can all fail when the recall option is permitted and U is strictly concave. In particular, it is shown that there may be no reservation wage in the sense that the acceptance set has gaps (Example 8.1), myopic policies need not be optimal (Example 8.2), the acceptance set depends on the individual’s wealth (Theorem 8.12), the recall option may be utilized (Example 8.1), an increase in the riskiness of F may increase the acceptance set (Example 8.3).

8.6 EXPECTED UTILITY MAXIMIZING SEARCH WITH RECALL In our analysis, we utilize dynamic programming techniques as well as several of Pratt’s (1964) concepts and results concerning risk aversion. For ease of reference, we present all of definitions and notation at this point. To begin, we say that the searcher is in state (M,w) if his current wealth (initial wealth minus search costs accumulated to date) is M and the best wage offer received to date is w. Let the random variable Y(M,w,U,i) be the individual’s final (accepted) wage less search costs associated with taking one more observation and thereafter following an optimal strategy corresponding to the utility function U and state (M,w) when at most i( ≤ + ∞) more offers can be received. For notational simplicity we will abbreviate and write Yi,YU, or YM in place of Y(M,w,U,i) when the nonsubscripted variables are being held constant and no confusion can arise. Next, define Vi(M,w) to be the maximum expected utility that can be achieved when the searcher is in state (M,w) and at most i more offers can be received. Then Vi(M,w) satisfies the recursive equation [V0(M,w) = U(M + w)].

230

The Economics of Search Vi(M,w) = max{U(M + w); EU(M + Y(M,w,U,i))}

(8.8)

= max{U(M + w); EVi − 1 (M − c,w ∨ Χi )} where x ∨ y ≡ max(x,y) and Y(M,w,U,0) ≡ w EU(M + Y(M,w,U,0)) = U(M + w) as desired. It is clear upon reflection that EU(M + Y(M,w,U,i)) ≥ EU(M + Z(w,i))

so

that

we

have

(8.9)

for all random variables Z that represent the individual’s final wage less search costs associated with taking one more observation and thereafter following any strategy when the initial state is (M,w) and at most i more offers can be received. From (8.7) we can immediately conclude that EU(M + Yi) ≥ EU(M + Yi − 1) so that Vi + 1(M,w) ≥ Vi(M,w).

(8.10)

In view of E(Χ1) < ∞, U concave, and (8.10), we see that V(M,w), the maximum expected utility attainable when the searcher is in state (M,w) and able to solicit an unlimited number of offers, satisfies9 V(M,w) = lim Vi(M,w)

(8.11)

V(M,w) = max{U(M + w);EU(M + Y(M,w,U))}

(8.12)

i→∞

and

= max{U(M + w);EV(M − c,w ∨ Χ1)}. Furthermore, this formulation of the optimal return informs us which wage offers to accept via (8.11) or (8.12); namely, accept w when wealth is M if and only if U(M + w) ≥ EU(M + Y(M,w,U,i)}.

(8.13)

Equation (8.13) merely says that the individual should search only if the expected utility from continued search is greater than the utility from accepting the best offer to date. Given M, U, and i, any wage w that satisfies (8.13) is referred to as an acceptable wage. Denote the set of acceptable wages by W(M,U,i) if i < ∞ and W(M,U ) otherwise. (When no confusion can arise, we will simply write W(M) or W(M,i) as appropriate.) We now investigate the behavior of W(M,U,i) as we vary each of the three arguments. Toward this end, let rU be the Arrow–Pratt measure of absolute risk aversion:

Expected utility maximizing job search rU (x) ≡ −U ″(x)/U ′(x), all x.

231 (8.14)

If rU is nonincreasing, then we say that the searcher has decreasing absolute risk aversion, and a searcher with utility function U* is said to be more risk averse than a searcher with utility function U if rU * ≥ rU . Recall that for a concave utility function u, Pratt (1964) defines the (nonnegative) risk premium π(x,Z,u) associated with initial wealth x and the random variable Z by Eu(x + Z ) = u(x + E(Z ) − π(x,Z,u))

(8.15)

Of course, the risk premium π ≥ 0 is simply chosen so that an individual is indifferent between receiving the certain amount x + E(Z ) − π and receiving the random amount x + Z. Pratt’s Theorems 1 and 2 assert that:10 πU * ≥ πU if rU * ≥ rU .

(8.16)

πy ≤ πx, if y ≥ x and rU is decreasing.

(8.17)

and

Furthermore, the inequalities in (8.16) and (8.17) are strict if rU * > rU and rU is strictly decreasing, respectively. As a direct consequence of (8.10), (8.11) and (8.13) we see that the searcher is more selective the greater his opportunities for search. Theorem 8.9

The acceptance sets satisfy

W(M,1) ⊃ W(M,2) ⊃ . . . ⊃ W(M), all M.

(8.18)

The next result validates intuition in its assertion that the more risk averse individual searches less (i.e., stops sooner) and, equivalently, is less selective in that he accepts all offers the less risk-averse individual accepts in addition to some that the less risk-averse individual does not accept. Nachman was the first to rigorously derive this result (see theorem 11 of Nachman 1972, and theorem 6 of Nachman 1975). He did so by developing a number of properties of stopping times in a broad setting. His method entails a rather sophisticated approach, whereas our proof merely utilizes the fact that, ceteris paribus, the more risk-averse individual has a higher risk premium. (Additionally, our version does not require E(Χ 21) < ∞ as does Nachman’s [1972: 18].) Theorem 8.10

If rU * ≥ rU , then the acceptance sets satisfy

W(M,U,i) ⊂ W(M,U *,i), for all M and all i ≤ + ∞.

(8.19)

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The Economics of Search

Proof Fix M and i ≤ + ∞ U(M + w) ≥ EU(M + YU ). Then

and

select

w ∈ W(M,U,i)

so

that

U *(M + w) = U * ° U −1 ° U(M + w) ≥ U * ° U −1 (EU(M + YU )) ≥ U * ° U −1 (EU(M + YU *)) = U * ° U −1 ° U(M + EYU * − πU ) ≥ U * ° U −1 ° U(M + EYU * − π* U ) = U *(M + EYU − π* U) = EU *(M + YU *) where the inequalities follow from U * ° U −1 nondecreasing, (8.9) and (8.16), respectively, Hence w ∈ W(M,U *,i). Q.E.D. Because rU ≥ 0 for a concave nondecreasing utility functions and rU = 0 for a linear utility function, an immediate consequence of Theorem 8.10 is that regardless of the searcher’s wealth or the number of additional observations that can be taken the acceptance set for a concave utility function contains the acceptance set for a linear utility function, that is,11 [ξ,∞) ⊂ W(M,i), all i ≤ + ∞.

(8.20)

While the acceptance set under utility maximization contains the acceptance set under income maximization, it is not as well behaved. In particular, the next example shows that the reservation wage property will not hold in general, that is, the optimal strategy at each stage of the search process cannot be represented by a single critical number such that all offers greater than or equal to it will be accepted and all others rejected. This means that the acceptance set may have “gaps”. Example 8.1 There may be no reservation wage. Let the distribution F of wages be given by P(Χ1 = 0.5) = P(Χ1 = 2) = 0.1 and P(Χ1 = e + 0.5) = 0.8. Let c = 0.5, M = 0 and U(x) =

冦ln(z) −∞

for z > 0, for z ≤ 0.

Regardless of what search strategy is adopted, if w = 0.5 there is a positive probability that continued search will result in a negative net worth, so that EV(−0.5,0.5 ∨ Χ1) = −∞ < ln(0.5) = U(0.5). Thus, an offer of 0.5 will be accepted. On the other hand, if w = 2 a strategy of searching once more and then stopping yields an expected utility of 0.20 ln(1.5) + 0.80 ln(e) which exceeds

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233

ln(2), the return from accepting w = 2 immediately. Consequently, it is optimal to continue to search when w = 2. Thus, a wage of 0.5 is accepted but a wage of 2 is rejected. Q.E.D. The explanation of this phenomenon lies in the fact that when searching with recall, the best offer received to date serves as a form of insurance against unsuccessful search in the future. If the searcher has a reasonably good wage offer to which he/she can return, then the loss in utility from unsuccessful continued search is limited. If, however, the maximum wage offer to date is poor, then searching for one more period and failing to get a better offer might entail a large loss of utility. A high wage offer, in effect, raises the searcher’s wealth to the point that he/she can afford to be more selective. That there might not be a reservation wage when sampling with recall is due, at least in part, to the fact that if the current wealth is M and an offer of w has been made, then M + w is the “true wealth” with w representing the accrued but as yet unclaimed portion of the searcher’s true wealth. In view of this, let µ and w denote the true wealth (as defined above) and the maximum offer received to date, respectively, and define P(µ,i) to be the set of acceptable offers when µ is the true wealth and at most i(≤ + ∞) additional offers can be received. The next result12 establishes the existence of numbers ξµ, such that P(µ,i) = (ξµ,i,∞). In this sense the reservation wage property is preserved and ξµ,i is the reservation wage. Theorem 8.11

Given µ and i there is a number ξµ,i with the property that

P(µ,i) = (ξµ,i,∞).

(8.21)

Moreover, ξµ,i ≤ ξµ,i + 1, i = 1,2, . . ., + ∞.

(8.22)

¯¯n(µ,w) be the optimal return when n observations remain. Proof Let V ¯¯n satisfies [V ¯¯0(µ,w) = U(µ)] Clearly V ¯¯n + 1(µ,w) = max{U(µ) : F(w)V ¯¯n(µ − c,w) V ∞

冮

¯¯n(µ − c + x − w,x)dF(x)} + V

(8.23)

w

≡ max{U(µ);V˜ n(µ,w)}. ∞

冮

Since V˜ 0(µ,w) = F(w)U(µ − c) + U(µ − c + x − w)dF(x), the fact that U is a w

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The Economics of Search

¯¯0(µ,w) is nonincreasing in w and nonnondecreasing function implies that V ¯¯n − 1(µ,w) so that V ¯¯n − 1(µ,w) is nondecreasing in µ. Assume it to be true for V increasing in w and nondecreasing in µ. Then w+ε

¯¯n − 1(µ − c,w) + V˜ n(µ,w) = F(w)V

冮 V¯¯

(µ − c + x − w,x)dF(x)

n−1

w

∞

+

冮 V¯¯

(µ − c + x − w,x)dF(x)

n−1

w+ε

w+ε

¯¯n − 1(µ − c,w + ε) + ≥ F(w)V

冮 V¯¯

(µ − c + x − w,w + ε)dF(x)

n−1

w

∞

+

冮 V¯¯

(µ − c + x − w,x)dF(x)

n−1

w+ε

∞

¯¯n(µ − c,w + ε) + ≥ F(w + ε)V

冮 V¯¯ (µ − c + x − w,x)dF(x) n

w+ε

≥ V˜ n(µ,w + ε), which completes the induction step. This establishes the existence of a reser¯¯n + 1 ≥ V ¯¯n, the optimal return vation wage ξµ,n for the n-period problem. Since V ¯¯ satisfies V ¯¯(µ,w) = lim V ¯¯n(µ,w) = max{U(µ),V˜ (µ,w)}, whence the function V n→∞

¯¯n + 1 ≥ V ¯¯n. existence of ξµ follows. The monotonicity of ξµ,i in i stems from V Q.E.D. As noted earlier, expected income maximizing search has the property that the optimal strategy is myopic. A myopic policy is, of course, particularly attractive, since the reservation wage and optimal value functions are easily dealt with, both computationally and analytically. In expected utility maximizing search, the myopic policy is defined in terms of ∆U(M,w), the expected increase in utility from searching exactly once more; that is, ∆U(M,w) = EU[M − c + (w ∨ Χ1)] − U(M + w).

(8.24)

The myopic decision rule accepts any wage w that makes ∆U(M,w) nonpositive. Example 8.2 reveals the unfortunate fact that in general the myopic policy is not optimal. Example 8.2 The myopic policy may not be optimal. Let the parameters be as in Example 8.1, but set M = 2 × 10−7. Compare the expected utility when in state (M,w) = (2 × 10−7,0.5) under the following three strategies:

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235

Strategy (a): Search exactly once more. Strategy (b): Search one more; then stop if an offer of 0.5 or e + 0.5 is received, but search exactly once more if an offer of 2 is received. Strategy (c); Do not search. Denoting the expected utility from using strategy (a) when in state (M,w) by Ea, we have Ea ≈ 0.10 ln(2 × 10−7) + 0.10 ln(1.5) + 0.80 ln(e) ≈ −0.7019, EaU(−0.5,2) ≈ 0.2 ln(1.0) + 0.80 ln(e − 0.5) ≈ 0.6374, Eb ≈ 0.10 ln(2 × 10−7) + 0.10 EaV(−0.5,2) + 0.80 ln(e) ≈ −0.6788, Ec ≈ ln(0.5) ≈ −0.6931. Since Eb > max(Ea,Ec), the myopic policy is not optimal. (The failure of the myopic policy to be optimal in this example is a direct consequence of the existence of gaps in the acceptance set.) Q.E.D. Example 8.1 indicates that an individual tends to be more selective in job search the higher his/her best offer to date. Similarly, one would expect that if the individual’s wealth also acts as insurance against unsuccessful search, then the set of acceptable wages would shrink as wealth increases. To ensure that wealth offers such insurance, we assume that the absolute risk aversion function rU is decreasing. Theorem 8.12 If rU is decreasing, then the set of acceptable wage offers is decreasing in wealth, that is W(M + ε,i) ⊂ W(M,i), all M,i ≤ ∞, ε < 0.

(8.25)

Proof Fix M,i ≤ + ∞, ε > 0 and pick w ∈ W(M,i). From (8.24) and then (8.25) we obtain U(M + w) ≥ EU(M + YM ) ≥ EU(M + YM − ε).

(8.26)

From (8.26) we see that there is a γ ≥ 0 such that U(M + w) = EU(M + YM − ε + γ).

(8.27)

Coupling (8.26) and (8.27) yields M + w = M + γ + EYM − ε − πM + γ. Utilizing (8.28), (8.17), and U ′ ≥ 0 yields

(8.28)

236

The Economics of Search U(M − ε + w) = U(M − ε + γ + EYM − ε − πM + γ) ≥ U(M − ε + γ + EYM − ε − πM − ε + γ) = EU(M − ε + γ + YM − ε) ≥ EU(M − ε + YM − ε).

Hence, w ∈ W(M − ε,i). Q.E.D. Decreasing absolute risk aversion also yields an increasing reservation wage with respect to true wealth. Theorem 8.13

If rU is decreasing, then

ξµ + ε,i ≥ ξµ − i, all i ≤ + ∞, ε > 0.

(8.29)

Proof Fix µ,i ≤ + ∞, and ε > 0. If ξµ + ε < ξµ, then ξµ + ε ∈ ℘(µ + ε) and ξµ + ε ∉ ℘(µ). Equivalently, ξµ + ε ∈ W(µ + ε − ξµ + ε) and ξµ + ε ∉ W(µ − ξµ + ε). But this contradicts (8.25). Q.E.D. As search proceeds, the individual’s wealth decreases by c with each offer received. Accordingly, an immediate consequence of Theorem 8.12 is the fact that the individual’s acceptance set grows as search continues. This is in marked contrast to the standard case of search with a linear utility function (Assumption 8.6) where the set of acceptable wages remains unchanged regardless of the duration of search or the individual’s wealth.13 In the case of adaptive search wherein the distribution function F is not known with certainty, the optimal strategy also entails an adjustment of the acceptance set as the search process proceeds. Under adaptive search, however, the specific values of all rejected offers influence the acceptance set whereas in our model the acceptance set only depends upon the current level of wealth and thus on the number of searches to date. More striking than the searcher’s changing acceptance set is the fact that the recall option will, on occasion, be utilized.14 For instance, the optimal strategy in Example 8.1 is to search from state (2,0), but after three consecutive unsuccessful searches it is optimal to accept the previously bypassed offer of 2. Again, this fact distinguishes the general model from the linear case where the recall option is never exercised. If there is a level of final net worth which is unacceptable in the sense of having utility equal to minus infinity, then not only will the search process be finite with probability 1, it will be bounded in length. We make this notion precise in the next theorem.15 Theorem 8.14

If E(Χ 21) < ∞ and there is a constant b such that lim

x → b+

U ′(x) = + ∞, then the search process is bounded. That is, given M there is an integer BM such that one of the first BM wage offers will be accepted.

Expected utility maximizing job search

237

Proof Let Z be a random variable having the same distribution as sup Yn, where Yi = max(Χ1,Χ2, . . .,Χi ) − ic. A direct application of the Borel–Cantelli lemma in conjunction with E(Χ 21) establishes that E(|Z|) < ∞. We now demonstrate that the return from any state (M,w) with M + w < b − E(Z ) is minus infinity. To begin, note that the final net worth from optimal search when the state is (M,w) is bounded above by M + w + Z, which in turn is less than b − E(Z ) + Z. Because Z is not degenerate (otherwise Z = Y1 and Χ1 is degenerate in which case all offers are accepted), we have P(M + w + Z ≥ b) ≤ P(Z − E(Z ) ≥ 0) < 1. Thus, there is a positive probability that the final net worth will be less than b so that the expected utility associated with state (M,w) is minus infinity. From (8.20) we know that any offer of ξ or more will be accepted. Therefore, at most BM = [M + ξ − (b − E(Z ))]/c offers can be received until: (i) a state with M + w < b − E(Z ) is reached, or (ii) an offer of ξ or more is received. Q.E.D. Using Theorem 8.14, it is an easy matter to verify that V(M,w) is jointly continuous on the set of state with M + w > b − E(Z ) where Z is defined in the proof of Theorem 8.16.16 If all wages are not acceptable (i.e., W(M) ≠ (0,∞)), then the continuity of V(M,w) and U(M + w) in w yields the existence of an “indifference wage”,17 that is, a wage wM satisfying U(M + w) = EU(M + YM,w). In the next example, we make use of the existence of an indifference wage to show that increasing the riskiness of the distribution F does not necessarily lead to a smaller set of acceptable wages as is true for linear utility functions (Kohn and Shavell 1974).18 Increasing the riskiness of F can lead to an increase or to a decrease in the set of acceptable wages. Before demonstrating that both changes are possible, we note that we are using the term riskiness in the sense of Rothschild and Stiglitz (1970). To wit, a distribution function G with density g is said to be riskier than F with density f if there is a mean preserving spread19 such that g = f + s where

 α −α s(x) = −β  β  0

for a < x < a + t for a + d < x < a + d + t for b < x < b + t for b + e < x < b + e + t for otherwise

(8.30)

a > 0, b > 0.0 ≤ a ≤ a + t ≤ a + d ≤ a + d + t ≤ b ≤ b + t ≤ b + e ≤ b + e + t, and βe = ad. Example 8.3 Let U have strictly decreasing absolute risk aversion (so U ′ is strictly decreasing), and choose M, c, F, and U so that W(M) ≠ (0,∞). Then there is an indifference wage wM . Now define V(M,w,f1,f2) to be the expected

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utility associated with initial state (M,wM ) and continued search when f1 is the density of the wage offer distribution but the searcher has adopted a search strategy that would be optimal if in fact f2 were the density of the wage offer distribution. First we show that if s is a mean preserving spread with g = f + s, then Wg(M) may not contain wM even though Wf (M) does, where Wk (M) is used to denote the acceptance set when the underlying distribution has density k. It suffices to demonstrate that V(M,wM,g,f ) > V(M,wM,f,f ),

(8.31)

for with (8.31) we would then have V(M,wM,g,g) ≥ V(M,wM,g,f ) > V(M,wM,f,f ) = U(M + wM ),

(8.32)

which reveals the desired conclusion that wM ∉ W g(M). By virtue of Theorem 8.11, wM ∈ Wf (M − c) so if f is thought to be the density of the wage offer distribution and one more offer must be taken, then the searcher will stop with the next offer. Thus, if s is selected so that b + e = wM, we obtain V(M,wM,g,f ) − V(M,wM,f,f ) = EgU(M − c + (Χ1 ∨ wM )) − EfU(M − c + (Χ1 ∨ wM )) wM

=

冮 U(M − c + w

M

)s(x)dx

0

∞

+

冮 U(M − c + x)s(x)dx

wM

= −βtU(M − c + wM ) b+e+1

+β

冮

U(M − c + x)dx > 0

b+e

where the inequality follows from the fact that U is strictly increasing. This verifies (8.12) and, hence, the fact that wM ∉ Wg (M). To demonstrate the more striking possibility, namely, w ∈ Wg (M) and w ∉ Wf (M) requires a more delicate argument. We begin by selecting s so that a = wM and verifying that (8.32) holds. To do so necessitates the use of the following inequalities which are an immediate consequence of U ′ > 0 and U ″ < 0: (y − x)U ′(y) < U(y) − U(x) < (y − x)U ′(x), for y > x

(8.33)

Expected utility maximizing job search

239

and y+z

(y − x)[U(y + z) − U(y)] <

y

冮 U(v)dv − 冮U (v)dv

x+z

(8.34)

x

< (y − x)[U(x + z) − U(x)], for y > x and z > 0. Utilizing (8.33), (8.34), b > a + d + t, and βe = ad yields20 V(M,wM,g,f) − V(M,wM,f,f ) ∞

wM

=

冮 U(M − c + w a+t

=

冮 U(M − c + x)s(x)dx

)s(x)dx +

M

0

wM

a+d+t

冮 U(M − c + x)dx − a 冮

U(M − c + x)dx

a+d

a

b+e+t

+β

冮

(M − c + x)dx − β

b+t

b+e

冮 U(M − c + x)dx b

< −at[U(M − c + a + d + 1) − U(M − c + a + d )] + βt[U(M − c + b + e) − U(M − c + b)] < −atdU ′(M − c + a + d + t) + βteU ′(M − c + b) < (−atd + βte)U ′(M − c + a + d + t) = 0. Thus, V(M,wM,g,f ) < U(M + wM ).

(8.35)

Without loss of generality we can assume that wM is the largest indifference wage; hence, [wM,∞) ∈ Wf (M). Consequently, we can employ (8.35) and the continuity of U and V to assert that there is an ε > 0 such that V(M,w,g,f ) ≤ U(M + w), for all w ≥ wM − ε.

(8.36)

Equation (8.36) states that stopping is preferable to taking exactly one more observation. Unfortunately this is not enough to imply that w ∈ W*g (M) for all w ≥ wM − ε, for Example 8.2 shows that myopic policies need not be optimal. However, (8.36) is equivalent to EU(x − c + (y ∨ Χ1)) ≤ U(x + y) for x ≤ M and y ≥ wM − ε,

(8.37)

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The Economics of Search

where Χ1 has density g. The fact that (8.37) holds for x < M stems from Theorems 8.12 and 8.16. Now define Vi (x,y) as in (8.8) but with Χ1 having density g rather than f. The virtue of (8.37) is that it enables us to begin an induction argument to verify that Vi (x,y) = U(x + y), for x ≤ M and y > wM − ε.

(8.38)

To begin, note that (8.38) holds for i = 1 by (8.37). Assume (8.38) holds for i. Then by hypothesis and (8.37), Vi + 1(x,y) = max{U(x + y),EVi (x − c,y ∨ Χ1)} = max{U(x + y),EU(X − c + (y ∨ Χ1))} = U(x + y). This completes the induction. Consequently, V(M,w) = lim Vi (M,w) = V(M + w) for w ≥ wM − ε. i→∞

Thus we see that it is optimal to stop from state (M,w) if w ≥ wM − ε, when Χ1 has density g. That is, w ∈ Wg(M) if w ≥ wM − ε. Q.E.D. We conclude this section by noting that the optimal return V need not inherit the properties of U. For example, V is not necessarily concave in the level of wealth. This raises the possibility that an individual pursuing an optimal strategy will accept some actuarially fair or even unfair gamble even though he is strictly risk averse (i.e., U ″ < 0). In the next example we demonstrate that this can indeed occur. Example 8.4 Unfair gambles may be accepted. Let the utility function, search costs, and distribution over wage rates be as in Example 8.1. Let wealth be zero and the best wage offer to date be 0.5. Suppose that the searcher is offered a bet such that he/she wins or loses the amount 2 × 10−7 with probability of 1/2 for each outcome. If the bet is not accepted then the searcher will accept the wage offer of 0.5 and will have the utility level ln(0.5) ≈ −0.6931 with certainty. On the other hand, if the offer is accepted then with probability 1/2 he will: (a) win, whence the expected utility will be at least −0.6788 (see Example 8.2); and (b) lose and accept the offer of 0.5 for a utility level of ln(0.5 − 2×10−7) ≈ −0.6931. Consequently, the expected utility from accepting the bet is at least −0.6860 as opposed to −0.6931 from rejecting it. The searcher will accept an actuarially fair gamble even though his utility function is strictly concave. Q.E.D.

Expected utility maximizing job search

241

8.7 EXPECTED UTILITY MAXIMIZING SEARCH WITHOUT RECALL In this section we assume that the recall option is not available. As will soon be revealed, this simplifies the analysis and induces the existence of reservation wages. When we say that the searcher is in state (M,w), we mean that his current wealth is M and the last offer (as opposed to the best so far) received is still available and equal to w. The random variable Y(M,w,U,i) is defined as in Section 8.6. Because w represents the value of the currently available offer and recall is not allowed, Y(M,w,U,i) does not depend upon w. Next, define V*i (M,w) to be the maximum expected utility that can be attained when the searcher is in state (M,w) and at most i more offers can be received. Then V*i (M,w) satisfies [V*0 (M,w) = U(M + w)]. V*i (M,w) = max{U(M + w);EU(M + Y(M,U,i))} = max{U(M + w); EV*i (M − c,Χ1)}

(8.39)

= max{U(M + w);V*i − 1(M)}. Because the obvious analogs of (8.9) and (8.10) hold, we can employ the concavity of U and the finiteness of E(Χ1) to ensure V *, defined by V *(M,w) = lim V*i (M,w),

(8.40)

i→∞

exists and that V * satisfies the functional equation V *(M,w) = max{U(M + w);EV *(M − c,Χ1)} = max{U(M + w);V˜ *(M)}.

(8.41)

Furthermore, a straightforward application of the Lebesque Dominated Convergence Theorem enables us to assert that V*(M) = lim V*i (M).

(8.42)

i→∞

Denote the set of acceptable wages by W *(M,U,i) for i < ∞ and by W *(M,U ) when the number of additional offers that can be received is unlimited. The monotonicity of Vi(M) in i and the fact that Vi(M) is not a function of the wage offer allows us to obtain sharper results than those in Theorem 8.9. Theorem 8.15 There are numbers w *(M) and w *(M,i), i = 1,2, . . ., referred to as reservation wages, which satisfy

242

The Economics of Search W *(M,i) = [w *(M,i),∞),

(8.43)

W*(M) = [w *(M),∞)

(8.44)

w *(M,i) ≤ w *(M,i + 1), i = 1,2, . . .,

(8.45)

w *(M) = lim w *(M,i).

(8.46)

and i→∞

A point of considerable interest is the relationship between the acceptance sets W (M,i) and W *(M,i). As anticipated, the reservation wage when recall is not permitted is no greater than the smallest acceptable wage when recall is allowed, that is, W (M,i) ⊂ W *(M,i). All that is needed to obtain this result is Vi(M,w) ≥ V *i (M,w),

(8.47)

which can be established with a simple induction argument. Theorem 8.16

Given M,

w *(M,i) ≤ wM,i i = 1,2, . . .,

(8.48)

w *(M) ≤ wM ,

(8.49)

and

where wM,i = inf{w: w ∈ W *(M,i)}, i = 1,2, . . .

(8.50)

wM = lim wM,i = inf{w : w ∈ W *(M)}.

(8.51)

and i→∞

Proof

Fix M and i < ∞ and pick w ∈ W *(M,i). Then by (8.47)

U(M + w) ≥ Vi(M,w) ≥ V *i (M,w).

(8.52)

This establishes (8.48). Next, pick w ∈ W *(M) so w ∈ W *(M,i) for all i by (8.18). This last fact coupled with (8.52) shows that w ≥ w *(M,i) for all i, whence w ≥ w *(M) by (8.46). Q.E.D. The proof of Theorem 8.2 suffices to establish:

Expected utility maximizing job search Theorem 8.17

243

* U

If r > rU, then the reservation wages satisfy

w *(M,U *,i) ≤ w *(M,U,i), all M and i ≤ + ∞.

(8.53)

Denote by ξ and ξ* the reservation wage when at most i more offers can be received in the presence of a linear utility function, with and without recall, respectively. It has been shown (Lippman and McCall, 1976a: 167f ) that ξ*i → ξ and ξi → ξ.

(8.54)

Notice that (8.19), (8.48), (8.49), (8.53) and (8.54) spell out the relationship between the set of acceptable wages under expected income maximization, expected utility maximization with recall, and expected utility maximization without recall; namely, (ξ,∞) ⊂ (M,i), i ≤ + ∞,

(8.55)

(ξ*i ,∞) ⊂ W *(M,i), i ≤ + ∞.

(8.56)

and

Alternatively, w *(M,i) ≤ wM,i ≤ ξi and w *(M,i) ≤ ξ*i . Evidently, we need not have a relationship which always holds between (ξ*i ,∞) and W *(M,i). The proof of Theorem 8.12 suffices to verify monotonicity of the reservation wage as a function of wealth in the presence of decreasing absolute risk aversion. Theorem 8.18 If rU is decreasing, then the acceptance sets W *(M,i) are nonincreasing in M; in particular,21 w *(M,i) ≤ w *(M + ε,i), all M,ε > 0, i ≤ + ∞.

(8.57)

In the presence of recall, we saw that increasing the riskiness of f via adding a mean preserving spread s could lead to a smaller or to a larger set of acceptable wages. It is easy to verify that both changes can occur without the recall option. We do so in a heuristic manner. First, from the argument immediately preceding (8.35) we obtain EgU(M − c + Y ) − Ef U(M − c + Χ ) < 0 if U ″ < 0, where Χ has density f and Y has density g = f + s. Consequently, the reservation wage when only one more offer can be received is higher under f. Clearly this result can be extended by selecting U and M so that even without any restriction on the number of offers that can be received from (M,w) it is optimal to accept w under g and take exactly one more offer under f. Conversely, if U is nearly linear, then it will be optimal to be more selective under g than under f. Finally, we note that typically the myopic policy will not be optimal. This

244

The Economics of Search

results from the fact that EV *(M − c,Χ1) can be significantly larger than EU(M − c + Χ1), so that basing the acceptance decision on the sign of U(M + w) − EU(M − c + Χ1) (which is what the myopic policy does) leads to an overly conservative decision rule. That is, the myopic policy’s acceptance sets are too large.

8.8 RISK AVERSION AND THE WAGE GAP Vesterlund (1997) studied the impact of risk aversion on the wage distribution and on the equilibrium labor force participation and equilibrium employment in a simple matching model. Vesterlund discovers that risk aversion may alter labor force behavior. For example, she finds that two equally qualified workers who search from an identical labor pool “will not receive the same wage if they differ in their attitudes towards risk.” Workers exhibiting more risk aversion are satisfied with lower quality matches and the lower wages associated with them. Vesterlund’s numerical solutions suggest that this wage gap is large. There is fairly solid evidence from psychology and economics that women are markedly more risk averse than men. Given the validity of these estimates, Vesterlund’s model implies that women receive lower pay than men even though they occupy equally productive jobs and when analyzing average productivity across gender. Women are less productive than men because they are more likely to be in lower quality jobs matches since they are more risk averse. Vesterlund observes that these model predictions are consistent with empirical evidence. The message of her paper is that a heightened risk aversion among women relative to men implies that the observed wage gap is compatible with profit and utility maximization.

8.9 CONSUMPTION COMMITMENTS AND LOCAL RISK AVERSION In concluding this chapter we briefly discuss the recent work of Chetty (2004) who has looked at the role that consumption commitments play in explaining some perverse behavior when viewed by the lens of the classic expected utility model. In particular, Chetty shows that with consumption commitments individuals may be more risk averse locally than globally. This occurs because for goods with consumption commitments (e.g. housing) small movements in wealth do not result in changes in the levels of consumption. However, this behavior leads to greater movements in noncommitted consumption goods (e.g. food) in response to the wealth shock. Hence, the utility function with respect to these noncommitted consumption goods appears to be more locally more concave.

Expected utility maximizing job search

245

In particular, goods with consumption commitments may only be changed if a transaction cost is borne. This transaction cost then results in the optimal response to have an (s, S) band of wealth such that the consumption of the good remains fixed if wealth varies within that band. The marginal utility wealth then diminishes more quickly over this smaller set of noncommitted set of goods within this (s,S) band of wealth. After imputing a global measure of risk aversion using existing labor supply estimates, Chetty than uses state-time variation in unemployment laws to estimate the local curvature within a dynamic search model. His estimates of the coefficient of local risk aversion are substantially greater than the estimates of global risk aversion.

9

Multi-armed bandits and their economic applications

9.1 INTRODUCTION In this chapter we show how the theory of multi-armed bandits can usefully be applied to many problems of optimal search. The multi-armed bandit (MAB) problem was first posed around the end of World War II and defied solution for three decades until it was finally solved by John Gittins (1979). Gittins showed that the optimal solution to this decision problem involved an index policy in which each bandit is assigned an index and the optimal policy is to sample the bandit that currently has the highest index. These indices are sometimes referred to as Gittins’ indices in honor of Gittins, and we will follow this tradition. The next section sets out a proof of the optimality of the index policy. The proof is based on that of Whittle (1980) who alternatively used a stochastic dynamic programming approach to prove Gittins’ result. Section 9.3 recasts the result in a more general framework and shows an alternative characterization of the index (which is actually its original characterization). Section 9.4 then describes the generalization of MAB to superprocesses and how an additional sufficient condition is needed to ensure an index solution. We then show how the BSM is simply a special case of the MAB problem and derive its solution using Gittins’ indices. The next several sections present applications of MAB theory to more general problems in search including the model occupational choice and job matching by Miller (1984) and the model of occupational matching by McCall (1991).

9.2 THE MULTI-ARMED BANDIT PROBLEM Suppose we have n independent bandits (projects) on which we can work. At a given time each bandit is in some state. If the state of the bandit is i and we decide to play (operate) that bandit, then we receive an expected reward R(i) and the next state becomes j with probability Pij. Moreover, the remaining n − 1 bandits which are played do not change state. In addition, we allow for the option of retiring which pays the fixed reward Z and the problem ends. To

Multi-armed bandits and their economic applications

247

solve for the optimal policy to this problem we first analyze the optimal policy for the single (one-armed) bandit case where n = 1. Consider a bandit that at any time is in some state. After observing the state the decision-maker must decide whether to play the bandit or retire. If the state is i and the decision is to continue playing, then a bounded reward R(i) is earned and the process moves to state k with probability Pik. If the decision to retire is made, then a terminal reward Z is received. Denote the optimal expected discounted return when the initial state is i and when retirement reward Z is available by V(i,Z ) where β denotes the discount factor and 0 < β < 1. Then V satisfies the optimality equation

冤

冥

V(i,Z ) = max Z,R(i) + β冱PikV(k,Z ) . k

(9.1)

We shall prove that if the retirement reward is optimal in a given state when the termination reward is Z then it is also optimal when the termination reward is greater than Z. Lemma 9.1

For fixed i, V(i,Z ) − Z is decreasing in Z.

Proof Let Vn(i;Z ) denote the maximal expected return when one is only allowed to continue for at most n stages before retiring. Then V0(i,Z ) = Z

冤 冥 V (i;M) = max 冤Z,R(i) + β冱P V

V1(i,Z ) = max Z,R(i) + β冱PikZ = max [ Z,R(i) + βZ ]

(9.2)

k

n

ik

k

冥

(k,Z ) .

n−1

We now show that Vn(i,Z ) − Z is decreasing in Z by induction on n. It is obvious for n = 0, so assume that it is true for n − 1, then

冤 = max 冤0,R(i) + β冱P (V

Vn(i,Z ) − Z = max 0,R(i) + β冱Pik(Vn − 1(k,Z ) − Z k

ik

k

n−1

冥

(k,Z ) − Z ) − (1 − β)Z

(9.3)

冥

and the result follows since Vn − 1(k,Z ) − Z is decreasing in Z. From Lemma 9.1 it follows that for fixed state i, if it is optimal to retire when the termination reward is Z, then it is also optimal to retire when the termination reward is Z + ε for ε > 0. Now Vn(i,Z + ε) − (Z + ε) ≤ Vn(i,Z ) − Z = Z − Z = 0.

(9.4)

But, Vn(i,Z + ε) ≥ (Z + ε) and so Vn(i,Z + ε) − (Z + ε) ≥ 0. This result in conjunction with (9.4) implies that Vn(i,Z + ε) = (Z + ε).

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The Economics of Search

Thus the optimal policy can be described as: Theorem 9.1 When in state i, it is optimal to retire if Z ≥ Z(i) where Z(i) = min[Z : V(i,Z) = Z ]. If Z < Z(i), then it is optimal to continue. We shall refer to Z(i) as the indifference value for the bandit in state i. We now consider the case where instead of a single bandit, there are n identical bandits available, each of which is in some state. After observing the state of each bandit, a decision must be made to play exactly one bandit or to retire. If a bandit is operated while in state i a reward R(i) is earned and the bandit moves to state k with probability Pik while all nonoperated bandits remain in the same state. If the decision to retire is made then the process ends and a termination reward Z is earned. Let V(i, Z ) denote the maximal discounted return when the initial states of the n bandits are i = (i1,i2,. . .,in). Then V satisfies

冤

冥

V(i,Z ) = max Z,Max{Vj(i,Z ) j

(9.5)

where Vj(i,Z ) = R(ij) + β冱V(i1,i2,. . .,ij − 1,k,ij + 1,. . .,in,Z )Pi k. k

(9.6)

j

The main difficulty in solving for V is that the number of states rapidly becomes large. If there are n projects each with K possible states, the multiarmed bandit problem has Kn states. It turns out, however, that the problem can be decomposed into n single bandit problems. We will show that the optimal policy is to calculate the indifference value, Z(i), for each bandit and then operate the bandit with the highest indifference value. Thus, it will be shown that the optimal policy for the multi-project (armed) bandit problem is to operate project j when in state i = (i1,i2,. . .,in) if Z(ij) = maxZ(ik) > Z and k

retire if Z > maxZ(ik). In order to prove that such a policy solves the optimalk

ity equation (9.5), we need to first establish some properties of V(i,Z ): Lemma 9.2

For fixed i, V(i,Z ) is an increasing convex function of Z.

Proof The increasing part is obvious given the definition of V. To prove convexity, consider a stationary policy π and for a given state let Tπ be a random variable which represents the retirement time under the policy π. Then, Vπ(i,Z ) = Eπ (discounted return prior to time Tπ + ZβT ). Hence, π

V(i;Z ) = max{Eπ (discounted return prior to time Tπ) + ZE(βT )}. π

π

Equation (9.7) is convex since the term in the brackets {} is linear in Z.

(9.7)

Multi-armed bandits and their economic applications 249 Lemma 9.3 For fixed initial state i let TZ be a random variable that denotes the optimal retirement time under the policy for terminal reward Z. Then, for all Z for which the derivative ∂V(i,Z )/∂Z exists, ∂ V(i,Z ) = E(βT |Χ0 = i). ∂Z

(9.8)

Z

Proof Fix Z and the initial state i. Let π* represent the optimal policy and let TZ be the number of stages of project operation before retirement. If we employ π* for a problem having terminal reward Z + ε , then we receive Vπ*(i,Z ) = Eπ* (discounted return prior to time TZ + (Z + ε)βT ). Z

(9.9)

Hence, using (9.7) and the fact that the optimal return when the terminal reward is Z + ε is at least as large as (9.9) we obtain the inequality V(i,Z + ε) ≥ Vn(i,Z ) + εE(βT ).

(9.10)

Z

Employing similar arguments it can also be shown that V(i,Z − ε) ≥ Vn(i,Z ) − εE(βT ).

(9.11)

Z

Thus, by (9.10) and (9.11) V(i,Z + ε) − Vn(i,Z ) ε

Vn(i;Z ) − V(i,Z − ε) ≤ E(βT ). ε

≥ E(βT ) and Z

Z

Taking the limit as ε goes to zero gives the result. Consider the indifference value Z(i) when only a single bandit can be played. Then we would continue to play the bandit as long as Z(i) > Z. Now, consider the multi-armed bandit case and suppose the state is i = (i1,i2,L,in). Then if Z(ij) > Z, it is clear that it would not be optimal to retire before playing bandit j since this would be true if j were the only bandit available. Although somewhat less obvious, but intuitive, we would never play bandit j if Z(ij) ≤ Z. It follows then that the optimal policy would be to retire in state i only if Z(ij) ≤ Z for all j. For a given state i, let T j denote the optimal retirement time when only bandit j is available, j = 1,2,L,N. Also, let T denote the optimal retirement time for the multi-armed bandit case. Because changes in the states of one bandit in no way affect the evolution of other bandits it n

follows that T =

冱T and since the T are independent: j

j

j=1

n

冢 冣 冲E(β n

T E(βT) = E β 冱 = j

Tj

j=1

j=1

).

(9.12)

250

The Economics of Search

Now, since convex functions are almost everywhere differentiable from Lemma 9.3, we have ∂ V(i;Z ) = ∂Z

N

∂

冲∂Z V(i ;Z ).

(9.13)

j

j=1

Suppose all rewards are bounded above by (1 − β)C. Then for Z ≥ C it is optimal to retire and V(i,Z ) = Z for Z ≥ C. Integrating (9.13) then gives C

C

冮

冮 冲∂Z V(i ,z )dz

∂ V(i,z )dz = ∂z Z Z

N

∂

j

j=1

or C

V(i,Z ) = C −

N

∂

冮 冲∂Z V(i ,z )dz j

Z j=1

Theorem 9.2

For Z ≤ C we have C

V(i,Z ) = C −

N

∂

冮 冲∂Z V(i ,z )dz j

Z j=1

and when in state i, the optimal policy retires if Z(ij) < Z for all j = 1,2,L,N and plays bandit j if Z(ij) = maxkZ(ik) > Z. C

Proof

Let Vˆ (i,Z ) = C −

N

∂

冮 冲∂Z V(i ,Z )dz. j

Z j=1

We shall demonstrate that Vˆ satisfies the optimality equation

冤

冥

Vˆ (i,Z ) = max Z,Max{Vˆ j(i,Z )} j

which by uniqueness of the value function then implies Vˆ = V. Fix j, j = 1,2, . . .,N and define

Pj(i;Z ) =

∂

冲∂z V(i ,Z ) k

k≠j

then

(9.14)

Multi-armed bandits and their economic applications

251

C

∂ Vˆ (i,Z ) = C − V(ij,z )Pj(i;z )dz ∂z Z

冮

(9.15)

or, integrating (9.15) by parts, C

冮

Vˆ (i,Z ) = Pj(i,Z )V(ij,Z ) + V(ij,z )dPj(i,z ).

(9.16)

Z

We also have Vˆ j(i,Z ) = R(ij) + β冱Vˆ (i1,i2,. . .,ij − 1,k,ij + 1,. . .,in,Z )Pi k

(9.17)

j

k

C

冮

冤

冥

= R(ij) + β冱Pi k Pj(i,Z )V(k,Z ) + V(k,z )dPj(i;z ) . j

k

Z

Equations (9.16) and (9.17) then imply Vˆ (i,Z ) − Vˆ j(i,Z ) = Pj(i,Z )[V(ij,Z ) − R(ij) − β冱Pi kV(k,Z )] k

j

(9.18)

C

冮

+ [V(ij,z ) − R(ij) − β冱Pi kV(k,z )]dPj(i,z ) ≥ 0 k

Z

j

since V(ij,Z ) − R(ij) − β冱Pi kV(k,Z ) ≥ 0.

(9.19)

j

k

Hence, Vˆ (i,Z ) ≥ Vˆ j(i,Z ). Now, note that (9.19) holds with equality when Z < Z(ij). Thus, from (9.18) we have C

Vˆ (i,Z ) − Vˆ (i,Z ) = j

冮 [V(i ;Z ) − R(i ) − β冱P j

j

k

Z(ij)

ij k

V(k,z )]dPj(i,Z ) if Z ≤ Z(ij).

Next, observe that

Pj(i,z ) =

∂

冲∂ZV(i ,z ) = 1 if z > maxZ(i ) k

k≠j

k≠j

k

which implies that dPj(i,z ) = 0 if z > maxZ(ik). So, we have Vˆ (i,Z ) = Vˆ j(i,Z ) k≠j

when Z ≤ Z(ij) = maxZ(ik). Also by (9.16) we have k

252

The Economics of Search C

冮

Vˆ (i,Z ) ≥ Pj(i,Z )V(ij,Z ) + V(ij,Z ) dPj(i;z ) = V(ij,Z ) ≥ Z.

(9.20)

Z

Now (9.20) holds with equality if Z ≥ maxZ(ik). Combining these results we k

have

冤

冥

Vˆ (i,Z ) = max Z,Max{Vˆ j(i,Z )} j

and so Vˆ = V. Thus, the Gittins index for bandit i can be calculated by the recursive equation Z = R(i) + β冱PikV(k,Z )

(9.21)

k

An alternative stopping time form of the index (indifference value) for each bandit is

  Z(i)(1 − β) = supτE   

τ−1

冱β R(i )  t

t

t=0

τ−1

冱β

t

t=0

  

where τ is a stopping time. To see that the two are equivalent define the stopping time λ* by the first time when it is optimal to choose Z: R(i′) − β冱Pi′kV(k,Z ) < Z. Then, k

λ* − 1

冦 冱 β R(i(t)) + β t

Z(i) = E

t=0

λ*

Z(i)

冧

(9.22)

or λ* − 1

冦 冱 β R(i(t))冧 t

E Z(i) =

t=0

1 − E{βλ*}

Multiplying both sides of (9.23) by (1 − β) gives

(9.23)

Multi-armed bandits and their economic applications λ* − 1

冦 冱 β R(i(t)) 冧

Z(i)(1 − β) =

λ* − 1

t

E

t=0

[1 − E{βλ*}] [1 − β]

253

冦 冱 β R(i(t))冧 t

E =

t=0

.

λ* − 1

(9.24)

冦 冱β 冧 t

E

t=0

We can then restate the optimal policy as follows: For each individual bandit look for the retirement time T whose ratio of expected discounted returns to T to the expected discounted time prior to T is maximal. Then consider the largest of these ratios. If it is greater than (1 − β)Z, then operate the corresponding bandit. Otherwise retire. This forward induction strategy can be used to derive the familiar recursive equation for the reservation wage in the basic search model with discounting. Let ξ denote the reservation wage. We know that the optimal value function equals ξ which then must equal Z(i)(1 − β). Hence, E [discounted return prior to T] ξ = max E[1 + β + L + βT − 1] T Denote the optimal stopping time (less one) by T*. This time is characterized by the first time a wage draw exceeds ξ. ξ=

=

E [discounted return until T*] E[1 + β + L + βT*] − cE[1 + β + L + βT* − 1] + E(βT*)E(w|w ≥ ξ) E[1 + β + L + βT*]

Now T* is a geometric random variable with probability equal to p = 1 − F(ξ). Thus, E(βT*) =

pβ , 1 − β(1 − p)

E[1 + β + L + βT* − 1] =

1 1 + βp and E[1 + β + L + βT*] = . 1 − β(1 − p) 1 − β(1 − p)

So, ξ= or

−c pβE(w|w ≥ ξ) + 1 + βp 1 + βp

254

The Economics of Search ∞

冮

[1 + β(1 − F(ξ))]ξ = −c + β xdF(x) ξ

which implies that ∞

冮

c = β (x − ξ)dF(x) − ξ = βH(ξ) − ξ

(9.25)

ξ

where ∞

冮

H(ξ) = (x − ξ)dF(x). ξ

Extending this model slightly, assume that the probability of an offer each period equals q. Then it is easy to show that (9.25) becomes c = βqH(ξ) − ξ.

(9.26)

Thus, the classic sequential search model can be put in a multi-armed bandit framework where there is but one project. Now consider that there are n markets in which an individual can choose to search with each market characterized by ci, qi, and Fi. Then, from (9.26) the index associated with each market is just ξi/(1 − β) where ξi satisfies ci = βqiHi(ξi) − ξi.

(9.27)

So when faced with N separate markets from which to search, it is optimal to search that market with the highest reservation wage. Next, suppose we think of each market as a different region we can use (9.27) to characterize the migration decision. Assume that individuals bear a “one-time” cost of entering a market and denote this cost by ki. The cost can be thought of as a relocation cost and we assume that if a person is currently residing in location j, kj = 0. It is straightforward to show that the index will be determined by ci + ki(1 − β) = βqiHi(ξi) − ξi

(9.28)

Thus, all else equal, the labor market where an individual already resides would tend to be favored. However, moving may be optimal if for example another region is characterized by a higher q. In McCall and McCall (1987) we use results from MAB theory to consider the case where migrants have different beliefs about the attributes of a particular region and hence may order the set of options. We allow for some uncertainty to be resolved only after migration has taken place. Again, sup-

Multi-armed bandits and their economic applications

255

pose that there is a one time cost of moving denoted by ki. Here, instead of moving before searching we assume that individuals get a wage draw from a city and then decide whether or not to move to that city based on the draw. Let Fi denote the distribution of wages in the city i. Now, however, it is assumed that individuals must decide to move before all information about a city’s attributes are known and cities may be characterized by more or less uncertainty (for example, uncertainty may depend on the distance away the city is or the number of acquaintances you have in the city). After a random period of time, information is received about the desirability of the attributes of the city. Let the arrival time of this information follow a geometric distribution with parameter si and suppose the information changes the return to living in the city by an amount αi where the distribution function for αi is denoted by Gi. Thus, in the first period an individual pays ci and receives information about their wage prospects wi. The individual must then decide whether or not to move to the city. If they decide to move the individual incurs a further cost of ki. For the next ˜t periods the individual works in the city and receives wi where ˜t is a random variable, after which they receive wi + αi. Of course, after all information has arrived for city i the Gittins index w i + αi associated with city i satisfies Zi = . After receiving wage information 1+β for city i the index satisfies t˜ − 1

冦 冱β + β

Zi(wi) = − ki + βE wi

t˜ + 1

r

r=0

= −ki +

冦

冢

EG max Zi(wi),

wi + αi 1−β

冣冧冧

β βpi wi + 1 − β(1 − p) 1 − β(1 − pi) wi + αi

冤 冦 1 − β |α ≥ Z (w )(1 − β) − w 冧

× EG

i

i

i

i

i

冥

+ Gi(Zi(wi)(1 − β))Zi(wi) or

∞

[1 − β(1 − pi)](1 − β){ − ki + (β/(1 − β))wi} + βpi Zi(wi) =

冮

αdGi(α)

Z(wi)(1 − β) − wi

[ (1 − β) − βpGi(Zi(wi)(1 − β) − wi) ](1 − β)

. (9.29)

If the next best alternative city has an index of Z* = maxj ≠ i{Zj}, then the reservation wage associated with city i, ξi which determines whether the individual will migrate to city i, is determined by the Zi(ξi) = Z*.

256

The Economics of Search

9.3 GENERAL MAB FRAMEWORK The MAB framework can be generalized to a non-Markovian, continuous state space, framework. Again we assume the special structure imposed on MAB problems that only one bandit can be played per period, bandits not played remain motionless (i.e., do not change states), and information is independent across bandits. Let t represent the total time an individual has been playing all N bandits and ti the total playing time of the ith bandit. N

Hence, t =

冱t . Now, let Ω represent the state space of the ith bandit, let F i

i

i

i=1

be a sigma-algebra defined on Ωi, and denote an element of Ωi by xi. Define xi(ti) as the state of bandit i after bandit i has been played for ti periods.1 Define the filtration representing the flow of information for the ith bandit by Fi(t),i = 1, 2, . . ., N. Note that for the ith filtration, time is indexed by ti which represents the playing time of the ith bandit. We assume that the conditional transition probability measures dPi(xi(ti)) | Fi(ti)) exists P − a.s. for all i. Similar to above we assume the when the bandit is played and the state is xi(ti), an expected return Ri(xii(ti)). At the outset, an individual knows the initial states of all bandits (x1(0), . . ., xN(0)), the return functions (Ri(xi(ti)), i = 1, . . ., N ), and the stationary probability transition kernels (dPi{xi(1) | Fi(0)}, i = 1, . . ., N ). An individual wishes to sequence the play of bandits so N as to maximize expected discounted returns. Let F (0) = ∨ Fi (0) (i.e., the i=1

smallest sigma algebra containing the Fi (0), i = 1, 2, . ., N). Then the MAB problem is ∞

冢冱β R t

maxi(t)EF

冣

(xi(t)(ti))

i(t)

0

t=0

(9.30)

where i(t) is a decision strategy which maps time into the set {1,2, . . .,N} and β is the discount factor, 0 < β < 1. This MAB problem as an index solution such that dynamic allocation index of the ith bandit at time ti, Zi(xi(ti) | Fi (ti)) or simply Zi(x ) is determined by t. Hence, i

Zi(xi) = Ri(xi) + β冮Ω V(y,Zi(xi))dPi(y|F i(ti)). i

The optimal policy is to play bandit i at time t if Zi(xi(ti)) = maxj = 1,. . .,N{Zj(xj(tj))}.

Multi-armed bandits and their economic applications

257

9.4 A MODEL OF JOB SEARCH WITH HETEROGENEOUS OPPORTUNITIES In this section we consider the search model initially described in McCall and McCall (1981) with its more complete analysis contained in McCall (1994a). In this model jobs are characterized by both inspection and experience uncertainty. Suppose that an individual has N different job prospects, i = 1, 2, . . . N, each of which is characterized by an inspection cost ci and two random variables w ˜ i and α˜i with joint cumulative distribution function (c.d.f.) Gi. Assume that for all i ≠ j, (w ˜ i,α˜i) is independent of (w ˜ j,α˜j). Also assume that one job is sampled (i.e., inspected or worked) per period, jobs that are not currently being sampled remain in the same state, and that recall is possible. After an initial inspection of job i, w ˜ i is revealed. The random variable thus characterizes the amount of inspection match (IM) uncertainty. At this point, the individual must decide whether to accept the job or sample another job. If the job is accepted, the individual receives a return of w ˜ i in his/her first period of work. More generally, w ˜ i can be considered the annuity equivalent of the expected return from working at job i permanently, given inspection information. This expected return calculation would incorporate the starting wage, expected wage growth, and the monetary equivalent of the non-pecuniary benefits derived from the job. Clearly, this model fits into the MAB framework. For convenience, assume that after one period of employment, all experience match (EM) uncertainty is resolved. This uncertainty is characterized by the random variable α˜i. Consequently, if an individual remains at job i beyond the first period, they will remain permanently with a net compensation of w ˜ i + α˜i (per period). Finally, assume that the time horizon is infinite and that individuals maximize expected discounted returns where β is the discount factor, 0 < β < 1. To simplify matters further (and to ensure the existence of a reservation wage), assume that, for all i, match uncertainty is of the form αiy˜ i where αi is a constant and y˜ i is a random variable, independent of w ˜ i, which takes either the value +1 or the value −1 with equal probability. Also, suppose that the random variable w ˜ i can be characterized by the absolutely continuous c.d.f. Fi, which admits a probability density function (p.d.f.) fi(w) with strictly positive support on [w,w ¯¯ ], where w ¯¯ > w > 0. The state space of job i, xi, is defined by the cross product [ w,w ¯¯ ] × {−1,0,1}. When xi = (0,0), the job has yet to be inspected. The state space of job i moves from (0,0) to (w,0) if an inspection of job i produces a wage offer of w. After the ith job is experienced for one period, the state space moves from (w,0) to (w,−1) or (w,1) depending on whether y˜ i = −1 or y˜ i = 1, respectively. Under these assumptions, it is not difficult to show that Vi((0,0),Z ) satisfies: Vi((0,0),Z ) =

Max w1i,w2i:w ≤ w1i ≤ w2i ≤w ¯¯

ER(w1i ,w2i ,αi,Fi,ci,Z )

(9.31)

258

The Economics of Search

where w2i

¯¯ w

冮

冤冮

冥

ER(w11,w21,αi,fi,ci,Z ) = −ci + β(1 − β)−1 wdFi(w)] + (1 − .5β) wdFi(w) 1

w2i

wi

+ .5β2(1 − β)−1[F(w2i ) − F(w1i )]αi

(9.32)

+ β[F(w1i ) + .5β{F(w2i ) − F(w1i )}]Z In equations (9.31) and (9.32), the value w1i denotes the minimal wage offer necessary to induce the individual to work job i for at least one period and the value w2i denotes the minimal wage offer which induces the individual to remain at job i after receiving unfavorable information (−αi). Choosing w1i and w2i to maximize expected discounted returns and denoting these optimal 2* values by w1* i and wi respectively, equations (9.21), (9.31) and (9.32), imply that the index for job i in state (0,0), Zi(0,0), satisfies: w2* i

¯¯ w

冤 冮 wdF (w)] + (1 − .5β) 冮 wdF (w)]冥

Zi(0,0) = −ci + β(1 − β)1

i

wi2*

i

wi1*

1* 1* + .5β2(1 − β)−1[F(w2* i ) − F(wi )]αi + β[F(wi )

(9.33)

+ .5β{F(w ) − F(w )}]Zi(0,0). 2* i

1* i

It can be shown that when job i is in state (w,0), its index is Zi(w,0) = (1 − β)−1w + β[(1 − β)(2 − β)]−1αi.

(9.34)

Once all information about job i has been revealed, the index is Zi(w,αi) = (1 − β)−1(w + αi)

(9.35)

Zi(w,αi) = (1 − β)−1(w − αi),

(9.36)

or

depending on whether the information revealed after one period of work is favorable or unfavorable, respectively. Suppose that job j currently possesses the largest index. Define Z*j or simply Z* to be the value of the index of the next best alternative. Thus, Z* = max{Zi(x)}. The next two lemmas characterize the dependence on αi. i≠j

Lemma 9.4

1* Zi(0,0) is increasing in i and strictly increasing in αi if w2* i > wi .

Proof Solving equation (9.33) for Zi(0,0) and differentiating with respect to i yields

Multi-armed bandits and their economic applications

259

.5β2(F(w ) − F(w )) >0 1* [1 − βF(w ) − .5β2(F(w2* i ) − F(wi ))][1 − β] 2* i

1* i

1* i

1* if w2* i > wi .

Lemma 9.5

Zi(0,0) is convex in αi.

Proof Let 0 < λ < 1 and let αi and αj be two arbitrary values of α. Define --α = λα + (1 − λ)α and Z ¯¯ = λZ(αi) + (1 − λ)Z(αj) where Z(αi) and Z(αj ) are i j the indices for jobs in state (0,0) when α = αi and α = αj, respectively. Denote by Z(α--) the index when α = --α. Equation (9.33) is of the form Z = k(w1*,w2*) + c(w1*,w2*)Z. Let w ¯¯ 1* and w ¯¯ 2* be the optimal values of w1 and w2 when α = --α and -M = Z(α). Then, Z(αi) ≥ k(w ¯¯ 1*,w ¯¯ 2*) + c(w ¯¯ 1*,w ¯¯ 2*)Z(αi) and Z(αj) ≥ k(w ¯¯ 1*,w ¯¯ 2*) + c(w ¯¯ 1*,w ¯¯ 2*)Z(αj) which in turn implies that ¯¯ ≥ k(w ¯¯ . Z ¯¯ 1*,w ¯¯ 2*) + c(w ¯¯ 1*,w ¯¯ 2*)Z But by definition, Z(α--) = k(w ¯¯ 1*,w ¯¯ 2*) + c(w ¯¯ 1*,w ¯¯ 2*)Z(α--). ¯¯ . Since c(w ¯¯ 1*,w ¯¯ 2*) < 1, this implies that Z(α--) ≤ Z For the remainder of this section assume, for simplicity, that 2* w < w1* ¯¯ , for all i. The next lemma shows that the optimal job i < wi < w acceptance policy is characterized by a reservation wage that decreases as αi is increased. Lemma 9.6 If job j is searched, then a wage offer of w will be accepted (temporarily) if, and only if, w ≥ ξj where ξj satisfies: ξj = Z*(1 − β) − β(2 − β)−1αj.2

(9.37)

Proof Job j will be accepted after it is inspected if and only if Zj(w,0) ≥ Z*. Substituting from equation (9.34), we have that a job is accepted if and only if (1 − β)−1w + β[(1 − β)(2 − β)]−1αj > Z* or

260

The Economics of Search w > Z*(1 − β) − β(2 − β)−1αj.

As a first step toward analyzing the behavior of reservation wages over an unemployment spell, assume that only αj differs across jobs. Label jobs from the largest to the smallest αj and assume that all jobs are initially in state (0,0). Hence, from Lemma 9.4, job 1 will be searched first, then job 2 (if the wage offer from job 1 is below ξj ), and so on. From Lemma 9.5 it is clear that if Z* is held constant ξj increases as αj decreases. However, with recall, Z* is a random variable which decreases almost surely as the unemployment spell lengthens (P(Z*i ≤ Z*j ) = 1 for j > i). Now, equation (9.37) implies for j > i,

冢

P(ξj ≥ ξi) = P Z*i − Z*j ≤

αi − αj . β(2 − β)(1 − β)

冣

The next two propositions give sufficient conditions for P(ξj ≥ ξi) = 1 for most j > i. Theorem 9.3 Assume that ci = c and Fi(w) = F(w) for i = 1,2, . . ., N. Furthermore, assume that α1 = α2 = L = αJ = α′ > αJ + 1 = αJ + 2 = L = αN = α″ for N − J ≥ 2. Then, ξJ < ξ1 = L = ξJ − 1 < ξJ + 1 = L = ξN − 1 and ξN < ξJ + 1 = L = ξN − 1 with probability 1. Proof Since αi = α′ for i ≤ J, Z(αi) = Z(α′) for i ≤ J. Hence, from (9.37), ξi = ξJ − 1 for i < J − 1. An analogous argument shows that ξi = ξJ + 1 for J + 1 < i < N − 1. Now, ξJ − 1 − ξJ + 1 = (Z(αj) − Z(αj + 2))(1 − β) − β(2 − β)−1(αJ − 1 − αJ + 1). But αJ − 1 = αJ = α′ > α″ = αJ + 1 = αJ + 2. So, ξJ − 1 − ξJ + 1 = (Z(α′) − Z(α″))(1 − β) − β(2 − β) − 1(α′ − α″). The convexity of Z(α) implies that ξJ − 1 − ξJ + 1 ≤ (Z′(α′)(α′ − α″)(1 − β) − β(2 − β)−1(α′ − α″)

Multi-armed bandits and their economic applications

261

Where Z′(α) denotes the (right) derivative of Z with respect to α. It is clear, however, that Z′(α) ≤

.5β2 (1 − .5β2)(1 − β)

which equals the derivative if α is sufficiently large that it always pays to work the job one period, irrespective of the wage, and then leave if experience information is negative. Substituting we have .5β2

冤(1 − .5β )(1 − β)(1 − β) − β(2 − β) 冥(α′ − α″) −1

2

.5β

=

冤(1 − .5β ) − (2 − β) 冥β(α′ − α″)

=

冤

.5β(2 − β) − 1 + .5β2 β(α′ − α″) (1 − .5β2)(2 − β)

=

冤(1 − .5β )(2 − β)冥β(α′ − α″) < 0

−1

2

冥

β−1 2

That P(ξJ < ξ1) = 1 follows because α1 = αJ and with a continuous wage distribution P(Z*1 > Z*J) = 1. An analogous argument shows that P(ξN < ξJ + 1) = 1. Reservation wages decrease only when the last job of each type is inspected, otherwise they remain constant or increase over an unemployment spell. Now, an individual may recall a past job offer before inspecting any job with lower EM uncertainty (i.e., a job with αi = α″). However, if a job with lower EM uncertainty is inspected, then the reservation wage corresponding to it is larger than the reservation wages of all previously inspected jobs. Theorem 9.3 can be extended to situations where there are more than two types of jobs as long as there are at least two jobs of every type. Theorem 9.4 Assume that ci = c and Fi(w) = F(w), i = 1, . . ., N. Furthermore, assume that there are k different types of jobs. Let Nk represent the number of jobs of type k and assume Nk ≥ 2, k = 1, . . ., K. Finally, assume that α1 = L = αN > αN 1

1

+1

= L = αN

1

+ N2

> L > αN

1

K

where N =

冱N . Then, k

k=1

ξN < ξ1 = L = ξN 1

1

−1

< ξN

1

+1

= L = ξN

1

,

+ N2 − 1

+ L + Nk − 1 + 1

= αN

262

The Economics of Search ξN

1

+ N2

< ξN

1

= L = ξN

+1

1

+ N2 − 1

< ξN

1

+ N2 + 1

= L = ξN

1

,

+ N2 + N3 − 1

M ξN + L + N < ξN = ξN + L + N − 1, k−1

1

1

1

+ L + Nk − 2 + 1

= L = ξN

1

+ L + Nk − 1 − 1

< ξN

1

+ L + Nk − 1 + 1

=L

k

and ξN

1

+ L + Nk

< ξN

1

+ L + Nk − 1 + 1

= L = ξN

1

+ L + Nk − 1

with probability 1. So far we have assumed that jobs can differ only in the amount of EM uncertainty. Jobs, however, may also differ in search costs and the amount of IM uncertainty. Applying equation (9.33) shows that Zi(0,0) decreases as inspection costs increase, and for i ≠ j Zi(0,0) > Zj(0,0) if Fi > 1Fj or Fi > 2Fj. Differences only in ci and Fi across jobs would cause the reservation wage ξ to fall over an unemployment spell. One implication of this analysis is that reservation wages may not behave in a monotonic fashion over an unemployment spell. Nevertheless, maxi{Zi(xi(t))} does decline monotonically over an unemployment spell. Since lingering uncertainty after a job inspection is an important determinant of the reservation wage, empirical studies of reservation wages which ignore it may be flawed.

9.5 MILLER’S MODEL OF JOB MATCHING AND OCCUPATIONAL CHOICE Robert Miller’s (1984) paper on job matching and occupational choice is a classic application of the theory of multi-armed bandits. In this section we review his theoretical model while in Chapter 12 on structural estimation, we describe how the structural parameters of the theoretical model were estimated. Let xmt be the return from working in the mth job at time t. Assume that xmt = ηt + ξm + σmεmt.

(9.38)

The variable ηt is a time trend representing the return to general experience. Miller assumes that an individual does not observe ξm directly. Rather, an individual only knows that ξm is drawn from a distribution, which Miller assumes is normally distributed: N(γm,δm2 ). Finally, εmt is a standard normal random error term. Let τmt be the accumulated experience in the mth job at time t. Define dms to be a dichotomous variable that equals 1 if an individual works at job m at time s and equals 0 otherwise. Then,

263

Multi-armed bandits and their economic applications t−1

τmt =

冱d

.

ms

s=0

Miller assumes that two jobs m and m′ belong to the same occupation if (γm,δm,σm) = (γm′,δm′,σm′). Standard results on Bayesian updating when sampling from a normal distribution with normal priors (see DeGroot, 1970, for example) implies that

冤

t−1

冱(x

γmt = δm− 2γm + σm− 2

冥

− ηt) (δ −m2 + τmtσm− 2)−1

ms

s=0

(9.39)

and −2 − δmt = (δ−2 m + τmtσm ) . 1 2

(9.40)

An individual’s objective is to maximize expected returns from jobs over his/ her lifetime by choosing a decision rule d which maps available information into his/her preferred job choice at each point in time. Now, d({γmt,δmt}m ∈ M) = × dmt({γmt,δmt}), m ∈M

where for all (m,t) ∈ M × T, dmt ∈ {0,1}, and 冱m ∈ M dmt = 1. The value function associated with this particular class of decision problems is characterized by ∞

冦冱 冱 β d

V({γmt,δmt}m ∈ M) = maxE0 d

t

x

mt mt

t = 0 m ∈M

冧

where Et represents the expectation operator conditional on information available at time t and β is a discount factor. This model fits into the multi-armed bandit problem. Recall that the optimal decision policy for the multi-armed bandit problem is characterized by an index policy. For this problem, the Gittins’ (dynamic allocation) index is defined by ν

Et冤冱βr − t(xmr − ηt)冥  r=t  Zm(γmt,δmt) = sup ν . υ>t  Et冤冱βr − t冥    r=t

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The Economics of Search

Consider a job match drawn from a standard normal distribution and assume that σm = 0. Denote the index by Z(0,0). Miller shows that if σm = 0, then Proposition 9.1

Z(γm,δm,0) = γm + δmZ(0,0)

Moreover, Proposition 9.2

Z(0,1) satisfies the recursive equation

Z(0,0) = {1 − βΦ[Z(0,0)]}−1βΦ′[Z(0,0)]. More generally, Z(γmt,δmt,σm) = γmt + δmtZ(αm + τmt,σm) σm2 is the information factor. Miller derives an algorithm for δm2 computing Z(αm + τmt,σm) which is based on the following result: where αm =

Proposition 9.3 Let F denote the space of bounded continuous real-valued functions taking their domain on the space (−∞,∞) × (0,∞). (a) For all f ∈ F define C, a mapping from F to itself as ∞

C [F(γ,α)] = β

冮 max0,冢(1 − β)冦 γ + ε[α(α + 1)] 冧 − 12

−∞

冦

+ f γ + ε[α(α + 1)] − ,α + 1 1 2

冧冣 dΦ(ε)

There exists a unique function g(γ,α) ∈ F satisfying g(γ,α) = C[ g(γ,α) ]. (b) Defining Ck(f ) recursively as C [Ck − 1(f )], for all f ∈ F, || Ck(f ) − g || ≤ (1 − β)−1 || Ck(f ) − Ck − 1(f ) ||. (c) For all (α,β) ∈ (0,∞) × (0,1), the standard index solves Z(αm,σm) = (1 − β)−1α ½m g[ − α −½ m Z(αm,σm),αm ]. Several results emerge from this model. First, the current job yields two benefits: payoff from expected reward for the period and information about future returns within that particular job. Second, the better one’s initial information about the suitability and hence the lower δm, the less one is prepared to forgo in order to find out more about the job. Third, greater variability in the payoff sequence (large σm) impedes learning about the underlying process and, so, the informational component is therefore less valuable. Fourth, specific experience τmt reduces Z(αm + τmt,σm) at a decreasing rate which implies that the value of additional information declines as it is

Multi-armed bandits and their economic applications

265

acquired. Additionally, matches about which a precise estimate can be ascertained more quickly lose their attractiveness as an information source. Finally, people who discount future payoffs more, value information less. One implication of these results is that quit rates of young workers are higher than those of older workers. This occurs because there is a lower probability that within a particular occupation a younger worker’s match will be better than an older worker’s: older workers have had more time to find one. Moreover, a greater percentage of young workers will work in occupations where high turnover is endemic. That is, since young workers know less about themselves than older workers they are more willing to experiment in activities that increase self-knowledge. 9.6 SUPERPROCESSES: ADDING DECISIONS TO BANDITS Many problems of optimal search might be modeled as a generalized bandit problem which is similar to the classic MAB problem except that within each bandit additional decisions are made which affect both the payoffs and evolution of the bandit. The question arises whether an index policy continues to characterize the optimal solution in this case. These one-armed bandits are referred to as superprocesses. So, each period an individual must determine both the best superprocess to “play” and how to best affect the motion of this superprocess given the control variables at his/her disposal. We continue to assume that superprocesses are independent and that unplayed superprocesses are motionless. Assume that there is only one control variable for the ith superprocess denoted by d where d ∈ Di(xi).3 The ith superprocess is characterized by the return function Ri(x,d), the state space Ωi(d ), and the transition kernel Pdi(x,B),B ⊂ B where B is the Borel sigma algebra and where for simplicity we assume a Markovian structure. Under this setup an optimal policy is an index policy, as above, when an additional restriction is satisfied. Let Vi(x,m,d ) be the optimal value function for the optimal stopping problem of continuing to play the ith superprocess and following policy d, or stopping and receiving the stopping reward m. Then Vi(x,m,d) satisfies Vi(xi,d,m) = max{Ri(xi,d ) + β

冮 V(y,d,m)P (x ,dy);m冧. d i

Ωi(d )

Define, d *(xi,m) = arg max{Vi(xi,m,d )} d ∈ Di(xi)

Then we have:

i

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The Economics of Search

Theorem 9.5 Suppose that for all i and x, d *(x,m) (= d *(x)) is independent of m for m ∈ [0,Zi(x,d *(x))], where Zi(x,d *(x)) is defined by the equation Zi(xi,d *(xi)) = Vi(xi,d *(xi),Zi(xi,d *(xi))).

(9.41)

Then the optimal sampling strategy is an index policy where the index for the ith superprocess is Zi(x,d *(x)). We refer to this sufficient condition as the W-condition after Whittle (1980) who first formulated and solved the superprocess problem. The Wcondition states that the optimal decision policy for the ith superprocess must be independent of the stopping reward m in the continuation region of the optimal stopping problem characterized by (9.41). Note that the W-condition is not innocuous. McCall (1991) applied results from superprocesses to study occupational choice. McCall assumed that associated with occupation i, i = 1,. . ., N, is a sequence of random variables αiYij, j = 1,2,3, . . ., with αi > 0 and where Yij takes only the values +1 or −1 depending on whether the matching outcome for the jth job is good or bad, respectively. The magnitude αi measures the degree of objective match uncertainty associated with an occupation (e.g. everybody knows that lawyers make either $30,000 or $500,000). For simplicity, the match information is assumed to be revealed immediately, so that the worker earns wi + αi or wi − αi in their first period of work, but that the worker must remain at the job for at least one period. Individuals are assumed to have an initial (at ti = 0) prior distribution over the probability of attaining a successful match (i.e. Yi1 = + 1) within the ith occupation, pi(0), which summarizes their subjective beliefs. Furthermore, pi(0) has a Beta(δi,δi) distribution with mean p¯ i(0) = 12 and precision hi(0) = 8δi + 4.4 The Beta distribution is a particularly convenient distribution to work with since it is a conjugate family of priors for samples from a Bernoulli distribution. It is not difficult to show (see DeGroot 1970: 160) that the posterior distribution associated with the occupation i after n jobs have been sampled, pi(n), has a Beta distribution with mean p¯ i(n,Si(n)) =

δi + Si(n) δi + n

and hi(n,Si(n)) =

(2δi + n)2 [(δi + n − Si(n))(δi + Si(n)]

n

where Si(n) =

冱Y . The precision of the prior probability distribution govij

j=1

erns the amount of learning that takes place through sampling. Occupations

Multi-armed bandits and their economic applications

267

associated with a low δi reveal information which is relatively occupationspecific whereas those associated with a high δi , reveal information which is relatively job-specific. Suppose that when employed in the ith occupation, a worker faces a constant risk, ui , of being laid off each period but only for the first “permanent” job (that is, the first job in which wi + αi is received). If an individual is laid off from this permanent job, assume that the risk of layoff from jobs subsequently worked in that occupation goes to zero. This assumption is made simply to keep numerical computations manageable. We will see that it is precisely the risk of layoff which tends to make occupations which reveal occupation-specific information more attractive. So, excluding it entirely is inappropriate. Finally, we assume that individuals sample occupations (and jobs within an occupation) so as to maximize expected discounted income. In this model, as workers gain occupational experience, they will update their beliefs about the likelihood of attaining a good match at subsequent jobs in an occupation in a Bayesian fashion. These beliefs, along with mobility costs and the worker’s opportunities in other occupations, determine whether or not an interoccupational or intraoccupational job switch is optimal. Given the simplifying assumptions of the model, if a worker is currently at a job with a good match, then a job switch would not take place voluntarily. However, when a well-matched worker is laid off, his/her subjective beliefs help determine the maximum number of jobs in the occupation that will be sampled in order to attain another good match. Owing to the complexity of the worker’s decision problem, it cannot be characterized as a simple MAB problem. It does, however, fall into the class of superprocess problems and so it is possible to analyze the worker’s decision problem using results from this theory. Each occupation, i = 1, . . ., N, can be represented by a superprocess with the state space symbolized by (Li, n, Si(n), Fi). Here, L = 0, 1, or 2 and represents whether the worker is currently out of work, working in a job with a good match or working in a poorly matched job, respectively. Fi is an indicator variable which indicates whether or not the worker has been laid off in the ith occupation. Finally, n (which represents the number of jobs sampled) and Si(n) (the number of good matches or successes in those n trials) are sufficient (along with δi) to summarize the posterior beliefs of the worker. The intraoccupational decision structure is of a particularly simple form; each state has at most two possible decisions associated with it. Let “c” denote the continue sampling decision and “s” the stop decision. Only when a worker is poorly matched at the current job will he/she need to decide about whether or not to sample a new job in the occupation. If a worker is laid off, then he/she must sample a new job. Of course, the worker is also simultaneously re-evaluating his/her occupational choice. Using the notation above for superprocesses, we have Di(0, n, Si(n), Fi) = {c}, Di(1, n, Si(n), Fi) = {s} and Di(2, n, Si(n), Fi) = {s,c}. The return structure is Ri(0, 0, 0, 0) = −c1i, Ri(0, n, Si(n), Fi) = −ci for n > 0, Ri(1, n, Si(n),

268

The Economics of Search

Fi) = wi + αi, and Ri(2, n, Si(n), Fi) = wi − αi. The transitions between states in the ith occupation are governed by a transition kernel which satisfies: pi((0, 0, 0, 0), (1, 1, 1,0)) = ½ pi((0, 0, 0, 0), (2, 1, 0, 0)) = ½ in the initial state, pi((0,n,Si(n),Fi),(1,n,Si(n),Fi) = p¯ i(n,Si(n)) pi((0,n,Si(n),Fi),(2,n,Si(n),Fi) = 1 − p¯ i(n,Si(n)) when the worker has had some occupational experience and is switching jobs, pi((1, n, Si(n), 0), (0, n, Si(n), 1)) = ui pi((1, n, Si(n), 0), (1, n, Si(n), 0)) = 1 −ui when a good match has been attained and the worker has never been laid off in the ith occupation, and pi((1, n, Si(n), 1), (1, n, Si(n), 1)) = 1 if the worker has previously been laid off in occupation i. When in states of the form (2, n, Si(n), Fi), a decision to continue sampling results in pci((2, n, Si(n), Fi ), (0, n, Si(n), Fi)) = 1 and a decision to stop gives psi((2, n, Si(n), 0), (0, n, Si(n), 1)) = ui psi((2, n, Si(n), 0), (2, n, Si(n), 0)) = 1 − ui if the worker has not been laid off and psi(2, n, Si(n), 1), (2, n, Si(n), 0)) = 1 if the worker has been laid off in occupation i. Appealing to results contained in Glazebrook (1982), it can be shown that the W-condition is satisfied for all states of the form (2, n, Si(n), 1). However, for states (2, n, Si(n), 0), it is possible for the W-condition to fail. Intuitively, this failure can occur because optimal decisions are no longer myopic when the probability of layoff is positive. McCall (1991) analyzed examples that were constructed so that the W-condition was satisfied for all states. Hence, equation (9.41) could be used to determine the optimal sampling policy.

Multi-armed bandits and their economic applications

269

Dynamic allocation indices were computed for combinations of δi = .5, 1, and 1.5, αi = 131.1 and 200, c1i = 1000, 1500, 2000, . . ., and 5000, ui = .01 and .02, wi = 1000, ci = 1000, and β = .95. The method of computation used by McCall (1991) was as follows. First, the optimal intraoccupational jobswitching policy for an occupation was solved by backward induction when the stopping option yielded 0. Next, Z(2, n, Si(n), 0) was determined for this intraoccupational job-switching policy. The invariance of the optimal intraoccupational job-switching policy to changes in the value of the retirement option over the interval [0, Z(2, n, Si(n), 0)], required by the Wcondition, was then checked. Finally, the index for an unsampled occupation was calculated by constructing a (finite) transition matrix and return vector given the optimal intraoccupational decision rule and employing numerical calculation techniques developed in Varaiya et al. (1985). For the example presented in McCall (1991), the optimal intraoccupational stopping rule varied considerably over the different parameterizations.5 For example, when α = 200 and u = .01, a worker behaving optimally was found to stop sampling jobs within an occupation after 3, 2 or 1 consecutive bad matches when the occupation is characterized by δ = 1.5, 1.0, or 0.5, respectively. As δ decreases, the amount by which occupational experience alters an individual’s prior beliefs (about the likelihood of obtaining a good job match in the occupation) increases. So the benefit of continuing to sample jobs in an occupation when the first job is a bad match falls with δ. When α = 200 and u = .02, a worker with initial beliefs δ = 1.5 would stop sampling jobs after only two consecutive bad matches; the sampling strategy of workers with δ = 1.0, or 0.5 would be unaffected by the increase in the layoff risk. When α = 131.1 and u = .01 a worker would stop sampling jobs after only one bad match for all values of δ. Table 9.1 presents the calculated values for Z(0, p(0), h(0), 0), or simply Z(0), in annuity equivalent form. As can be seen from the table, the dynamic allocation index, Z(0), associated with δ = .5 exceeds that for δ = 1.5 at low levels of c1. However, for sufficiently high levels of c1 this result is reversed. These results suggest that occupation-specific information is more attractive only when occupational entry or training costs are relatively low. As training costs rise, individuals tend to prefer, ceteris paribus, occupations where match information is job-specific. Actually, this result is not surprising. When training costs for an occupation are sufficiently high, an individual will never leave the occupation once it is sampled. This occurs because the index attached to the occupation never falls below its initial value (which is higher than any other occupation index since it was optimal to sample the occupation in the first place) when an individual follows his/her optimal intraoccupational job-switching policy. Hence, all possible outcomes must be taken into account when evaluating that occupation. A poor match at the first job sampled results in a larger drop in p¯ for occupations with less precise prior beliefs. This large drop in p¯ makes it more likely that the worker would forgo further job sampling. Therefore,

962.7 914.2 871.1 836.8 805.5 779.7 754.7 729.7 704.7

Training costs (c1) 1000 1500 2000 2500 3000 3500 4000 4500 5000

964.0 915.5 867.5 829.4 796.1 771.1 746.1 721.1 696.1

1.0

966.7 918.3 870.3 822.4 787.1 762.1 737.1 712.1 687.1

0.5

902.2 861.2 836.2 811.2 786.2 761.2 736.2 711.2 686.2

1.5

903.5 861.3 836.1 811.1 786.1 761.1 736.1 711.1 686.1

1.0

α = 131.1, u = .01

Notes: w = wage (fixed at 1000) c = intraoccupational job switching cost (fixed at 1000) c1 = occupational training cost α = match specific uncertainty (net wage = w ± α when this uncertainty is resolved) u = exogenous firing or layoff risk δ = parameter of the prior distribution for the probability of a successful match (Beta(δ,δ)) β = discount rate (fixed at .95)

1.5

Priors(δ)

α = 200, u = .01

Table 9.1 Dynamic allocation index of occupation (Z(0)(1 − β))

905.3 862.2 835.4 89.4 785.4 760.4 735.4 79.4 685.4

0.5

941.7 893.0 849.5 815.1 787.7 762.7 737.7 712.7 686.7

1.5

944.1 895.3 847.1 808.3 781.3 756.3 731.3 706.3 681.3

1.0

α = 200, u = .020

950.8 903.2 855.5 807.9 779.8 754.8 729.8 704.8 679.8

0.5

Multi-armed bandits and their economic applications

271

one attaches a lower likelihood to ever attaining a good job match within occupations of this type. On the other hand, the smaller is δ, the larger the upward revision in p¯ when the first job worked results in a good match. Even though a worker would never voluntarily leave such a job, this information is still valuable because of the risk of layoff. When training costs are low, so in fact occupation “shopping” is observed, this latter effect dominates.6 When entry costs are relatively high, objective match uncertainty, measured by α, is of less importance in determining the overall attractiveness of an occupation. Table 9.1 also shows that the drop in the index for a unit increase in occupational training costs can be greater than one, ∆Z(0)/∆c1 < −1 at low levels of c1.7 This reflects the fact that the higher are occupational training costs, the less valuable is the occupational-switching option. So it is clear that job switching or transaction costs play an important role in determining the overall sampling strategy of a worker. When training and job-switching costs are low relative to the amount of matching uncertainty, a significant premium is attached to objective match uncertainty and, when layoffs are probable, to occupations in which individuals learn considerably from sampling. This occurs because both the occupation- and job-switching options are viable. However, if training costs are considerable, then objective match uncertainty is of less importance and workers prefer matching information which is relatively job-specific. Finally, the model developed by McCall (1991) offers an explanation for why occupational exit rates vary substantially across occupations. Low exit rates occur in occupations characterized by relatively high entry costs and/or where match information tends to be job-specific. Exits from occupations of this type would be primarily by workers with considerable occupational experience. High exit rates, on the other hand, would be observed from occupations in which entry costs are low and information is relatively occupation-specific. These exits would occur after relatively little occupational experience. In McCall (1990) some of the implications of the theory of optimal sampling of occupations are tested. McCall constructs a superprocess model of occupational matching where there are two types of information, occupationspecific and job-specific, and returns are linear in these two types of information. Within an occupation both types of information arrive according to a geometric distribution and when the occupation or job match information arrives its quantity is drawn from a given distribution. McCall shows that under these conditions, if occupational-specific information is important then the impact of tenure in a previous job on the job separation hazard from the current job should be smaller when the occupation of the previous job is different. Using data from the National Longitudinal Survey of Youth (NLSY) he finds some support for the theory. Neal (1999) also tests the theory under the further assumption that occupation information arrives immediately.8 Under these circumstances, individuals will first find a suitable occupational match before seeking a suitable

272 The Economics of Search job within an occupation. Using data from NLSY Neal finds that, while not perfect, individuals do tend to look for suitable occupations first, in that employer switches that take place early in an individual’s career are more likely to involve a change in occupation than employer switches that take place later in an individual’s career. More recently Pavan (2005) estimated the structural parameters of a model of job mobility where career choice is important. He found that a model with both career and job-matching components fits the data better than a model with just either a job-matching component or career-matching component.

9.7 DISCOUNTED RESTLESS BANDITS We now consider a somewhat broader class of decision problems which contain the MAB problem as a special case. This is the restless bandit problem considered by Whittle (1988).9 In this model there are N bandits. However, more than one bandit can be played at any time and the states of “inactive” bandits as well as “active” bandits can change states. The process governing the change of states, however, can depend on whether the bandit is active or inactive. Suppose that there are N projects. Let the state space for project i be denoted by Χi, i = 1, . . ., N. We will denote a particular state by xi and represent the vector of states by x and the cross-product of state spaces across bandits by X. At each time t, any individual can activate M < N projects. Let ai(t) = 1 if bandit i is active at time t and ai(t) = 0 if it is inactive, i = 1, . . ., N. Let a(t) denote the vector [ai(t), i = 1, . . ., N]. If bandit i is active and in state xi, then the reward R1i (xi) is received and the bandit changes states according to the transition matrix P1i (x′i,xi). If bandit i is inactive and in state xi, then the reward R0i (xi) is received and the bandit changes states according to the transition matrix P0i (x′i,xi). It is assumed that the bandits evolve independently from one another and that the reward of bandit i depends on xi. Let R0(x) and R1(x) denote the vector of rewards when in state x and either all bandits are inactive or all bandits are active, respectively. An individual’s problem is to choose the set of active and inactive bandits each period in ∞

冤冱β {(1 − a)′R t

order to maximize E

0

冥

+ a′R1} where 1 is an N-vector of ones

t=0

subject to a′1 = M. If we let Pa(x,x′) represent the transition matrix of the all bandits, the associated value function V(x) for this stochastic dynamic decision problem satisfies the recursive equation

冦

V(x) = maxa a′R1(x) + (1 − a)′R0(x) + β

冱P (x,x′)V(x′)冧. a

x′

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273

A computationally tractable solution to this problem may be impossible due to the curse of dimensionality when the number of bandits and state space are large. In this case, it may be pragmatic to give a heuristic which reasonably approximates the optimal solution. Before turning to such a heuristic note that in the special case where M = 1 and R0(x) and P0i (x′i,xi) = 0 for x′i ≠ xi and P0i (x′i,xi) = 1 for x′i = xi, this problem becomes the classic MAB problem and, hence, an index policy is optimal. The heuristic is due to Whittle (1988) who proposed an index heuristic for the class of restless bandit problems that pass an indexability test. Indexability and indices are traits of individual bandits so we can focus on the discussion on a single bandit. Suppose that the reward in the passive state is subsidized by an amount W. Then the Whittle indexability criterion is satisfied if the set of states in which it is optimal to choose the passive action is increasing in W. If we let Λ(W) be the set of states for which it is optimal to choose the passive action then

冦

Λ(W) = x ∈ Χ | R0(x) + W + β

冱 P (x,x′)V(x′,W) ≥ R (x) 0

1

x′ ∈ Χ

+β

冱 P (x,x′)V(x′,W)冧. 1

x′ ∈ Χ

Definition 9.1 A bandit is indexable if for all W ′ > W, Λ(W ′) ⊇ Λ(W ). A Whittle index then is defined by Definition 9.2 Whittle index is the smallest value W such that choosing the passive action is optimal: W *(x) = inf {W | x ∈ Λ(W )} Now, the Whittle heuristic states that if all bandits are indexable, then when the process is in state x activate the M bandits with the M highest Whittle indices, W *(x). The remaining N − M bandits are inactive. In a recent paper, Glazebrook and Ruiz-Hernandez (2005) demonstrate how to use the results for restless bandits to compute approximate solutions to the switching cost problem. The switching cost problem starts with the classic multi-armed bandit problem and then relaxes the assumption that it is costless to switch from project i to project j. In general, switching costs can depend on both i and j. Denote switching costs by kij(x). When switching costs are independent of the arm to which (from which) the switch is made, costs will simply be denoted by ki(x) (kj(x)). Banks and Sundaram (1994) showed that, even in the case where switching costs are independent of the arm to which the switch is made, the Gittins’ index policy is in general not the optimal policy. While general dynamic programming methods could be used

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to solve such a problem, when there are a large number of arms the problem is numerically intractable. In analyzing the switching cost problem, Glazebrook and Ruiz-Hernandez (2005) first demonstrate that the general problem where switching costs depend both on the arm from which an individual is switching and the arm to which an individual is switching (or using Glazebrook and Ruiz-Hernandez’s terminology involves both “tear down” and “setup” costs) the problem can be equivalently formulated as a problem in which switching costs depend only on the arm to which an individual is switching (kj(x)). The key to the analysis of this problem is to expand the state space to include an indicator variable for whether or not the same arm was played in the previous period. Let ai = 1 if arm i was played last period and equal 0 otherwise. The new state space then equals the cross-product of pairs (ai, xi) = 1,. . .,N. The value functions used to determine whether this problem determines the indexability condition then satisfies V(1,x) = max{R(x) + βE(V(1,x′);W + V(0,x)}

(9.42)

V(0,x) = max{−k(x) + R(x) + βE(V(1,x′);W + V(0,x)},

(9.43)

and

where the first term in the brackets of equations (9.42) and (9.43) are the returns for choosing the active action and the second term the returns for choosing the passive action. Equation (9.43) can be simplified to

冦

V(0,x) = max −k(x) + R(x) + βE(V(1,x′);

W . 1−β

冧

(9.44)

Consider the Gittins index associated with each component of (9.43) and (9.44). Let Zaj (x) denote the index of the jth component of action a, j = 1,2 a = 0,1. It is clear that Z 21(x) = Z 20(x) =

W and Z 11(x) > Z 10(x). 1−β

Thus the optimal policy can be written as Theorem: (Optimal Policies for the W problem) W then the active policy is optimal in states (0,x) and (1,x). 1−β W (b) If Z 11(x) > > Z 10(x) then the active option is optimal in state (1,x) 1−β and the passive option is optimal in state (0,x). (a) If Z 10(x) >

Multi-armed bandits and their economic applications (c) If Z 11(x) <

275

W then the passive option is optimal in states (0,x) and 1−β

(1,x). W then either option is optimal in the state (1,x) and the 1−β passive option is optimal (0,x). W (e) If Z 10(x) = then the active option is optimal in state (1,x) and either 1−β option is optimal in the state (0,x).

(d) If Z 11(x) =

From this result it is easy to see that the indexability requirement is satisfied. Theorem 9.6 The Whittle indices are given by W(1,x) = (1-β)Z 11(x) and W(0,x) = (1 − β)Z 10(x), and the bandits are indexable. Define the set of states for which it is optimal to take the passive action given W by Λ(W ). Then by Theorem 9.6 we have

冦

W

W

冷1 − β ≥ Z (x)冧 ∪ 冦(0,x)冷1 − β ≥ Z (x)冧

Λ(W ) = (1,x)

1 1

1 0

which is clearly increasing. Glazebrook and Ruiz-Hernandez (2005) provide several numerical simulations that assess the accuracy of the Whittle heuristic. The Whittle heuristic could be used to examine a generalization of the sequential migration model presented in McCall and McCall (1987). In that model we assumed that there were N different locations each characterized by a wage distribution Fi , belated information characterized by the random variable Yi = αiB where B is a random variable which equals +1 or −1 with equal probability, a search cost ci and a migration cost ki. It was assumed that first an individual pays ci to observe the wage in the ith city. Then the individual decides whether or not to move there. If the individual migrates, then additional cost ki is incurred. After one period of residing in the city (and earning wi) belated information Yi is received reflecting perhaps updated information on the non-pecuniary aspects of the city. In order to satisfy the assumptions required for the Gittins index, we assumed that migration costs were incurred only the first time an individual moved to the city. While there are certainly such fixed costs (finding a doctor, etc.) to migrating to a particular city surely there are also costs that are borne each time an individual relocates there. While ki could certainly depend on whether it was the first time an individual moved to a city (as well as what city an individual is migrating from), for simplicity, assume that an individual incurs cost ki every time that they relocate to city i. While this problem no longer satisfies the assumptions required for Gittins’ indices to characterize the optimal solution it does fall into the class of switching cost problems

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analyzed above. In fact, the heuristic would yield a reservation wage strategy similar to McCall and McCall (1987) as the sampling strategy. That is, move to a new city i if wi ≥ ξi = Z *(1 − β) + ki −

β(ki + αi) (2 − β)

(9.45)

(as long as αi is sufficiently large) where Z * = maxj ≠ i{Zj(x)}. It is obvious from (9.45) that the reservation wage is decreasing in αi and increasing in ki. Return migration to city i would occur when wi ≥ Z *(1 − β) + ki + αi. Of course the larger ki or αi, all else equal, the lower the likelihood that an individual will return to a city that he/she once resided in.

10 A sample of early responses to Diamond’s paradox and Rothschild’s complaint

Search theory as exposited in Stigler (1961), but also including BSM, generated several provocative critiques, the most illuminating and illustrious being that of Rothschild (1973), which we call Rothschild’s complaint.1 At the heart of the complaint is a version of Diamond’s paradox. Levine and Lippman (1995) have a succinct description of this paradox. Diamond showed how sensitive price dispersion is in equilibrium: without heterogeneity somewhere in the model there can be no price dispersion. (Butters, 1977, which we cite shortly, has an excellent exposition of this argument.) Furthermore, there will be market failure if each act of search, including the visit to the first store, is costly: the firms’ attempt to extract all surplus causes the marginal shopper to drop out of the market which then propagates by a process similar to that discussed in Akerlof ’s (1970) lemons model. Some heterogeneity must be introduced so that price dispersion becomes the mechanism enabling price discrimination.

10.1 INTRODUCTION2 In Stigler’s (1961) work on optimal consumer search, he assumes that the distribution of prices is known by the searcher and not influenced by the searcher’s actions. In order to perform the partial equilibrium analysis of Marshall, the response of sellers must be specified and the ensuing equilibrium price distribution determined. Rothschild (1973) was the first to raise this important point. That its solution is not trivial is clearly stated by Butters (1977: 465) What is an appropriate concept of equilibrium? Under what conditions will equilibrium consist of a non-trivial distribution of prices, as opposed to the classical unique price? What parameters determine the level of prices and the degree of price dispersion? What kind of externalities may arise when consumers search? Can any welfare judgments be made? These questions have all been inadequately treated in the literature.

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The Economics of Search One difficulty in developing the analysis is that once the problem of information is introduced, one must face the fact that there are many types of markets, varying according to the structures of information flow, the degree of centralization, the homogeneity of the good traded, the types of services provided jointly with the good, the number of actual or potential buyers and sellers, the volatility of parameters affecting supply and demand, the time span of the market, and the geographical location of the market. The interaction of these factors and others is involved in determining price dispersions, and it is not obvious which factors and interactions should be singled out as the building blocks of a model. As a start, it is natural to take one particular market structure as given and to derive an equilibrium for that market. This approach is certainly not the whole story, however, because the market structure itself is endogenous to a fuller model.

Of course, Stigler (1961: 220–3) recognized the importance of price dispersion (this is what he was explaining!), the necessity of its persistence, the effect of advertising, and the response of sellers. Indeed, his informal discussions are an indispensable guide for the formal analysis of equilibrium distributions. The maintenance of appreciable dispersion of prices arises chiefly out of the fact that knowledge becomes obsolete. The conditions of supply and demand, and therefore the distribution of asking prices, change over time. There is no method by which buyers or sellers can ascertain the new average price in the market appropriate to the new conditions except by search. Sellers cannot maintain perfect correlation of successive prices, even if they wish to do so, because of the costs of search. Buyers accordingly cannot make the amount of investment in search that perfect correlation of prices would justify. The greater the instability of supply and/or demand conditions, therefore, the greater the dispersion of prices will be. In addition, there is a component of ignorance due to the changing identity of buyers and sellers. There is a flow of new buyers and sellers in every market, and they are at least initially uninformed on prices and by their presence make the information of experienced buyers and sellers somewhat obsolete. The amount of dispersion will also vary with one other characteristic which is of special interest: the size (in terms of both dollars and numbers of traders) of the market . . . Price advertising has a decisive influence on the dispersion of prices. Search now becomes extremely economical, and the question arises why, in the absence of differences in quality of products, the dispersion does not vanish. And the answer is simply that, if prices are advertised by a large portion of the sellers, the price differences diminish sharply. That

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they do not wholly vanish (in a given market) is due simply to the fact that no combination of advertising media reaches all potential buyers within the available time. Assuming, as we do, that all sellers are equally convenient in location, must we say that some buyers are perverse in not reading the advertisements? Obviously not, for the cost of keeping currently informed about all articles which an individual purchases would be prohibitive . . . The seller’s problem is even greater: he may sell two thousand items (a modest number for a grocery or hardware store), and to advertise each on the occasion of a price change—and frequently enough thereafter to remind buyers of his price—would be impossibly expensive . . . The search rule used by Stigler was a fixed sample size rule. The consumer calculated the optimal number n* of price observations before he began his searching activity. An alternative “optimal stopping” approach is to calculate a “reservation price” pr, such that a purchase is made whenever p ≤ pr. In this sequential procedure, the sample size N is a random variable. In the remainder of this introductory section, we assume that the consumer uses an optimal stopping rule.

10.2 PRICE DISPERSION The consumer’s attitude toward dispersion of prices corresponds to his attitude with respect to demand curves. In all of his experience he has never observed a positively sloped demand curve. He assumes that other consumers will continue to be rational and, therefore, concludes that his buying plans always can be based on this “empirical fact.” Similarly, the price searcher always observes price dispersion. Once again he assumes that all the other economic actors will continue to play the economic game in a rational manner. Price variability is also regarded as an “empirical fact,” which he should exploit by searching until its expected marginal gain equals its expected marginal cost. Rothschild’s complaint was not aimed at “empirical facts.” What he required was a theoretical justification for price dispersion that was comparable to the Slutsky equation for demand functions. Simply put, the requirement is: the price distribution F must persist in the wake of rational consumer search and the rational producer response elicited by search. More formally, the requirement is that there exist a fixed point, the equilibrium distribution F, to the search process in which both consumers search according to optimal stopping rules3 and firms maximize expected profits. For good or ill, there are many ways of responding to Rothschild’s complaint. Clearly, if a nondegenerate probability distribution is to emerge as a fixed point of any search process, the overall process including consumer and firm behavior must contain persistent variability. There are several ways of introducing persistent

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variability. The first is through the production process: firms charge different prices because their costs of production are variable. The second is to assume that consumers are heterogeneous,4 and this heterogeneity can be captured by treating the cost of search as a random variable. The third is to assume that some consumers are better informed than others about F and, knowing this, the firms engage in price discrimination policies that give rise to persistent price variability. One can also assume that markets are geographically separated thereby inducing persistent price differences. Uncertainty could also be modeled by the birth or death of both consumers (firms) with different costs of information (costs of production). Underlying vicissitudes in market price and the lagged adjustment to them by both consumers and firms is another candidate as the source of persistent variability.

10.3 MATCHMAKING Throughout our analysis it is important to remember the goal of both consumer and firm—they wish to achieve a stable match where the cost of search can be substantially reduced. For example, the supermarket is looking for a reliable buyer who will remain attached to it in the face of modest price fluctuations; the firm is looking for suppliers of inputs (workers) who will remain steadfast in spite of fluctuating prices (wages); marriage partners remain attached in spite of minor adversities. Thus search is a prelude to matchmaking and as such is marked by a continuous flow of participants into and out of the search state. The population of searchers at any time is composed of new entrants, i.e., those seeking their first match and those who have re-entered the search market after discovering that their current match was undesirable for one reason or another. The behavior of these two groups could be quite different, but we do not study these differences here.

10.4 SEARCH TECHNOLOGY It is also important to distinguish among the markets in which search occurs. The most familiar is that in which participants on both sides of the market are trying to find someone on the other side with certain characteristics. This is the mutual search activity that characterizes the job market, the commodity market, and the marriage market. The intensity of search by participants in these markets is rarely treated symmetrically. Stores seek out customers by their advertising campaigns, whereas customers look for stores by either sequential or nonsequential search. The intensities of search are controllable by both parties. This is also true in the labor market and the marriage market. There is a second class of search models in which the search activity is only undertaken by one side of the market. This includes models of research and development in which firms are searching for new techniques, professors are

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281

looking for new theorems, and firms are looking for new locations or industries where there are no barriers to entry. The price distribution corresponds to the random variable denoting the return from the research and development activity. While these research and development processes have been modeled, no one has studied the persistence of F. In the third class of search models one party is searching while the other is evading. The class of models comprises the predator–prey models of biology and anti-submarine warfare. They would also include industrial organization models in which firms are seeking entry into industries that are trying to deter entry. Clearly the intensity of search and the intensity of evasion are crucial in these models and will determine the persistence of “price” variability, where price is the cost of deterrence.

10.5 PRINCIPAL–AGENT PROBLEMS AND DURABLE MATCHES A tangential but pertinent comment is worth making at the outset. There has been an enormous amount of research on the problems of moral hazard and adverse selection. This is especially true of the employer–employee match, but is also true for other matches (see Radner, 1982, for an excellent survey). The point is that most of the paradoxical results of the principal/agent paradigm are based on one-period models—single encounters between buyer and seller. But most relationships studied herein are enduring; otherwise why spend all this time and effort searching for the right one. In the repeated play of these matches, the problem of moral hazard and adverse selection should diminish. While they may be characterized by multiple equilibria and other nasty mathematical problems (see Radner), the history of behavior will be known to both sides and the problems of asymmetric information should vanish.

10.6 EQUILIBRIUM MODELS OF PRICE DISPERSION There has been an enormous amount of research addressing the existence of an equilibrium price dispersion. There are many possible sources of variability and this accounts for the variety of models. At first, the results were disappointing in that the only equilibrium achievable was degenerate (concentrated at a single price). As more structure was added, the results became more interesting—that is, nondegenerate distributions were obtained. The purpose of this section is to review these models and their underlying assumptions. Table 10.1 partitions several of the most important early models by their assumptions. Some models assume that firms possess identical technologies and that consumers are homogeneous with respect to tastes and preferences. In this homogeneous environment, both sequential and nonsequential sampling procedures are studied. The nonsequential includes both

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The Economics of Search

Table 10.1 Taxonomy of equilibrium dispersion models Firms Identical (technology)

Consumers

Sequential sample search Burdett–Judd Axell, Pratt et al. Berninghaus Fixed-sample search Burdett–Judd Wilde–Schwartz “0–1” + search Salop–Stiglitz Varian Braverman

Note:

+

Nonidentical (technology) Reinganum MacMinn Carlson–McAfee MacMinn

The 0,1 denotes uninformed and partially informed respectively.

the fixed-size sample rule and “0–1” information structure where some fraction of the searchers have no information about prices, the “0” group, whereas the remainder do have some price information, the “1” group. For each of these models, we describe consumer behavior by specifying the consumer informational assumption, the decision rule used by consumers, and the demand function. The consumer sector is then determined by the number of consumers, the distribution of search costs and the expected market demand function. An identical procedure is followed for firms. Their behavior is determined by the informational assumption, the decision rule describing their search activity, and the firm-specific demand function. The firm sector is then determined by the number of firms and the technological assumptions: the level and distribution of technology among firms. Given these assumptions about firms and consumers, market behavior is characterized by the equilibrium concept employed. All these together determine whether an equilibrium dispersion exists. For each model, we describe the equilibrium and its properties. We of course, also discuss other aspects of a particular model that strike us as being especially interesting like comparative static results and conclusions that differ from those obtained from search models in a nonequilibrium setting. Finally, we will note aspects of the model (assumptions and/or implications) that are unusual and either conflict with common sense or conventional economic wisdom.

10.6.1 The degenerate equilibrium model (the Diamond paradox) Suppose that many consumers are searching for the same commodity and each wishes to purchase exactly one unit. At prices above p¯ , no consumer is willing to buy. The firms have identical cost curves. The searchers know the

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283

distribution of p, but they are unable to attach prices to firms. To accomplish this they must pay a fee to discover the price at a particular firm. Assume they search sequentially. In this environment, it has been shown5 that when firms know search costs and the prices of competitors, there is no Nash equilibrium in pure strategies that will support more than one price. To see this, consider a possible equilibrium distribution with the minimum price below p¯ . Firms offering prices below p¯ can raise their prices by ε < c and retain all their customers. The minimum price will continue to rise until p¯ is hit. Hence, the pure distribution is degenerate with all of the mass concentrated at the monopoly price p¯ . But if there is some positive cost of entering the market, consumers will not pay p¯ . No equilibrium exists. In order that an equilibrium price distribution emerges, the model must be altered. Some have assumed that the cost of search is either zero or that the distribution of search is not bounded away from zero. The model of Salop– Stiglitz makes this assumption.

10.6.1(a) Salop–Stiglitz model (1982) In their model Salop and Stiglitz assume that the distribution of prices is known, but that the location of the prices is unknown. They assume that for a fixed price ci shopper i learns the prices at all stores, i = 1,2,. . . . The shopper may also choose to buy no information about prices. These are the only two types of shoppers permitted: the perfectly informed and the ignorant. From these assumptions it should be clear that there are only two possible equilibrium prices. Firms are assumed to have perfect information about their demand curves and earn zero profits. Each consumer demands only one unit. The fraction of type 1 customers (c = c1) is λ whereas the fraction of type 2 customers (c = c2), c2 ≥ c1 is (1 − λ). Consumers must decide whether to enter the market and if they do how much information to buy. Finally, they check to see if a firm could increase its profits by charging a different price. If the firm prefers the original distribution, then it is an equilibrium distribution. Suppose there were a single price equilibrium po at which each of the n firms obtains zero profits and equal sales. Let a single firm raise its price to (n − 1)po + (po + ε) ε po + ε. The average market price increases to p¯ = = po + . n n The expected gain from buying perfect information is Eg = p¯ + ε/n − p¯ . Thus consumer i purchases the information if and only if ci ≤ Eg. If c2 ≥ c1 > 0, there is an ε > 0 such that this firm loses no customers—the increase does not cause any customer to purchase perfect information. But his new price is higher and the original position could not have been a Nash equilibrium. If c1 = 0, the firm can lower its price by ε > 0 and group 1 buys perfect information and purchases from the price cutting firm. Thus po is not a Nash equilibrium.

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Salop and Stiglitz show that a single price equilibrium exists at the competitive price if c1 = c2 = 0, and either c2 is small or the AC curve is steep or the size of group 1 is large. A single price equilibrium exists at the monopoly price if both c1 and c2 are large enough. For other values of c1 and c2, either no equilibrium exists or a two-price equilibrium obtains. In this case the higher priced store gets no informed buyers, but 1/n uninformed buyers. Nevertheless, 1/n suffices for achieving the zero profit condition. 10.6.1(b) The Varian equilibrium model (1980) Varian analyzes temporal as opposed to spatial price dispersions. Stores vary prices over time and hence any cross-section of prices would display dispersion. However, since the composition of stores charging differing prices is itself fluctuating over time, consumer learning is not possible and the dispersion persists. The rationale for Varian’s temporal dispersion is the existence of sales. Some of the causes of this behavior include: price discrimination between informed and uninformed buyers, inventory costs, cyclical fluctuations and advertising. Consumers buy at most one unit and pay no more than r, their reservation price. Informed consumers I > 0 know the store charging the lowest price and they shop there. Uninformed shop at first store they visit provided p < r. The number of informed and uninformed consumers are I and U respectively and n is the number of stores, so that u = U/I is the number of uninformed shoppers per store. Each week stores take a random draw from f (p), the distribution of prices. The store with the lowest price has I + u shoppers, whereas all other stores attract only u shoppers. The stores have identical, decreasing average c(q) cost curves, . Entry occurs until profits are zero, the average cost associq c(I + u) . Thus the ranated with the maximum number of customers is p* = I+u dom variable p will always lie in the interval (p*,r). Varian also shows that there are no point masses in the equilibrium pricing strategies. If there were, a deviant store could charge a price p − ε with the same probability that other stores charge p. It would lose profits of order ε, but gain a fixed amount of profits when the other stores tied. Hence for ε > 0, profits would be positive, a contradiction. The expected profit of a representative store is r

冮 {πs(p)(1 − F(p))}

n−1

p*

where

+ πf (p) {1 − (1 − F(p))n − 1]} f (p)dp,

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285

πs(p) = p(u + I ) − c(u + I ) πf (p) = pu − c(u ). Hence the firm chooses f (p) to maximize expected profits subject to r

f (p) ≥ 0,

冮 f (p)dp = 1.

p*

Thus if f (p) > 0, πs(p)(1 − F(p))n − 1 + πf (p)(1 − (1 − (1 − F(p))n − 1] = 0, on which rearrangement yields the equilibrium c.d.f. 1 − F(p) =

πf (p)

1/n − 1

冢π (p) − π (p)冣 f

s

πf (p) is strictly decreasing in p. Furthermore, prices πf (p) − πs(p) near p* and r have positive density and there is no gap (p1,p2) such that f (p) = 0, p ∈ (p1,p2). Thus the equilibrium density is given by f (p) for all p ∈ (p*,r) and f (p) = F′(p), where Observe that

F(p) = 1 −

πf (p)

1/n + 1

冢π (p) − π (p)冣 f

.

s

Varian gives no direct existence proof for an equilibrium price dispersion. His procedure is as follows: At first he establishes many “necessary” conditions for an equilibrium (mixed) strategy. From all these necessary conditions he can deduce an explicit price dispersion. But this does not yet assure that the resulting dispersion is an equilibrium dispersion. He proves this by demonstrating that deviating from this dispersion does not pay. There are two conceptual problems attending this concept of an equilibrium dispersion. First, the dispersion has to be continuous. But what does it mean for a single firm to vary prices continuously? Second, the result of Varian cannot be regarded as an “approximation” (for the case of discrete distributions) or as a mathematically simplified version (continuous distributions are easier to manage). There is clearly a discontinuity problem. That is, discontinuity of demand makes the existence of Nash-equilibria improbable. The crucial point is the existence of “ties.” But Varian eliminates all ties by a mathematical “trick.”

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10.6.1(c) The equilibrium model of Wilde and Schwartz (1979) In their shopping model Wilde and Schwartz assume that there are two groups of consumers in the market. Those in the first group do not shop at all, buying at the first store they visit. They refer to this group as A1. Those in the other group, An, visit n stores and purchase at the minimum price. In the beginning shoppers know neither the prices charged nor the distribution of these prices. Assume that there are N stores in the market, 1 ≤ n ≤ N. Observe that if n = N, the number of firms in the market, the members of the set An A1 have complete information. By setting large (small) and/or n small A1 + An (large) the model posits high (low) information costs. Notice that the An consumers can be viewed as using a search strategy that is a mixture of sequential and fixed sample rules. This is compatible with the search strategy used most frequently in the marketing literature. Wilde and Schwartz show that the number of firms sampled by members of An has no effect on the existence of an equilibrium price distribution. Its existence depends on the firms cost functions, the limit price, and the fraction An/(A1 + An) of shoppers. For n large, the equilibrium is concentrated via a mass point at the competitive price. For modest values of n, an equilibrium price distribution exists. Even when the equilibrium collapses to the competitive price, consumers continue to shop—a rather bizarre result. By replacing the sequential search assumption with one in which there are two classes of searchers with fixed length of search, one long and one short, Wilde and Schwartz achieve a plausible specification of real-world search behavior and generate nondegenerate market equilibria with substantive policy implications. Essentially, the paper establishes that if the percentage of price conscious searchers (i.e., long search duration) is high enough they effectively police the market, imposing the competitive prices on all sellers and exerting a positive monetary externality on those searchers who are not (and do not choose to become) informed. The policy implications are compelling, and the model has received much attention in the legal/policy areas. 10.6.1(d) The Butters model Another prominent goal of the Stigler paper was to explain the role of advertising. Within his information-theoretic framework he viewed advertising as a method for transmitting information about products to potential consumers. A somewhat contrary view had been expressed by Knight and, of course, others have a very negative attitude towards advertising. Stimulated by Stigler’s perspective, Butters (1977) developed an equilibrium model of sales and prices, where the sole objective of advertising is to convey information. We now present a brief summary of this fine paper.

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287

Assumptions of the Butters model 1 There is a single homogeneous good. 2 There are many identical buyers and sellers. 3 Sellers can send advertising messages informing buyers of their price and location. Sellers advertise different prices among the buyers. 4 Buyers have no other source of information about sellers. They receive advertisements. At the end, they either buy exactly one unit or nothing. 5 All buyers have the same limit price, m, namely, the maximum they are willing to pay for the good. 6 The cost to every seller of sending out each advertisement is b, and each advertisement reaches a potential buyer. There are no economies of scale in advertising. 7 Advertisements are allocated randomly among buyers and independently of one another. Buyers receive advertisements freely and cannot affect their probability of receiving them. 8 All sellers have the same constant cost of production po. 9 All sellers know the buyers’ limit price and the distribution of prices advertised by other sellers. 10 Sellers maximize their expected profits given the behavior of the other sellers, while buyers minimize the price they pay, subject to the above restraints. For mathematical simplicity, Butters further assumes that there are an infinite number of sellers and buyers. The criterion for equilibrium is as follows: for a given advertised price distribution, the expected profit of each seller is zero, and no seller would expect higher profits at a different price and/ or at a different level of advertisement. Let π(p) be the probability that an ad at price p will result in a sale. Let β(p) be the number of ads per (potential) buyer sent at a price no greater than p. Using the analogy that buyers are empty urns and the number of ads per buyer, β(p) is like the ratio of balls to urn, the probability, π(p), of a successful ad, i.e., an advertisement resulting in a sale, is equal to the expected fraction of empty urns. Feller (1970) has shown that when the ratio of balls to urn remains constant while their number goes to infinity, the expected fraction of empty urns is the exponential minus the ratio. So, π(p) = exp(−β(p)).

(10.1)

In equilibrium, there is no price for which the expected revenue can be greater than the expected cost, because of free entry. Thus, from the assumptions, we get directly π(p) = b/(p − po).

(10.2)

Also, if one advertises a price k equal to the cost of production plus the

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cost of an ad, he generates a sale with certainty, since no one else can offer a lower price without taking a loss. Obviously no price greater than the reservation price m of buyers will be advertised. In addition, assume there is an interval of prices (p1,p2) belonging to (po + b,m), such that no price in that interval is advertised; it will therefore pay the seller who advertises at p1 to increase his price to p2, as this will lead to no loss of sale and increased revenue on each sale, a contradiction. So, all prices are advertised in the following interval. m ≥ p ≥ po + b = k.

(10.3)

Then, from equations (10.1, 10.2, 10.3), the density of advertisements λ(p) and sales, δ(p), is obtained: p ≤ po + b

0 δ(p) = 1/(p − po)

for

(10.4)

m≤p

0

p ≤ po + b

0 λ(p) = b/(p − po)

2

0

po + b ≤ p ≤ m

for

po + b ≤ p ≤ m

(10.5)

m≤p

Integrating (10.4), the total number of ads per person, β(m), is found to be equal to ᐉn((m − po)/b). Finally, a measure of the welfare gain, W, resulting from the existence of a market is defined as the value of the items bought minus the sum of the production and advertising costs. Since, when β advertisements per buyer are already sent at a price lower then m, the probability of an additional successful advertisement is π = exp(−β); thus the fraction of buyers served is (1 − exp(−β)). It then follows that: W = (1 − exp(−β)) (m − po) − bβ. The optimal amount of advertising, β*, is found by differentiating W with respect to β and setting the expression equal to zero, yielding β* = ᐉn((m − po)b) = β(m). The free market generates an optimal amount of advertising and the maximum possible welfare given the information technology. 10.6.1(e) The Reinganum equilibrium model In her simple equilibrium model of sequential search, Reinganum (1979) imposes two conditions that yield an equilibrium price distribution. The first is that consumers have identical elastic demand curves. The second is that the

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289

marginal cost curves of firms differ. Buyers know the price dispersion, but are ignorant of firm prices; the expected number visiting each store is the same. Sellers behave as monopolists, setting marginal cost equal to marginal revenue. The variation in costs gives rise to an equilibrium distribution with a point mass at the common reservation price. An odd consequence of this model is that there is no sequential search in equilibrium—searchers buy at the first store they visit.6 Positive profits also persist in equilibrium! Consumers follow a sequential search strategy. The most innovative feature of Reinganum’s model is her assumption that consumers have demand curves d(·): ⺢ + → ⺢ + with constant elasticity (ε < −1). The demand function is derived from an indirect utility function that is additive in income. Let the consumer be characterized by the reservation price R. Then the demand curve is given by d *(p),

p≤R

0,

p > R,

q(p) =

and the expected purchase is f (p)d(p) p≤R d F(p) = F(R) g

0

p>R

There are an uncountable number of identical consumers each with the same search cost and demand functions. The expected market demand at p is not considered explicitly, but apparently corresponds to MacMinn (1980b). Firms know the consumers reservation price R and the price dispersion. They maximize expected profits πF(p)

π Fp =

(p − k)d(p) 0,

1 , p≤R F(R) p > R,

where k denotes marginal production costs. There are uncountably many firms with random production costs k. The random variable k is distributed over [k,k¯ ] according to G, a continuous, differentiable, distribution function. Definition 10.1 An equilibrium is a reservation price R* and a dispersion FR* over [p*,p¯ *] such that 1

FR*(·) is continuously differentiable (a.e.) and has positive density over [p*,p¯ *].

290 2 3

The Economics of Search When consumers are faced with FR*(·), R* is the optimal reservation price. Given R*, the expected profit maximizing prices of the economy’s firms generate FR*(·)

The main result is contained in the following: Theorem 10.1

Suppose k¯ ≤ k

ε

冢1 + ε冣.

Then there is an equilibrium R*,

FR*(·) over [p*,p¯ *], where p* = k

ε

冢1 + ε冣, p¯ * = R*

and 1+ε , ε

冢冢

G p FR*(p) =

冣冣

p < p < p¯ * = R* .

1,

p = R*

10.6.1(f) MacMinn’s nonsequential search model In this equilibrium model MacMinn (1980) assumes that consumers know only the dispersion F of prices prevailing in the market. As usual, they are unable to assign prices to stores. The consumer pursues a fixed sample size search strategy. Thus he/she calculates n* the optimal number of stores to visit by minimizing the expected cost of buying one unit of the desired good, Min {E [min(p1, . . ., pn) − cn]} n

where c > 0 is the cost of visiting each store. The consumer buys at that store posting a price min(p1, . . ., pn*). MacMinn assumes that the set Γ of consumers is [0,1], so there is an uncountable number of them and they are identical, all facing the same cost of search c. Each has the same inelastic demand curve d(p) where 1 d(p) =

for p ∈ [0,L] ,

0

for p ∉ [0,L]

and L denotes the common limit price.

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The market expected demand function is derived from F, the price distribution F has support [p,p¯ ] and a continuous density f (p). The expected demand -∆n(p¯ ) at prices p ≤ p¯ for n consumers is given by -∆n(p¯ ) = Prob{min[p1, . . ., pn] ≤ p¯ } = 1 − (1 − Fn(p¯ ))n. Hence, the expected demand at price p, ∆n(p), is ∆n(p) =

-d∆n(p¯ ) = f(p) n(1 − F(p))n − 1. dp

Firms know ∆n(p) and F. Each firm sets its price p* to maximize expected profit πFn(p), πFn(p) = (p − k)δFn(p), where k is the constant per unit production cost and δFn(p) is the expected firm specific demand, defined by δFn(p) =

∆n(p) . fp

Intuitively, f (p) represents the number of firms setting price equal to p. The definition that each store has the same probability of being visited by a consumer. There are uncountably many firms and they are designated by the set S = [0,1]. The random production costs k have cumulative distribution function H(·) and density h(·) with support [k,k¯ ], where k¯ < L and

h(k) =

1 , k¯ − k

k ∈ [k,k¯ ]

0,

k ∉ [k,k¯ ].

Definition 10.2 A dispersion F* is called an equilibrium dispersion if and only if it generates an optimal sample size n* of consumers such that the firms’ profit maximizing behavior yields a price dispersion F*. MacMinn’s basic result is given in the following: 1 ¯ (k − k). Then there is a nondegenerate 12 equilibrium dispersion F* such that

Theorem 10.2

Suppose c <

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The Economics of Search

F *(p) =

p−p , p¯ − p

p ∈ [p,p¯ ]

1,

p > p¯

0,

p
n* + 1 k. n* Observe that c must be sufficiently small so that n* = 1 does not occur in which case the equilibrium dispersion collapses at p* = L. It should also be noted that F* is a linear transformation of H, where the transformation is defined by the mapping which associates to each firm its profit maximizing price. MacMinn also presents some interesting comparative statics. He shows ¯ that the mean of F*, µ* F increases if c decreases or if (k − k) decreases, whereas 2 the variance of F*, σ* F increases if c decreases or σH increases. Recall that 2 Stigler conjectured that σF would decrease if c decreased. Notice that decreasing c does imply increasing n*. But only the lower bound of the support of F* decreases, the upper bound remaining fixed. This induces a greater price variability. where p¯ = k¯ , and p =

1

,

冢n* k¯ + 冢

冣 冣

10.6.1(g) MacMinn’s sequential equilibrium model MacMinn’s (1980b) sequential search equilibrium model assumes that consumers know the price dispersion F, but not the prices at individual stores. They use the sequential search rule which is completely characterized by the reservation price R, the solution to R

c=

冮 (R − p)dF(p),

(10.6)

−∞

where c > 0 is the cost of visiting a store. The sequential rule is given by: buy at price p if and only if p ≤ R. There are uncountably many consumers contained in the set Γ = [0,1]. The sampling cost (cost of visiting a store and learning p) c > 0, is a uniform random variable on the interval [0,K]. Only one unit of the commodity is purchased when the consumer encounters a store selling at p ≤ R. The price dispersion F coupled with the distribution of c induces a reservation price distribution G. The market expected demand function ∆F (p) is given by ∆F (p) =

冮 f (R) dG(R)dR. f (p)

(10.7)

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The expected demand at prices p ≤ p¯ of a consumer with reservation price R is F(p¯ | p ≤ R). The expected demand at p of a consumer with reservation price R is

d Fg(p) =

dF(p| p ≤ R) = dp

f (p) , p≤R F(R) . 0,

p>0

“Adding up” all of these consumers who accept price p gives (10.7). Firms are assumed to know G and F. They choose that price p* which maximizes expected profit, πF (p) = (p − k) δF (p). Per unit production costs k are constant. The firm-specific demand δF (p) is defined by ∞

∆F(p) 1 δ (p) = = dG(R).7 f (p) F(R) p

冮

F

Firms never stock out—all demands are satisfied. There are uncountably many firms belonging to the set S = [0,1]. The production costs k are random variables with distribution function H(·). The support of H is [ k,k¯ ] and its continuous density is H′ = h(·). Definition 10.3 A dispersion F* is called an equilibrium dispersion if it generates a reservation price distribution G such that the firm’s profit maximizing prices are distributed according to F*, that is, f* is a “fixed point.” MacMinn’s major result is given in the following: Theorem 10.3 If k¯ ≤ 2K + µH, then there is an equilibrium dispersion F* characterized by the density function 2k(2p − 2K − µH)

p∈I ,

f*(p) = 0, 1 1 where I = K + µH + k, 2 2

冤

p ∉ I, K+

1 1 µH + k¯ . 2 2

冥

Remark 1 The assumption of the theorem guarantees a nonnegative profit to the highest cost firm. This is not a sufficient existence condition.

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Remark 2 F* is generated by a linear transformation of H. This transformation associates to each firm with production costs k a unique profit maximizing price. MacMinn also obtains the following comparative statics results. If no firm makes losses, i.e., K is sufficiently large, then µF* increases if either K or µH increases. Furthermore, σF*2 increases if σH2 increases (σ*F is independent of K). An increase in K can be interpreted as a decrease in search intensity. If K is allowed to vary arbitrarily, so firms may exit the industry, MacMinn shows that: if h is nonincreasing in p, then µF* and σF*2 increase with K. It should be noted that the assumptions concerning search and production cost distributions are less restrictive than in MacMinn’s nonsequential search model. This characteristic is shared by other sequential models when compared with their nonsequential counterparts. 10.6.1(h) Axell’s equilibrium models Axell (1977) developed one of the first sequential search equilibrium models. While his most recent paper does contain a dynamic adjustment process, the mathematical discussion is rather preliminary and incomplete. Therefore we will interpret his model as being “static,” that is, a stationary solution of the adjustment process. Axell’s assumptions about consumer behavior and their informational state are identical to MacMinn (1980b). He makes the following demand assumptions: the first is the MacMinn inelastic demand function; the second assumes that each consumer has the same demand function d(·): ⺢ + → ⺢ + . In this case the optimal rule is: “stop and buy d(p) units of the commodity from the first firm in the sample with p ≤ R.” There are uncountably many consumers and the heterogeneity among them is due to differing search costs c, a random variable with density γ(·): ⺢ + → ⺢ + . The density γ has the following properties: 1

γ ∈ C2 γ′ < 0, γ″ > 0 c → ∞ implies γ(c) → 0 c → 0 implies γ(c) → ∞.

2

γ*(c) ≡

3

lim γ(c) = −

(γ(c))3/2 is decreasing in c. γ′(c)

c→∞

4

√B . 2

lim γ(c) = 0. c↓0

The expected market demand at p is identical to MacMinn (1980) in the inelastic case.

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In the elastic case ∞

冮 F(R) dG(R). f (p)

∆ (p) = d(p) F

p

Firms are assumed to know G and F. They maximize expected profits. Axell’s definition of an equilibrium is the following. Definition 10.4 A price dispersion F* is an equilibrium dispersion if and only if it generates via consumer behavior a firm-specific demand such that profit maximization of the firms may reproduce the price dispersion F*. Axell’s main result is contained in the following. Theorem 10.4 Suppose consumers have inelastic demands. Then there is at least one equilibrium price dispersion. In particular, if B

γ(c) = c + 2

冢

冪

c2 c+ 4

, 2

冣

then

f *(p) =

2 , (p − k + 1)3 0,

p ∈ (k,∞) p≤k

.

Remark 1 Since firms are identical, the profit function must have a particular shape to induce an equilibrium dispersion. Axell shows that given γ(·), the expected profit function is constant and positive over the interval (k,∞). Thus, any subjective mapping between the set of firms and the interval (k,∞) generates an equilibrium price dispersion. Remark 2 Axell also gives the conditions for a degenerate price dispersion and shows that the degenerate price equals the monopoly price. Note that it is intuitively obvious that a constant profit function requires a specific demand curve generated by a very special search cost distribution. Axell’s equilibrium dispersion is one in which all prices from k to ∞ are charged. (Recall that this is a partial market.)

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10.6.1(i) The equilibrium search models of Pratt, Wise and Zeckhauser The Pratt et al. (1979) paper contains several models and a wealth of interesting examples. Here we concentrate on the model which seems to be the most original. Consumers are assumed to know the price dispersion or have, at least, a priori information about it. They engage in sequential search which is standard when F is known but quite arbitrary and probably not optimal when there is only a priori information about F. In any case the consumer’s decision is translated into a mathematical expression for the “probability of stopping at price p.” When he/she stops the consumer purchases one unit, that is, he/she has an inelastic demand curve. While not stated explicitly, the number of consumers is uncountable and the search cost is a random variable with finite support. The expected demand curve at p, ∆F (p) is assumed continuous in F, but an explicit expression is not stated. Firms know ∆F (·) and F. They are assumed to maximize expected profits F π (p), πF (p) = (p − k) δF (p), over the finite set {p1, . . ., pn}. The firm-specific demand δF (pi) is given by δF (pi) =

∆F(pi) , i = 1, . . ., n, f (pi)

and k denotes the unit production costs. Definition 10.5 Their definition of an equilibrium dispersion is identical to Axell’s. This definition is sensible only if F* generates a function πF* which has several maximizers. All possible equilibrium dispersions are discrete and are subsets of the (n − 1) dimensional unit simplex in Rn. Their major result is contained in the following: Theorem 10.5 Assume each πF (p) (p ∈ (p1, . . ., pn)) is a continuous function of ( f1(p), . . . fn(p)). Then there is an F* such that πF*(pi) = Max πF*j , if f *(pi) > 0. j ∈ {1, . . ., n}

The expected profit must be the same for each firm and no deviations are profitable. Notice that the proof uses a fixed point theorem, which yields considerable insight into the nature of the equilibrium dispersion problem.

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The authors also present an example (with three prices) where a nondegenerate equilibrium dispersion exists. 10.6.1(j) The equilibrium model of Carlson–McAfee In the Carlson–McAfee (1983) model, consumers know F, follow the standard sequential search rule, and have inelastic demands. Their definition of the reservation price Re is distinctive and is called the effective reservation price: Re = sup{p| p ∈ support (F*) and p ≤ R} where R is the “usual” reservation price. This definition implies that a discrete price dispersion gives rise to a discrete reservation price distribution, even when the search cost is continuously distributed. There are uncountably many consumers and consumers’ search costs are random and distributed uniformly G(x) = x/s over the closed interval [0,T]; the total number of buyers is T/s. Firms know the distribution of Re and F and maximize expected profits πF (p) = (p − c(δF (p))) δF (p) where c(q) = αq + βq2 and δF (p) is defined directly.8 There are n firms and their technology is explicitly specified cj(q) = αjq + βq2, α1 < . . . < αn, and β > 0. The equilibrium concept is identical to MacMinn’s (1980b). Their major result is given in the Theorem 10.6

冢

1 αn − n

Suppose

n

冱α 冣 < j

j=1

2n − 1 + γn 2β(n − 1) Γ, where γ = . n−1 sn2

Then there is a nondegenerate price dispersion F* with each p*j ∈ support (F*) given by (1 + γ) n−1 1 p*j = αj + T+ n−1 2n − 1 + γ n

冤

n

冢 冱α − α 冣冥. i

j

i=1

The proof of the theorem is based on the following reasoning. The profit maximizing conditions for the n firms comprise a system of n equations

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(linear in p*j ). As δF (p) can be expressed as explicit functions of the pj, the equation system can be solved (in p*j ) using Cramer’s rule. The important contribution of the Carlson–McAfee model is that it gives an explicit solution to the equilibrium dispersion problem and therefore comparative statics conclusions can be extracted. They show that 1 2 3 4 5

p*j increases as γ increases. σF* decreases as γ increases. Note that γ increases if β increases or s decreases. And s decreases if the search cost dispersion increases. 1 σF*2 = f (n)σ2α, where f (n) increases in n and lim f (n) = , coinciding with 4 n→∞ MacMinn’s result. A per unit tax is passed on in higher prices with σF* unchanged. Firms are unable to pass on all of a proportional tax and σF*2 increases.

10.6.1(k) The Burdett–Judd search equilibrium models In contrast with most of the previous models, the Burdett–Judd analysis does not rely on ex ante heterogeneity to produce equilibrium dispersion in both “noisy” (sequential) and nonsequential search models. Firms’ costs are identical, consumers are identical, that is, the cost of search is constant across consumers. Nor is the equilibrium dispersion supported by a flow of poorly informed “new” searchers into the market. What is crucial is the “ex post” variability in consumer information. In this way their models resemble the earlier work of Butters (1977) and Wilde (1977). The study concentrates exclusively on rational expectations equilibria, that is, in equilibrium consumers know F and in equilibrium firms know the search rules used by consumers. Burdett–Judd examine two search rules: the nonsequential fixed-sample size rule and a “noisy” sequential rule. By “noisy” they mean that two or more prices may be revealed when the searcher is looking for a single price. The possibility of dispersion arises because some consumers know more than one price. The firm’s information about consumer search consists of qk, the probability that the randomly arriving consumer knows k prices and R, the reservation price. The basic assumptions of both models are: 1

2 3

Consumers are identical, demand a single unit, know F, but do not know the price charged by any arbitrary firm. This information is obtained by search with c being the cost of visiting any store. Both the number of firms and consumers is given. Firms have identical constant average cost curves.

Definition 10.6 Given {< qn > n ∞= 1,p˜ }, a firm equilibrium is a pair (F(·),π) where F(·) is a distribution function and π is a scalar, such that

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(a) π = π(p), for all p in the support of F(·), and (b) π ≥ π(p), for all p. Condition (a) implies that all firms earn the same profit in equilibrium and condition (b) means that there is no incentive to deviate from this strategy. Nonsequential search Definition 10.7 The triple (F(·),π, < qn > n ∞= 1) is a market equilibrium with nonsequential search if and only if for fixed p* and cost of search c > 0, (a) (F(·),π) is a firm equilibrium given (< qn > n ∞= 1 p*) and (b) (< qn > n ∞= 1) is generated from the expected cost minimizing strategies of consumers given F(·). If F is concentrated at p*(R) it is called a monopoly (competitive) price equilibrium. If F is not concentrated at any price, it is called a dispersed price equilibrium. Noisy (sequential) search Definition 10.8 For any given < qn > n ∞= 1 and any cost of search c > 0, a market equilibrium with noisy search is a triple (F(·),π,p¯ ) where (a) (F(·),π) is a firm equilibrium given (< qn > n ∞= 1, p¯ ) and (b) p¯ is the effective reservation wage given F(·). The effective reservation price p¯ is the “obvious”9 generalization of R, when a cost c is incurred and a random number of prices is observed. The main results of the Burdett–Judd analysis are contained in three theorems, which we summarize in Table 10.2. Table 10.2 Number of different market equilibria, by search method Market equilibrium

Non-sequential

Noisy sequential

Monopoly price (price = reservation price) Competitive price (price = average cost)

Unique q1 = 1 Impossible If all firms charge same price, q1 = 1 → monopoly price 1 or 2 q1 > 0, q1 + q2 = 1 Number of equilibria depends on search costs

Unique q1 = 1 Unique q1 = 0

Dispersed price p ~ F(p)

Note: qk = probability a random consumer observes k prices.

Unique 0 < q1 < 1

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Observe that in the nondegenerate sequential search equilibrium consumers engage in no search, a result similar to that obtained by Reinganum (1979).

10.7 GENERAL STRUCTURE OF EQUILIBRIUM MODELS A general approach to equilibrium dispersion concentrates on the fixed point property of the equilibrium and prices when a nondegenerate equilibrium dispersion exists.10 Consumers differ only with respect to their search costs. Each consumer is thereby described by a search cost k > 0, and the corresponding reservation price z*, which is the solution to z*

冮 (z* − p)F(dp) = k.

−∞

The density of k is denoted by m(·). The firm sector is assumed to be finite rather than “atomless.” Firms are assumed to maximize expected profits. The expected profit function of the ith firm is given by: πi(p¯ (i),pi) = 冮 (p − ci)D(p¯ (i),εp,R)pi(dp). D

where R is the reservation price distribution, εp is the pure strategy of the ith firm concentrated at p and (p¯ (i),pi) is the activity vector representing the exact association between firms and their optimal strategies. Given p¯ (i), firm i chooses the strategy pi ∈ µ(D), the space of probability measures on the set of admissible prices D, to maximize πi(p¯ (i),·). The novel concept with respect to the firm is the definition of firm-specific expected demand. The expected demand d(p,z) of a consumer following an optimal stopping rule with reservation price z is simply the stopping probability at p, that is ∞

冱(1 − F (z))

n−1

p

d(p,z) =

∆p(p)

if p ≤ z

n=1

0

if

p>z

where Fp is the distribution function generated by the price dispersion, ∆p. Aggregating d(p,z) over z gives

Early responses to Diamond’s paradox and Rothschild’s complaint d(p) =

∞

∞

−∞

p

301

∆p(p)

冮 d(p,z)R(dz) = 冮 F (z) . p

To obtain the firm-specific demand from d(p), the effects caused by other competitors charging p must be eliminated. This is achieved by dividing d(p) by the normalizing factor ∆p(p), the probability of sampling price p given the price dispersion ∆p. Replacing R(dz) by its density, m(hF (·))Fp(·), the firmspecific demand curve is p

z¯ (p)

d(p) = D(p¯ ,εp,R) = ∆p(p) (i)

冮 m(h

(z))dz

Fp

p

where z¯ (p) is the maximum reservation price generated by the consumer sector given the price dispersion ∆p. The key assumption underlying the analysis is that each firm knows both the actions taken by the other firms and the “activity mapping” H, of the consumers sector. Each firm has the same probability of being contacted by a consumer. The firm sector is represented by an ordered N-tuple. C − (c1, . . ., cN) where ci is the constant marginal cost of the ith firm. Let (p¯ (i)) be the set of profit maximizing strategies for the ith firm given the actions of its competitors. To each N-tuple p¯ = (p1, . . ., pN) ∈ (µ(D))N, we can associate the N-tuple f¯ = ( f1(p¯ (1), . . ., fN (p¯ (N )) ∈ (µ(D))N. This association is denoted by H2 and is the “best reply” correspondence of the firm sector. Definition 10.9 A price dispersion ∆ ∈ µ(D) is called an equilibrium price dispersion if and only if it is a fixed point of the correspondence H2. Theorem 10.7 Every fixed-point of a correspondence H2 generates a nondegenerate price dispersion. This is the major result of Berninghaus (1984). The papers reviewed in the previous section are special cases of this existence proposition, with the models of MacMinn and Carlson–McAfee being the closest relatives. There will exist many open problems in the theory of equilibrium dispersions. Clearly a restrictive assumption is the inelastic consumer demand. In her stimulating article, Reinganum (1979) has constructed a

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nondegenerate equilibrium dispersion where the consumers’ sector consists of identical consumers each with an elastic demand function. However, the demand function is based on a specific utility function (over commodities). It is difficult to see how the results could be extended to a more general class of utility functions. Extension of the “classical” search paradigm to this general type of search problem is crucial. Without such an extension it is difficult to analyze the existence problem in a general equilibrium framework. In this framework one could dispense with the (“ad hoc”) concept of opportunity costs of information which could then be “internalized” in a multi-market search process. A first step in the direction of a multidimensional search process which surely must precede a general equilibrium theory has taken place recently by Burdett and Malueg (1981). Another promising method might be the “multiarmed bandit” theory as it has been developed by Gittins (1979), Whittle (1980). It also should be noted that the models concerning equilibrium dispersions are essentially static. Firms adjust to only the long run effects of the consumers’ search process. They are not allowed to change their strategies in response to realized consumers’ purchases. But if one considers a “time-path of searching” the reservation price distribution obviously changes in every search period. For “high cost” consumers will stop searching at first which results in a “shrinking” of the support of the reservation price distribution. It is easy to see that an extension of our static model to more “realistic” market adjustment processes will involve many mathematical and conceptual complications. A promising step has been taken by Mirman and Porter (1974) who analyzed a particular adjustment process for price dispersions. One could use the sequential character of the consumers’ search process to build “dynamic elements” into the static models. But it is not at all clear how to model firms’ behavior in a dynamic framework. In the Mirman–Porter model, for example, firms use an “ad hoc” price adaptation rule which can only be regarded as a first step in constructing a dynamic theory of price dispersion. Finally, the manner in which firms locate consumers and consumers locate firms should be modeled explicitly. In most of the models surveyed here, with Butters a notable exception, the consumer is the active searcher. But surely both parties to a potential match engage in search of some kind. This contact technology should also be explicitly modeled, as well as the bargaining process that occurs when potential partners do meet. Research on this topic is contained in the papers by Mortensen (1978, 1982), and Diamond and Maskin (1979). Some of these matching models are discussed in the next chapter.

11 Equilibrium search after the Diamond–Mortensen– Pissarides breakthrough

When we planned this project we decided to partition the voluminous equilibrium model literature into two parts. The first part would include those articles constructed before the matching innovation of Diamond, Mortensen, and Pissarides, and those designed after this innovation would comprise the second part. The first part was to be presented in Volume I, while the second part would be considered in Volume II. As time has passed, this division now seems inappropriate. Volume I includes an introduction and description of several key articles in the second part as well as those articles written before the matching breakthrough. Volume II will include a sample of the remaining matching literature together with more nonmatching research.

11.1 INTRODUCTION As frequently occurs in science, the idea of a matching equilibrium was sparked almost simultaneously in the minds of three prominent economists: Peter Diamond, Dale Mortensen, and Christopher Pissarides. These illuminations happened around 1980. In their excellent survey, Mortensen and Pissarides (1999a) note that the ensuing match equilibrium literature is composed of two distinct branches. The objective of the first branch is to provide an explanation of worker and job flows and unemployment in a setting where frictions accompany the stochastic process that matches jobs and workers. This matching approach studies the incentives to invest in search, recruitment, job learning, and all the forms of specific human capital which produce equilibrium levels of employment and unemployment. The second branch investigates the impact of market frictions on wage formation assuming that employers are able to post wages, whereas workers search for the best wage. The search friction is the time that workers consume in their search activity. These two branches have generated an enormous and sophisticated literature which we only skim in this chapter. The next section of this chapter describes two-sided search and wage determination as formulated by Mortensen and Pissarides. The following section exposes the matching technology which is basic to this new approach.

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As part of the exposition, we present an urn model that exemplifies the properties of this important function. Section 11.4 is a brief description of the search equilibrium. The properties of equilibrium unemployment are then presented for this version of the matching model. The best way to comprehend the Diamond–Mortensen–Pissarides (DMP) innovation is to see how it operates in a relatively simple setting. Hence, Section 11.5 shows a clever use of this innovation including some new ramifications. The model presented in this section has been devised by Burdett and Mortensen. Perhaps the most important feature of the matching equilibrium analysis is its welfare properties. Using a criterion developed by Hosios, Section 11.5 also describes the razor-like condition required for efficiency. The Diamond paradox (1971) is resolved by Burdett and Mortensen. In brief, buyers and sellers are identical, respectively, and have the same search cost. In this setting sellers charge the monopoly price. On the other hand, if buyers make take-it-or-leave-it offers and sellers engage in costly search for buyers, then the competitive price clears the market. This perplex has preoccupied search theorists for the past 35 years. One way of cutting the Gordian knot is to introduce heterogeneity. Butters did this in decisive fashion in his 1977 article. What is remarkable about Burdett and Mortensen is that the conundrum is solved when there are ex-ante identical buyers and identical sellers as postulated by Diamond. On-the-job-search is the key to the Burdett–Mortensen resolution. This model is outlined in Section 11.6. In the Burdett–Mortensen model wage-posting and the inability of workers and firms to bargain is the key to the equilibrium solution. Section 11.7 considers an alternative equilibrium search model of Postel-Vinay and Robin (2002) which instead allows firms to make counteroffers to employees who receive job offers from other firms. It will be surprising to some to see directed search model exhibit efficiency. This is Moen’s contribution and it is elaborated in Section 11.8. The following section presents the seminal island model by Lucas and Prescott. Here we have undirected search as in the basic sequential search model. This combination of undirected sequential search and the Lucas–Prescott model was instigated by Alvarez and Veracierto. It is their model which we describe in Section 11.9 together with its empirical implications. In Section 11.10 we review recent work by Ljungqvist and Sargent who have used search models to explain what may be termed as the European unemployment puzzle: with relatively stable social insurance programs over the period between the 1960s and the 1990s in both the U.S. and Europe, European unemployment was initially lower than in the U.S. but then became considerably higher after the late 1970s. Ljungqvist and Sargent show how changes in economic turbulence may be the key to solving this puzzle. The next three sections shift to a class of matching models originated by Gale and Shapley (1962). Brock has shown that this model and the illustrious human capital marriage model by Becker are similar. In Section 11.11, we review the two-sided search model with matchmakers by Bloch and Ryder

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305

(2000) that extends the work by Collins and McNamara (1990). Section 11.12 briefly reviews the work by Eeckhout (1999) which shows the link between these models and Gale and Shapely (1962) while Section 11.13 reviews the work by Chade (2001) on two-sided search with fixed search costs. Section 11.14 describes some important recent work by Rocheteau and Wright (2005) which in many ways unifies the previous research and describes a search-theoretic foundation for monetary theory, which was composed by Kiyotaki and Wright. The final section summarizes some recent research by Shimer (2005) and Hall (2005). As the recent survey by Rogerson et al. (2005) has pointed out, many of these models can be classified by the assumptions they make about how agents meet and how wages are determined.

11.2 TWO-SIDED SEARCH AND WAGE DETERMINATION: THE MORTENSEN–PISSARIDES APPROACH The key problem in labor and marriage markets is the formation of cooperating coalitions with two or more agents of different types. In the labor market, the cooperating entity is the worker–employer match. When an unemployed worker searches his expected return U is U=

b−a r+λ

+

λ 冮max{w,U}dF(w), r+λ

(11.1)

where r is the discount rate, a is the cost of search, b is flow of income given unemployment net of search costs, W is a random variable denoting per period return from working, F is the c.d.f. of W, and λ is the Poisson arrival rate of offers. When an employer has a job vacancy a similar problem arises. Let c be cost of recruitment and η the rate at which searching employees are encountered. The value of maintaining a job vacancy is the expected present value of future profit. It solves V=

−c η + 冮Max(V, j)dG(j) r+η r+η

(11.2)

where G is the c.d.f. of the r.v. J, the value of filling the vacancy. In the case of transferable utility (TU), the total value of the match to the partners is the sum of shares, W + J = Χ. In market equilibrium, the value of each share is determined by the wage. By

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rationality of the agents, the shares must exceed the option of continued search. Letting these values (U,V) comprise a threat point, a general solution to this problem gives a fraction β of the net surplus Χ − (U + V) to the worker and the remainder to the entrepreneur. W − U = β(X − U − V), β ∈ [0,1].

(11.3)

Mortensen and Pissarides note that a necessary and sufficient condition for formation of a match under TU and individual rationality is Χ − (U + V) ≥ 0.

11.3 MATCHING TECHNOLOGY A matching technology describes the relation between inputs, search and recruiting, and the output of the match process, the rate at which new matches are generated. The assumption that each searching worker meets prospective employers at rate λ implies that the expected aggregate rate of meetings between searching workers and vacancies equals λu, where u is the measure of unemployed workers. Similarly, for η the frequency at which each vacancy is met by searching workers, the aggregate rate at which vacancies meet applications is ηv where v is the measure of vacancies. These two flows are identical, and the identity requires that the arrival rates be functions of the participation measures, u and v. The matching function, m(v,u), which represents the aggregate meeting rate resolves the problem in that λu ≡ m(v,u) ≡ ηv

(11.4)

implies that λ = m(v,u)/u and η = m(v,u)/v are the average rates at which unemployed workers and vacancies meet partners. In the applied literature, a Cobb–Douglas matching function is usually assumed, i.e., m(v,u) = mo v1 − α uα, 0 < α < 1.

(11.5)

11.3.1 Important matching urn models In their scholarly survey of matching, Petrongolo and Pissarides (2001) note that the first matching function had its roots in a simple urn model.1 In this model urns are firms and workers are balls. An urn is unproductive until a ball enters. Clearly, with an equal number of urns and balls with random placement of balls in urns the number of matches (at least one ball in an urn is a match), will not equal the number of balls and urns. This inequality is a measure of coordination failure. In a labor market setting, if only one worker can be in each job, there will be “overcrowding” in some jobs and no entrants

The Diamond–Mortensen–Pissarides breakthrough 307 in the remaining jobs. Petrongolo and Pissarides label this “unemployment,” whose cause is the paucity of information regarding the actions of other workers. The authors present three versions of the process where U workers know the location of V vacancies and send one application to each. If a vacancy is the recipient of one or more applications, it selects one at random. The matching function for this process, given that each vacancy receives an application with probability 1/V and there are U applicants implying a probability of (1 − 1/V)U that a vacancy receives no applications, is given by 1

U

冤 冢V冣 冥.

M=V 1−

(11.6)

As V → ∞, M is approximated by M = V(1 − e−U/V).

(11.7)

Note that this matching function has constant returns to scale and also is increasing in each argument and concave. In the first version of (11.6) workers do not know which firms have vacancies and simply apply at a randomly chosen firm. The probability that a vacancy obtains no applications is given by 1

冢1 − (N + V) 冣, U

with N equal to the level of employment. If the labor force size is assumed equal to L, N = L − U, the matching function is approximated by

冤

M=V 1−e

−U

冢L − U + V冣冥.

(11.8)

This function has increasing returns to scale in U and V. The second version of (11.5) assumes that every worker is not acceptable for the available vacancies, but workers do not know which vacancies are acceptable. Letting K be the fraction of acceptable workers for a randomly chosen vacancy, the probability that a vacancy is not contacted by a worker remains equal to 1/V, but only KU workers can take the job. This gives the following generalization of the matching function M = V(1 − e−KU/V)

(11.9)

where K − 1 is an index of mismatch between available jobs and workers. The authors’ third version introduces a search intensity parameter. A

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fraction 1 − s of the unemployed do not look for work. This is a rotating fraction in that each worker misses an application out of every (1 − s)−1 rounds. The probability that a particular vacancy has no applicants during a given round of applications is (1 − 1/v)sU. This yields the following matching function M = V(1 − e−sU/V).

(11.10)

Both (11.8) and (11.9) satisfy the properties of (11.6) given K and s. They make it possible to model mismatch as well as the frequency of applications. This renders (11.10) susceptible to the available data for estimation and tests. An intensive and careful empirical study of matching is conducted by the authors. Two of their major conclusions with respect to future research are: Future work needs to elaborate a number of issues. The search for microfoundations needs to continue, and rigorous tests of plausible alternatives done. . . . Currently, the most popular functional form, Cobb–Douglas with constant returns to scale, is driven by its empirical success and lacks microfoundations. The most popular microeconomic models, such as the urn-ball game, do not perform as well empirically. Yet, different microeconomic matching mechanisms have different implications for wage determination and other types of behavior in markets with frictions, and can help in the design of optimal policy toward unemployment and inequality. On the empirical side, on-the-job search and search out of the labor force need to be more carefully measured and their implications for unemployed search and matching studied. The meaning of constant returns also needs to be studied further. Although constant returns in the numbers involved in matching are supported, there have been no rigorous tests of the plausible property that the quality of matches is better in larger markets, on the grounds that participants have more choices.2 These urn matching models are important for two reasons: (1) Butters uses urn models in his explanation of Diamond’s paradox; (2) As we will see in our companion volume, Rothschild’s Bayes’ adaptive search model is based on Polya’s urn model. Thus the urn models of Petrongolo and Pissarides in a matching environment suggests that they may contribute to a unified model of macroeconomics.

11.4 SEARCH EQUILIBRIUM The matching technology determines the arrival rates. Therefore equations (11.1), (11.2), (11.3), and (11.4) imply that the value of unemployed search U and job vacancy V satisfy

The Diamond–Mortensen–Pissarides breakthrough 309 b+ U=

m(v,u) β 冮 max(Χ,U + V)dF(Χ) u m(v,u) r+ u

(11.11)

and −c+ V=

m(v,u) v

(1 − β) 冮 max(Χ,U + V )dF(Χ ) r+

m(v,u) v

.

(11.12)

The matching function is increasing in u and v, so the unemployed workers and vacancies are complements in that an increase in measure of one increases the value of participation for the other. On the same side of the market, a congestion effect exists, i.e., a larger number of the same type reduces their own participation value.

11.5 EQUILIBRIUM UNEMPLOYMENT (MORTENSEN AND PISSARIDES, 1999a) Let there be identical workers and identical employers who are sure about the match product. The matching function m(v,u) is the rate that new jobs are generated with a fixed rate of job destruction. The matching function exhibits constant returns to scale (homogeneous of degree 1) and is increasing and concave. Jobs are assumed to have the same productivity p. A wage is established by employer–worker negotiation at their first meeting. They continue to produce until the match is destroyed by an idiosyncratic shock. When “destruction” occurs the firm exits the market and the worker enters unemployment from which job search is initiated. The constant rate δ is the arrival rate of shocks. The evolution of unemployment is given by u˙ = δ(1 − u) − m(v,u).

(11.13)

With constant returns the unique equilibrium solution for each vacancy rate v: u=

δ δ = . δ + m(v/u,1) δ + λ(θ)

(11.14)

The ratio of vacancies to unemployment θ = v/u is a measure of the “tightness”

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of the market. The re-employment hazard is given by λ(θ) = m(v/u,1). The market tightness gives the duration of unemployment. The equilibrium market tightness is a consequence of profit maximization given the bargained wage. Firms incur a flow cost c of recruitment for retaining a job vacancy. Searching workers arrive at rate m(v,u)/v, denoted by η(θ) =

m(θ,1) λ(θ) = . θ θ

(11.15)

Observe that λ(θ) is increasing and concave while η′(θ) < 0 and has elasticity 1 − θλ′(θ)/λ(θ) which is in the interval (0,1). When an unemployed worker meets a vacancy, bargaining over wage begins. The ensuing wage w divides the quasi-rents associated with a worker– vacancy match. For an arbitrary wage w the value of a filled job J is a solution to the asset pricing equation rJ = p − w − δJ,

(11.16)

with p denoting the output of the match. Since the value vanishes at destruction, equation (11.16) is the same as Bellman’s continuous time version: J=

1 1 + rdt

((p − w)dt + (1 − δdt)J ),

(11.17)

where δdt is the probability of destruction in a small time interval dt. Similarly, the value of the job to the worker is given by rW = w − δ(W − U ).

(11.18)

Note that in the worker’s case a destroyed job generates unemployed search with value U. The generalized Nash bargain yields a wage outcome solving w = argmax(W − U )(J − V )1 −  = argmax

w − rU



p − w − (r + δ)v r+δ

冢 r+δ 冣 冢

1−

冣

(11.19)

where V is the employer’s value of keeping a job vacant and U is the value of continued search. Maximization implies that  is the worker’s share of match surplus. That is, W − U = (W + J − U − V ).

(11.20)

The Diamond–Mortensen–Pissarides breakthrough 311 Substitution from (11.17) and (11.18) into (11.20) gives the wage equation w = rU + [p − rU − (r + δ)V ]

(11.21)

The value of unemployment U solves rU = b + λ(θ)(W − U ),

(11.22)

where b is value of leisure less the cost of search. Similarly, the value of a new vacancy V is the solution to rV = − c + η(θ)(J − V ),

(11.23)

where c represents the flow cost of recruitment. Profit maximization and free entry require all rents from a vacancy creation be zero. Thus, the job creation condition is given by V = 0.

(11.24)

Mortensen and Pissarides note that a steady state search equilibrium for this economy is a vector (u,w,θ,V,U) satisfying (11.14), (11.21), (11.22), (11.23) and (11.24). Substitution from (11.17) and (11.24) into (11.23) gives the alternative job creation condition c η(θ)

=

p−w . r+δ

(11.25)

Since (11.23), (11.22), (11.20), and (11.24) imply rV = − c +

(1 − )(rU − b) = 0, θ

substitution into (11.22) for rU gives the equilibrium wage equation w = (1 − )b + (p + cθ).

(11.26)

Mortensen and Pissarides conclude that the equilibrium is completely described by the wage and tightness pair (w,θ) which satisfy (11.25) and (11.26). These conditions have a single intersection which implies that the equilibrium is unique: p−b=

c[r + δ − θη(θ)] (1 − )η(θ)

.

(11.27)

Davis et al. (1996) have shown that an extension of the model regarding p

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as a stochastic process characterizing an aggregate shock is consistent with the time series characteristics of their job creation and destruction series.

11.5.1 Hosios (1990) analysis of welfare The social planner’s allocation problem, i.e., choosing the measure of vacancies vt and next period’s employment level nt + 1 is given by: ∞

max

冱 β [pn + b(1 − n ) − cv ], t

{vt,nt + 1} i = 0

t

t

(11.28)

t

subject to nt + 1 = (1 − δ)nt + η

vt

冢1 − n 冣 v , given n . t

o

t

Forming the LaGrangian and solving for y − z gives p−b=

c[ r + δ − θη(θ) ] (1 − )η(θ)

(11.29)

When the matching function has a Cobb–Douglas form such as (11.5), Hosios compared (11.29), the social optimum, with the private solution in (11.27) and showed that the decentralized matching equilibrium is efficient only if  = α. Hosios also showed that this efficiency condition is the general one for this matching environment. Hosios (1990: 280) observes that: “Though wages in matching-bargaining models are completely flexible, these wages have nevertheless been denuded of any allocative or signaling function: this is because matching takes place before bargaining and so search effectively precedes wage-setting.” Ljungqvist and Sargent (1995) note that in Davis’s (1995) matching model this Hosios problem is exacerbated further by the presence of heterogeneous workers. To avoid this additional problem Davis ingeniously partitioned jobs ex ante into separate submarkets. Different wages in the diverse submarkets would allocate workers across these submarkets, with the equilibrium wage in each submarket being set by bargaining after matching. Ljungqvist and Sargent verify that the social optimum, using Davis partitioning, is achievable when  is set equal to α. Ljungqvist and Sargent observe socially optimal wages ( = α) imply wage differentials for ex ante similar workers. This does not represent inefficiency, but rather reflects a situation where higher paid employees are such because their unemployment spells in their submarkets are on average lengthier.

The Diamond–Mortensen–Pissarides breakthrough 313 11.6 WAGE DIFFERENTIALS, EMPLOYER SIZE, AND UNEMPLOYMENT In an important paper, Burdett and Mortensen (1998) show that persistent wage differences are compatible with strategic wage formation in an environment with market frictions. This result obtains when either heterogeneity or homogeneity is observed among both workers and employers.3 The existence of wage differences among industries (Krueger and Summers 1988) and across firms (Abowd et al. 1999) has been shown by empirical research. These differences cannot be explained by a corresponding diversity in observed job attributes. The following basic question is raised by these empirical findings: can the phenomenon of workers of equal ability receiving different pay when they perform similar tasks be explained by standard competitive market theory? Applied economists suggest that there are unobservable aspects of employee’s ability and/or job requirements which when properly partitioned can resolve the conundrum. Economic theorists have designed a myriad of equilibrium sequential job search models attempting to show how different wages can exist as an equilibrium phenomenon. This theoretical work was stimulated by Diamond’s paradox and Rothschild’s critique of one-sided search models. Diamond (1971) shows in his model that all seller’s charge the monopoly price. Both buyers and sellers are identical with the former having the same search cost. The paradox arises when sellers incur the search cost and buyers are discovered to announce take-it-or-leave-it offers and the competitive price is market clearing. Rothschild (1973) asked: how could the selling of homogeneous goods at different prices be an equilibrium phenomenon? In terms of job search, Rothschild’s paradox is: how can the equilibrium wage distribution be nondegenerate when firms and job-searchers are identical with search costs positive? Over the past 30 years many ingenious models have been designed to resolve these perplexes. The Burdett–Mortensen model is one satisfactory resolution of this paradox. Suppose a continuum of employers and workers comprise a labor market. The measure of workers is m, while the measure of employers is uniform over (0,1). The workers are identical as are the employers. The distribution of wages is F. At random intervals the worker, who is either employed (state 1) or unemployed (state 0), is informed of a new job opportunity. The arrival process is Poisson with arrival rate λi, with i = 0 or 1. The arriving offer is a random draw from F. Workers respond immediately and there is no recall. In this model with on-the-job search, employed workers may move to other jobs and unemployed workers may move to employment. Workers may also be discharged or quit into unemployment. Job–worker matches are dissolved at rate δ. Unemployed workers receive a benefit flow of b per unit time. The discount rate is r. The expected discounted return V0 when a worker is currently unemployed solves the Bellman equation

314

The Economics of Search V0 = b/r +

λo [ 冮max{V0,V1(x)}dF(x) − V0]. r

(11.30)

The expected discounted return V1(w) when the worker is currently employed and earning w is given by: V1(w) = w + λ1[冮max{V1(w),V1(x)] − V1(w)]dF(x) + δ[V0 − V1(w)]. (11.31) Clearly, V1(w) increases in w and V0 is constant, so there is a reservation wage R such that V1(w) ≥ V0 iff w ≥ R. This inequality coupled with (11.30) and (11.31) gives ∞

R − b = [λ0 − λ1]

冮 (V (x) − V )dF(x) = [λ 1

0

R

∞

0

− λ1]

F¯¯(x)

冮 r + δ + λ F¯¯(x) dx. R

1

(11.32)

Let r be small compared to λ0 so that r/λ0 → 0. Then (11.32) can be rewritten ∞

R − b = [k0 − k1]

F¯¯(x)

冮 冤1 + k F¯¯(x)冥 dx R

(11.33)

1

where k0 = λ0/δ and k1 = λ1/δ. Let u be steady-state unemployment. In steady-state the flow of workers into employment, λ0(1 − F(R))u, equals the flow from employment to unemployment, δ(m − u). Therefore, u=

m . 1 + k0F¯¯(R)

(11.34)

The number of employed workers receiving a wage greater than or equal to w at t, G(w,t)(m − u(t)) can be obtained, where G(w,t) is the proportion of employed workers at t receiving a wage η greater than or equal to w, and u(t) is the measure of unemployed at t. Then dG(w,t)(m − u(t))/dt = λ0 max{F(w) − F(R),0}u(t) − [δ + λ1F¯¯ (w)]G(w,t)(m − u(t)).

(11.35)

The Diamond–Mortensen–Pissarides breakthrough 315 The first term on the right-hand side of (11.35) is the flow at t of unemployed workers to firms offering a wage between w and R; the second term is the flow to unemployment and to higher paying jobs, respectively. The unique steady-state distribution of wages for employed workers is G(w) =

[F(w) − F(R)]/F¯¯ (R) ∀ w ≥ R. 1 + k1F¯¯ (w)

(11.36)

The steady-state number of workers earning wages in [w − ε,w] is [G(w) − G(w − ε)](1 − u) and the measure of firms offering wages in the same interval is F(w) − F(w − ε). Hence, the measure of workers per firm earning w is given by G(w) − G(w − ε) (m − u). ε → 0 F(w) − F(w − ε)

ᐉ(w|R,F ) = lim

This can be rewritten ᐉ(w|R,F ) = mk0

[1 + k1F¯¯(R)/[1 + k0F¯¯(R)] [1 + k1F¯¯(w)][1 + k1F¯¯(w − )]

(11.37)

provided w ≥ R, where F(w) = F(w−) + V(w) given that V(w) is the mass concentrated at w. If R > w, ᐉ(w|R,F ) = 0. Note that ᐉ(·|R,F ) is increasing in w, continuous except at the mass point for F, and strictly increasing on the support of F while constant on any connected interval off the support. For firms, let p be the flow of revenue produced per worker, so the firm’s steady-state profit given w is (p − w) ᐉ(w|R,F ). Given R and F each firm announces a wage that maximizes steady-state profits, i.e., π = max (p − w) ᐉ(w|R,F ).

(11.38)

w

An equilibrium solution to the search and announcement game is described by (R,F,Π) such that R, the reservation wage of unemployed workers satisfies (11.33), Π satisfies (11.38), and F is given by (p − w) ᐉ(w|R,F ) = Π, ∀ w on support of F

(11.39)

(p − w) ᐉ(w|R,F ) ≤ Π, otherwise. To establish existence of a unique equilibrium, assume ∞ > p > b ≥ 0 and ∞ > ki > 0, i = 0,1. Let w and w ¯¯ be the inf and sup of an equilibrium F and suppose w ≥ R. Noncontinuous wage offer distributions are not permissible. Recall that ᐉ(·|·) is discontinuous at w = w ˆ iff w ˆ is a mass point of F and w ˆ ≥ R. This

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The Economics of Search

implies that a firm posting a wage slightly larger than w ˆ , where R ≤ w ˆ < p, has a much larger steady-state labor force and only a slightly smaller profit per worker than a firm offering w ˆ . Thus, any w slightly larger than w ˆ gives a larger profit. If a mass point of F were at w ˆ > p, all firms offering it earn nonpositive profit. Yet, any firm with w slightly below p makes a positive profit because it continues to attract a positive steady-state labor force. Thus w ˆ cannot maximize profits according to (11.39). This conclusion rules out a single market wage as an equilibrium possibility. Now, (11.37) implies that ᐉ(w|R,F ) = mk0/(1 + k0)(1 + k1)

(11.40)

independent of w provided w ≥ R. Hence, a firm maximizes profits iff w = R.

(11.40′)

Every equilibrium offer must produce identical steady-state profits given by Π = (p − R)mk0/(1 + k0)(1 + k1) = (p − w) ᐉ(w|R,F ), ∀ w in the support of F.

(11.41)

Now w = R, so equations (11.37) and (11.41) give F(w) =

冤

p−w

1 + k1 k1

1/2

冥 冤1 − 冢p − R冣 冥.

(11.42)

Substituting (11.42) into (11.33) gives R−b=

冤

k0 − k1 k1

冥 冤w¯¯ − R +

2(p − R) 1 + k1

p−w ¯¯

1/2

冢冢p − R冣

冣冥

−1 .

(11.43)

Observe that F(w ¯¯ ) = 1, so rearranging (11.42) yields p−w ¯¯ = (p − R)/(1 + k1)2

(11.44)

so that R=b+

(k0 − k1)k1

冤 (1 + k ) 冥(p − R) 2

1

or R=

(1 + k1)2b + (k0 − k1)k1p . (1 + k1)2 + (k0 − k1)k1

(11.45)

The Diamond–Mortensen–Pissarides breakthrough 317 Equations (11.43) and (11.44) imply that support of the only equilibrium possibility, [R,w ¯¯ ], is nondegenerate and lies below p. Hence, profits, π, on the support are strictly positive. Equations (11.40′), (11.42), (11.43), and (11.44) characterize the unique equilibrium. This follows by showing that no wage off the support of F yields higher profits. Profits less than those on the support attract no employees, yielding zero profits. An offer larger than w ¯¯ attracts no more workers than ᐉ(w ¯¯ |R,F ), and therefore yields a lower profit. Q.E.D. The competitive Bertrand solution and Diamond’s (1971) monopsony solution are limiting cases. As k1 → 0 (11.42) implies the highest wage converges to R. Further, (11.44) implies an unemployed worker’s reservation wage R converges to b. Thus, Diamond’s solution is this limiting case. The paradox is resolved by on-the-job search which allows workers to move to higher paying jobs and the fact that employers do not know whether a searcher comes from the pool of unemployed workers or from those searching on the job. Also, as frictions vanish, k1 → ∞, G, the equilibrium wage distribution, goes to a mass point at p. If k0 → ∞ as well, the steady-state unemployment rate goes to zero and Bertrand’s competitive solution emerges. Equilibrium with heterogeneous workers and equilibrium with job productivity differentials can be considered as problems. Note that Nash bargaining plays no crucial role in obtaining the equilibrium result: wages are posted, ex ante, and search is random with on-the-job-search being the key.4

11.7 EQUILIBRIUM SEARCH WITH OFFERS AND COUNTEROFFERS In this section of the chapter we discuss the recent equilibrium job search model of Postel-Vinay and Robin (2002). The main difference between this model and Burdett’s and Mortensen’s original model resides in the wagesetting mechanism. It is implicitly assumed in the Burdett and Mortensen (1989) model that firms have incomplete information on the reservation wages of job applicants. The optimal wage-setting mechanism in this case is a take-it or leave-it wage offer. In the Postel-Vinay and Robin model the alternative hypothesis is that firms compete in a complete-information environment. Firms, however, still make the first offer and thus retain some bargaining power; the firm will offer the minimum wage needed to attract a worker given the firm’s productivity type. Second, in the Burdett–Mortensen model incumbent firms are assumed to be entirely passive to outside wage offers made by competitors to the firm’s employees while in the Postel-Vinay and Robin model the incumbent firms are allowed to make counteroffers to a raiding firm’s wage offer. In response the raiding firm can make a new wage offer and this process continues in a Bertrand competitive fashion.

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The Economics of Search

Rewarding outside offers by matching may lead to incentives for on-the-job search, although in the Postel-Vinay and Robin model job offers arrive at an exogenous rate. Postel-Vinay and Robin note that wages may more generally include promotion possibilities and non-pecuniary aspects of the job. This may be important since workers of identical ability will be paid different wages and such perks could be easier to hide from other employees out of fairness concerns. To see the intuition of the Postel-Vinay and Robin model first consider the simple example. All workers have the same reservation wage when unemployed (i.e. the same opportunity costs of employment) and all firms have the same marginal productivity. In this circumstance, it can easily be shown that the equilibrium would consist of two mass points: one at the common reservation wage and the other at the common marginal productivity of the worker. The intuition is as follows: an unemployed worker will be offered their common reservation wage. This wage persists until a competing firm makes a wage offer. In the Bertrand competition that follows the worker will be paid the common marginal productivity.5 Additional heterogeneity is required in order to generate an equilibrium wage distribution that resembles actual labor market wage distributions. Postel-Vinay and Robin assume exogenous worker heterogeneity in the opportunity costs of working and heterogeneity in the employee marginal productivities across firms. Initially, Postel-Vinay and Robin analyze the case of an exogenous distribution of marginal productivities but then allow a firm to choose their capital investments making the marginal productivity of workers across firms endogenous. One implication of the Postel-Vinay and Robin model is that workers may actually accept lower initial wages at firms where their productivity is higher. The intuition here is that subsequent wage offers would yield large counteroffers (and hence, assuming the worker remains at the firm) larger future wage growth since a firm will never pay an employee more than their marginal productivity. Next we will describe in more detail the Postel-Vinay and Robin (PVR) model. For simplicity, we focus on the case where a firm’s productivity is exogenously given. Denote the number of workers in the economy by M and the number of firms by N. For a given firm the marginal productivity of its employees equals p. Unemployed workers are assumed to be contacted by a firm at the exogenous rate λ0 while employed workers are assumed to be contacted by a “raiding” firm at the exogenous rate λ1. Also, for each firm there is an exogenous layoff rate given by δ. A worker’s opportunity cost of work (value of leisure) is denoted by b. Denote by w the wage that a worker receives at a firm. Workers are assumed to die at the rate µ but are immediately replaced by a new worker with the same value of leisure. These new entrants to the labor market must subsequently seek employment. The aggregate unemployment rate in this economy will be represented by u.

The Diamond–Mortensen–Pissarides breakthrough 319 Assumption 11.1 PVR assume that firms have perfect information about the characteristics (opportunity costs of work) of employees. Employer “knows” each workers b. The implication of this is 1 2

Firms vary their wage according to the characteristics of a particular worker instead of making the same offer to each. Incumbent employers counter the offers that employees receive from raiding firms instead of being totally passive.

Denote the marginal productivity of the raiding employer by p′ and the productivity of the incumbent employer by p. Then if p′ > p the raiding employer is ultimately successful at luring away the employee and pays the employee (in the limit) p. If p′ < p then the incumbent employer manages to retain the employee. However if w < p < p′ the worker will get a raise. Assumption 11.2 Firms have heterogeneous marginal productivities which are distributed over [ p,p¯ ] according to the continuous cumulative density Γ -where Γ ≡ 1 − Γ. Assumption 11.3 Workers are heterogeneous with respect to their value of leisure b > 0 which is distributed across workers according to the cumulative density H0 on the interval [b,b¯ ]. Assumption 11.4 Workers are risk neutral and maximize the present discounted sum of expected future income flows. Let V0(b) represent the maximum lifetime discounted expected utility of an unemployed worker with value of leisure equal to b and let V(p, w, b) be the maximum lifetime discounted expected utility of an employed worker with value of leisure equal to b who works in a firm with marginal productivity p and receives wage w. Define ξ0(b,p) to be the minimum wage that makes a worker indifferent between working and not working (i.e., the individual’s reservation wage). Then V[b,ξ0(b,p),p] = V0(b).

(11.46)

The innovation in the Postel-Vinay and Robin model is the inclusion of the firm’s productivity. Since more productive firms are more attractive to workers, the minimum wage at which an unemployed worker would be willing to work at the firm is decreasing in p. The best offer a firm can make, and still break even is p. Thus, an employee will move to a raiding firm if p′ > p. Defining ξ(p,p′) to be the minimal acceptable wage that induces an employee to switch employers, we have: V[b,ξ(p,p′),p′] = V0(b,p,p)

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The Economics of Search

Any offer less than ξ(p,p′) would be successfully countered by the incumbent firm. Now the Bellman equation for V0 satisfies -(ρ + µ + λ0)V0(b) = b + λ0Γ(b) × Ep{V[b,ξ0(b,p),p]|p > b} + λ0Γ(b)V0(b). (11.47) Substituting (11.46) into (11.47) and solving yields V0(b) =

b . ρ+µ

(11.48)

If an employed worker earning w receives an outside offer, three things can occur. If ξ(p,p′) < w than the worker stays with the incumbent employer and does not receive a raise; if w < ξ(p,p′) < p then the worker remains with the incumbent employer but receives a raise; if p < p′ then the worker switches employers and receives ξ(p′,p) from the new employer. Define the threshold productivity by q(p,w) such that ξ[q(w,p),p] = w. It is clear that if p′ < q(p,w) then the employee remains with the current employer and receives no wage increase. The Bellman equation of an employee, V(b,w,p), satisfies -(ρ + δ + µ + λ1Γ [q(w,p)])V(b,w,p) = w + λ1[Γ(p) − Γ(q(w,p))]Ep′{V(b,p′,p′)|q(w,p < p′ ≤ p} -+ λ1Γ(p)V(b,p,p) + δV0(b)

(11.49)

where the second term on the right-hand side of (11.49) indicates the probability of getting an offer that results in the worker staying at the incumbent firm but receiving a raise times the expected value of such an offer while the third term represents the probability of the worker receiving an offer that leads to a switch times the value of such an offer. Using the fact that V(b,w,p) = V(b,w,q(w,p)), the threshold productivity q(w,p) satisfies

q(w,p) = w +

λ1

p

冮 Γ--(x)dx.

ρ + δ + µq(w,p)

(11.50)

Also, λ1 ξ(p,p′) = p + ρ+δ+µ

p′

冮 Γ--(x)dx.

(11.51)

p

From (11.50) and (11.51) it can be shown that equation (11.49) is equivalent to

The Diamond–Mortensen–Pissarides breakthrough 321 -(ρ + δ + µ + λ1Γ [q(w,p)])V(b,w,p) p

x + δV0(b) -- p + δV0(b) = w + λ1 d1Γ(x) + λ1Γ(p) + δV0(b). ρ+δ+µ ρ+δ+µ q(w,p)

冮

(11.52)

Finally, it can be shown that the reservation of an unemployed worker receiving an offer from a firm of productivity p, ξ0(b,p), satisfies λ1 ξ0(b,p) = b − ρ+δ+µ

p

冮 Γ--(x)dx

(11.53)

b

Since p is the upper limit that a worker can make at a firm, and larger future salary increases are more desirable, ξ0(b,p) is a decreasing function of p. In order to examine the equilibrium for this model we must first examine the flows into and out of states (w,p). Let L(w|p) be defined as the number of employees paid a wage less than or equal to w in a firm of type p where p ≥ w . Since the highest paid worker in a firm earns p, the level of total employment in a firm of productivity p is denoted L(p) = L(p|p). Thus, L(w|p) × NdΓ(p) workers are paid less than w by type p firms and the flow into unemployment at rate δ or die at rate µ . Employees of this type either get a raise or leave their -employer at rate λ1Γ(q(w,p)). Turning to the inflow side, consider new employees of firms with productivity p who are paid w. These employees are either hired from a firm with productivity less than q(w,p) or they come from the pool of unemployed workers. In the former case, the number of workers hired equals q(w,p)

-λ1 × dΓ(p) ×

冮 L(x)Nd Γ(x).

(11.54)

p ¯

Unemployed workers for less than w are those whose value of leisure, b, is such that ξ0(b,p) ≤ w which occurs if and only if b ≤ q(w,p). If we denote the distribution of b among the unemployed as H, then imposing the stationarity condition that inflows equals outflows of L(w|p) implies -(δ + µ + λ1Γ [q(w,p)])L(w|p)N =

λ0uMH[q(w,p)] q(w,p)  λ0uMH[q(w,p)] + λ1N 冮 L(x)d Γ(x)  p

if w ≤ ξ(p,p) if w > ξ(p,p)

(11.55)

¯

Note that the stock of workers at firms of productivity p or lower equals

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p

N

冮 L(x)d Γ(x). This stock can only be replenished from the unemployed since p

no¯ employee from a firm with p′ > p can be lured away. The outflow of -workers from this firm is equal to δ + µ + λ1Γ(p). The flow of workers into p

firms of productivities less than p is given by λ0uM

冮 H(x)dΓ(x). Equating

p ¯

inflows with outflows then gives p

λ0uM

p

N

冮 L(x)d Γ(x) =

p ¯

冮 H(x)dΓ(x) p

¯ . -δ + µ + λ1Γ(p)

(11.56)

Implicitly differentiating (11.56), integrating by parts and rearranging then yields p

λ0uM × L(p) = N

冮 [δ + µ + λ Γ--(x)]dH(x) 1

b ¯

-[δ + µ + λ1Γ(p)]

.

(11.57)

In steady state, the inflow and outflow into unemployment must be equal. This implies u=

δ+µ b

δ + µ + λ0

(11.58)

冮 Γ--(b)dH(b) b ¯

The two endogenous components on the right side of (11.57) are u and H, while equation (11.58) shows the relationship between these two components. To determine how H and H0 are related, let He denote the distribution of b among employed workers, then a stationary equilibrium implies the following two flow-balance conditions: -(µ + δ)(1 − u)dHe(b) = λ0Γ(b)udH(b)

(11.59)

-[µ + λ0Γ(b)]udH(b) = δ(1 − u)dHe(b) + µdH0(b).

(11.60)

and

Eliminating dHe(b) from the two equations produces

The Diamond–Mortensen–Pissarides breakthrough 323 -udH(b)[µ + δ + λ0Γ(b)] = (µ + δ)dH0(b) (11.61) or -[µ + δ + λ0Γ(b)]dH(b) b

= dH0(b).

(11.62)

-δ + µ + λ0 Γ(b)dH(b)

冮 b ¯

Equations (11.58) and (11.62) show how u and H depend on the exogenous components of the model. Turning to the equilibrium wage distribution, it is easiest to determine the number of workers in a firm earning less than w and then sum across the particular type of firm (characterized by p). Moreover, it is necessary to partition the support of the productivity distribution Γ. Case 1 ξ0(b,p¯ ) ≤ w ≤ ξ0(b,p). In this situation wages are so low that the worker with the lowest value of leisure is not attracted to the least productive firm. Define s0(w,b) by the equality ξ0(b,s0(w,b)) = w, only firms with p > s0(w,b) can hire workers for less than w. With ξ0(b,p¯ ) ≤ w ≤ ξ0(b,p), then s0(w,b) > p so that not all firms will be able to pay employees less than w. Thus, G(w) is given over that range by p¯

p¯

(1 − u)M G(w) = L(w|p)dΓ(p) = L(q(w,p))dΓ(p) N s (w,b) s (w,b)

冮

0

冮

0

¯

¯

Case 2 ξ0(b,p) ≤ w ≤ p. Here, all firms are productive enough to employ at least some workers by an offer of w. In this case G(w) is simply given by p¯

p¯

冮

冮

¯

¯

(1 − u)M G(w) = L(w|p)dΓ(p) = L(q(w,p))dΓ(p). N p p Case 3 w > p. All firms have employees paid w or less but only firms with productivities higher than w have workers paid more than w. Here, G(w) satisfies w

p¯

冮

冮

(1 − u)M G(w) = L(p)dΓ(p) + L(w|p)dΓ(p) N p w w

¯ ¯p

冮

冮

= L(p)dΓ(p) + L(q(w,p))dΓ(p). p ¯

w

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The implications of the Postel-Vinay and Robin model is that equilibrium firms with higher productivities have more employees and that the equilibrium wage distribution can have thin tails. One undesirable feature of the Burdett–Mortensen “wage-posting” model is that, in equilibrium, the wage distribution has an upward sloping density which is at odds with the empirical data. Postel-Vinay and Robin relax the assumption of an exogenous productivity distribution by allowing firms to choose capital levels and solve for the explicit equilibrium in simulations illustrating that these properties are similar when allowing productivity to be endogenously determined.

11.8 COMPETITIVE SEARCH EQUILIBRIUM Moen (1997) challenges the widely held view that the employment market does not produce a socially efficient unemployment rate because search externalities are not reflected in the wage rate (Hosios 1990; Mortensen 1982; Pissarides 1990). Moen introduces a competitive search equilibrium and shows that the corresponding equilibrium allocation is socially efficient. He also shows how frequently observed market phenomena lead to the existence of the competitive search equilibrium. Moen builds an equilibrium for markets with functions that captures properties usually associated with a competitive equilibrium. A basic assumption is that a market-maker or broker can partition the market into submarkets. Each submarket contains a subset of unemployed workers and firms with vacancies. Workers and firms search for one another within each submarket. Matching technology is identical across submarkets and the arrival rate of job offers to workers (applicants for vacancies) thus depends positively (negatively) on the ratio of searching workers to vacancies. The broker also sets wages. In each submarket, all jobs pay the same wage, but wages differ across submarkets. Entry into submarkets is freely chosen by unemployed workers and employers with vacancies. Moen assumes identical workers, so all submarkets must give unemployed workers the same expected utility. The ratio of searching workers to vacancies (labor market tightness) is high in markets with low wages and vice versa. A firm’s entry decision to a particular submarket is based on a tradeoff between wage costs and search costs. Firms are heterogeneous. High-productivity firms generally enter submarkets with higher wages than do low-productivity firms. The market-maker establishes wages so that it is not possible to create more submarkets. Moen finds that the marginal rate of substitution between market tightness and wages is identical for workers and firms and implicitly determines a market price of search time. Agents maximize income given this price, so social and private returns from entry are equalized. The equi-allocation of resources is socially efficient with respect to both the

The Diamond–Mortensen–Pissarides breakthrough 325 distribution of searching agents on submarkets and the number of agents entering the market. The basic assumptions of the Moen model are that workers have information about wages before they search or at an early stage in the search process. Firms with vacancies publicly announce wage offers. Workers only apply for a subset of jobs. In equilibrium, they are indifferent among subsets. Workers who apply for wages announced by firms form a submarket, while firms choose a wage to maximize expected profits and the arrival rate of workers. 11.9 LABOR MARKET POLICIES IN AN EQUILIBRIUM SEARCH MODEL (Lucas–Prescott in action)

11.9.1 Introduction Alvarez and Veracierto (1999) modify the Lucas–Prescott island model and study how differences in employment and unemployment among economies, namely, the U.S. and Europe, are explicable by differences in labor market policies. This paper is a good vehicle for expositing the seminal equilibrium model by Lucas and Prescott (1974) and testing its policy implications with real data. Their general findings are: (1) minimum wages have almost negligible effects; (2) firing taxes have effects comparable to those discovered in frictionless general equilibrium models; (3) unions have substantial negative impact on employment, unemployment, and welfare; and (4) as unemployment benefits increase there is a significant decline in welfare and unemployment rises markedly.

11.9.2 The model The model incorporates the basic sequential search model in a general equilibrium production economy thereby altering the Lucas–Prescott model to accommodate undirected search and nonparticipation in the labor force. Production occurs in a large number of distinct places (called islands). Labor is the only input to the production function which has decreasing returns to scale. There are fixed numbers of firms living on each island and all are receiving the same sequence of productivity shocks. A Markov process is the source of the shocks, which are independent and identically distributed (IID) across islands. At the start of each period, the employees are allocated among islands according to a given distribution. Given a shock, agents may leave their island and enter unemployment or they can remain in their initial jobs. Unemployed workers decide whether to search or conduct home production. If search is chosen, a worker is randomly assigned to another island, where it takes one period to complete the assignment. Alvarez and Veracierto note that this is the sense in which search is undirected. Within each island labor markets are competitive. That is, both firms and

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employees take the market for spot wages as given. Workers have access to a complete set of state contingent securities, which are indexed by the shocks to each island. Given this market structure, workers and firms maximize expected discounted earnings. The model does not consider insurance aspects of labor market policies.6 Alvarez and Veracierto note that their model is general equilibrium for the following reasons: 1 2 3 4

Wages are market clearing on each island. Cross-sectional distribution of wages and employment is endogenous. The endogenous distribution of search among islands is compatible with the incentive to search. Aggregate employment is consistent with the number of searchers and the total labor supply.

Their model operates on three margins: (1) the employment decision of firms yielding firm dynamics; (2) home market production decisions yielding an analysis of labor force participation; and (3) the search decisions of firms, which permits the analysis of unemployment.

11.9.3 Basics The economy consists of a measure one of ex-ante identical agents with preferences ∞

E

c1t − γ + ht , 1−γ

冱 冤冢 βt

t=0

冣

冥

where ct is consumption of market goods, ht is consumption of home goods, γ ≥ 0 and 0 < β < 1. There is a continuum of islands. Each has the following production technology yt = F(zt,gt) ≡ ztgαt , where yt is output, gt is labor input, zt is an idiosyncratic shock and 0 < α < 1. The production shock zt follows an AR(1) process ln zt + 1 = a + ρ ln zt + εt + 1 where εt + 1 ~ N(0,σ2), and 0 < ρ < 1. Realizations of zt are independent over islands. The corresponding transition function for zt is denoted by Qt. The marginal product of labor is f(gt,zt) = ∂F(zt,gt)/∂gt. If an agent engages in nonmarket (home) activity, he receives wh units of the home good for every period of time invested in home production.

The Diamond–Mortensen–Pissarides breakthrough

327

An island cannot employ more than the total number of workers xt on the island at the beginning of the period. Those who remain on the island produce market goods and begin the next period in the same location. Those who leave the island become unemployed. The unemployed have two options: initiate home production in the current period or remain unemployed in the following period. If the agent chooses the search option he/she accrues zero home production during the current period and is randomly assigned to an island at the beginning of the next period. Agents have no control over their island assignments. Search is undirected and searchers arrive uniformly over all islands. Agents engaged in home production are said to be “out of the labor force.” Agents searching are “unemployed,” while those working in the island sector are “employed.” Alvarez and Veracierto show that stationary allocations are feasible. Feasibility requires that the islands level of employment, g(x,z) is less than the number of workers initially available. g(x,z) ≤ x. The number of agents at the beginning of the following period, x′, is given by x′ = U + g(x,z), where U is total unemployment in the economy. The law of motion for x and the Markov process for z produce an invariant distribution µ µ(Χ ′, Z ′) =

冮

Q(z, Z ′)µ(dx × dz)

{(x,z):g(x,z) + U ∈ Χ ′}

for all Χ′ and Z′. Aggregate employment N is given by N = 冮g(x,z)µ(dx × dz) and aggregate consumption is c = 冮F(g(x,z),z)µ(dx × dz). These aggregates are obtained by summing the corresponding magnitudes across all islands. The number of agents remaining out of the labor force must be nonnegative. 1 − U − V ≥ 0.

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11.9.4 Laissez-faire competitive equilibrium Alvarez and Veracierto assume that their island model has a competitive equilibrium with complete markets. As a first pass, the market good and home good are perfect substitutes, that is, γ = 0. In these circumstances, each agent maximizes expected discounted value of wages and home production. Each island has competitive spot labor markets so that the marginal products of labor equals wages. Suppose an agent is located on an island of type (x,z) and decides whether to stay or leave given the island employment level g(x,z) and aggregate unemployment. A stay decision earns the competitive wage f(g(x,z),z) and the agent begins next period on the same island. If the leave decision is made, the worker transits to unemployment and receives a value θ (see below). The agent’s problem is characterized by the following Bellman equation: v(x,z) = max{θ, f(g(x,z),z) + β冮v(g(x,z) + U,z′)Q(z,dz′)},

(11.63)

where v(x,z) is the expected discounted value of starting period on a type (x,z) island. In equilibrium, the employment rule g(x,z) must be compatible with individual decision-making. That is, (a) If v(x,z) > θ (a stay is superior to a leave) g(x,z) = x.

(11.64)

(b) If v(x,z) = θ (the indifference point between staying and leaving) g(x,z) = g¯ (z)

(11.65)

where g¯ (z) satisfies θ = f(g¯ (z),z) + β冮v(g¯ (z) + U(z′)Q(z,dz′)).

(11.66)

The unemployed agent confronts the following decision. Engage in home production or initiate job search. If he chooses the former, he receives wh of home goods during the current period and remains unemployed in the next period. If search is chosen no home product is received during the current period and a new draw is made from µ, the invariant distribution of islands, at the start of the next period. Formally this problem is encapsulated in the Bellman equation θ = max{wh + βθ,β冮v(x,z)µ(dx,dz)}.

(11.67)

If the first term in the braces on the right-hand side is less than the second term, no one engages in home production and employment feasibility becomes:

The Diamond–Mortensen–Pissarides breakthrough 329 U + 冮g(x,z)µ(dx × dz) = 1.

(11.68)

If the two terms in the parentheses are equal, this indifference means that some agents may remain at home and employment feasibility is now: U + 冮g(x,z)µ(dx × dz) ≤ 1.

(11.69)

Finally, if the first term exceeds the second, then U = 0, which is inconsistent with equilibrium. Thus, θ = β冮v(x,z)µ(dx,dz).

(11.70)

The authors show (in Alvarez and Veracierto 1999) that despite search frictions, the economy is such that the welfare theorems obtain: laissez-faire competitive allocations are coincident with the stationary solutions of the Pareto problem. Furthermore, they prove the existence and uniqueness of a stationary competitive economy. The proof is constructive in that it provides an algorithm for calculating the unique steady state equilibrium. If γ > 0 market and home goods are imperfect substitutes. Hopenhayn and Rogerson (1993) make this assumption in their study of firing taxes. They assume that agents can diversify the income risk of search and employment through employment lotteries and financial markets.7 They show how this can be accomplished, but in their policy simulations they assume γ = 0.

11.9.5 Labor market policies Alvarez and Veracierto analyze the following labor market policies: minimum wages, unions, firing taxes, and unemployment insurance. We focus only on the empirically relevant findings that follow their formal development and give only brief summaries of these findings. Minimum wages Alvarez and Veracierto discovered extremely small welfare effects of minimum wages. For example, when the minimum wages are 90 percent of average wages, the welfare cost is only approximately 0.2 percent in terms of consumption. These results are invariant to the application of an insideroutsider model. These results are similar to the in-depth study of minimum wages by Card and Krueger (1995). Unions The authors consider two different union models. The following story is a good metaphor of them. Suppose an economy is composed of a large number of piers, where ships unload, and the arrival of ships is random. Workers are

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allocated across piers and it takes one period to move from one pier to another. There is a gate at each pier where workers are hired in a competitive spot market. In the coalition model the gate is controlled by all workers present at the beginning of the period. In the union-boss model the gate is controlled by the union boss. The coalition model The authors find that the effects of unions on unemployment is quite large. For example, when 60 percent of the islands are unionized the unemployment rate increases from 5.3 percent (the rate when there are no unions) to 12.5 percent. They also discover a large welfare cost associated with unions. For example, when γ = 1 and 60 percent of the islands are unionized, the welfare loss is 3.5 percent of consumption. The union boss The union boss retains all monopolistic rents, with workers being paid their opportunity cost. Thus, average wages fall as the union sector grows. With lower average wages, both union boss and firms hire more workers— unemployment rates decrease in each sector. Due to the composition effect which does not dominate, unemployment rates fall so rapidly in each sector as unionization increases that economy-wide unemployment declines. For example as the fraction of unionized islands rises to 60 percent, the unemployment rate drops from 5.3 percent to 3.5 percent. Alvarez and Veracierto observe that in the coalition model, union members receive higher wages than workers in the competitive sector. The reverse is true for the union boss model. The evidence by Card (1996)8 revealed a union wage premium of 15 percent for the U.S. economy. The sign of the premium supports the coalition model. To obtain a wage premium as large as that of Cards, 20 percent of the islands must be unionized yielding a 12.5 percent premium. Firing taxes The effects of firing taxes are between three and twelve months of average wages. Alvarez and Veracierto observe that firing taxes induces two changes in firm behavior: (1) firms are now less willing to fire workers and (2) they are less willing to hire workers (to avoid future taxes). These two effects reduce the incidence of unemployment and increase its average duration, respectively. Thus, the unemployment rate may increase or decrease depending on the strength of these two effects. When home and market goods are perfect substitutes (γ = 0), the number of searchers drops 40 percent. Employment decreases by 11.9 percent, the fall in labor force participation increases home output by 47.3 percent and market output declines by 12 percent.

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Unemployment insurance As UI benefits increase, workers are more amenable to changing islands after a negative shock. This increases the incidence of unemployment. On the other hand, the average duration of unemployment is impacted by two forces. Because they become eligible for UI benefits, employment rises and average duration of unemployment declines. Second, agents search until they are indifferent between working in the market and remaining at home, which increases because UI has increased. This causes the average duration of unemployment to increase. Alvarez and Veracierto find that the general equilibrium effect dominates in that larger UI benefits cause average duration of unemployment to increase. A present value of UI benefits equal to one model period of wages increases the unemployment rate to 11.9 percent from 5.3 percent. The welfare costs of introducing UI are substantial. For a present value of UI benefits equal to one model period of wages, welfare drops by 2.5 percent in terms of consumption under γ = 1. There have been many studies of these four categories by other researchers. Alvarez and Veracierto find that their results are roughly compatible with recent research. It is natural to ask whether the policy differences are significantly different for a search model like Alvarez and Veracierto and a matching model with Nash bargaining like Mortensen and Pissarides (1999b). We turn to this next. 11.10 THE SEARCH CONTRIBUTIONS OF LARS LJUNGQVIST AND TOM SARGENT Over the past twenty-five years, Tom Sargent has been a prolific composer of search articles. While his emphasis has been on recursive macroeconomic models, his creative compositions include many original contributions to search theory. Recently, he has collaborated with Lars Ljungqvist and they have written a series of very important search articles.9 They also have a survey on search in their textbooks on recursive macroeconomics.10 These surveys and articles exemplify a startling clarity that is sometimes absent in the search literature. In this section we review some of their work that focuses on explaining the European unemployment experience.

Lungqvist and Sargent (2005c) An abstract of Ljungqvist and Sargent (2005c:1) states: Before the 1970s, similarly short durations but lower flows into unemployment meant that Europe had lower unemployment rates than the United States. But since 1980, higher durations have kept unemployment rates in Europe persistently higher than in the U.S. A general equilibrium search model with human capital explains how these outcomes arise from the way Europe’s higher firing costs and more generous

332

The Economics of Search unemployment compensation make its unemployment rate depend on a parameter that determines a worker’s loss of valuable skills after an involuntary job loss. An increase in that skill loss parameter after the 1970s explains microeconomic findings that indicate that workers experienced more earnings volatility then. Our model also explains why, especially among older workers, hazard rates of gaining employment in Europe fall sharply with increases in the duration of unemployment, and why displaced workers in Europe experience smaller earnings losses and lower re-employment rates than those in the United States. The effects of layoff costs on unemployment rates depend on equilibrium proportions of frictional and structural unemployment that in turn depend on the generosity of unemployment benefits and the skill loss parameter that confronts displaced workers.

Many of us believe that the high unemployment currently afflicting Europe, relative to the corresponding unemployment rates in the United States, has been a longstanding phenomenon and relatively easy to explain. In fact, unemployment rates during the 1950s and 1960s were lower in Europe and only became persistently higher than in the U.S. with the commencement of the 1970s. Ljungqvist and Sargent (LS) explain this perplex using a general equilibrium version of a McCall (1970) search model in which frictional unemployment comprises workers actively searching for jobs with high expectations of success and structural unemployment comprising discouraged workers who neither engage in intensive search nor do they expect much success. The LS model reveals how Europe’s employment protection (EP) measures supress frictional unemployment by lowering the rate into unemployment and how this policy was followed in the 1950s and 1960s when there was virtually no structural unemployment in Europe. Section 2 of the paper reviews the empirical puzzles regarding European unemployment and early efforts to resolve these perplexes. Section 3 describes the dynamic program used to explain riddles. It is an equilibrium version of the basic search model with aging workers, stochastic skill transitions that take place when workers are employed, unemployed, and displaced, respectively; employment protection (EP) in the form of a layoff tax; and unemployment benefits (UI) indexed to earnings on the previous job. In Sections 5 and 7 quantitative outcomes of the LS model are calibrated for welfare state (WS) milieu with high EP and UI and a laissez-faire (LF) milieu without EP or UI. Section 8 concludes by discussing whether the main forces of the LS model would also emerge in other models of the labor market. After their intensive study of these alternative models of the labor market Ljungqvist and Sargent give an affirmative answer to this robustness query. In Section 2.1, LS describe the employment–unemployment history in Europe during the 1950s and 1960s. This history is clearly a success story.11 During the past twenty-five years, Europe has been ravaged by persistently high unemployment. See Table 1 in LS (2005) which reveals that long-term

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333

unemployment is the essence of Europe’s unemployment problem. Hazard rates decline dramatically with increases in the length of unemployment spells. While some Europeans depart from unemployment quickly, a significant fraction become trapped in the long-term unemployment state. This trapping phenomenon was much less severe in the 1960s. Ljungqvist and Sargent also address two major mysteries: why do some (usually older) workers transit to long-term unemployment, while the economy at the same time is actively creating jobs for new workers? And why did Europe’s unemployment rates become persistently higher than in the U.S. after the 1970s, when the opposite was true in earlier decades? Macroeconomic shocks Instead of seeking macroeconomic shocks, LS judiciously concentrate on microeconomic disturbances. Ljungqvist and Sargent (1998) imputed part of the increased variability to shocks incurred by the worker’s human capital. According to this model, generous benefits produce sizeable long-term unemployment in Europe. This followed from the increased probability of human capital loss at the time of a firing. Increased earning variability The papers by Gottschalk and Moffitt (1994), Dickens (1994), and Katz (1994) displayed great interest in the documented increased earning instability. Research by Katz and Autor (1999: 1495) summarize the state of knowledge in the Handbook of Labor Economics: A consistent finding across studies and data sets is that large increases in both the permanent and transitory components of earning variation have contributed to the rise in cross-section earnings inequality in the United States from the late 1970s to early 1990s. The increase in the overall permanent component consists of both the sharp rise in returns to education and a large increase in the apparent returns to other persistent (unmeasured) worker attributes. The rise in cross-sectional residual inequality for males (controlling for experience and education) in the 1980s seems to consist of approximately equal increases in the permanent and transitory factors. Ljungqvist and Sargent seek to interpret these findings of increased earnings variability in terms of a parameter used to measure turbulence. Turbulence refers to the instantaneous obsolescence of valued skills of a displaced worker. It is one of the central players in the Ljungqvist–Sargent drama. Individual workers may experience turbulence in the LS sense without changes in the economy’s sectoral composition. As reported by Davis et al. (1996), about one in ten manufacturing jobs disappears in the U.S. in an average year, and a comparable number of new manufacturing jobs is

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The Economics of Search

created. Two thirds of job creation and job destruction are concentrated at plants that expand or contract by 25 percent in one year. Workers laid off in the job destructive process might not qualify for new jobs created in the same industry. Earnings losses of displaced workers The evidence showing the effects of turbulence in labor markets is revealed by studies of displaced workers, individuals with established work histories who have involuntarily left their jobs. Jacobson et al. (1993) found that long-tenured workers incurred large losses. Displaced workers’ relative earnings begin to decline three years before their displacement, drop sharply when they are discharged, and then improve rapidly during the next six quarters. After this, recovery proceeds at a niggardly pace. Next we present in more detail three different models that LS developed to explain these puzzles. One model is contained in Ljungqvist and Sargent (1998) while the other two are contained in Ljungqvist and Sargent (2005a)

11.10.1 Ljungqvist and Sargent (1998) In this paper LS formulate a general equilibrium search model in which workers’ skills depreciate during spells of unemployment and unemployment benefits are determined by workers’ past earnings. Simulations demonstrate the sensitivity of the equilibrium unemployment rate to the amount of skills lost at layoffs. Ljungqvist and Sargent model human capital as a state variable that in part depends on the work and in part depends on the job. In European countries, displaced workers receive high amounts of unemployment benefits resulting in especially high replacement ratios when measured as a fraction of expected earnings in the new job. In such situations welfare losses are not great from generous UI in tranquil times but become severe when the economy is turbulent. It is assumed that an unemployed worker searches with intensity st that produces a disutility of c(st). The probability of a job offer is π(st) which is increasing in st. If an unemployed worker receives an offer they get a wage draw from a distribution F(w). A worker gets a wage for each period he/she works until laid off (which occurs with probability α) or dies (which occurs with probability λ). There is a finite number of skill levels denoted by h with transition probabilities between skill levels represented by µe(h,h′) while employed, µu(h,h′) while unemployed, and µl(h,h′) when transiting from employment to unemployment (i.e., incurring a layoff). All newborn workers begin with the lowest skill level and when a worker dies they are immediately replaced with a new worker.

335

The Diamond–Mortensen–Pissarides breakthrough

A worker observes his/her new skill level at the beginning of the period before deciding to accept a new wage offer, choose a level of search intensity or quit a job. The objective of the worker is to maximize the expected value ∞

Et

冱 β (1 − α) y i

i

t+i

i=0

where E is the expectation operator conditional on the information available at time t, β is a subjective discount factor, and yt + i is the worker’s income net of both taxes and the disutility of searching at time t + i. Ljungqvist and Sargent assume that workers who are laid off are entitled to unemployment compensation benefits b(I) where I is the earnings in the lost job. Benefits are terminated if a worker turns down a suitable wage offer where a suitable wage offer is determined by the function Ig(I). Newborn workers and job quitters are not entitled to unemployment compensation benefits. All income is taxed at the flat rate τ and the government must set b(I), Ig(I) and τ so that, in equilibrium, income taxes cover unemployment compensation expenditures. The value function V(w,h) associated with being employed satisfies: V(w,h) = max

冦(1 − τ)wh + (1 − α)β 冤(1 − λ)冱µ (h,h′)V(w,h′) e

accept, reject

h′

冱µ (h,h′)V (wh,h′)冥,V (h)冧.

+λ

l

0

b

h′

Thus a worker receives net earnings (1 − τ)wh and if they survive until next period they remain employed with probability 1 − λ where the human capital moves according to µe(h,h′) or they are laid off with probability λ in which their human capital moves according to µl(h,h′). Finally, they have the option of quitting which yields V0(h). The value function Vb(I,h) associated with being unemployed and receiving unemployment compensation satisfies:

冱µ (h,h′)× 冤[1 − π(s)]V (I,h′)

冦

Vb(I,h) = max − c(s) + (1 − τ)b(I) + (1 − α)β s

+ π(s)

冢 冮

冤

× (1 − λ)

冮

V(w,h′)dF(w) +

w ≥ Ig(I)/h′

+λ

u

b

h′

w
max accept, reject

冦(1 − τ)wh + (1 − α)β

冱 µ (h′,h )V(w,h ) ″

e

″

h″

冱 µ (h′,h )V (wh′,h )冥, V (I,h′ )冧dF(w)冣冥冧. u

h″

″

b

″

b

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The Economics of Search

Here, − c(s) + (1 − τ)b(I) represents income net of taxes and search effort, and if the unemployed worker survives until next period their human capital changes according to µu(h,h′). With probability [1 − π(s)] no job offer is received and with probability π(s) the unemployed worker receives a job offer. If the job offer then exceeds the level set by the government Ig(I)/h′ the unemployed worker is forced to accept. Otherwise, the worker can either voluntarily accept the offer or reject the offer. The value function V0(h) associated with being unemployed and not receiving unemployment compensation satisfies

冱µ(h,h′) × 冢[1 − π(s)]V (h)

冦

V0(h) = max − c(s) + (1 − α)β s

0

h′

冮

冣冧

+ π(s) V(w,h′)dF(w) . Ljungqvist and Sargent study properties of the stationary equilibria of this economy where a steady state is defined in terms of the set of government policy parameters search intensities and reservation wages when unemployed and receiving unemployment compensation (s¯ b(h),w ¯¯ b(h)) and search intensities and reservation wages when no receiving unemployment compensation (s¯ 0(h),w ¯¯ 0(h)). For the government policy parameters, LS fix b(I) = 0.70I and Ig(I) = .7(I); that is, unemployment compensation benefits replace 70 percent of earnings from the lost job and you are required to accept a job that pays at least 70 percent of the previous job’s earnings. The stationary equilibrium is then found by determining a fixed point for the tax rate τ that balances the government’s budget. Ljungqvist and Sargent perform a number of numerical simulations to determine how such an economy would respond to changes in economic turbulence. This is then compared to a laissez-faire economy that has no unemployment compensation insurance. In tranquil times the welfare economy performs reasonably well compared to the laissez-faire economy with average unemployment spells less then two weeks apart (13.3 in the welfare economy versus 11.8 weeks in the laissez-faire economy). In this baseline situation of tranquil times there is no skill loss at the time of job loss, although skill levels trend downward as the unemployment spell lengthens. Reservation wages in this economy exhibit a U-shape with respect to current skill levels. The intuition is that at high skill levels the value of further skill accumulation is low compared to workers at intermediate levels so workers instead focus on obtaining a high wage offer whereas at low skill levels workers have less to lose then those at intermediate skill levels in terms of further skill loss from extending their unemployment spells and therefore also focus on high wage offers. Turbulence is introduced into the model by allowing the initial skill loss to be determined by a distribution and then examining the properties of the stationary equilibrium as the distribution becomes successively less favorable

The Diamond–Mortensen–Pissarides breakthrough 337 for job losers. Although both the welfare and laissez-faire economies are worse off relative to the baseline case when turbulence is introduced, the welfare economy is made relatively worse off. This occurs because once skills fall below a certain level, the value of re-employment is relatively low compared to remaining unemployed and receiving 70 percent of former wages that were based on a higher skill level. These reduced incentives are compounded by the uncertainties associated with future skill accumulation and the increased tax burden borne by workers to fund the unemployment compensation payments. Thus, LS demonstrate how a system with generous welfare benefits such as is the case in many European countries might exhibit relatively low unemployment rates when times are tranquil but perform rather poorly if the economy becomes turbulent to exogenous changes. In the Ljungqvist and Sargent (2005c) paper, the above model is modified by allowing: (a) workers to age stochastically and behave differently at different ages; (b) while working an employee’s wages (per unit of human capital) follows a Markov process; (c) there is employment protection in the form of a tax imposed on all job separations. These embellishments are done because an LS earlier model had difficulty accounting for the fact that prior to the 1970s European unemployment rates were actually lower than those in the U.S. and the long-term unemployment observed in Europe after the 1970s was concentrated among older workers.

Ljungqvist and Sargent (2005a) This paper looks at the impact of economic turbulence in three separate general equilibrium frameworks: a matching model in the spirit of Mortensen and Pissarides (1999), a search-island model which is a variant of Alvarez and Veracierto (2001) and an employment lotteries model with economywide insurance arrangements (see Hansen 1985a, 1985b; Prescott 2002; Rogerson 1988). The general setup in three models is as follows. In the matching model workers are risk neutral and have no asset accumulation decisions. The allocative role of wages is attenuated while there is a salient role for the ratio of vacancies to unemployed workers. The model features adverse congestion effects unemployed workers impose on each other and firms with vacancies impose on each other. There is a wage-bargaining process and waiting times serve the role of equilibrating signals that reconcile the decisions of firms and workers. In the search-island model workers are risk averse and their decisions about how intensively the search when unemployed depend on their skills, benefit entitlements and accumulations of a risk-free asset, which is the only manner in which they can save. The model has incomplete risk sharing via self-insurance against unemployment risks and uncertain life spans in retirement. Wages are determined competitively but are constrained to remain

338

The Economics of Search

fixed for the duration of an employment spell even though there are idiosyncratic productivity shocks. The lotteries model features risk-averse agents who have complete markets in history-contingent consumption claims at their disposal. Labor contracts are identical to those in the search-island model. An indivisibility exists in the worker’s choice set that forces him/her to either work a fixed number of hours or not at all. This potential non-convexity poses no difficulties, however, since in such a representative-agent economy lotteries which assign fraction of workers to either work or consume leisure convexifies the production opportunity set. Such employment lotteries result in large aggregate labor supply elasticities that makes aggregate employment highly sensitive to tax wedges or UI benefits. Ljungqvist and Sargent show that in the lotteries model introducing a generous unemployment insurance produces the unrealistic result that the economy virtually shuts down. So we focus only on the matching and search-island models. Matching model The matching model contains the matching function M(v,u) where the probability of a firm being visited by an unemployed worker with characteristics h and b, λf(h,b), satisfies λf(h,b) =

M(ν,u¯ ) u(h,b) u(h,b) × = m(θ) ν u¯ ν

and the probability of a worker of type h and b, λw(h,b) finding a vacancy satisfies λw(h,b) =

M(ν,u¯ ) = m(θ). u¯

In this model there are two possible skill levels indexed by h ∈ {0,H}. Newborn workers enter the economy with skill level 0. An employed worker with skill index h faces a probability of transiting to skill level h′ next period of pn(h,h′) conditional on no exogenous job loss. In an event of an exogenous job loss the transition probability is po(h,h′). Skill levels remain unchanged during an unemployment spell. A new job opportunity is associated with an initial productivity, z, which is modeled as a draw from the distribution of productivities, Q0h(z). The productivity of an ongoing job is governed by the transition probabilities Qh(z,z′) where for any productivity levels z and z″ with z < z″, Qh(z″,z′) stochastically dominates Qh(z,z′). Moreover, QH0 (z) stochastically dominates Q00(z) and for any z, QH(z,·) stochastically dominates Q0(z,·).

The Diamond–Mortensen–Pissarides breakthrough 339 Let u(h,b) denote the number of unemployed workers with skill level h and skill during previous employment of b, then the total number of unemployed workers is u¯ = 冱u(h,b). Ljungqvist and Sargent assume that the government h,b

pays unemployed workers unemployment benefits equal to a replacement rate, η, times a measure of past income. Moreover, a layoff tax Ω is imposed on every endogenous job separation. The amount of benefits is a function of skill in the lost job and is denoted by b˜ (b). Workers are assumed to be risk neutral and make decisions to maximize ∞

E0

冱 β (1 − α) c t

t

t

t=0

where β is a discount factor and (1 − α) is the probability of surviving to the next period. Denote the match surplus by Now, if an unemployed worker with skill level h and benefits b meets a firm with a vacancy, the firm–worker productivity is determined by a draw from Qoh(z). If we denote the initial match surplus by So(h,z,b) then max {(1 − τ)z − [1 − β(1 − α)]W(h,b) + β(1 − α)

So(h,z,b) =

{stay, depart}

冱p (h,h′)Q (z,z′)S(h′,z′)冥,0冧

冤

× − π0Ω (1 − π0)

n

h′

h′,z′

where W(h,b) is the worker’s outside value and S(h,z) represents the surplus of an ongoing match. The value of the worker’s outside option in turn satisfies

冱ψS (h,z,b)Q (z)冥.

冤

W(h,b) = b˜ (b) + β(1 − α) W(h,b) + λw(h,b)

o

b h

z

Free entry makes the firm’s outside value 0. The match surplus is divided between worker and firm through Nash bargaining with the outside values as threat points. Let ψ denote the worker’s share of the surplus. Matches are formed when there is positive surplus thus the reservation value of productivity, z¯ o(h,b), satisfies So(h,z¯ o(h,b),b) = 0. Thus surplus of a continuing match, S(h,z), satisfies S(h,z) =

max {continue, break up}

冤

冦(1 − τ)z − [1 − β(1 − α)W(h,h) + β(1 − α) 冱p (h,h′)Q (z,z′)S(h′,z′)冥, − Ω冧.

× − π0 Ω + (1 − π0)

n

h′

h′,z′

Since the government imposes a tax of Ω on matches that are broken a match will dissolve only when the surplus falls below − Ω . Thus the reservation productivity z¯ (h) below which matches are dissolved satisfies S(h,z¯ (h)) = − Ω.

340

The Economics of Search

In equilibrium firms expect to break even when posting a vacancy. Let µ be the firms expected gain from posting a vacancy. Then, in equilibrium

冱λ (h,b)S (h,z,b)Q (z) = 0.

µ = β(1 − ψ)

o h

0

f

h,z,b

The zero profit condition determines the equilibrium level of market tightness measured by θ. Ljungqvist and Sargent assume that once a match is formed the firm’s threat point shifts from 0 to − Ω since now firms must pay a layoff tax if the match is dissolved. This results in a two-tier wage structure, one wage for the newly matched worker determined by

冦 冱 p (h,h′)W(h,h′)

w ˜ o(b) = W(h,b) + ψS o(h,z,b) − β(1 − α) π0

o

h′

+ (1 − π0)

冱 p (h,h′)Q (z,z′)(ψ[S(h′,z′) + Ω] + W(h′,h′)冧 n

h′

h′,z′

and one for continuing workers satisfying

冦 冱 p (h,h′)W(h,h′)

w ˜ (h,z) = W(h,h) + ψ[S(h,z) + Ω] − β(1 − α) π0

o

base

冱p (h,h′)Q (z,z′)(ψ[S(h′,z′) + Ω + W(h′,h′)冧.

+ (1 − π0)

n

h′

h′,z′

Search-island model The dynamics for productivity and human capital are the same in the searchisland model. Workers preferences are described by ∞

E0

冱 β (1 − o ) 冤log(c ) + A t

t

t

t

t=0

(1 − st)γ − 1 γ

冥

where st represents a worker’s search intensity at time t and (1 − ot) represents the probability of survival where ot = α if an individual is in the labor market and ot = σ if a worker is retired. Bellman equations for an existing firm satisfy Vf(h,z) = max{V˜ f(h,z), − Ω} where

The Diamond–Mortensen–Pissarides breakthrough

341

V˜ f(h,z) = max{zkφ(1 + h)1 − φ − w*(1 + h) − (i + δ)k} k

+

1−α − π0Ω + (1 − π0) 1+i

冤

冱 p (h,h′)V (h′,z′)Q(z,z′ )冥. n

f

h′,z′

Here i is the risk free interest rate and (i + δ) is the price of capital. The exogenous rate of job termination equals π0 and the firm’s production function is assumed to have the form zkφ(1 + h)1 − φ. The first order condition for the maximization problem is zφkφ − 1(1 + h)1 − φ = (i + δ) or zφ i+δ

冤

k(h,z) =

冥

1 1−φ

(1 + h).

Associated with the firm’s optimization problem is a reservation level of productivity z¯ (h) below which the worker is laid off where V˜ f(h,z¯ (h)) = − Ω. Define the indicator function Λ(h,z) by Λ(h,z) =

z ≥ z¯ (h) . 冦1,0, ifotherwise

The break-even condition for a new firm is µ=

1 1+i

冱 max{(1 − ω)V˜ (0,z) + ωV˜ (H,z),0}Q (z) = 0 f

f

o

(11.71)

z

where ω is the fraction of high-skilled workers among new hires. Equation (11.71) is used to determine the reservation productivity level for a firm to hire a worker z¯ o where (1 − ω)V˜ f(0,z¯ o) + ωV˜ f(H,z¯ o) = 0. Define the indicator function Λ(h,z) by Λo(h,z) =

z ≥ z¯ . 冦1,0, ifotherwise o

The productivity distribution of firms who hire new workers is Γ(z) =

Λo(z)Qo(z)

. Λo(z′)Qo(z′)

冱 z′

The household’s problem is characterized by three value functions, Vn(a,h,z)

342

The Economics of Search

for an employed worker, Vu(a,h,b) for an unemployed worker and Vr(a) for a retired worker. The state variables are last period’s assets a, the skill index h, the firm’s current productivity level, z, for an employed worker and the worker’s benefit entitlement for an unemployed worker. The value function for an employed worker satisfies

冢 冱 p (h,h′)V (a′,h′,h)

冤

Vn(a,h,z) = max log c + β(1 − α) π c,a′

+ (1 − π0)

o

u

h′

冱 p(h,h′)冦V (a′,h′,z′)Λ(h′,z′) n

h′,z′

冧

冣冥

+ V (a′,h′,h)[1 − Λ(h′,z′)] Q(z,z′) u

(11.72)

subject to c + a′ ≤ (1 + i)a + (1 + h)w and c,a′ ≥ 0 where the worker loses the job exogenously with probability π0 and endogenously with probability (1 − π0)[1 − Λ(h′,z′)]. The solution of the optimization problem in (11.72) determines the policy functions: c¯ n(a,h,z) and a¯ n(a,h,z). For an unemployed worker the value function Vu(a,h,b) satisfies

冤

Vu(a,h,b) = max log c + A c,a′,s

(1 − s)γ − 1 + βαVr(a′) + β(1 − α) γ

冢

× (1 − sξ)Vu(a,h,b) + sξ

冱 V (a′,h,z′)Γ(z)冣冥 n

z′

subject to c + a′ ≤ (1 + i)a + η(1 + b)w and c,a′ ≥ 0,s ∈ (0,1) where the maximization in (11.73) gives rise to the policy functions ¯cu(a,h,b), a¯ u(a,h,b) and s¯ (a,h,b). Finally, for a retired worker we have Vr(a) = max{log c + β(1 − σ)Vr(a′)} c,a′

subject to c + a′ ≤ (1 + i)a and c,a′ ≥ 0 with the associated policy functions c¯ r(a) and a¯ r(a).

(11.73)

The Diamond–Mortensen–Pissarides breakthrough 343 Steady state Let N(h,z) represent a time-invariant measure that describes the number of firms operating with workers of skill level h and productivity level z. For skill levels 0 and H these steady state functions satisfy N(0,z′) = vQo(z′)Λo(z′)(1 − ω) + (1 − α)(1 − π0)Λ(0,z′) ×

冱 p (h,0)N(h,z)Q(z,z′) n

h,z

and N(H,z′) = vQo(z′)Λo(z′)ω + (1 − α)(1 − π0)Λ(0,z′) ×

冱 p (h,H)N(h,z)Q(z,z′) n

h,z

where v is the number of newly created firms. The steady-state number of employed, unemployed and retired individuals are described by the functions yn(a,h,z), yu(a,h,b), and yr(a), respectively. For yn(a,h,z) we have

冤

yn(a′,h′,z′) = (1 − α) (1 − π0)Λ(h′,z′)

冱

pn(h,h′)yn(a,h,z)Q(z,z′)

a,h,z:a¯ n(a,h,z) = a′

冱

+ Γ(z′)

冥

s¯ (a,h′,b)ξyu(a,h′,b)

a,b,z:a¯ u(a,h,b) = a′

(11.74)

where the first term on the right side of (11.74) inside the brackets represents the number of workers that remained employed next period while the second term represents the number of individuals who find jobs by next period. The steady state function yu(a,h,z) satisfies

冤

yu(a′,h,b) = (1 − α) π0Λ(h′,z′) n

冱

po(b,h)yn(a,b,z)

a,z:a¯ (a,b,z) = a′

冱

+ (1 − π0)

pn(b,h)yn(a,h′,b){1 − Λ(h,z′)}Q(z,z′)

n

a,z,z′:a¯ (a,b,z) = a′

+ u

冱

冥

a:a¯ (a,h,b) = a′

yu(a,h,b){1 − s¯ (a,h,b)ξ} + I(h,b)σ

冱

yr(a)

a:a¯ r(a) = a′

(11.75)

where I(h,b) is an indicator variable that equals one if h = b = 0 and zero otherwise. The first term in the square brackets of (11.75) represents the inflow to unemployment and the second term represents the number of continuing unemployed. The final term in equation (11.75) represents the inflow of new workers which equals the number of retirees who die.

344

The Economics of Search

Lastly, the steady-state function for the number of retirees satisfies

冱

yr(a′) = (1 − σ)

冤 冱

yr(a) + α

冱

yn(a,h,z) +

a,h,z:a¯ n(a,h,z) = a′

a:a¯ (a) = a′

冥

yu(a,h,z) .

a,b,z:a¯ u(a,b,z) = a′

In equilibrium, the government is required to balance its budget every period

冱(1 + h)N(h,z) + ΩD − ηw 冱

0 = (w* − w)

yu(a,h,b)

a,h,b:a¯ u(a,h,b) = a

h,z

where D represents the amount of job destruction and satisfies

冦 冱N(h,z) + (1 − π ) 冱 p (h,h′)[1 − Λ(h′,z′)]N(h,z)Q(z,z′)冧.

D = (1 − α) π0

0

h,z

n

h,h′,z,z′

The goods’ market clearing condition is c¯ + δk¯ + µν =

冱N(h,z)zk(h,z) (1 + h) ω

1−ω

h,z

where aggregate consumption c¯ satisfies c¯ =

冱c¯ (a,h,z)y (a,h,z) + 冱c¯ (a,h,b)y (a,h,b) + 冱c¯ (a)y (a) n

n

u

a,h,z

a,h,z

u

r

r

a,h,z

and aggregate capital, k¯ , equals

冱N(h,z)k(h,z).

k¯ =

h,z

The number of new firms that hire workers of type h must equal the number of workers of type h who find jobs. This implies that (1 − α)

冱s¯(a,h,b) y (a,h,b) ξ u

a,h,b

ν=

冱Λ (z)Q (z) o

o

z

and

冱s¯(a,H,b) y (a,H,b) ξ u

ω=

a,b

冱

. s¯ (a,h,b)ξyu(a,h,b)

a,h,b

The Diamond–Mortensen–Pissarides breakthrough

345

Finally, the household’s aggregate demand for assets a¯ satisfies a¯ =

冱ay (a,h,z) + 冱ay (a,h,b) + 冱ay (a) n

u

a,h,z

r

a,h,b

a

which in equilibrium equals the supply of assets which, in turn, equals the capital stock, k¯ plus the total value of claims to all the firms in the economy:

冱[zk(h,z) (1 + h) φ

a¯ = k¯ +

1−φ

− w*(1 + h) − (i + δ)k(h,z)]N(h,z) − µν − ΩD

h,z

i

Ljungqvist and Sargent compute the equilibrium outcomes for various values of the underlying parameters. For both the matching and searchisland models increases in the replacement rate increases the unemployment rate, although for the search-island model the impact is relatively less at lower values of η and greater at higher values of η than the matching model. By increasing the value of a worker’s outside option and, hence, decreasing the value of surplus for a newly matched worker, an increase in UI decreases market tightness and reduces the expected amount of time that firm’s need to fill a vacancy just enough to maintain the zero profit condition for creating new jobs. In a search-island model there are no congestion effects and an increase in UI acts to decrease an unemployed individual’s search intensity thereby lengthening unemployment spells and increasing the unemployment rate. Increased turbulence leads to substantial increases in the unemployment rates of a welfare state economy while having little effect on the unemployment rate of the laissez-faire economy regardless of whether the underlying equilibrium is generated by a matching model or search-island model. Thus, both models do well at explaining the European unemployment experience.

11.11 TWO-SIDED SEARCH, MARRIAGE, AND MATCHMAKERS12

11.11.1 Introduction The objective of this section is to model explicitly the presence of matchmakers in two-sided search. Two-sided matching has developed along two separate tracks. The Gale–Shapley marriage model has stimulated a large literature in cooperative game theory and operations research, which has concentrated on analysis and design of centralized matching mechanisms. At the same time, McNamara and Collins (1990) and Smith (1997) have extended the classic one-sided job search model. They have developed models of decentralized search. Bloch and Ryder integrate these separate approaches by analyzing how centralized matching agencies operate in a decentralized

.

346

The Economics of Search

matching setting. Two strong assumptions are made regarding the matching surplus. There is no utility transfer between match partners and there are no complementarities between the qualities of the matching partners. The first model of Bloch and Ryder is of decentralized search where men and women meet randomly each period and decide at each meeting whether to marry. Search costs take the form of a discount factor. To preserve stationarity, married men and women are replaced by men and women of the same quality. The model is formally the same as the McNamara–Collins (1990) model of two-sided search. The equilibrium is characterized so that it yields similar results. The key to both models is to show that in equilibrium there is perfect segregation: the set of men and women can be partitioned into subintervals so that agents belonging to the same subinterval follow the same threshold strategy. The role of marriage brokers is studied in the second part of the paper. At the start of the game, agents decide whether to use a broker or search in the decentralized market. An agent’s decision creates externalities. The existence of the matching broker depends on how many agents choose centralized search. Also searchers who leave the decentralized market affect the expected value of search of agents who remain in the decentralized market. These externalities may cause multiple equilibria in the participation game. Bloch and Ryder consider two pricing schemes for the matchmaker. First, the broker must charge a uniform participation fee to all agents. The authors show that there is a unique participation equilibrium and matches are only made between highest quality agents. Low-quality agents shop in the decentralized market. In the second model, the broker charges a commission after the agent’s quality is revealed. There may be multiple equilibria. But under all equilibria, only the highest quality agents shop in the decentralized market. Next we present the equilibrium.

11.11.2 Equilibrium of two-sided search Two populations, one of men and one of women, are considered and indexed on [0,1]. The quality indices of men and women are denoted x and y and are distributed according to F(x) and G(y) with full support on [0,1]. The respective densities are f(x) and g(x). Let λm and λw be the measure for men and women. All women have identical preferences over the set of men. The utility of a match with a man of quality x is simply x: uy(x) = x ∀y ∈ [0,1]. A similar assumption holds for men: ux(y) = y ∀x ∈ [0,1]. Time is discrete: t = 0,1, . . ., + ∞. At any date t, unmarried men and

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women are matched by a simple random matching technology. In particular, it is assumed that given the measures λm and λw, the measure of matched agents is M(λm,λw), where M increases in both arguments, and M(λm,λw) ≤ min(λm,λw). The matching technology is efficient if M(λm,λw) = min(λm,λw). The probability of being matched for any man is M(λm,λw)/λm. Assume women are randomly drawn from G. The probability for any man being matched with a woman of quality less than y is given by M(λm,λw)G(y)/λm. Also, the probability for any woman being matched with a man of quality less than x is given by M(λm,λw)F(x)/λw. Agents have a common discount factor δ. Thus, the utility of a woman who marries a man x after searching for T periods is given by U = δTx. At each meeting, the man and woman assess one another and simultaneously decide whether to reject or accept. If both accept, they marry. If either rejects, both remain unmarried and wait for a new match next period. Following a match (x,y), there is an inflow of agents of quality x and y so the distribution is invariant over time. Bloch and Ryder consider stationary Markov strategies. Thus, a woman y has a strategy σy which is a function from [0,1] to (yes, no). Woman y’s payoff depends on her strategy σy and on the profile strategies of men σx. The payoff for a man is similarly defined. Thus, for any stationary strategy profile σx,σy, the expected utility of woman y is EVy(σx,σy). Definition 11.1 A search equilibrium is a stationary strategy profile (σx,σy) such that (a) For almost all men x in [0,1] and all strategies σ′x for man x, EUx(σx,σy) ≥ EUx(σ′x, σy). (b) For almost all women y in [0,1] and all strategies σ′y for women y, EUy(σy,σx) ≥ EUy(σ′y, σx). The search equilibrium McNamara and Collins (1990) prove that there is a unique search equilibrium characterized by the formation of a sequence of classes such that men and women marry if and only if they are members of the same class.13 Shimer and Smith (1996) notes that perfect segregation is a nontransferable utility (NTU) phenomenon in that no matching set disconnectivity can occur in a transferable utility model. In NTU models bargaining does not occur and there may be unexploited matching rents.

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The perfect segregation theorem (McNamara and Collins 1990) There is a unique search equilibrium in the two-sided marriage model. This equilibrium is characterized by two sequences of subintervals In = (an,an − 1) and Jn = (bn,bn − 1) of [0,1] such that all men x in (an,an − 1) follow the same search strategy σx = (bn,1) and all women y in (bn,bn − 1) follow the same search strategy σy = (an,1). The subintervals are defined recursively:

• •

ao = 1 For any n ≥ 1, an is the solution to M(λm,λw) an(1 − δ) = δ λw

an − 1

冮 (x − a )f(x)dx n

n

a

and

• •

bo = 1 For any n ≥ 1, bn is the unique solution to M(λm,λw) bn(1 − δ) = δ λw

bn − 1

冮 (y − b )g(x)dx. n

n

b

The expected value of search for any woman y in (bn,bn − 1) is given by V(y) = an/δ and for any man x in (an,an − 1) by V(x) = bn/δ. Perfect segregation is a natural generalization of the simple sequential search model. Men and women are partitioned into a collection of classes such that each member of the same class follows the same reservation quality price. Bloch and Ryder give an intuitive explanation of the perfect segregation theorem. Consider the problem confronting the highest quality male. Since he is accepted by any woman, he can search the entire [0,1] interval and choose a reservation level b1 < 1, which represents the highest quality women he rejects. Any woman in (b1,1] is accepted by the highest quality man and hence by all men in [0,1]. Therefore, all women in (b1,1) adopt a search policy with reservation quality a1. Thus the first cluster for high quality agents is determined by the two reservation qualities a1 and b1 and equals (a1,1) × (b1,1). Now focus on the highest-quality woman still in the market b1 and highest quality man a1. Woman b1 is accepted by any man in [0, a1] and man a1 by any woman in [0, b1]. Hence, arguing as before, the second cluster is determined by four reservation qualities and equals (a2,a1) × (b2,b1). Continuing in this fashion gives the entire partition. Comparative statics results are given by Proposition 11.1 (Bloch and Ryder 2000) As the discount parameter δ increases, the cluster bounds an and bn increase. Also, as matching becomes

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more efficient, an and bn increase. And as λ, the measure of men, rises, so too does each upper bound of men’s clusters an. Proof

See Bloch and Ryder (2000).

Matchmakers In Bloch and Ryder (2000) the matchmaker proposes to match agents according to a fixed, centralized mechanism where single matches are arranged between the two prospective partners. Before the market opens the matchmaker proposes to match according to identity matching and states the fee for his services. Agents who contract with the matchmaker are forbidden to search in the decentralized market. Bloch and Ryder also assume that it is costless for the matchmaker to discover the quality of the prospective partners. They note that externalities are introduced by the matchmaker’s exclusive contracting and by the fact that, given nonparticipating agents, the matchmaker cannot guarantee “best” matches to the partners of the nonparticipants. Brokers may charge a commission or a uniform participation fee to all agents. Bloch and Ryder show that a broker who charges a uniform fee only reaches the upper end of the market. Low-quality agents do not use brokers, but instead search in the decentralized market. It is interesting that the reverse occurs when the broker charges different commissions to different agents. In this situation, the broker is not hired by high-quality agents, who search in the decentralized market.

11.12 BILATERAL SEARCH AND VERTICAL HETEROGENEITY Eeckhout (1999) shows that provided the distribution of singles is stationary, a (Nash) equilibrium exists and is unique. This is true for any specification of the utility function. The distribution of types is endogenously partitioned for preferences that are multiplicatively separable14 (Eeckhout 1999: 877, Proposition 2). Eeckhout shows that his equilibrium model is a generalization of the perfect matching model, which was first described in Gale and Shapley (1962). Gale and Shapley’s formulation was for a finite number of agents and for any set of preferences. Eeckhout extends it to a continuum of agents and preferences display vertical heterogeneity, which links his specification to Becker (1973, 1974). A major difference between his model and the perfect matching model is the presence of search frictions in the former. Eeckhout notes that another major difference between his model and perfect matching is that the former was a noncooperative equilibrium concept, while the latter employs a

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cooperative equilibrium. What he shows is that the noncooperative search equilibrium gives the same outcome as the cooperative stable matching without friction when the search friction is infinitesimal, that is, as the arrival rate of matches (which is Poisson and denoted by β) tends to infinity. This is all encapsulated in Eeckhout’s (1999) Proposition 3 Equivalence: the Gale– Shapley–Becker perfect matching model is the limit of the search model when trading opportunities arrive instantaneously (i.e., lim β → ∞).

11.13 TWO-SIDED SEARCH WITH FIXED SEARCH COSTS There is a large number of agents that at the beginning belong to one of two disjoint populations. In each population agents differ in their characteristics. It is costly to discover potential partners from the other population, but is preferable to search rather than remain single. Each period agents randomly meet with members of the other population, and a decision is made to mate forever with the other agent or continue their costly search. The goal of each agent is to match with the best possible partner. Agents differ and the value of search is not the same for all. Similarly each has a different set of acceptable mates. In this decentralized environment with costly search and heterogeneity Chade (2001) studies the following: (1) In equilibrium who matches with whom? (2) Do agents with superior qualities search longer before discovering an acceptable mate? (3) Does the structure of search frictions matter? These questions are answered in a framework with a continuum of agents of different types in both populations, with nontransferable utility that is additively separable in types, and where agents incur a fixed cost of search in each period to locate possible mates. Given these assumptions, Chade shows that if utility from a partner is increasing in her type, there is a unique matching equilibrium in which agents partition into classes and matching occurs only among agents who belong to the same class in the partition. He obtains a sufficient condition that orders the sizes of the classes as a function of the type they contain, thereby providing a simple way to calculate the duration of search as a function of the attributes of each agent. Chade provides an intuitive explanation clarifying why additive separability and multiplicative separability are sufficient conditions for perfect segregation in the fixed search cost and discounted cases, respectively. Additionally, he gives a set of sufficient conditions that generalizes perfect segregation to a richer class of cost functions that include discounting and fixed costs as special cases. He also exemplifies the relationship between decentralized search just outlined and centralized matching presented in Roth and Sotomayor (1990).

The Diamond–Mortensen–Pissarides breakthrough 351

11.13.1 Reservation-type strategies Suppose a woman of type y is accepted by men who belong to Ωj(y) = {x:y ∈ Am(x)}, where Am(x) is stationary. Her problem is to discover an acceptance rule such that the expected utility is maximized. Let Φ*f (y) be the expected value of her search under an optimal strategy. From optimal stopping theory, it follows that there is an optimal acceptance rule with the following threshold property: Accept iff α1(x) + α2(y) ≥ Φ*(y), where α1(x) is nonnegative and strictly increasing and α2(·) is an arbitrary realvalued function. Let x(y) be the unique solution to α1(x) + α2(y) = Φ*(y). The monotonicity of α1(x) and the acceptance rule gives Lemma 11.1 In equilibrium the set of males that a woman of type y accepts is Af (y) = [x(y),1]. The reservation policies in Lemma 11.1 imply that the set Ωi, i = f,m, is larger for higher types. Hence, we expect them to be more selective in their acceptance decisions. This is formalized in Lemma 11.2 In equilibrium, the reservation-type x(y) is increasing in y. And similarly for males. Consider a woman of type y; the monotonicity of y(x) implies that there is a type x¯ (y) such that y is accepted by men with types x ≤ x¯ (y) and is rejected otherwise. The threshold x¯ (y) is defined by x¯ (y) = sup{x ∈ [0,1]: y ≥ y(x)}. ¯ Hence, for all y ∈ [0,1], Ωf (y) = [0,x¯ (y)]. And likewise Ωm(x) = [0,y¯ (x)] for all x ∈ [0,1]. The expected value of search for a type y woman who behaves optimally is x¯ (y)

−c+ *f (y) = max

0 ≤ a ≤ x¯ (y)

冮 (α (x) + α (x))f(x)dx 1

2

a

F(x¯ (y)) − F(a)

(11.76)

Lemma 11.3 If x¯ (y) > 0, for each y there is a unique x(y) ∈ [0,x¯ (y)] that ¯ solves the optimization problem in (11.76). Analogous results hold for males. Thus, optimal strategies are characterized by functions x(y) and y(x) which when ¯ ¯ x(y)

y(x)

冮 α (x) f (x)dx > c and 冮 y (y)g(y)dy > c, 1

0

1

0

352 The Economics of Search are implicitly and uniquely defined by x¯ (y)

冮 (α (x) − α (x(y)) f (x)dx = c 1

1

x(y) ¯

and y¯ (x)

冮 (γ (y) − γ (y¯ (x))g(y)dy = c 1

1

y(x) ¯

and are equal to zero otherwise. Theorem 11.1 (Chade 2001) There is a unique matching equilibrium characterized by the formation of a finite number of disjoint classes of men and women that partition the two populations. Marriages occur only between men and women of the same class. Comment Within each class there is a mixing between agents of different types. This is caused by search frictions.

11.13.2 The role of search costs: an optimal stopping approach Consider the optimal stopping problems solved by agents in equilibrium. Under fixed search costs, a woman of type y solves *f (y) = maxE [uf (xτ ,y) − τyc | Ωj(y)], y

τy

where the maximization is over the class of stopping times for which E is welldefined. If uf (x,y) is additively separable, then *f (y) = α2(y) + maxE [α1(xτ ) − τyc | Ωj(y)], y

τy

and the optimal stopping time τ*y depends on y only through Ωj(y). Similarly for τ*x. Since optimal stopping times are characterized by x(y) and y(x), perfect segregation follows. Now assume discounting and multiplicative separability. Then Φ*f (y) = maxE [βτ uj(xτ ,y) | Ωj(y)] y

j

τy

= α2( y)maxE [βτ α1(xτj) | Ωj(y)]. y

τy

The Diamond–Mortensen–Pissarides breakthrough 353 Thus, the optimal stopping rule depends on y only through Ωj (y) and similarly for men. These results show that when deriving properties like perfect segregation (or more general results of assortative matching), it is important how search costs are modeled. This is clear from Chade’s Theorem 3.

11.14 ROCHETEAU–WRIGHT MODELS OF SEARCH AND MONEY Using the rich, search-theoretic monetary framework developed by Lagos and Wright (2005) and others, the major assumption of this framework is that in addition to actions taken in decentralized markets, Rocheteau and Wright (2005) examine the efficiency implications of three different market structures in an environment where money has an essential role. In their model agents operate part of the time (at night) in a decentralized market. Agents also have periodic access to centralized competitive markets (during the day). Agents are assumed to be anonymous. These assumptions produce an essential role for money. Centralized markets simplify the analysis. When combined with the assumption that preferences are quasi-linear, it implies that all agents of a particular type bring identical amounts of money into the decentralized market. This simplifies the distribution of money holdings making the model easier to analyze relative to comparable models without centralized markets. The authors explain that the source of this simplicity comes from the fact that quasi-linearity eliminates wealth effects on the demand for money, and hence eliminates dispersion in money holdings based on trading histories. While we believe that the wealth/distribution effects from which we are abstracting are interesting . . . the goal here is to focus on other effects that have not been analyzed previously. To introduce these new effects we extend existing monetary models by adding a generalized matching technology and a free-entry condition. These extensions can be thought of as adaptations from labor market models like those discussed in Pissarides (2000). Their role here is to allow us to discuss the effects of inflation on the extensive margin (the number of trades) as well as the intensive margin (the amount exchanged per trade), and to discuss search “externalities”, i.e., the dependence of the amount of trade on the composition of the market. (Rocheteau and Wright 2005: 176) The authors’ major result is showing that different market structures have remarkably different implications for both the nature of equilibrium and the effects of policy. In search equilibrium (bargaining), they show that the amount of trade and entry are inefficient. For this model inflation implies a

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The Economics of Search

first-order welfare loss, and Friedman’s rule (that the rate of inflation equal the discount rate) is optimal but impotent with respect to correcting inefficiencies on the intensive margin (amount exchanged per trade) and the extensive margin (number of trades). In competitive equilibrium (price taking), the Friedman rule is efficient along the intensive margin, but inefficient along the extensive margin. Here the effects of policy are ambiguous: inflation that is greater than the Friedman rule may be required. In competitive equilibrium (posting), Friedman’s rule is first best. Here inflation diminishes welfare, but with a second order effect. We believe that important future work in equilibrium search and monetary theory must begin with this article as a point of departure. We now present a summary that is extracted from their concluding section. Rocheteau and Wright (2005) studied three market structures for monetary economics: (1) search equilibrium with bargaining,15 (2) competitive equilibrium with price taking,16 and (3) competitive search equilibrium with price posting and directed search.17 The authors discovered that efficiency and the policy impact “depend crucially on market structure.” (1) Under bargaining, trade and entry are inefficient with inflation producing first-order welfare losses. (2) When price taking occurs, Friedman’s welfare rule solves the first inefficiency but not the second; inflation may enhance welfare! (3) When price posting takes place, Friedman’s rule is first best, with inflation generating second-order welfare losses.

11.15 RECENT DISCOVERIES While research on equilibrium search models continues unabated, and we have certainly not covered all the extant research, we nevertheless must stop. So to conclude we briefly describe a couple of recent discoveries. Shimer (2005) notes that “in recent years, the Mortensen–Pissarides (MP) search and matching model has become the standard theory of equilibrium unemployment.” He references Pissarides (2000).18 Shimer then argues that this “standard” model is unable to explain the cyclical motion of both unemployment and vacancies, both of which display great variability and are strongly negatively correlated in U.S. data. Shimer notes that “equivalently, the model cannot explain the strong procyclicality of the rate at which an unemployed worker finds a job.” Readers are urged to study this article and the large related literature which is nicely summarized by Shimer (2005) on pages 44–5. One of Shimer’s striking comments is that his work is not attacking the search models of labor markets. Instead, his attack is aimed at the Nashbargaining assumption in its role in determining the wages. Shimer suggests a number of alternative modifications that may save the M–P edifice. He notes that an alternative wage determination mechanism that

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produces rigid wages in new jobs measured in terms of present value may increase the impact of productivity shocks on the vacancy–unemployment ratio. This would bring theory and empirics closer together. Shimer also suggests eliminating some of the informational assumptions in the standard search model. Bargaining is conducted so that the usual link between average labor productivity and the equilibrium vacancyunemployment ratio is severed. Whether or not Shimer’s striking suggestions work is an important topic for future research. In a similar article, Hall (2005) adopts the matching function of the DMP model. Hall observes that Shimer (2005) and Veracierto (2003) have stressed that the DMP model does not provide plausible explanations of observed unemployment fluctuations. Once again, the reader is encouraged to study this important article. Hall’s basic contribution is the introduction of “sticky wages” into the DMP model: The resulting model makes recruiting effort, job-finding rates, and unemployment remarkably sensitive to changes in determinants. A small decline in productivity results in a slump in the labor market. With wage stickiness, these changes depress employer returns to recruiting substantially. The immediate effect is a decline in recruiting efforts, a lower jobfinding rate, and a slacker labor market with higher unemployment. (Hall 2005: 51) In his concluding remarks, Hall links his sticky wage hypothesis to the recent intensive empirical study by Bewley (1999), who discovered that trust and morale discouraged employers from cutting wages. Howitt (2002: 129), in his review of Bewley, reports that the direct interview method used by Bewley and others “exposed an empirical regularity that has escaped other types of investigation . . . If morale is not in fact a serious problem discouraging firms from cutting wages then what accounts for these facts?”

12 Structural estimation methods

12.1 INTRODUCTION There have been several excellent surveys on structural estimation in economics, two of which are books: the volume by Devine and Kiefer (1991), which presents commentaries on some of the early research that employs structural estimation; and the monograph by Wolpin (1995), an illuminating portrait of structural estimation. The survey articles include Eckstein and Wolpin (1989), which we summarize in Section 12.2,1 and the splendid articles by Rust (1994a, 1996), which expose the over-ambition in structural estimation. This exposure centers on the computational complexity of dynamic programming which we discuss in Section 12.9. Finally, the recent survey by Eckstein and Van Den Berg (2007) describes some recent research involving the estimation of equilibrium search models which we discuss in Section 12.10. The concluding section discusses some methods developed in operations research which may be potentially useful in ameliorating, to some extent anyway, the bane of computational complexity.

12.2 EARLY MODELS (KIEFER AND NEUMANN, 1979a, 1979b) The early attempts to incorporate the implications of the BSM into empirical work used selection correction methods (Heckman 1979) to adjust for the fact that accepted wages were not random draws from the underlying wage distribution but were wages that exceeded an individual’s reservation wage (Kiefer and Neumann 1979a, 1979b).2 Let ξ denote the reservation wage of an individual and w0 a wage offer. Then the offer is accepted only if w0 > ξ. Now, assume that the wage offer satisfies w0 = x′β + ε where x is a vector of regressors, β a vector of parameters and ε is a mean zero, standard deviation σε normally distributed random variable with

Structural estimation methods

357

E(ε|x) = 0. The reservation wage is modeled as ξ = z′α + ν where the vector of regressors z may contain x but has at least one additional variable not in x for identification purposes, α is a vector of parameters and ν is a mean zero, standard deviation σν, normally distributed random variable with E(ν|z) = 0. The offered wage is accepted then if w0 = x′β + ε > ξ = z′α + ν or ε − ν > z′α − x′β. The expected value of the accepted wage wa then has the form x′β − z′α σ E(wa) = x′β + E(ε|w0 > 0) = x′β + ρσε +τ x′β − z′α Φ σ

冢

冣

冢

冣

φ

where E(τ) = 0 and σ = {σ2ε + σ2ν − 2σεν} with σεν representing the covariance between ε and ν. Thus, not accounting for the selectivity induced by optimal search results in biased estimates of the impact of the regressors, x, on the expected value of the offered wage, w0. Later empirical work chose to embed all the restrictions of optimal job search, or more generally dynamic models of discrete choice, into the estimation. We turn to some illustrations of these in the next section. 1 2

12.3 ILLUSTRATIONS OF STRUCTURAL ESTIMATION In this section we present some illustrative examples of structural estimation. We review briefly the models of Gotz and McCall (1984) which looked at job retention in the U.S. Air Force, Miller (1984) which analyzed occupational matching, and Christensen and Kiefer (1994) which looks at the prototypal search model.

12.3.1 Gotz and McCall (1984) One early application of structural estimation methods had to do with an individual’s decision of when to leave a firm.3 Individuals typically do not know whether or how their career at a particular firm will develop and must

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revise their choices as information is revealed. For example, individuals do not know whether or not they will be promoted over a certain time horizon or whether they may be slotted into the fast track for top managerial jobs. As this information is revealed an individual will then compare their current and expected future circumstances with alternative opportunities at other firms. One large organization where the structure of promotion opportunities is well known by the employees is the United States Air Force. This is the organization that was analyzed by Gotz and McCall. In particular, Gotz and McCall analyzed optimal stay/leave decisions of Air Force officers using a stochastic dynamic programming model and estimated the underlying parameters of the model using structural estimation methods.4 Gotz and McCall assume that there is a time-constant unobserved taste parameter for military service, γ, as well as independently identically distributed transient returns εt,t = 1,2,. . . . In the Air Force, an officer’s pay is a function of their grade (captain, major, etc.), whether they are regular or reserve status, and at what year of service they were promoted to the current grade. These combinations form the state space in this model. Let Pijt be the probability of transiting from state i to j at year of service t. Many of the Pijt are zero as it is impossible to be promoted from, for example, a captain to a colonel. Denote the external labor market opportunities of an officer in state i who leaves the military after t years of service as Wt(i). More specifically, Wt(i) is the expected present discounted value of moving into the civilian labor market. The discount factor is denoted by β. Finally, after twenty years of service an officer is eligible for retirement pay. For officers in the Air Force retirement pay equals rt = min{c,ftI(t ≥ 20)} × mit where f is the fraction of pay per year of service and equaled 0.025 and c equals the cap which in the 1970s equaled 75 percent of pay. We will denote the optimal value function for a person in state j at time t who has the preference γ and the current transient return ε by Vt( j,γ,ε). Letting stj be the probability of surviving until year j given survival until t, the value of leaving the Air Force at time t, Lt(i), is then ∞

Lt(i) = min{c,ftI(t ≥ 20)} × mit

冱sβ tj

j−t

+ Wt(i).

j=t+1

Under the circumstances, the expected return of remaining in the Air Force, Rt[j,γ,εt], one more period satisfies: J

冱s

Rt[j,γ,εt] = εt + β

t,t + 1

Pijt{γ + mj,t + 1 + Eε[Vt + 1( j,γ,ε)]}.

j=1

The expectation Eε[Vt + 1( j,γ,ε)] in equation (12.1) satisfies

(12.1)

Structural estimation methods

359

∞

冮V

Eε[Vt + 1( j,γ,ε) ] =

( j,γ,ε)dF(ε)

t+1

−∞ ∞

=

冮

−ct + 1( j,γ)

[ε + At + 1( j,γ)]dF(ε) + Ut + 1

−ct + 1( j,γ)

冮

dF(ε)

−∞

∞

=

冮

εdF(ε) + At + 1( j,γ)[1 − F(−ct + 1( j,γ))] + Ut + 1F(−ct + 1( j,γ))

−ct + 1( j,γ)

where At + 1( j,γ) represents the return from staying net of the transient return and is equal to J

冱s

At + 1( j,γ) = β

Pj,l,t + 1{γ + ml,t + 2 + EεVt + 2(l,γ,ε)}

t + 1,t + 2

l=1

and − ct + 1( j,γ) is the threshold value of εt + 1 such that the officer quits the Air Force next period if εt + 1 < ct + 1( j,γ). It is clear that the optimal value function then satisfies Vt( j,γ,ε) = max{Lt( j),Rt( j,γ,ε)}.

(12.2)

Gotz and McCall assume that there is some distribution of the tastes γ and we will denote its c.d.f. by G and its p.d.f. by g. It is interesting to consider how the distribution of the unobserved taste variable changes with an officer’s job tenure in the Air Force. First, consider that there had been some selection into the reserve versus regular categories based on γ. Let Pr(Reg|γ) be this exogenously determined probability. Then by an application of Bayes theorem we have g*(γ|Reg) =

Pr(Reg|γ)g(γ) 冮Pr(Reg|γ)g(γ)dγ

g*(γ|Res) =

Pr(Res|γ)g(γ) 冮Pr(Res|γ)g(γ)dγ

and

where Pr(Res|γ) = 1 − Pr(Reg|γ). More generally, we can think of the conditional probability of γ given the history up to time t, Ft, which we denote by g*(γ|Ft). The data used to estimate the model will consist of the sequence of states an officer transits through

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until they leave the Air Force or are right-censored. Thus, we need to calculate the probability that an officer stays in the Air Force at time t + 1 given that he/she stayed until time t, the state at time t was i, and the state at time t + 1 was j. This probability, Pr{St + 1(it + 1)|St(it)}, satisfies ∞

∞

冮冤 冮 Pr{St + 1(it + 1)|St(it)} =

冮

dF(ε)

− ct(it,γ)

冥

dF(ε) g*(γ|Ft − 1)dγ

− ct + 1(it + 1,γ)

.

∞

冮 冮

dF(ε)g*(γ|Ft − 1)dγ

− ct(it,γ)

To compute g*(γ|Ft) we repeatedly apply Bayes theorem. So, g*(γ|F1) =

Pr(ε > c1(i1,γ)) × g*(γ|Reg) [1 − Fε(c1(i1,γ))] × g*(γ|Reg) = 冮Pr(ε > c1(i1,γ))g*(γ|Reg)dγ 冮[1 − Fε(c1(i1,γ))]g*(γ|Reg)dγ

g*(γ|Ft) =

Pr(ε > ct(it,γ)) × g*(γ|Ft − 1) [1 − Fε(ct(it,γ))] × g*(γ|Ft − 1) = 冮Pr(ε > ct(it,γ))g*(γ|Ft − 1)dγ 冮[1 − Fε(ct(it,γ))]g*(γ|Ft − 1)dγ

or t

冲[1 − F (c (i ,γ))] × g*(γ|Reg) ε

g*(γ|Ft) =

j

j

j=1 t

冲[1 − F (c (i ,γ))]g*(γ|Reg)dγ

冮

ε

j

j

j=1

for regular officers with a corresponding posterior probability function for reserve officers. Now for a given γ the probability of leaving at time t equals ∞

冢

Pr{Lt(it)|γ} = 1 −

冮

t−1

∞

冣 冲 冮

dF(ε) ×

− ct(it,γ)

dF(ε)

h = 1 − c (i ,γ) h

h

t−1

= (F(−ct(it,γ))) ×

冲[1 − F(−c (i ,γ))]. h

h

h=1

To obtain the unconditional probability we integrate out the variable γ:

Pr{Lt(it)} =

冮冦

t−1

(F(−ct(it,γ))) ×

冲[1 − F(−c (i ,γ))]冧dG(γ) h

h=1

h

(12.3)

Structural estimation methods

361

or

Pr{Lt(it)} =

t−1

冮冦

(F(−ct(it,γ);θ)) ×

冲[1 − F(−c (i ,γ))]冧dG(γ;δ) h

h

(12.4)

h=1

where θ and δ are vectors of parameters of the distributions F and G, respectively. These probabilities then form the basis of the likelihood function. The functions ct(it,γ) in (12.4) are computed by solving the dynamic programming problem in (12.2). The parameters θ and δ are estimated by maximum likelihood estimation and can then be used to simulate the impacts of changes in Air Force pay and promotion policies.5

12.3.2 Miller (1984) Recall from Chapter 9 that each occupation has associated with it a Gittins index, Z (γmt,δmt,σm), and that optimal strategy is to sample that occupation with the largest Gittins index. Now since there are many jobs in an occupation and information is assumed to be independent across jobs within an occupation, one implication of Miller’s model is that an individual will continue to sample jobs (if the information from the previous job is unfavorable) in the same occupation until all jobs are tested. The underlying parameters to be estimated by the model are the triplets (γm,δm,σm) for each occupation, m = 1,. . ., M. Now the hazard function λt for switching jobs in the same occupation equals λtm = Pr(Z (γmt,δmt,σm) < Z (γm,δm,σm)) or, using the results from Propositions 9.1 through 9.3: λtm = Pr(γmt + δmtZ (0,αm + τmt) < γm + δmZ (0,αm)). The component of the likelihood function for an individual with completed spell ti in occupation m then equals for individual i ti − 1

Lim = λmt

冲(1 − λ

).

ms

i

s=1

These components are constructed for each job spell for each individual and the log likelihood is maximized with respect to the 3M parameters and the discount rate. To estimate the model, Miller further transforms the model so that the probability of leaving depends only on the time at the job, the information factor, αm and the discount factor β. Moreover, he assumes that these are both functions of employment category and education. Using data from

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The Economics of Search

the National Longitudinal Survey of Youth (NLSY79) Miller estimates the model and finds evidence that the speed of learning does depend on an individual’s education and employment group.

12.3.3 Christensen and Kiefer (1994) The Christensen and Kiefer paper describes how to structurally estimate the BSM model (see also Flinn and Heckman 1982, and Wolpin 1987). In each of the discrete time periods t = 0, 1, 2, . . ., an unemployed worker receives a wage offer with probability p. Jobs are distinguished here only by the value of the wage. The event of receiving an offer and the value of the offer are independent. Unemployed workers receive unemployment income b, net of search costs, until they accept an offer w. After this they expect to receive w every period. Workers maximize the expected discounted value of their income stream. The value function for a worker with outstanding offer w is V(w) = max{V e(w),V u} =

w

冦1 − β, b + βEV 冧,

where β is the subjective discount factor, assumed constant, 0 < β < 1 and EV denotes the expected value function. Since the value V e of employment increases in the wage offer, and the value V u of continuing search is constant, a reservation wage strategy is optimal. The worker accepts w if and only if it exceeds ξ defined by the indifference between unemployment and employment, ξ = (1 − β)(b + βEV ). If offers are distributed according to the density f, the conditional probability of acceptance given an offer has been received is ∞

Π=

冮 f (w)dw.

(12.5)

ξ

The expected value is now given as EV =

1 {pΠE(w|w > ξ) + (1 − pΠ)ξ}. 1−β

(12.6)

The first term in the brackets of (12.6) equals the probability of receiving an acceptable offer times the expected wage given that it is acceptable while the second term equals the probability of an unacceptable offer times the value of continued search. Thus the density of accepted offers is g(w) =

f (w) 1{w ≥ ξ} Π

where 1{·} is the indicator function.

(12.7)

Structural estimation methods

363

Suppose that the aim is to estimate the unknown parameters θ ∈ ⺢k based on a random sample (w1, w2,. . ., wN ) where wi is the accepted wage of the ith individual. Here, the vector θ comprises variables parameterizing the economic model, including the offer distribution f. The likelihood based on (12.7) is N

L(θ) =

冲 i=1

f (wi;θ) I{wi ≥ ξ(θ)}. Π(θ)

(12.8)

This is only positive for θ ∈ Θ + = {θ ∈ Θ : wm ≥ ξ(θ)} where Θ ⊂ ⺢k is the full parameter space and wm = min{wi}. The maximum likelihood estimator (MLE), θˆ = arg maxθL(θ), is typically (exactly or asymptotically) to be found along the boundary ∂Θ + = {θ ∈ Θ : wm = ξ(θ)}. Thus, there is a one-dimensional restriction on the estimator, θˆ, that is determined by the minimum observation. In other words, wm determines a k − 1 dimension sub-manifold of Θ+ to which belongs. This gives rise to nonstandard asymptotics. Christensen and Kiefer show that under a wide set of assumptions N−1/2(θˆ − θ) → Normalk(0,v(θ)) where rank v(θ) = k − 1. The prototypal specification for the offer distribution is the shifted exponential distribution f (w) = γexp−γ(w − c), w ≥ c, with γ > 0 and c ≥ 0 and E(w) = c + γ−1. This specification is empirically relevant in that it allows clustering close to the reservation wage. In this case (12.5) becomes Π = e−γ(ξ − c) and the density of accepted wages is then g(w) =

γe−γ(w − c) 1{w ≥ ξ} = γe−γ(w − ξ)1{w ≥ ξ}. e−γ(ξ − c)

Thus, the likelihood function (12.8) where θ = (γ, ξ) is given by L(θ) = N

γ exp(− γ N

冱(w − ξ))1{w i

m

≥ ξ}. The derivative of the log likelihood or score

i=1

N

冱冤w − 冢ξ + γ冣冥 and s

function is given by sγ(θ) = −

1

i

ξ

> 0.

i=1

In the prototypal model, E [w|w ≥ ξ ] = ξ + γ−1, so E(sγ) = 0 at the true parameter value. However, over Θ + , ξˆ = wm, where wm is the minimum wage observed in the sample, and thus γˆ =

N N

.

冱(w − ξˆ ) i

i=1

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The Economics of Search

Without measurement error the asymptotics of the estimates are nonstandard. In particular, the estimate of ξ is Op(N−1) instead of Op(N−1/2). This inference result depends not only on wm being the exact wage offered to and accepted by individual j but also on the exactness of the sξ(θ) = Nγ measurements of wi, i ≠ j. However, as a practical matter Christensen and Kiefer note that data on wages contains measurement error and so develop a model that incorporates measurement error. Christensen and Kiefer again assume that the true accepted wages are distributed according to a shifted exponential. Thus the true accepted wages are distributed according to g(w) = γe−γ(w − w )1{w ≥ ξ}. Thus, no wage will be accepted below ξ. However with measurement error, the wage observed by the econometrician, we, may be below ξ. Assume that we = wm where m > 0 is a (multiplicative) random measurement error factor. We assume that the measurement error m is distributed independently of w. Denote the p.d.f. of m by h. Then, r

∞

冣 冮冢

x x = P w ≤ | m h(m)dm. P(w ≤ x) = P(wm ≤ x) = P w ≤ m m 0

冢

e

冣

Since w and m are independent we can ignore the conditioning and x/ξ

P(we ≤ x) =

冮 冢1 − e

(

x

− γ m −ξ

冣

) h(m)dm.

0

(12.9)

To determine the density of observed wages, (12.9) must be differentiated by x x/ξ

λ

冣 冢 冣 冮 me

x 1 x −ξ fe(x) = 1 − e−γ( x/ξ ) h + wr ξ

冢

( x − ξ)

−γ m

h(m)dm

0

x/ξ γξ

= γe

冮 mh(m)e 1

−γmx

dm.

0

This expression forms the basis for all applications of the prototypal search model with measurement errors. From the worker’s perspective, however, the model is unchanged. Assume that the measurement error has the following distribution h(m) = α1m−(α

2

Then,

+ 2)

, m > 0.

Structural estimation methods

365

e

w /ξ

冮αm

γξ

fe(w ) = γe e

1

−(α2 + 3)

e

− (α2 + γx m

dm

0

αα2 + 1Γ(α2 + 2,ξ(γ + wα )) (α2 + γwe)α + 2Γ(α2 + 1) 2 e

2

= γeγξ

(12.10)

2

where Γ(x,y) is the upper incomplete gamma function and Γ(x)/Γ(x,0). Equation (12.10) can be alternatively written as fe(we) = γeγξ

α2α + 1 (α2 + γwe )α 2

2

+2

(α2 + 1)S(ξ(γ + wα );α2 + 2) 2 e

(12.11)

where S(x;b) denotes the survivor function for Γ(b,1). From (12.11) the log-likelihood function for the unknown parameter vector θ = (α2,γ,ξ) based on a random sample w = (w1, w2, . . ., wN) of observed wages is ᐉ(θ) = N {lnγ + γξ + (α2 + 1)lnα2 + ln(α2 + 1)} N

N

α2 + lnS ξ γ + ;α2 + 2 − (α2 + 2) ln(α2 + γwi). wi i=1 i=1

冱 冢冢

冣

冣

冱

With measurement error the score of the likelihood with respect to ξ exists and thus conventional MLE can be used to obtain an estimate of θ = (α2,γ, ξ). 12.4 THE GENERAL MODEL In the next few sections we will develop the general methodology of structural estimation for discrete choice models. These sections closely follow Eckstein and Wolpin (1989). Consider a model with I discrete choices over T periods of time, where T is finite or infinite. In each period an individual chooses one of the possible I alternatives where di(t) = 1 if alternative i is chosen at time t and di(t) = 0 otherwise. Alternatives are mutually exclusive so i

冱d (t) = 1. i

i=1

The objective of the individual at any time t, t = 0,1, . . .,T is to maximize T

E

冤冱β j=t

j−t

冥

Σ Ri( j)di( j)|F (t)

i∈t

(12.12)

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The Economics of Search

where 0 < β < 1 is the individuals discount rate, E(·) is the mathematical expectations operator, F(t) is the individual’s information set at time t, which can include all past and current realizations of variables that directly or indirectly influence the value of equation (12.12), and Ri(t) is a random variable representing the individuals reward if alternative i is chosen at time t. The maximization of (12.12) is accomplished by the choice of the optimal sequence of control variables {di(t)}i ∈ 1 for t = 0,1, . . .,T, which are functions of the information that is available when the decision is made. Define the maximal expected value of the reward at time t by T

V(F(t)) = sup E {di(t)}i ∈ 1

where R( j) =

冤冱β

j−t

冥

R( j)|F(t)

j=t

(12.13)

冱R ( j )d ( j) is the actual reward at time j. The function V depends i

i

i ∈1

only on the information set at time t and obeys the dynamic programming equation V(F(t) = max {LiV(F(t))}

(12.14)

i∈I

where LiV(F (t)) = Ri(t) + βE [V(F(t + 1))|di(t) = 1)], t = 0,1, . . .,T.

(12.15)

It is not difficult to show that the basic job search model can be put in this framework. In this case, there are two alternatives, i = 1, 2. Let d1(t) = 1 if the individual is not employed (searching) and d1(t) = 0 otherwise, and d2(t) = 1 if the individual is employed and equals 0 otherwise. The reward function is given by R1(t) = b R2(t) =

冦R w(t) (t − 1) 2

if d1(t − 1) = 1 if d2(t − 1) = 1

where b is the net income the individual receives while searching (unemployment compensation-search costs) and w(t) is the wage at time t. If the individual is not employed at time t, a wage offer is drawn with known probability p from a time-independent distribution function which is also known. With a probability equal to 1 − p no wage offer is received and net income is equal to b.

Structural estimation methods

367

12.5 SOLUTION METHODS In order to estimate the parameters of these type of models a closed form solution (not necessarily analytical solution) is required for the optimization problem characterized by (12.12) The actual computation of the solution may be technically different for finite horizon and infinite horizon models. In the finite horizon case, the method is a backwards sequential solution of the Bellman equation. Specifically, equation (12.15) can be written as follows: LiV(F (t)) = Ri(t) + βE[V(F (t + 1))|di(t) = 1)], t = 0,1, . . .,T − 1, i ∈ I

(12.16)

LiV(F (T)) = Ri(T), i ∈ I.

(12.17)

and

The solution for LiV(F (t)) is obtained by substituting recursively from T. There exists a unique optimal policy which is characterized by a sequence of stochastic reservation values, R*i (t) t = 1, . . .,T. For the infinite horizon case we seek a time-independent value for Li(V(F(t)) for each i ∈ 1 that satisfies LiV(F) = Ri(t) + βE [V(F ′)|di = 1], i ∈ I.

(12.18)

12.6 FINITE HORIZON (T < ∞) SEARCH MODEL In the search model LiV is the value function when d1(t) = 1, i.e., the individual is not employed and looking for work, and L2V is the value function when d2(t) = 1, i.e., the individual is employed. In the last period, T, the problem becomes L1V(T) = b, L2V(T) = w(T). It is clear that d2(t) = 1 if and only if w(T) > b. For t < T equation (12.16) can be written as L1V(t) = b + βE[V(t + 1)|di(t) = 1],

(12.19)

L2V(t) = w(t) + βE[V(t + 1)|d2(t) = 1].

(12.20)

and

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The Economics of Search

Because the simple search problem is an optimal stopping problem, the solution is of the form d2(t + 1) = 1 if d2(t) = 1; moreover, R2(t + 1) = w(t) if d2(t) = 1, so that (12.20) becomes T

冱β

s−t

L2V(t) = w(t)

=

s=1

w(t)(1 − βT + 1 − t) . 1−β

The solution to the search problem consists of a sequence of reservation wages, ξ(t), such that the job is accepted if the offered wage equals or exceeds ξ(t) and is rejected otherwise. The reservation wage at t, ξ(t), solves the equation T

冤冱β 冥 = L V(t) s−t

ξ(t)

1

s=1

which implies that ξ(t) = b. For t < T we can solve sequentially for L1V(t) in terms of ξ(s), T ≥ s > t. That is using (12.20) we obtain L1V(t) = b + βE{pmax[L1V(t + 1),L2V(t + 1)] + (1 − p)L1V(t + 1)} = b + βpL1V(t + 1)Pr(w(t + 1) < ξ(t + 1))) T

冢冱β

冣 E(w(t + 1|

s − (t + 1)

+β

s=t+1

|w(t + 1) ≥ ξ(t + 1)Pr(w(t + 1) ≥ ξ(t + 1))) + β(1 − p)L1V(t − 1), where ∞

冮

Pr(w(t + 1) < ξ(t + 1)) = f (w(t + 1))dw(t + 1), 0

∞

E(w(t + 1)|w(t + 1) ≥ ξ(t + 1)) =

冮

f (w(t + 1))dw(t + 1)

ξ(t + 1)

Pr(w(t + 1) ≥ ξ(t + 1))

and L1V(t + s), s ≥ 0 is given by (12.19). 12.7 ESTIMATION METHODS Consider a panel of M individuals or households, in which, at a minimum, the choice of each alternative i is observed for each individual for Tm periods.

Structural estimation methods

369

Thus let the decision set for household m be d m(t) = [d m1 (t),d m2 (t), . . ., d ml(t)] and d m = [d m(1),d m(2), . . ., d m(Tm)]. Full information maximum likelihood can be employed as the method of estimation. In general, M

L=

冲

m=1

M

PT(d m) =

Tm − 2

冲 冤 冲 Pr(d

m=1

(Tm − r)

m

t=0

冥

|d m(Tm − (r + 1)),d m(Tm − (r + 2), . . ., d m(Tm − (r + Tm + 1))) Pr(dm(1)). The solution of the dynamic programming model can be used to provide a likelihood value for the parameters of the model conditional on the data dm, m = 1, 2, . . ., M. The parameters which maximize the likelihood function can then be found by a numerical nonlinear optimization algorithm.

12.8 STRUCTURAL ESTIMATION OF EQUILIBRIUM JOB SEARCH MODELS Equilibrium models of job search have also been structurally estimated. In this section we review the structural estimation of three such models; the estimation of an extension of the Albrecht and Axell equilibrium model by Eckstein and Wolpin (1990) which was the first structural estimation of an equilibrium job search model, the structural estimation of a Burdett– Mortensen type equilibrium model by Van Den Berg and Ridder (1998) and a recent study by Cahuc et al. (2006) which structurally estimates an equilibrium model of job search with wage bargaining and on-the-job search.

12.8.1 Eckstein and Wolpin (1990) Eckstein and Wolpin’s (1990) paper structurally estimated an extended version of Albrecht and Axell’s (1984) labor market search equilibrium model. In this model workers are homogenous in productivity but differ in their tastes for leisure. It is this heterogeneity that gives rise to an equilibrium wage distribution. In the Eckstein and Wolpin model, during each period infinitely lived workers enter and exit the labor market. The probability of leaving the labor market is assumed to be a constant and is denoted by τ. While in the labor market an individual is either working or searching for work. Once a wage

370

The Economics of Search

offer is accepted the worker remains at the job until they exit the labor market. Eckstein and Wolpin assume that once an individual leaves the labor market they do not return and they receive a lifetime income of R. Let w be the accepted wage. Then the expected present discounted value of accepting a wage of w equals V a(w) =

w βτR + 1 − (1 − τ)β 1 − (1 − τ)β

(12.21)

where β denotes the discount factor. The key heterogeneity in this model is Eckstein and Wolpin’s assumption that there are n + 1 types of individuals each with a different value of per-period nonmarket time, zj, j = 1,. . ., n. The value of not working for individual j equals V rj = zj + β{(1 − τ)[pEmax(V a(w),V rj) + (1 − p)(V rj) ] + τR}

(12.22)

where p is the probability of receiving a job offer. Recall that with no heterogeneity, workers would be offered their reservation wage. The Nash equilibrium wage distribution with n types of individuals turns out in this model to be a distribution with n different wages where each distinct wage is the reservation wage of a particular type of individual. Ordering these types from the lowest to the highest value of nonmarket time and denoting the reservation wage for type j by ξj, the proportion of firms offering that wage by γj, j−1

and defining γ = j

冱γ we have i

i=0

n

Emax[V (w),V ] = γ V + a

r j

j

r j

冱γ V (w ). a

i

i

(12.23)

i=j

After substituting (12.21) and (12.23) into (12.22) the (reservation) wages are found to satisfy ξj = (zj + b)

1 − β(1 − τ) 1 − β(1 − τ)(1 − p + pγ j + 1) n

β(1 − τ)p + γiξi, 1 − β(1 − τ)(1 − p + pγ j + 1) i = j + 1

冱

for j = 1,. . ., n − 1 and ξn = zn for j = n. Firms are assumed to have linear production technologies y = λl

(12.24)

Structural estimation methods

371

where l equals the number of employees and λ is a “productivity index.” The productivity index, is assumed to come from a continuous distribution A(λ) with A(0) = 0. Each firm’s productivity index is private information that only the firm knows and the firm’s objective is to maximize profits π(w,λ) = (λ − w)l(w) where l(w) is the labor supply function to the firm. For any wage less than λ the firm would like to hire as many workers as possible but expects that in equilibrium the number of workers available declines as w is lowered. Suppose that wages w0,. . ., wn are equilibrium wages offered by firms. Then there is a critical value of the productivity index, λj which makes a firm indifferent between offering wj and wj + 1. The solution to this critical value is λj =

wjl(wj) − wj − 1l(wj − 1) . l(wj) − l(wj − 1)

(12.25)

A firm with λ between λj and λj + 1 will offer a wage wj and λ0 = w0. Note that in equilibrium it would make no economic sense for a firm to deviate from offering one of the j + 1 wages since raising the wage a little bit would not increase the firm’s labor supply and lowering the wage a little bit results in a discontinuous drop in the firm’s labor supply. In order to establish the equilibrium it is necessary to determine the supply of workers per firm offering wage wj, l(wj) for all j = 0,1,2,. . .,n. Let ωj represent the proportion of type j workers and k represent the total number of workers in the market. The number of workers of type j that join the market each period is then τkωj. Denoting the number of potential firms by F, the number of workers per active firm, µ(λ0), satisfies µ(λ0) =

k F(1 − A(λ0))

(12.26)

and the proportion of total workers per firm offering wj (= ξj ) is ∞

j

l(wj) =

冱冤τµω 冦冱(1 − τ) 冧 × p 冦冱((1 − p) + pγ ) 冧冥 r i

r

r=0 j

=

∞

i

i=0

(12.27)

i=0

µωrp

冱1 − ((1 − p) + pγ )(1 − λ) r

r=0

Equation (12.26) shows that l(wj) increases with j so that at higher wages there is greater market supply. Moreover, this result guarantees that λj in (12.25) is positive.

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The Economics of Search

The probability of receiving an offer p is assumed to be an increasing function of the number of active firms per worker and so satisfies p=G

1

冢µ(λ )冣,G′ > 0,

(12.28)

0

where G is a technological function that maps R to [0,1]. Eckstein and Wolpin then establish the existence of a Nash equilibrium where is a Nash equilibrium is a probability density function such that wj( = ξj) has probability γj satisfying (12.24), the firm’s strategy is to offer wj if λ ∈ (λj,λj + 1) where λj is determined by (12.25) and (12.26) and is increasing in j, l(wj) is determined by (12.27), p is given by (12.28) and the probabilities, γs, are equal the proportion of each active firm offering each wage: γj =

A(λj + 1) − A(λj) for all j = 1, 2, . . ., n − 1 1 − A(λ0)

with n

冱γ = 1. j

j=0

Given that the equilibrium distribution of wages exists, the model generates an equilibrium distribution of unemployment durations for a cohort of new entrants. Let D be a random variable representing the duration of search unemployment, then n

Pr(D = d + 1) =

冱(1 − τ) [pγ + (1 − p)] p(1 − γ )ω ,d = 1, 2,. . . d

j

d

j

j

j=0

and the hazard rate function, h, in period d + 1 satisfies hd + 1 =

fd + 1

, d = 1, 2,. . . .

d

(1 − τ)d

冲(1 − h ) k

k=1

The equilibrium unemployment rate is then n

s=

τω j[pγ j + (1 − p)] j j = 0 1 − (1 − τ)[pγ + (1 − p)]

冱

Structural estimation methods

373

while the mean accepted wage in the market is

 wjγjl(wj)  n  ¯¯ = 冱wjηj = 冱 w  . j=0 j = 0 冱γjl(wj) j = 0  n

n

Eckstein and Wolpin use data from the NLSY79 to estimate the model. Letting ci be a variable that equals 1 if an individual spell is complete and 0 if it is right censored and di the duration of the spell, the log likelihood function equals n

logL =

冱(1 − c )log 冤冱(1 − τ)

di − 1

i

冥

(pγj + (1 − p))d ωj j

j=1

i

(12.29)

n

+

冱c log 冤冱(1 − τ) (pγ + (1 − p)) p(1 − γ )ω 冥 di

i

i

j

dj

j

j

j=1

where the parameter p and the n + 1 γs are functions of the n + 1 zs, β, the n + 1 ωs, τ, the distribution parameters of A(λ) and the parameters of the function G. One can consider the parameters in (12.29) as the unrestricted version while the equilibrium model imposes further restrictions on p and the γs. The estimates from the NLSY79 clearly reject the restrictions of the structural equilibrium model. Eckstein and Wolpin also estimate a model that includes the wage data under the assumption that wages are measured with error and that the measurement error follows a normal distribution. This allows them to free up one restriction on the zs needed for identification. While the estimates using the wage data does slightly better at fitting the first 50 periods of unemployment, overall it performs more poorly than the model without wage data.

12.8.2 Van Den Berg and Ridder (1998) As mentioned above, Van Den Berg and Ridder estimate a version of the Burdett Mortensen equilibrium model. The basic assumptions of the model are as follows: Assumption 12.1 There are a continuum of workers and firms with mass m and 1, respectively. Assumption 12.2 Workers receive job offers at rate λ0 if unemployed and λ1 if employed. Job offers are assumed to be i.i.d. draws from a wage offer

374

The Economics of Search

distribution with c.d.f. F(w). An offer is either accepted or rejected upon arrival (no recall). During the tenure of a job the wage is constant and the utility flow from being employed at wage w is w. Assumption 12.3 Job worker matches break up at an exogenous rate δ. The utility from being unemployed equals b. Assumption 12.4 Firms have linear production technologies and the marginal product equals p. A firm pays all its workers the same wage. Assumption 12.5 Workers maximize their expected wealth and firms maximize their expected, steady-state, profits. Under these conditions the reservation wage with a zero discount rate can be shown to solve ∞

ξ = b + (λ0 − λ1)

F¯¯ (w)

冮 δ + λF¯¯(w)dw

(12.30)

ξ

where F¯¯ = 1 − F. In addition, an employed worker accepts a wage offer only when it exceeds his/her current wage. Under these circumstances a worker never quits to search for a better job. However, an employed worker may become unemployed and this occurs at rate δ. Define the earnings distribution G as the distribution of wages earned by those employed. Then there are G(w)(m − u) workers who receive wage w or less where u represents the number of unemployed workers. Now, the flow of employed workers into jobs that exceed wage is equal to λ1F¯¯ (w)G(w)(m − u) and the flow of employed workers into unemployment who earned w or less equals δG(w)(m − u). In steady state this outflow from jobs that pay w or less must be balanced by an equal inflow. This inflow only occurs from unemployment and equals λ0(F(w) − F(ξ))u where F(ξ) = 0 since firms offering less than the reservation wage will not attract any workers. Hence in steady state the following relation holds G(w) =

F(w) λ0u × . δ + λ1F¯¯ (w) (m − u)

(12.31)

The steady-state unemployment rate is determined by taking the limit of (12.31) as w → ∞ which gives u δ = . m δ + λ0 Substituting this back into (12.31) then yields

Structural estimation methods G(w) =

375

δF(w) . δ + λ1F¯¯ (w)

In this framework, a match between a worker and employer generates a flow of net revenue p − b. At the prevailing wage w, these rents are split between the firm, p − w, and worker w − b. Note that the wage is posted prior to firm contacts with searching individuals and there is assumed to be no bargaining over the wage. Exploring firm behavior in more detail, first note that the steady-state level of production is determined by the size of the available workforce l which, in turn, depends on wage set by the firm, the reservation wage set by the unemployed individuals, and the distribution of wages, F, set by the other firms that compete for the same pool of labor (i.e. l(w;ξ,F)). For firms the steady state profit flow π then equals π = (p − w)l(w;ξ,F). Given this setup, the equilibrium wage distribution must be absolutely continuous and by implication not degenerate. Suppose not and that there were a mass of firms offering wage w. Then by offering a wage slightly higher than w, a firm could significantly increase its labor force while suffering only a second-order loss on per worker profit. Hence, total profits would increase. Thus in equilibrium F and G have probability density functions f and g with support [ξ,w ¯¯ ] where w ¯¯ < p. So, the measure of workers earning w equals g(w)(m − u)dw and the measure of firms offering w equals f (w)dw. This implies that l(w;ξ,F) =

g(w)dw mλ0δ(δ + λ1) (m − u) = on [ξ,w ¯¯ ]. f (w)dw (δ + λ0)(δ + λ1F¯¯ (w))2

In steady state, then, the equilibrium profit of a firm paying ξ equals π(ξ;ξ,F) = (p − ξ) ×

mλ0δ

冤(δ + λ )(δ + λ )冥. 0

(12.32)

1

In equilibrium, all higher paying firms must earn the same profit as in (12.32). Thus mλ0δ(δ + λ1) mλ0δ (p − w) = (p − ξ) × 2 (δ + λ0)(δ + λ1F¯¯ (w)) (δ + λ0)(δ + λ1)

冤

冥

or

F(w) =

δ + λ1 × 1− λ1

冢

p−w

冪 p − ξ 冣 on [ξ,w¯¯ ]

(12.33)

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with density f (w) =

δ + λ1 1 × . 2λ1√p − ξ √p − w

The reservation wage follows from (12.31) and (12.33) and satisfies ξ=

b · (δ + λ1)2 + p · (λ0 + λ1) (δ + λ1)2 + λ1 · (λ0 − λ1)

while ¯¯ = w

δ 2 δ ·ξ + 1 − δ + λ1 δ + λ1

冢

冣

冦 冢

2

冣 冧·p

and the equilibrium earnings density is g(w) =

δ√p − ξ 1 × . 2λ1 (p − w) 3 2

Finally, the distribution of job tenures for jobs paying w is exponential with parameter δ + λ1F¯¯ (w), and the distribution of unemployment durations is exponential with parameter λ0. One drawback to this model is the implication that the earnings density distributions are upward sloping which is at odds with the data. To circumvent this, Van Den Berg and Ridder assume that there are heterogeneous markets that differ in p. Thus, labor markets are divided into segments depending on observed characteristics x and unobserved characteristic v. More specifically, p is modeled as p = v exp(β′1x). Van Den Berg and Ridder (1998) also assume that wages are measured with error and that the measurement error distribution follows a log-normal distribution. The model is empirically estimated using Dutch panel data which contains employment to employment, employment to unemployment, and unemployment to employment spells. Based on the empirical estimates, Van Den Berg and Ridder are able to decompose the total variation in observed wages into components due to search frictions (22 percent), variation in x (33 percent), and variation due to measurement error (23 percent) and variation due to unobserved differences in productivity (21 percent). Thus, unlike the findings of Eckstein and Wolpin most of the variability in the wage distribution is due to factors other than measurement error.

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12.8.3 Cahuc, Postel-Vinay, and Robin (2006) In this paper the authors estimate an equilibrium model of job search in which there is strategic wage bargaining and on-the-job search. Unlike the model of Van Den Berg and Ridder (1998), which assumes that wages are posted and no wage bargaining occurs, the model in this study allows for a three-way bargaining process between the worker and the incumbent and raiding employers when a worker receives an outside wage offer. This paper considers a labor market with measure M workers and measure l of employers. Workers have different abilities. A given worker’s ability is denoted by ε and the distribution of abilities among workers in the population is assumed to be exogenously determined with continuous cumulative distribution H over [ε,ε-] and associated p.d.f. h. A firm’s productivity is denoted by p. Firms differ in productivities with the continuous distribution of productivities denoted by J over [p,p¯ ] that admits the p.d.f. j. Cahuc et al. assume that the marginal productivity of an ε-worker with a p-firm to be pε. Moreover, an unemployed ε-worker is assumed to receive bε in benefits. As in Van Den Berg and Ridder (1998), the rate of job offers while employed is exogenous fixed at the constant λ1 while the rate of job offers while unemployed is exogenously fixed at the constant λ0. The market unemployment rate is denoted by u. Letting ρ be the discount rate, workers are assumed to maximize the expected discounted flow of income. When a worker receives an offer from a firm, it is assumed to be a random draw from a wage distribution with c.d.f. denoted by F (with F¯¯ ≡ 1 − F) and p.d.f. denoted by f. Wages are bargained over by workers and employers in a complete information context. In particular, all job and wage offers are perfectly observed and verifiable. Renegotiations are also assumed to be costless. Let V0(ε) denote the expected lifetime utility of an unemployed ε-worker and V(ε,w,p) denotes the expected lifetime utility of an ε-worker employed at a p-firm at wage w. So, the match surplus is V(ε,εp,p) − V0(ε). Cahuc et al. show that the outcome of a Rubenstein (1982) infinite-horizon game of alternating wage offers is a generalized Nash-bargaining solution where a worker receives a constant share α of the rents to the match. So, the bargained wage on a match between an unemployed ε-worker and p-firm, denoted by (ε,p) satisfies V(ε,(ε,p),p) = V0(ε) + α [V(ε,εp,p) − V0(ε)]. While in principal α may depend on other factors such as the time between offers, discount rates, etc., it is shown that α is approximately independent of the other structural parameters in the model and so is treated as a structural parameter to be estimated. First consider the case where an employed worker gets an offer from a p′firm with p′ > p. Competition will, in equilibrium, result in the employee moving from the p-firm to the p′-firm. What will be the wage? Here the

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outside option is the worker’s alternative of working at the p-firm so the wage at the p′-firm satisfies V(ε,(ε,p,p′),p′) = V0(ε) + α[V(ε,εp′,p′) − V0(ε)]. Renegotiation takes place only if it is in the interest of the worker. The question then arises whether renegotiation would take place if a worker received an offer from a p′-firm where p′ < p. This would occur if (ε,p′,p) > w. So there exists a threshold value of p such that: 1 2 3

If p′ < q(ε,w,p), then the worker continues to work at the p-firm for wage w. If p ≥ p′ > q(ε,w,p), then the worker continues to work at the p-firm for wage (ε,p′,p). If p′ > p, then the worker moves to p′-firm for wage (ε,p,p′)

Cahuc et al. (2006) show that the wage of an ε-worker currently working at a p-firm whose last wage offer was from a p′-firm satisfies p

冢

(ε,p′,p) = ε· p − (1 − α)

ρ + δ + λ1F¯¯ (x)

冮 ρ + δ + αλ F¯¯(x) dx冣 for p′ < p. 1

p′

While for an unemployed ε-worker p

ρ + δ + λ1F¯¯ (x)

冮ρ + δ + αλ F¯¯(x)dx冣

冢

0(ε,p) = ε· p − (1 − α)

1

p ¯

where p is the lowest viable marginal productivity of labor that makes the worker indifferent between employment and unemployment. Now, V(ε,εp, p) = V0(ε) if and only if p¯

F¯¯ (x)

冮ρ + δ + λ αF¯¯(x)dx.

p = b + α(λ0 − λ1)

p ¯

1

It is important to note that in these models workers may move to firms that pay less than their current pay (or if unemployed, less than b) because part of the benefit of being at a firm with higher p is the better opportunity that this firm provides for future wage increases. In steady state the following must be true

Structural estimation methods u=

δ , δ + λ0

L(p) =

with ψ1 =

l(p) =

379

(12.34) F(p)

(12.35)

1 + ψ1F¯¯ (p)

λ1 , δ 1 + ψ1 f (p), [1 + ψ1F¯¯ (p)]

(12.36)

l(ε,p) = h(ε)l(p),

(12.37)

and G(w|ε,p) =

冢

2 1 + ψ1F¯¯ (p) 1 + ψ1L(q(ε,w,p)) 2 = . 1 + ψ1F¯¯ (q(ε,w,p)) 1 + ψ1L(p)

冣 冢

冣

(12.38)

The estimation strategy in this paper consists of several steps. First, firmlevel productivity data identifies p. Next, worker data on job and employment durations are used to draw inferences on the job destruction rate δ and job offer arrival rate λ1. The model implies that the likelihood function for job spell durations, after integrating out the distribution of p, satisfies ∞

n

∞

δ(1 + ψ1) e−x e−x L(t) = dx − dx . ψ1 x x i=1 δt δt (1 + ψ )

冲

冮

冢冮

i

i

冣

1

The empirical distribution of the firm’s p type among workers then identifies the distribution of firm types across employees L(p). Finally, the intercept and slope parameters of the regression of log wages on log productivity by occupation and industry identify mean worker ability and bargaining power. The rational for this regression comes from first noting that for any integrable function of w, T(w) ε-

E(T(w)|p) =

εp

冮冦 冮

冧

T(w)G(dw|ε,p) + T(0(ε,p)G(0(ε,p)|ε,p h(ε)dε (12.39)

ε (ε,p,p) ¯ ε-

冮

p

ε-

冮冦冮T ′((ε,q,p)h(ε)dε冧

= T(εp)h(ε)dε − [1 + ψ1F¯¯ (p)]2 ε-

p ¯

ε-

(1 − α)[1 + (1 − σ)ψ1F¯¯ (q)] × dq [1 + (1 − σ)ψ1F¯¯ (q)][1 + ψ1F¯¯ (q)]2

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where σ=

ρ ρ+δ

.

Next, the distribution of p is derived from French firm panel data on output and capital and labor inputs. Let r be the cost of capital. Using a time series of firm-based data on labor, L and capital K along with the assumption of a Cobb–Douglas production function Yjt = θjK jtυ Ljtκexp(ηjt) where θj is a firm-specific productivity parameter and ηjt is a zero-mean stationary productivity shock independent of θj. Then pjt ≡

Yit = Lit

1−κ θjexp(ηjt) . r

冤冢

冣

1 κ

冥

Finally, substituting (12.38) into (12.39) produces

冢

E(w ¯¯ jt| p) = κµε pj −

1−α [1 + ψ1L(pj)]2

p

[1 + ψ1(1 − σ) + σψ1L(q)][1 + ψ1L(q)]2 dq 1 + αψ1(1 − σ) + (1 − α + ασ)ψ1L(q) p

冮

×

冣

¯

which when taking logs of both sides demonstrates how the underlying structural parameters can be derived from such a log-wage regression. Using matched employee–employer data from France, Cahuc et al. find that between-firm wage competition matters a lot in the determination of wages and is quantitatively more important than Nash wage bargaining in raising wages above a worker’s reservation wages.

12.9 STRUCTURAL ESTIMATION AND MODEL COMPLEXITY In his superb essay, Wolpin (1995) defines structural estimation as an estimation process whose major objective is to recover the basic parameters of the theory. Thus structural estimation presupposes that the theory has been formulated as an optimization problem. On the other hand, empirical work that proceeds without an explicit optimal theory is called a reduced form estimation. Wolpin uses the job search environment to illuminate the distinction between structural and reduced form.

Structural estimation methods

381

Wolpin observes that while there is not a conceptual problem in conducting structural estimation with large choice sets and large state spaces, there are problems associated with achieving computational feasibility. Let us see how Rust (1987) introduced simplifications to obtain computational simplicity. Rust assumes that: (1) the reward functions are additively separable in the unobservables; (2) each unobservable is attached to a mutually exclusive choice; (3) the observables are conditionally independent, that is, given the observable state variables, the unobservables are serially independent; and (4) unobservables have a multivariate extreme value distribution. These assumptions have two significant implications with respect to obtaining solutions and feasible estimation: (A) The Emax(·) functions given by V(S(t),t) = max {Vk(S(t),t}, k ∈K

where Vk(S(t),t) is the expected lifetime reward associated with the kth alternative or simply put, this is Bellman’s equation, has a closed form solution K

¯¯k(S ¯¯ (t),t)) exp(V , τ k=1

冦冱

E[V(S(t),t] = γ + τln

冧

¯¯k is the expected value of the alternative-specific value functions where V and γ is Euler’s constant. (B) The choice probabilities are multinomial logit with τ normalized to unity, ¯¯ (t)) = P(dk(t) = 1 |S

¯¯k(S ¯¯ (t),t)) exp(V . Σ exp(Vj(S(t),t))

j ∈K

By (A), multivariate numerical integration is avoided in obtaining a solution to the DP, and by (B) likelihood estimation also does not entail multivariate numerical integrations.6 Both Rust (1994a) and Wolpin (1995) discuss the complexity problems inherent in structural methods. Rust’s presentation of complexity and structural estimation is remarkably trenchant. He describes three different oneway avenues by which complexity blocks structural estimation. He first observes that “there is a deeper set of problems concerning model complexity that may have no easy solution” (Rust 1994a: 123). Computer technology has permitted researchers to estimate a class of structural models that was forbidden a decade ago. Certain two-state stopping problems that characterize almost all search theory have enlarged the set of estimable models—their threshold structure being a direct consequence of the 0–1 structure. Indeed, for MDPs with finite state space and action spaces, the policy iteration algorithm is equivalent to solving the primal LP problem by the simplex method.

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But Rust maintains that “the computational complexity of most dynamic structural models is exponential in any relevant measure of the size or realism of the model” (1994a: 123; emphasis added). The second form of complexity is “estimation complexity.” This may be more restrictive than computational complexity. It occurs because preferences and beliefs have “very high-dimensional unknown parameters.” Rust proclaims that the estimation complexity of beliefs and preferences is also exponential. Rust (1994a: 123) explains the meaning of this proclamation to be that the number of observations “to reliably estimate a finite-state approximation to a dynamic choice problem increases exponentially in any relevant measure of the size or realism of the choice problem.”7 The third form of complexity is related to, but distinct from, the second— Rust calls it “data complexity.” This problem concerns the data quality that is essential for estimating a structural model. Rust mentions a host of data problems that afflict all empirical research. The seasoned empirical researcher resolves these problems in a reasonable manner. It seems unnecessary to characterize this problem as one of complexity. While Rust (1992) places his complexity appraisal of empirical structural research at the beginning of his paper, Wolpin (1995) places his complexity evaluations at the end of his study. Wolpin observes that a major limitation in conducting empirical research using discrete dynamic programming models is their computational complexity. He mentions several ways in which this problem has been “accommodated” in empirical studies.8 He notes that by limiting the state space and the number of choices to two, dichotomous decision-making was used by Rust himself in his classic (1987) study. Concluding paragraphs by Wolpin (1995: 71) are especially discerning: The empirical literature on labor force dynamics has made enormous progress over the last twenty years. Regardless of their limitations, the initial pioneering attempts simply to relate unemployment duration data to observable characteristics with economic theory in mind were critical to that progress. Statistical advances in the use of duration data have led to a revolution in the empirical implementation of the reduced form approach to the study of labor force dynamics. Finally, the development of methods to estimate structural parameters of dynamic discrete choice models has begun what in my view is another revolution in the way as stated on the Econometric Society’s constitution, “the unification of theoretical and factual studies in economics” is achieved. There are few empirical literatures in economics that have been as successful. However, there is still much to be done. We know far too little to make confident policy statements. Significant progress will require that we take a more active role in the collection of data. . . . Almost none of the variables that are used in the analysis of labor market transition decisions are collected with the intention of matching them to the parameters of the behavioral models that underlie the empirical work.

Structural estimation methods

383

One potential avenue of reducing such computational complexity comes from recent work by researchers in operations research. As mentioned in Chapter 9, the MAB, which is a special class of semi-Markov decision processes, has an optimal policy that can be stated in terms of a forwards induction policy. Glazebrook et al. (2002) observe this and use it in their response to the computational complexity of DP. That is, optimal policies of forward induction minimize the rate at which costs are incurred up to an optimally selected stopping time. Glazebrook et al. (2002) proposed a simple class of forward induction heuristics based on cost rates. In these heuristics, successive decision horizons are a priori selected by the decision-maker and then actions are taken to minimize a suitable cost rate. By fixing these horizons computational complexity is reduced but at the cost of nonoptimality. The question of interest is then how close is the nonoptimal solution to the optimal solution. Glazebrook et al. (2002) study the behavior of this class of forward induction rules and its association with the evolutionary speed of the process. Evolutionary speed is naturally present in semi-Markov decision processes derived from Bayesian sequential decisions, where the system state is the current posterior for an unknown parameter. Both are candidates for the application of forward induction heuristics. As another route to reducing computational complexity, in the next section we describe the achievable region method and the accompanying notion of general conservation laws.

12.10 DACRE, GLAZEBROOK, NINO-MORA (1999) Optimal dynamic control of queues and other stochastic systems has relied primarily on dynamic programming. One important result that emerged from the dynamic programming enterprise was Gittins’ (1979, 1989) discovery that an index policy is optimal for the MAB. As we saw in Chapter 9, the optimal solution for the MAB proceeds by assigning an index to each bandit and laying the bandit with the highest index. The development of this index method has till now taken place within the dynamic programming framework. Dacre et al. (1999) survey a new setting called the achievable region approach which is based essentially on linear programming. This approach is relatively free of the complexity problems associated with the DP procedure. The achievable region method searches for solutions to stochastic optimization problems by: 1 2

identifying the space of all possible behaviors of the stochastic system and choosing the best behaving system in this space.

Frequently, the achievable region is a polyhedron with a particular structure, which frequently gives rise to a linear program (LP). The authors intend

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to develop the analyses in a projected space of natural performance variables with much smaller dimension than the space of state-action frequencies. The authors present the basic ideas associated with the achievable region approach in the setting of multiclass queuing systems. These systems are models for computer and communication systems and have been applied to stochastic economic processes. Let E = {1,2, . . .,N} be the set of customer classes. Customers demand service provided by a set of servers. A control u is a rule revealing how the servers should be allocated to the customers. U denotes the set of admissible controls. Controls are non-anticipative (based on the process’ history) and non-idling (servers are never idle when there are waiting customers). To each control u there is a system performance vector xu = (xui , . . ., xNu ), where xui is the class i performance, i ∈ E. The authors let xui be the expected value of a quantity associated with class i. Typically, xui , denoted Eu(Ni), is the long-run average number of class i customers in the system when u is the control. The collection of possible performances is called the performance space and given by X = {xu,u ∈ U}. A cost c(xu) is incurred when the system runs under control u. The cost depends on control only through its performance. The stochastic optimization problem is given by Zopt = inf {C(xu)}.

(12.40)

u ∈U

The u associated with the infimum is labeled uopt. Equation (12.40) can also be expressed as Zopt = inf {c(x)},

(12.41)

x ∈Χ

when Χ is known. Denote the x associated with this infimum by xopt. They assume that c(x) = cTx, where c is a cost vector and Χ is a convex polyhedron making equation (12.41) an LP. The question is: can a control uopt be discovered giving xopt? Dacre et al. summarize the achievable region approach as a method of searching for optimal solutions in (12.40) by (a) identifying the performance state Χ, (b) solving a math programming problem as stated in (12.41) and processing a feasible space Χ, and (c) identifying controls giving the best performance. They present the following example to clarify this approach: a two-class M/M/1 queuing system first studied by Coffman and Mitrani (1980). Customers of class k arrive at a single server according to independent Poisson streams of rate λk with service requirements (independent of each

Structural estimation methods

385

other and of the arrival streams) which are exponentially distributed with mean µ--k1, k = 1,2. The rate at which work arrives in the system is λ1/µ1 + λ2/µ2 which is assumed to be less than 1 (the available service rate) to guarantee stability, i.e., the time-average number of customers in the system is finite. Controls for the system must be non-anticipative and non-idling and priorities between customer classes may be imposed pre-emptively (i.e., a customer whose requirements have not yet been fully met may be removed from service to make way for another customer of higher priority). The goal is to choose a control u to minimize a long-term holding cost rate, i.e., Zopt = inf [(c1Eu(N1) + c2Eu(N2)].

(12.42)

u ∈U

In expression (12.42) ck is a cost rate, Nk is the number of class k customers in the system and Eu denotes an expectation taken under the steady state distribution when control u is applied. The achievable region approach solves the stochastic optimization problem (12.41) by proceeding through the steps (a)–(c). The authors present steps (a)–(c) and the reader is encouraged to study these. In this particular example the achievable region P satisfies

冦

P = (x1,x2);x1 ≥

ρ1µ1− 1 ρ2µ2− 1 ρ1µ1− 1 + ρ2µ2− 2 , x2 ≥ , x1 + x2 = 1 − ρ1 1 − ρ2 1 − ρ1 − ρ2

冧

and in particular P = Χ. From these steps, they suppose that the optimization problem is given by Zopt = inf ( Σ cjxuj) = min( Σ cjxj). u ∈U j ∈ E

(12.43)

x ∈P j ∈E

The LP on the right-hand side of (12.43) is solvable by the performance x = xu where uG is a control based on a Gittins index policy. The Gittins policy uG assigns an index Gk to each customer class. Then by observing the priorities among customers classes are imposed by the Gittins indices, with the maximal index class receiving first-priority. The indices can be calculated from an adaptive greedy algorithm AG(V,c).9 Using this approach, Gittins policies can be proven optimal for a relatively general class of single-server models as they dynamically allocate stochastic projects and multiclass queues called branching bandits.10 Dacre et al. (1999) develop generalized conservation laws (GCLs) such that from these laws the performance space must belong to a convex polyhedron, P. In general, P⊆Χ so that G

Zopt = inf ( Σ cjxuj) = min( Σ cjxj) ≤ min( Σ cjxj) u ∈U j ∈E

x ∈Χ j ∈E

x ∈P j ∈E

(12.44)

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and thus the solution to the right-hand side minimization problem of (12.44) may not be the solution to the minimization problem in the middle of (12.44). Dacre et al. (1999) show how bounds may be obtained that will enable one to assess the performance of a heuristic control based on P. So, even when the performance space cannot be identified exactly, GCLs may be used to assess the performance of some heuristic controls. Glazebrook et al. (2002) use these GCLs to assess the performance of index-based policies in the restless bandit problem described in Chapter 9.

13 The ubiquity of search

13.1 HISTORICAL PRELUDE Thomas Hobbes (1588–1679) is “the grandfather of game theory”1 and one of the first to consider risk and its generation.2 Hobbes maintained that no man is secure against the natural “covetousness, lust, anger, and the like” of his companions and therefore must predict his remaining life to be “nasty, brutish, frustrated and full of contention.” Oakeshott (1962) is an excellent discussion of the plight of humans in a world with uncertainty and a paucity of trust. Even in this state, men . . . are capable of making contracts, agreements, covenants, etc., with one another, but these, so far from substantially modifying the condition of insecurity, are themselves infected with this insecurity. And this is specially the case with covenants of mutual trust. For in these, one of the covenanters must perform his part of the bargain first, but he who does so risks being bilked; indeed, the risk that he whose part it is to be the second performer will not keep his promise . . . must always be great enough to make it unreasonable for any man to consent to be a first performer. (Oakeshott 1962: 345; emphasis added) To obtain peace and some degree of certainty and trust, Hobbes claims that a necessary condition for peace is a Sovereign who is authoritative and powerful. Oakeshott (1962: 349) continues: The authority of this Sovereign can derive only from a covenant of mutual trust of every man with every man in which they transfer to him their natural right to govern themselves and in which they own and acknowledge all his commands in respect of those things which concern the common peace and safety as if they were their own. Hobbes’ famous claim is: “This is the generation of that great Leviathan . . . to which we owe under the immortal God, our peace and defense.”

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We will study briefly throughout this chapter how social contracts can appear without assuming “that great Leviathan.” The first of these contracts is exchange. According to Vico: It is in the Athenian agora or marketplace . . . that laws originate. . . . As the open space of traffic and transactions, the marketplace . . . traces the circle of all possible material exchanges, and it brings to the fore the existence of separate worlds with their possibly colliding viewpoints and interests. . . . One thing is clear: in this space laws are devised to rationalize, counter, and limit the possible chaos of exchange. . . the solemn formulas of the law seek to establish a dialogical relation between strangers.3 Livingston (1998) has a substantial treatment of the similarities between Vico and Hume. Skinner (1993) compares Smith, Hume, and Galiani, who many regard as a disciple of Vico. Skinner observes that Hume’s essays do not comprise a coherent treatise but do display a systematic treatment.4 Skinner maintains that perhaps the most important single feature of this treatment is to be found in the use of history and the historical method: Hume consistently sought to link economic relationships with the environment and the state of manners. (Skinner 1993: 243; original emphasis) This links Hume to the American Institutionalists. For Smith, “history is the preface to political economy rather than integral to the treatment” (Skinner 1993: 248). Some claim that Smith made no use of the historical method in his economic analysis. Skinner (1993: 248–9) cites Schumpeter’s description of Galiani’s research as applicable to Hume. Schumpeter writes: One point about his [Galiani’s] thought must be emphasized . . . he [Galiani] was the one 19th century economist who always insisted on the variability of man and on the relativity, to time and place, of all policies; the one who was completely free from the paralyzing belief . . . in practical principles that claim universal validity; who saw that a policy that was rational in France at a given time might be quite irrational, at the same time, in Naples. Skinner concludes his Introduction by stating his belief that the enormous analytical success of economics had some unfortunate consequences. The dominant classical orthodoxy made it possible to think of economics

The ubiquity of search

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as quite separate from ethics and history, thus obscuring the true purposes of Smith and Hume. Hutchinson makes the telling remark that Smith was unwittingly led by an invisible hand to promote an end no part of his intention, that “of establishing political economy as a separate autonomous discipline.” We believe with Jack Hirshleifer that: There is only one social science. . . . While scientific work in anthropology and political science and the like will become increasingly indistinguishable from economics, economists will reciprocally have to become aware of how constraining has been their tunnel vision about the nature of man and social interactions. Ultimately, good economics will also have to be good anthropology and sociology and political science and psychology. (Hirshleifer 1985: 53)

13.2 INTRODUCTION Search drenches economics and transforms a static set of definitions into a dynamic, interrelated construct. To see this, we consider several important topics in economics: exchange, liquidity, economic growth, house selling, the natural rate of unemployment, insurance, real options, the evolution of money, property rights, division of labor, and trust. A proper explanation of each of these topics entails a description of how the particular topic interacts with search. We now present a brief description of each section of this chapter. The previous section of this chapter presented some pertinent historical information which may cast some light on the remaining sections. Actually, this historical information is transmitted throughout the entire chapter. The main purpose of this section is to alert the reader to the contents of the subsequent sections. The third section traces the evolution of money beginning with a barter economy and progressing to the money economies now operating in almost every country. The presentation leans on the excellent article by Robbie Jones which emphasizes the role of sequential search and the work by Kiyotaki and Wright, which centers on the matching models of Mortensen, Diamond, and Pissarides. Section 13.4 surveys liquidity from a search perspective. The general setting is described first. This is followed by alternative definitions of liquidity and their compatibility with a search definition. Liquidity is then discussed vis-à-vis the thickness of markets, predictability and flexibility. In Section 13.5 we analyze the interplay between the housing market and liquidity. The section is drawn from the article by Krainer and LeRoy. Both the demand and supply of housing are studied. An equilibrium is derived and

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characterized. Under certain conditions, the market has the martingale property. Section 13.6 alerts the reader to the equivalence between the basic search model (BSM) and the house-selling problem.5 A brief description of the natural rate of unemployment is contained in Section 13.7. We emphasize its historical development beginning with its almost simultaneous discovery by Ned Phelps and Milton Friedman in 1967. Section 13.8 presents a modern approach to the natural rate of unemployment based on Rogerson’s 1997 article. Rogerson describes the natural rate in terms of the matching methods devised by Diamond, Mortensen, and Pissarides, and presents a critical assessment. After briefly considering Smith’s views on economic dynamics, Section 13.10 discusses the Ball–Romer model that illustrates the role coordination plays in inflation. Section 13.11 describes a search/matching approach to growth in an environment of technological advance. We expose the recent research on this topic by Jones and Newman (1995). Their article models this process of adaptation and readaptation to ongoing shocks emanating from continuous technological changes, by taking a search/matching perspective. Their central idea is that each state of technical knowledge reveals a set of feasible activities and also reveals a set of potential matching opportunities which agents can exploit. Beneficial matches improve technology, but are not realized immediately. Their realization requires a learning process in which agents search over and/or experiment with the novel opportunities. Learning in this manner is an investment in adaptive information. A key assumption is that each advance triggered by technical improvement increases potential rewards to feasible matches, but its immediate effect is to lower matching efficiency as it diminishes the usefulness of adaptive information acquired in earlier states. This is Schumpeter’s creative destruction—the enhancement of one aspect of informational capital destroys the usefulness of its adaptive counterpart. The following section examines the close relationship between auctions and sequential search. Arnold and Lippman show that there are at least three connections. The analysis first generalizes BSM to encompass the sale of several units. With respect to the auction regime, they next demonstrate that the expected return per unit is increasing when the valuations of bidders are random variables with exponential or uniform distributions. Their third discovery shows that the preferred mechanism depends on the number of units for sale. More precisely, they prove that there is a crossing point n* such that BSM is preferred if n < n* and an auction is more desirable if n ≥ n*. The predictions of Arnold and Lippman are confirmed in a careful empirical study of livestock markets. Section 13.13 is the comprehensive and illuminating study of auctions and bidding by McAfee and McMillan. Their analysis begins with a set of eight assertions based on their own paper, together with the insights of Akerlof regarding asymmetric information. McAfee and McMillan believe that asymmetric information lies at the core of auction theory. In the next section we explore search with posted prices in which a seller sets the price of a good

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and then buyers arrive and determine whether or not to purchase the item according to whether their valuation exceeds the posted price. Section 13.15 concentrates on middlemen, market-makers, and electronic search. Each of these topics deserves a separate book, so our treatment is necessarily incomplete. We begin with Shevchenko (2004) on middlemen, who cites Rubenstein and Wolinsky (1987) as a commencement to his study. Shevchenko’s fundamental idea is that in an economy with many goods and heterogeneous tastes, there is a definite role for agents who hold inventories of a broad spectrum of items.6 This critical tradeoff facing intermediaries is similar to that confronting individual firms when they are deciding on the level of inventory to hold for each good.7 Shevchenko obtains the steady-state equilibrium number of middlemen, together with their size, and the distributions of inventories and prices. This section also surveys the article by Rust and Hall (2003). They study two types of competitive intermediaries— middlemen (dealer/broker) and market-makers (specialists). They show that with free entry into market-making and search and transactions costs going to zero, the bid–ask spreads are driven (competitively) to zero. Thus, a fully efficient Walrasian equilibrium is obtained asymptotically. Their findings imply that middlemen and market-makers survive in this competitive environment with each allocated half of the total trade volume. Section 13.16 considers electronic search. Autor (2001) obtains three consequences of the labor market internet. First, in their job search, workers use many sources of job information: friends, co-workers, private employment agencies, newspaper advertisements, etc. Second, the Internet labor markets may facilitate the acquisition of new skills, and third, e-commerce has a potent effect on labor outsourcing and specialization. A brief summary of the recent article by Ellison and Ellison (2005) comprises Section 13.16. They enumerate several important lessons about markets which have been taught via the Internet. They address two basic questions: how has the Internet affected commerce and how can it generate frictionless commerce? They explain how the Internet has revealed new insights with respect to the operation of markets with negligible search costs. Empirical analysis can be conducted where only search costs differ. The Internet has also decisively improved the ability to conduct field experiments. The Ellisons note that some observations coming from an economic analysis of the Internet challenge the basic economics of search costs and product differentiation. The next topic related to BSM is learning from experience. We present the research of Jones and Newman, Muth, and Rothschild. The recent article by Muller and Ruschendorf (2001) is then covered. They show that the “main tool” for a constructive proof of Strassen’s theorem is the BSM. This is important because this classic theorem links the BSM to the max-flow-min-cut theorem of Ford and Fulkerson and then to a host of graph-theoretic results, including matching and networks that have assumed importance in the economic literature.

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Section (13.17) is an introduction to real options. After an introduction in subsections 13.17.1 and 13.17.2, subsection 13.17.3 then presents the essence of real options and their link with BSM. The following subsection studies the origin of real options. Subsection 13.17.5 demonstrates briefly the vital role of real options in ecology, that is, resource management techniques, while the final subsection looks at real options’ role in organizations. The final section (13.18) is devoted to a presentation of the famous assignment theorem by Derman, Lieberman, and Ross (DLR) (1972b). This is also closely related to the BSM and other important “empirical” theorems. 13.3 THE EVOLUTION OF MONEY The general use of media of exchange is so common in our everyday economic activities that it is taken for granted, like breathing. Nevertheless, it also is a vital activity driven by specialization which has evolved from some very crude exchange processes like barter. Adam Smith (1776: 19) regarded the “propensity to truck, barter, and exchange one thing for another” to be a basic component of human nature and hence a primary feature of every society. Marx (1857–58) understood that exchange was intimately related to both the division of labor and private property. He maintained that “If labor is once again to be related to its objective conditions as to its property, another system must replace that of private exchange” and Engels (1972: 175) agreed wholeheartedly: “No Society can permanently retain the mastery of its own production and control over the social effects of its process of production unless it abolishes exchange between individuals.” Media of exchange allow society to circumvent the striking inconvenience, or even impossibility, of discovering the “double-coincidence of wants” required for direct barter exchanges (Jevons, 1875: 3). The money perplex was lucidly described by Carl Menger: every economic unit in a nation should be ready to exchange his goods for little metal discs apparently useless as such, or for documents representing the latter, is a procedure so opposed to the ordinary course of things. . . . The problem, which science has here to solve, consists in giving an explanation of a general, homogeneous course of action pursued by human beings when engaged in traffic, which, taken concretely, makes unquestionably for the common interest, and yet which seems to conflict with the nearest and immediate interests of contracting individuals. (Menger 1892: 239–40) Jones (1976: 758) observes that The literature on monetary exchange has a recurring theme. A role for money can be found in the fact that exchange is decentralized, either in the sense that individuals trade with each other rather than with a central

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“market,” or in the sense that there are separate markets for each pair of goods exchanges; and that compensation for goods given up is guaranteed by a quid pro quo from the individual or market to which they are delivered. One branch of the literature is concerned with the feasibility of attaining a competitive allocation of goods in a decentralized context. Ostroy (1973) shows that when individuals meet sequentially and cannot give up goods they have not yet acquired or base trades on information that is available only to traders not yet encountered, trading rules which satisfy all excess demands with just one meeting between individuals are generally not compatible with quid pro quo at each exchange. As Jones (1976: 759) notes Ostroy envisages money as an abstract “record-keeping device,” permitting enforcement of budget constraints without requiring quid pro quo at every exchange, and thus allowing excess demands to be satisfied with the minimum number of meetings. If everyone is endowed with a quantity of one common good sufficient to pay for all their intended acquisitions, Ostroy and Starr (1974) indicate that the less abstract “record-keeping device” of using this good as a medium of exchange with quid pro quo can achieve the same result. Feldman (1973) interprets quid pro quo to mean that bilateral exchanges are mutually utility increasing rather than balanced at previously determined prices. He proves that the existence of a good which everyone desires and holds in positive quantities, which might be interpreted as money, guarantees that such utility-increasing trades lead the economy arbitrarily close to a Pareto-optimal allocation of goods. While these papers show how money can overcome the logistical problems of achieving an efficient allocation they do not give insight into how a monetary pattern of trade could evolve without centralized decision-making. Other work by Niehans (1969, 1971) and Karni (1973) focus on the role of transaction costs in determining optimal patterns of exchange and the conditions on these transaction costs that give rise to monetary trade equilibriums. The paper by Jones then considers why these conditions should be so prevalent. In his model, Jones assumed: voluntary bilateral exchange, exchanges bilaterally balanced at given prices, exchange is costly, and there is limited information about other individuals. Search and Bayes adaptive learning are crucial in Jones’s model. To achieve a given exchange of good i for good j, the individual enters the trading market and searches for a trader who is looking for someone to exchange good j for good i. The transaction cost is the search time required to locate a trading partner. Assuming a constant rate of search, the searcher chooses a trade sequence so that the expected number of encounters with other individuals is minimized. The trader is assumed to be a Bayesian, basing his trading plans on subjective

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estimates of the probabilities that an encountered trader wishes to buy or sell a particular good. The trader also believes that what a trader offers to trade and what he wishes in return are statistically independent. The subjective probability that a randomly encountered trader wishes to exchange j for i is pipj, where pi is the probability he wishes to buy or sell the ith good. The mean number of encounters is 1/pipj, this is the expected number of encounters to exchange good i for good j. Jones’s use of search is similar to the search theory associated with the BSM. He also develops an equilibrium model and a learning model. His learning process is Bayesian and resembles Rothschild’s (1974) model. We discuss this gem in detail in a companion volume. Jones concludes his article with a summary: The use of search costs to guide individuals through a market with neither trading posts nor clearing houses has yielded plausible implications for the pattern of exchange. With certain assumptions about the costs of finding traders with whom a desired exchange could be made, we have demonstrated why an individual may prefer to execute a given ultimate exchange indirectly rather than directly. Two trades can be less costly than one. However, given that we have opened up the possibility of lengthy indirect trade sequences, we have also demonstrated by sequences of more than two trades would not rationally be chosen. Three or more trades are always more costly than two. Further assuming that all individuals form their beliefs about market parameters by sampling from the same market, we have an explanation for how all traders could independently, without centralized decisions or agreements, settle on the same good as a medium of exchange. Thus individual optimization implies that the pattern of trade must always be some combination of direct barter and monetary exchange. (Jones 1976: 774) A commodity used as a medium of exchange and not in consumption or production is called commodity money. While this first transition may strike us as crude, it was actually quite essential in streamlining trade and rendering barter obsolete. Barter, of course, was the method of trade before money was introduced. To trade with another person required the resolution of Jevons’s “double coincidence of wants” problem. This conundrum is resolved only if one trader owns what the other trader wants and the other trader owns what the first trader desires. This problem was mitigated by the evolution of marketplaces, where traders were usually able to search and find another trader so that the double coincidence was satisfied. Obviously, market trading without money was an awkward and costly mechanism. But it did relieve the stark and dreary self-reliance that preceded barter in a marketplace. Gradually, commodity money emerged in the marketplace. Now traders need not produce or acquire a variety of goods to insure

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against barter failures. They could specialize in producing that for which they were most proficient. The onset of specialization or division of labor was certainly a milestone in the development of a civilized society. As society becomes more homogeneous and trusting, “fiat money” replaced commodity money. “Fiat money” has no intrinsic value and corresponds to the paper money we use today. Thus fiat money evolved from barter and gave the holder a general command over society’s produce. The search time required to find a particular good has declined dramatically from the time to complete a barter trade. It is the goal of all consumers and producers to minimize search costs. In a civilized society with fiat money, search is minimized because the discovery process has been transformed from a highly uncertain activity to one that is almost sure. The formal study of these evolutions into commodity money and/or fiat money is not trivial. For a superb analysis of how sequential matching (see the article on matching by Mortensen 1982) is used to derive these monies endogenously, see Kiyotaki and Wright (1989). A splendid exposition of Kiyotaki and Wright (1989) is contained in Ljungqvist and Sargent (2004b). In his concluding paragraph, Lucas (2000) envisages Kiyotaki and Wright (1989) as containing the matching models needed for proceeding along the road leading to microeconomic foundations for monetary theory: Successful applied science is done at many levels, sometimes close to its foundations, sometimes far away from them or without them altogether. As Simon (1969) observes, “This is lucky, else the safety of bridges and airplanes might depend on the correctness of the ‘Eightfold Way’ of looking at elementary particles.” The analysis of sustained inflation illustrates this observation, I think: Though monetary theory notoriously lacks a generally accepted “microeconomic foundation,” the quantity theory of money has attained considerable empirical success as a positive theory of inflation. Beyond this, I have argued in this survey that we also have a normative theory that is quantitatively reliable over a wide range of interest rates. There are indications, however, that the level of the models I have reviewed in this paper is not adequate to let us see how people would manage their cash holdings at very low interest rates. Perhaps for this purpose theories that take us farther on the search for foundations such as the matching models introduced by Kiyotaki and Wright (1989) are needed. (Lucas 2000: 271–2; emphasis added) By holding a stock of money, the agent has liquidity by which both future opportunities can be exploited as they become manifest and the impact of future hazards are greatly mitigated. When an unforeseen option with a high expected return is discovered, the agent who has maintained a sufficient level

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of liquidity can seize the opportunity.8 Uncertainty is the mother of search. In a certain world, search is unnecessary! An individual with a liquid portfolio also has an insurance policy which reduces the pain and suffering of untoward events. A full-fledged insurance policy undoubtedly evolved from this primitive policy. Liquidity, in brief, is a potent mechanism for reacting to both positive and negative information, which is revealed in the sequential episodes that comprise life as we know it. Bayesian learning teaches us how to use liquidity in an uncertain, sequential milieu. This learning is the onset of civilized behavior in an open stochastic economy. Inflation is the great challenge to the benefits flowing from money in a noninflationary economy. A significant inflation erodes almost all of the benefits listed above. In their excellent study, Heymann and Leijonhufvud (1995: 148) note that “although high inflations show signs of widespread currency substitution, a tax of one percent per day on cash holdings . . . will not drive domestic money totally out of existence.” The authors conjecture that social conventions are such that the evolution of a new exchange system does not occur. “The point remains that we do not have a clear picture of what ultimately supports the demand for domestic money” (1995: 149). The Kiyotaki–Wright model implies that a severe prolonged inflation causing “bad money has no hope of serving as a medium of exchange” (Kiyataki and Wright 1992: 20). But Heymann and Leijonhufvud (1995) show that the facts don’t accord with this prediction! They do show that high inflations induce coordination failures. “Intertemporal supplies and demands are not reconciled and intertemporal valuations are not made consistent” (p. 168). Their concluding section observes that It is odd that economists routinely make obeisance to the point when discussing the division of labor. . . . In that context, the representative agent is not supposed to know and master everything. In the theory of exchange in contrast, cognitive limitations have been ignored. And monetary (and finance) theory—has been pursued altogether within the framework of exchange theory, with production and (therefore) specialization of knowledge eliminated as inessential complications. . . . High inflations destroy social institutions . . . that would not have existed in the first place if the rationality of economic agents was unbounded. By insisting on the assumption that individual decisionmakers are not subject to any relevant cognitive limitations, we fail to find a solid foundation for monetary theory and fail to understand the social and economic consequences of inflation and, therefore, the benefits of stability. (Heymann and Leijonhufvud 1995: 184; emphasis added) In spite of these contradictory findings, the Kiyotaki–Wright article is a tour de force and when joined by the earlier series of innovative articles on

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the evolution of money by Menger (1892), Clower (1967), Jones (1976), Lucas (1978), Ostroy and Starr (1990), and Oh (1989), we do seem to have the foundations of potent monetary theory.

13.4 LIQUIDITY AND SEARCH John Hicks (1974: 41–2; emphasis added) has a verbal definition of liquidity that captures the essence: For though there are many kinds of flexibility which are relevant to economic decisions, there is one that is outstanding. It is the flexibility that is given by the market. A firm which acquires a non-marketable asset—say a new factory, designed and equipped for its own particular purpose—has committed itself to a course of action, extending over a considerable time. . . . It has “given hostages to fortune.” The acquisition of an easily marketable asset, on the other hand, can be revoked . . . the firm is in a position that is almost as flexible, after the acquisition, as before it. That, I suggest, is precisely what we mean by saying that the marketable asset possesses liquidity. In the Treatise, John Keynes (1930: 67) defined liquidity as follows: one asset is more liquid than another if it is “more certainly realizable (that is to say convertible into money) at short notice without loss.” Hicks (1974: 43) interprets Keynes’s remarks to mean that “if the price is very variable, the asset is still imperfectly liquid—because of the risk that at a date chosen at random . . . the price at which the asset can be sold will be abnormally low.” Hicks also observes that assigning liquidity to an asset presupposes that there is a stable price level. Lippman and McCall (1986: 43) formalize some of these verbal insights together with the crucial observations by Hirshleifer that liquidity is “an asset’s capability over time of being realized in the form of funds available for immediate consumption or reinvestment in the form of money.” Lippman and McCall present a definition of liquidity in terms of what they regard as its most significant property—the time until an asset is exchanged for money. We present several sections of their article and show how this definition illuminates the evolution of liquidity. Kenneth Boulding says that “liquidity is a quality of assets which . . . is not a very clear or easily measurable concept” (1955: 310; emphasis added). According to John Maynard Keynes: There is, clearly, no absolute standard of “liquidity” but merely a scale of liquidity—a varying premium of which account has to be taken . . . in estimating the comparative attractions of holding different forms of wealth. The conception of what contributes to “liquidity” is a partly

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Similarly, Helen Makower and Jacob Marschak observes that: “ ‘liquidity’ has so often been used to cover all properties of money indiscriminately that it seems better not to use it for any of the separate properties of money. We thus resign ourselves to giving up ‘liquidity’ as a measurable concept: it is, like the price level, a bundle of measurable properties” (1938: 284). However, they also note that the term liquidity suggests “the fact that money is easily transformable (on the market) into other assets and is thus an effective instrument for maneuvering” (1938: 284). Closely related is the notion of liquidity due to Jack Hirshleifer who said that liquidity is “an asset’s capability over time of being realized in the form of funds available for immediate consumption or reinvestment—proximately in the form of money” (1968a: 1).9 The notion of liquidity presented here most closely resembles Hirshleifer’s. Lippman and McCall present a precise definition of liquidity in terms of this most important characteristic—the time until an asset is exchanged for money. They then show that this definition is compatible with several other useful notions of liquidity. While academic economists do not possess a definition of liquidity as a measurable concept (though they do mention an assortment of its attributes), other workers in the area casually respond that liquidity is the length of time it takes to sell an asset (i.e., convert into cash); thus cash is considered the most liquid asset, while stocks listed on the New York Stock Exchange (NYSE) are viewed as more liquid than collectibles, precious metals, jewels, real estate, and capital goods. The problem with this view of liquidity is the lack of precision and casual reference to “the” length of time it takes to convert the asset into cash. This length of time is a function of a number of factors including frequency of offers (i.e., difficulty in locating a buyer), impediments to the transfer of legal title (namely, the time it takes to verify legal ownership as in a title or patent search and the right to dispose of the asset as in a leasehold interestdealership, or letter stock), the costs associated with holding the asset, and, most importantly, the price at which you (the owner) are willing to sell. If your minimal price is too dear, then it might never be sold. On the other hand, if the price is exceedingly lower (and legal niceties such as proof of ownership are readily established), then the asset might be sold in a very short period of time. Any thoughtful response to clarify the meaning of liquidity must incorporate the idea that the price demanded be “reasonable.” The approach they suggest incorporates this idea as it consists in embedding the sale of the asset in a search environment, discerning a sales policy that maximizes the expected discounted value of the net proceeds associated with the sale, and

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defining the asset’s liquidity to be the expected time until the asset is sold when following the optimal policy.10 Clearly the concepts of liquidity and money are intimately connected. As defined here an asset’s liquidity is the optimal expected time to transform the asset into money. A distinguishing characteristic of money is its role as a medium of exchange.11 From this perspective, money is desirable because of the ease with which it can be exchanged for other commodities. If we rank commodities by their liquidity, our definition is equivalent to money being the most liquid asset. An exchange commodity i for commodity j is accomplished most swiftly by first trading i for money and then trading money for j.12 The expected time to go from i to money corresponds to our measure of i’s liquidity. The expected time to go from i to j measures the liquidity of the (i,j) transaction. The crucial point is that in going from commodity i to money, the individual follows an optimal selling policy, and in going from money to j, the individual pursues an optimal buying policy. This approach by Lippman and McCall is novel in that rational behavior under uncertainty, as exhibited by adherence to optimal stopping rules, is the defining characteristic of liquidity. This perspective illuminates both the demand for money,13 and portfolio analysis.14 The environment in which the sale of the asset occurs is presented in Subsection 13.4.1; there we define the expected time of sale as our measure of an asset’s liquidity. The compatibility of this measure with other notions of liquidity is demonstrated in Subsection 13.4.2. In particular, we show that the Lippman and McCall definition is compatible with Keynes and that liquidity increases with the market interest rate (Subsection 13.4.3), the thickness of a market with brisk trading (Theorem 13.1), and (Subsection 13.4.4) the predictability of offers (Theorem 13.2). Furthermore, this last result is consonant with the concept of an efficient market. One expects that any change in a parameter which leads to an increase in liquidity also would lead to a decrease in the discount associated with a quick sale. Theorem 13.3 fulfills this expectation for the search cost c. In the final subsection we present the Lippman and McCall model of liquidity when there exists a future golden investment opportunity. Theorem 13.2 reveals that the choice of a more liquid initial investment enhances the investor’s ability to profit from the arrival of the golden opportunity: as per our definition, liquidity provides flexibility.

13.4.1 The setting Search is the fundamental feature of the arena in which the sale of the asset is to take place; the setting is similar to the standard job search or house selling model. The search environment is characterized by four objects: ci, ti, xi, and β. First, there are the costs of owning/operating the asset as well as the cost of attempting to sell the asset. In the discrete time framework, the net operating and search cost for period i is ci.

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Second, one offer arrives at each time in the set (Si: i = 1,2, . . . ) of arrival times. The random arrival times Si satisfy: i

Si =

冱T , j

j=1

where the integer-valued random variables Tj ≥ 0 need not be either independent or identically distributed. The ith price offered is a nonnegative random variable Χi. In the standard search paradigm the Χi are independent, identically distributed, and independent of {Ti}. None of these three assumptions is invoked here. As evidence in equation (13.1) below, this formulation can be structured so that either it does not permit the seller to accept any offer other than the one most recently tendered so recall of past offers is not allowed, or it does permit the seller to accept any of the tendered offers so recall is allowed. Finally, all expenditures and receipts are discounted at the rate β so that the present value of a dollar received in period i is βi. The seller seeks to maximize the expected discounted value of his net receipts. More formally, the discounted net receipts R(τ) associated with a stopping time τ is given by τ

τ

R(τ) = β YN(τ) −

冱β c

i i

(13.1a)

i=1

Yi =

Χi

if no recall

冦max(Χ , . . .,Χ ) if recall allowed 1

(13.1b)

i

where N(τ) = max{n:Sn ≤ τ) is the random number of offers that the seller observes when employing the decision rule τ and the random variable YN(τ) is the size of the accepted offer. Consequently, the seller chooses a stopping rule τ* in the set T of all stopping rules (we do not require P(τ < ∞) = 1) such that ER(τ*) = max{ER(τ): τ ∈ T }.

(13.2)

The value of the asset is V * ≡ ER(τ*), and the length of time it takes to realize the asset’s value and to convert the asset into cash is the random variable τ*. (In making this statement we are implicitly assuming that there is no lag between the time an acceptable offer is made and the time that the seller is paid.) Lippman and McCall propose to use Eτ* as the measure of an asset’s liquidity with an increase in Eτ* corresponding to a decrease in liquidity. The key point is that for any given asset (with its concomitant cost function ci, arrival times {Si} and offers {Χi}), there is an optimal policy τ* of the asset. The asset’s liquidity is determined by τ*.

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13.4.2 Compatibility with other notions of liquidity This measure of liquidity is not only internally consistent but also compatible with a good deal of what economists have said. In regard to its consistency we note that Eτ* = 0 for money so that money is perfectly liquid: it is the most liquid asset. Second, an illiquid asset is one that cannot be sold, or rather one with Eτ* = ∞. This can occur when there are informational asymmetries or structural constraints that induce the potential buyers to undervalue the asset: that is, its worth to the current owner exceeds its assessed or actual worth to any potential buyer. Informational asymmetries arise in the context of a business in which there are many cash transactions and the company’s books are not a reliable guide to revenues. Structural constraints such as tax considerations in which only some assets “may be burdened by transaction duties” (Hirshleifer, 1972: 137) provide another example of an impaired marketability which can render an asset totally illiquid.15 To analyze how this might come about, suppose that ci, the net search and operating cost per period, is c < 0 for all i. In addition, suppose that no buyer is willing to offer more than −cβ/(1 − β). The policy τ* of never accepting an offer at or below −cβ/(1 − β) yields the owner an expected discounted value of ∞

−c

冱β = −cβ/(1 − β) i

i=1

so τ* is indeed optimal. Moreover, τ* ≡ ∞ so Eτ* = ∞, and the asset is illiquid. The “standard” search paradigm is utilized extensively in the ensuing analysis. It entails: (1) a constant search cost so ci ≡ c, (2) one offer tendered each and every period so Ti ≡ 1, (3) independent offers Χi drawn from the same known probability distribution F, and (4) recall of past offers. With these assumptions the existence of an optimal rule τ* is guaranteed, and τ* has the following representation: τ* = min{n: Χn ≥ ξ}, where ξ is referred to as the reservation price. Thus, the seller accepts the first offer greater than or equal to his reservation price ξ; consequently τ* is a geometric random variable with parameter P(Χ1 ≥ ξ), the probability that a given offer is successful in effecting the asset’s sale. Furthermore, it is clear upon reflection that ξ = V *. According to Keynes one asset is more liquid than another if it is “more certainly realisable at short notice without loss” (1930: 67). In the context of the standard search paradigm, Keynes’s definition is equivalent to ours if we interpret “at short notice,” “more certainly realisable,” and “without loss” to mean “in one period,” “has a higher probability p of being sold in one period,” and “in accord with the optimal policy.” To see this, recall that the asset is sold if and only if the offer price is ξ or larger, and merely observe that p = P(Χi ≥ ξ) is related to Eτ* via Eτ* = 1/p so that liquidity increases with p. Continuing with the standard search paradigm, recall that the reservation

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price ξp is a function of the discount factor β. The asset’s reservation price satisfies c = H(ξ) − ξ(1 − β)/β,

(13.3)

where ∞

冮

H(x) = (y − x)dF(y). x

Differentiating the first-order condition (13.3) with respect to β yields dξ/dβ = ξ/β 2[1 − F(ξ) + (1 − β)] < 0.

(13.4)

Hence ξ is a strictly increasing function of β. Consequently, an increase in β leads to an increase in Eτ* as Eτ* = 1/P(Χ1 ≥ ξβ), with ξβ strictly increasing in β. That is, an increase in the market interest rate or in the asset holder’s time preference (a lower value of β), leads to an increase in the asset’s liquidity. This demonstrates two facts. First, because more impatience (as might arise from increased consumption needs that can only be satisfied via the expenditure of wealth in the form of money) leads to a more liquid asset, impatience and liquidity preference are commensurate in that they vary directly. As expected, an increase in liquidity preference leads to an increase in liquidity itself. Second, because an asset’s liquidity depends upon the discount factor, it is a property of the asset holder as well as an intrinsic property of the asset itself. Nevertheless, to the extent that dispersion of the offer price distribution, rate of receipt of offers, and relative costs of soliciting offers are common across all sellers, the ranking of assets in terms of liquidity will be similar across individuals, regardless of their degree of impatience. With this view we return to the notion that liquidity is determined by characteristics of the asset; characteristics of the seller have virtually no impact.

13.4.3 Liquidity and thickness of markets When there are many transactions per day of a homogeneous asset such as wheat or long-term Treasury bonds, the market for the asset is thick. On the other hand, the more idiosyncratic the asset, as is the case if it is one-of-akind (a work of art or a castle), or has a limited set of uses (a germ-free refrigerated warehouse or a special purpose lathe), the thinner the market becomes. The number of transactions in a market is a function of several factors, including the frequency of offers. Theorem 13.1 An increase in the frequency of offers causes the expected time of sale to decrease (and liquidity to increase) if either the interest rate is near zero or the frequency of offers is very high.

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Proof To begin the analysis, suppose offers arrive according to a Poisson process with rate λ, and let α be the continuous time interest rate. Then βλ, the one-period (a period is the time interval until the next offer) discount factor, satisfies βλ = λ/(α + λ). Define Tλ and τ* λ to be the time of sale and the number of offers received until the sale, respectively, when offers arrive at rate λ. As 1/λ is the expected time between offers, we have ETλ = Eτ* λ /λ. Differentiating with respect to λ and utilizing (13.4) and Eτ*λ = 1/[1 − F(ξλ)] yields dETλ/dλ = −Eτ*λ /λ2 + λ− 1 dEτ*λ /dλ = −[λ2(1 − F(ξλ ))] − 1 + f (ξλ)[1 − F(ξλ)]−2λ − 1 × (dξλ/dβλ) · (dβλ/dλ) = − [λ2(1 − F(ξλ))] − 1 + αξλ f (ξλ){λ3[1 − F(ξλ)]2 × [1 − F(ξλ) + α/λ]}− 1. Noting that both the reservation price ξ and the expected time [1 − F(ξ)] − 1 until an acceptable offer is received are bounded for α near zero and λ large, the above expression for dETλ/dλ reveals that its sign is negative when α is near zero and also when λ is large. The expected time of sale is the product of the expected time between offers and the expected number of offers received until the asset is sold. A decrease in the expected time 1/λ between offers, the first term in the product, causes the discount factor βλ to increase. This leads to an increase in the reservation price and hence to an increase in the number of offers received, the second term in the product. Theorem 13.1 asserts that the net effect is negative if discounting tax has little impact (because the time to sale is short or the interest rate is small). In a practical sense, Theorem 13.1 implies that liquidity, as defined here, increases with thickness for all of the familiar highly organized markets (such as the NYSE) characterized by brisk trading: Theorem 13.1 is uninformative as regards thin markets.

13.4.4 Liquidity and predictability In Jacob Marschak’s view, the word liquidity “denotes a bundle of two measurable properties and is therefore itself not measurable” (1938: 323). The two properties he refers to are “plasticity,” that is, “the ease of maneuvering into and out of various yields after the asset has been acquired,” and “the low variability of its price.” A version of this view of liquidity might provide the following definition: an asset is liquid if it can be sold quickly at a predictable price. By this definition, liquidity is a two-dimensional attribute.

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Consider commodities such as wheat and long-term Treasury bonds. The market for both assets is nearly perfect in that the attempt to sell even as much as one million dollars worth of these assets will have only a minute effect upon “the market price.” Moreover, there is a ready (and highly organized) market for both assets with a multitude of transactions taking place each weekday. The transaction can be effected in a matter of minutes. Consequently, it is indisputable that these assets can be sold quickly. On this dimension they would be seen to be near-money.16 However, interest rates have been highly volatile; fluctuations of as much as 9 percent in a single day (see Federal Reserve Chairman Volcker’s announcement of October 6, 1979) have occurred. And the wheat market has a long history of volatility. Thus, neither of these assets rates high on the dimension of “predictable price.” Predictability, we maintain, is an expression of concern with adverse events or downside-risk, that is, safety. As such it ignores and fails to account for the occurrence of favorable events or upside-risk. Our measure of liquidity implicitly utilizes both the adverse and the favorable events by requiring that the asset be sold at its “fair market price” where the price is derived from the seller’s optimization (see equations 13.1 and 13.2). To see the relation between predictability and our measure of liquidity, let Wi = Χi − µx so EWi = 0 and parameterize predictability by the following representation of the offers: Χi = µx + εWi. Naturally, a decrease in ε is interpreted as an increase in predictability. An increase in ε is a mean-preserving increase in risk of the sort that might properly be labeled a dilation. We shall limit our investigation to dilations because other mean-preserving increases in risk are less regular in that the concomitant change in liquidity they induce can be either an increase or a decrease. The seller’s problem is to choose a stopping rule τε in the set T of all stopping rules to maximize E[βτ(µ + εWτ) − c(β + . . . + βτ)] = − (cβ/(1 − β)) + Eβτ × [µ + (cβ/(1 − β)) + εWτ] = − (cβ/(1 − β)) + εEβτ × [(µ + (cβ/(1 − β)))/ε + W τ ]. Equivalently, the seller needs to maximize Eβτ(µτ + Wτ),

(13.5)

where µτ = (µ + (cβ/(1 − β)))/ε. When there is total predictability, that is when ε = 0, low offer correctly has no impact upon the owner’s opinion (an opinion shared by the market) of the asset’s value, for the offer does not constitute new information. On the other hand, the source of variation in an efficient markets setting is the arrival of new information. As new information arrives,

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say in the form of a low offer, each agent, including the asset owner, simultaneously revalues the asset. In short, risk is the embodiment of heterogeneous preferences in one analysis and the arrival of new, commonly shared information in the other. In view of this discussion, it is clear that financial assets traded in a thick, efficient market will be exceedingly liquid.

13.4.5 Liquidity as flexibility17 An investment of funds today obviously reduces the range of options open to the agent tomorrow. This flexibility aspect of liquidity is implicit in Section 4.3. Both the search paradigm, in general, and the liquidity search model, in particular, are based on the opportunity cost doctrine—the cost of holding one asset is the return that could be achieved by investing in the next best asset. Most of search theory employs models in which these opportunity costs are constant. However, if the agent’s future opportunities differ from his current opportunities, he may eschew commitments that yield an inflexible or illiquid portfolio. In terms of our definition, he may avoid investments with large values of Eτ*. The key feature of the simple search model we propose is the existence of a single golden investment opportunity that becomes available at some future date. This feature of the investment environment causes the constant opportunity costs to vanish. By specifying the functional form of the offer distribution F, we are able to demonstrate the investor’s preference for a more liquid/flexible current investment. At time 0 the investor’s endowment is ξ, all in the form of cash. A set of assets parameterized by λ > 0 is available for purchase at cost ξ, where asset λ has the associated offer distribution Fλ and search cost cλ. For simplicity in presentation, assume that each asset’s reservation price ξλ satisfies cλ = Hλ(ξλ) rather than (13.3), the discounted form of the first-order condition. The value ξλ of asset λ equals its purchase cost ξ, and each asset generates a constant flow of income at the rate α > 0 is the continuous-time interest rate (i.e., e−α plays the role of β). Assume that the investor purchases exactly one asset at time 0 and consumes its income stream as it flows in. Thus, after selecting an asset, say λ0, for purchases at time 0, the investor’s endowment is λ0 at each point in time. At one point T in time in the future, the investor will be presented with the opportunity to invest in the golden asset. He/she will have enough time to solicit exactly one offer for the asset he/she owns in order to generate cash to purchase the golden asset. For the purposes of our analysis it does not matter if T is a random variable with known distribution, deterministic and specified in advance, or uncertain in the sense of Frank Knight (1921). What is important is that there be time to generate but one offer. (This will be the case if T is a geometric random variable.) The golden asset is divisible and has constant returns to scale. Each dollar invested in the golden asset generates a constant flow of income at the rate r

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per unit time. As implied by its name, the income flow associated with an investment of ξ dollars in the golden asset exceeds αξ; that is, r > α. Let G(λ) be the expected gain to search at time T when asset λ was purchased at time 0. If the observed value x of the offer Χλ were to be invested in the golden asset, the resulting cash flow would be rx. This investment is made only if rx > αξ; otherwise, the investor foregoes investment in the golden asset and retains asset λ. Hence max{rXλ/α;ξ} − ξλ is the return to search, and G(λ), the expected gain, is given by G(λ) = Emax{rXλ/α;ξ} − cλ − ξ =

r E max{Χλ;αξ/r} − cλ − ξ α ∞

r αξ αξ = Fλ + xdFλ(x) − cλ − ξ α r r αξ/r

冦

冢 冣 冮

冧

(13.6)

∞

r αξ αξ = + x− dFλ(x) − cλ − ξ α r αξ/r r

冦

冮冢

冣

冧

= (r/α)Hλ(αξ/r) − cλ, where Hλ is the usual H function associated with the offer distribution Fλ. Recalling that r > α and Hλ is a nonincreasing function, we observe that G(λ) > Hλ(αξ/r) − cλ ≥ Hλ(ξ) − cλ = 0, so search is profitable for each asset λ. Most of us believe that liquidity is sought to provide flexibility—be it to meet special consumption exigencies or special (golden) investment opportunities. In order to test the validity of this conventional wisdom in the context of our simple search model with a golden opportunity, we shall assume further that the offer distributions Fλ are exponential: Fλ(x) = 1 − e−λx. Consequently, Hλ(x) = e−λx/λ and cλ = e−λξ/λ. With exponential offers we have −1 Eτ* = 1/e−λξ λ = [1 − Fλ(ξλ)]

so Eτ* λ increases with λ: liquidity decreases as λ increases. From (13.6) and Fλ exponential we obtain G(λ) =

r 1 −λξα/λ −λξ e − e /λ αλ

and then, because r/α > 1,

The ubiquity of search dG(λ) r λξα − λξα/r = − e − e − λξα/r dλ α r

λ2

冦

407

冧

= {−λξe−λξ − e−λξ} = (λξ + 1)e − λξ − (λξ + (r/α))e − λξα/r < 0. Thus, the expected gain to search also decreases with λ. More formally, we have established the following theorem. Theorem 13.2 In the simple search model (with timeless search), a set of initially available assets with exponential offer distributions, and a subsequently available golden asset, the risk-neutral investor improves his/her expected return by selecting a more liquid asset for his/her initial investment. As conjectured, the choice of a more liquid initial investment does indeed enhance and facilitate the investor’s ability to profit from the arrival of the golden opportunity. We view this result as providing an endorsement of John Hicks’s (1974: 43–4) remark that “by holding the imperfectly liquid asset the holder has narrowed the band of opportunities which may be open to him . . .” By choosing a less liquid asset, the investor has more nearly “locked himself in.” In particular, note that the probability (1 − exp{− λξα/r)} of not investing in the golden asset—being locked in—increases as the asset’s liquidity decreases. Although the Lippman and McCall model is not an equilibrium model and the extent to which this result is robust remains to be investigated, the analysis of the simple search model with a golden asset is tantalizingly suggestive of a broad range of macroeconomic phenomena that might successfully be treated with this approach to liquidity and liquidity preference. We conclude by quoting Robert Jones and Joseph Ostroy’s (1984: 26) remarks concerning the profession’s long but spotty treatment of flexibility. The difficulty of defining flexibility in such a way as to have universal application, and the difficulty of obtaining formal results without modelspecific qualifications, may account for its limited role in conventional microeconomics. From a macroeconomic perspective, however, the tantalizing prospect of portraying the connection between business cycles and public confidence as a relation between flexibility induced shifts in asset demands (away from capital investments and towards more liquid assets, especially money) and uncertainty is too compelling to be ignored.18 13.5 THE HOUSING MARKET AND LIQUIDITY (KRAINER AND LEROY) Krainer and LeRoy (2002) develop an equilibrium model of illiquid asset valuation based on search and matching. In their model illiquidity arises because the asset is heterogeneous, asset quality can only be assessed by

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costly search which leads to noncompetitive markets, the acquisition of the asset involves a cost that cannot be completely recouped if the asset is subsequently sold, and the asset is indivisible. In their model agents consume two goods: housing services and some other “background” good. Agents are risk neutral, have an infinite horizon and a common discount rate, β. Agents’ endowment of the background good is zero, hence, any consumption of the background good at a given date equals the negative of expenditures on housing. Agents can consume housing services only when owning a house. Housing services can be consumed only from one house even though agents are not constrained in the number of houses they can own. An agent who lives in a particular house is said to have a “match” and the flow of housing services provided by this house is labeled the “fit.” The fit is denoted by ε and is assumed to be uniformly distributed over the [0, 1] interval. Agents do not search for new homes unless they do not currently own a home or if they own a home and the match fails which occurs at an exogenous fixed rate π. When the match fails the house becomes a financial asset that is to be sold in an optimal fashion. Krainer and LeRoy first study an unmatched agent who simultaneously has a house to sell and is searching for a new house. First, consider the agent as house buyer. They assume that his strategy set is a linear function expressing his reservation fit ε- as a function of current price: ε- − ε-* = δ(p − p*).

(13.7)

Clearly, ε- and p are decision variables. They express reservation fit and price as deviations from their equilibrium values. (The superscript * denotes equilibrium values of the variables.) The parameter δ measures the impact of deviations from the equilibrium price on the buyer’s optimal reservation fit. Since the agent is unmatched, he has an asset consisting of the right to search for a house. Let s be the value of this right. Then s is given by ε- + 1 − p* + β(1 − µ)s*, 2

冢冢

s=µ v

冣

冣

(13.8)

where µ is the probability of a sale, β is the agent’s discount factor, and v(ε) is the value of a house that has fit ε. The value v(ε) satisfies v(ε) ≡ βε + βπv(ε) + β(1 − π)(q + s),

(13.9)

where q is the value of the house to the owner after his match is lost. The argument of v in (13.8) equals the expectation of ε given that ε ≥ ε-. Thus, ε- + 1 v − p* equals the expectation of the buyers surplus given a fit that is 2 greater than the reservation fit ε-. Solving (13.9) for v(ε) gives

冢

冣

The ubiquity of search v(ε) ≡

βε + β(1 − π)(q + ε) . 1 − βπ

409

(13.10)

The buyer’s problem is: find the value of ε- that maximizes s in (13.8) for any p. Given a symmetric Nash equilibrium, the buyer takes the value of s on the right-hand side of (13.8) as a given. This reflects the assumption that the buyer will set future values of ε- at the equilibrium level in deciding whether to buy now. Furthermore, the future values of p confronting the buyer will equal the equilibrium value. The buyer derives µ from µ = 1 − ε-,

(13.11)

which follows from the assumption that ε is uniformly distributed on the unit interval. Substitute (13.11) into (13.8) and use (13.10) to obtain the first-order condition for a maximum of s with respect to ε-, which is β(1 − ε-) 2(1 − βπ)

−v

ε- + 1 + p* + βs* = 0. 2

冢

冣

(13.12)

The first order condition can be written in a form that is easier to interpret, using v

冢

ε- + 1 β(1 − ε-) = v(ε-) + 2 2(1 − βπ)

冣

(13.13)

which follows from (13.10) and (13.12) becomes v(ε-) = p* + βs*.

(13.14)

Krainer and LeRoy give the following interpretation of (13.14). It states that at the reservation fit, the expected utility of owning the house equals its price plus the discounted value of search. This reveals that the buyer must surrender the option to search if he chooses to buy the house. The value of εsolving (13.14) is the equilibrium value ε-*. v(ε-*) = p* + βs*.

(13.15)

They derive the value of δ in (13.7) after relaxing the assumption that the current value of p in (13.15) is its equilibrium value. The optimal value of εfor arbitrary p satisfies v(ε-) = p + βs*.

(13.16)

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Now they subtract (13.15) from (13.16) and solve for ε- − ε-* using (13.10). This yields ε- − ε-* = (β−1 − π)(p − p*)

(13.17)

which gives δ* = (β−1 − π).

(13.18)

Krainer and LeRoy now address the seller’s problem. The seller owns an asset, the unsold house, with value q. His problem is to derive the optimal price p. The wholesale price q and retail price p are related as follows: q = µp + β(1 − µ)q*,

(13.19)

which on using (13.11) yields q = (1 − ε-)p + βε-q*

(13.20)

The first-order condition when q is maximized in (13.20) with respect to p is dεdε(1 − ε-) + (βq* − p) = µ + (βq* − p) = 0. dp dp¯ The term including

(13.21)

dε-

shows that the seller realizes that his choice of p dp influences the buyer’s reservation fit. Using (13.17) they obtain dε= β − 1 − π. dp

(13.22)

Hence, the first order condition can be rewritten µ + (β − 1 + π)(βq* − p) = 0.

(13.23)

The values of µ and p solving the first-order condition are equilibrium values: µ* + (β − 1 − π)(βq* − p*) = 0.

(13.24)

Krainer and LeRoy note that the model has five equations. The first three: ε-* + 1 − p* 2

冢冢

s* = µ* v

冣冣 + β(1 − µ*)s*

q* = µ*p* + β(1 − µ)q*

(13.25) (13.26)

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411

and µ* = 1 − ε-*

(13.27)

are the equilibrium counterparts of (13.8), (13.19), and (13.11), respectively. The others are equilibrium versions of the first-order conditions (13.15) and (13.24). There are five unknowns: q*, p*, µ*, ε-*, and s*. Note that the solution to this equation system is a stationary symmetric Nash equilibrium. The equations are nonlinear, but easily solved. Krainer and LeRoy observe that the model is not quite complete in that the number of agents has not been specified. If there are a finite number of agents, then one agent could possibly own all the houses in the economy. In such a situation, the question arises of which house he inspects if his match fails. The authors ignore these possible events. Over any finite period they occur with a low probability if there is a large number of both houses and agents. They next study the problem of determining the existence of a solution. They start by using (13.27) to eliminate ε-* in (13.15) and (13.25). They note that if µ* is fixed, the remaining equations are affine (and linearly independent). Therefore, p*, q*, and s* are uniquely determined as functions of µ*. Hence, the equations of the model define a map—called ψ—from the unit interval to itself. Since ψ is continuous one uses the Schauder fixed point theorem to prove existence. Krainer and LeRoy make an important point when they observe that assuming risk neutrality, excess returns on liquid assets are martingales, that is, fair games. The conditional expected return on any asset less the interest rate is zero. It is supposed by many that the fair game model describes returns only in markets that are perfectly liquid. This supposition is based on the fact that the simplest rationale for the fair game model does require market liquidity. This justification consists of the observation that if there were some asset with an expected return different from the interest rate then a single risk neutral investor could obtain a gain in expected utility by borrowing and buying the mispriced asset, or the reverse. Trade would continue until fair game prices were established. In the case of illiquid assets, transactions costs usually deter the investor from bidding away the return differentials. The argument concludes that one would not culminate with a fair game. Krainer and LeRoy show that this argument is invalid. Necessity is confused with sufficiency. They note that if it is correct that markets are liquid, then one can justify the fair game model by appealing to the actions of a single risk-neutral investor. The argument also fails for illiquid markets. But these facts do not imply that perfect liquidity is necessary for the martingale model. Asset returns in liquid markets behave as they do, not because, if they did not, a single agent could arbitrage profitably, but because otherwise the optimal trading rules of agents collectively would be mutually incompatible.

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They conclude that they have no argument either way about whether returns on illiquid assets are martingales. Krainer and LeRoy study the nature of the equilibrium distribution of gross returns on an asset.19 In their study there are three origins of wealth: (1) any agent whether matched or not can own one or more houses that he does not occupy. Each unoccupied house is always for sale at p*. Before selling all unoccupied houses are valued at q* per house. (2) An agent whose match with his occupied house is ε has an asset worth v(ε). (3) The search option with value s* is owned by an unmatched agent. Krainer and LeRoy analyze the returns on each of these assets. 1

The equilibrium distribution of the return on a house that is being sold is p*/βq*,

r* =

冦1,

with probability µ* with probability

(13.28)

1 − µ*.

By the author’s convention, the returns from a sale are received by the seller in the current period rather than the subsequent period. The nextperiod value of the payoff given a sale of the house is β − 1p*. If the house is not sold, its next-period value is q*. The current value of the house is q*. Therefore, the return distribution is as displayed in (13.28). The expected return is E(r*) = µ*λ*p*/βq* + (1 − µ*)

(13.29)

Apply (13.26) and (13.29) becomes: E(r*) = β−1. 2

(13.30)

The expected return is equal to the investor’s time preference. The distribution of returns to an owner-occupied house is ε/v(ε) + 1,

r* =

3

with probability π

冦(ε + q* + s*)/v(ε),with probability

1 − π.

(13.31)

Observe that the values of an owner-occupied house to its owner is v(ε), not p* or p* + βs*. The next period payoff on the house is ε + v(ε) if the match is preserved and is ε + q* + s* if the match is severed. Take the expectation and apply (13.10) to yield the expected rate of return as presented in (13.30). Matchless agents own search options each of which have a current value of s*. The expected return on the search option conditional on buying or not buying is -

-

 ε + 1) + πv ε + 1 + (1 − π(q* + s*)) − p* with probability µ* 冤冢 2 冣 冢 2 冣 β冥 r* =  1 with probability (1 − µ*).

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413

Observe that p* is multiplied by the interest rate because returns are defined as next period payoff divided by current period value. Thus (13.30) follows from (13.3) and (13.25). Krainer and LeRoy note that in all three cases, excess returns r = β−1 are fair games—the expected excess returns given that the values of any or all of an agent’s state variables are zero. The authors conclude by noting the difficulty of testing the martingale hypothesis for illiquid assets. They make a key observation: “A more promising research strategy is to test the model by determining its predictions for return and price variables that one can measure, rather than by trying to construct a proxy for the theoretically correct return measure” (Krainer and LeRoy 2002: 241). This type of approach, which is ongoing by Krainer (1997), is compatible with a strict Bayesian analysis that is currently showing a wide range of applicability. One of the Bayesian key commandments is “avoid unobservables” by redesigning your model in Bayesian terms. We will study this problem extensively in Volume II. 13.6 THE HOUSE-SELLING PROBLEM: THE VIEW FROM OPERATIONS RESEARCH In the desperate wartime environment Abraham Wald developed the theory of sequential analysis. This stimulated an enormous research undertaking by statisticians and probabilists. An important outcome was the formulation and solution of the house-selling problem by MacQueen and Miller (1960). This model is equivalent to the basic sequential search model presented in 1970. A concise statement follows: The House-Selling Problem (MacQueen and Miller 1960). The presentation is from Ross (1970). Offers arrive daily for an asset we own. Each offer, independent of others, equals j with probability, Pj ≥ 0. Offers may be accepted or rejected; for each day the asset remains unsold a maintenance cost c, c ≥ 0 is incurred. The wish is to find a policy that maximizes expected return, where return is the value of accepted offer less the total maintenance cost incurred. Suppose we are able to recall any past offer, so the state at any time is the maximum offer received by that time. Hence, transition probabilities are given by

 0,  i Pij = 冱Pk, k = 0 Pj,

if j < i if j = i. if

j>i

The stopping set for the one-step look-ahead policy is

414

The Economics of Search ∞

i

冱P + 冱 jP − c冧

冦

B = i:i ≥ i

j

20

j

j=0

j=i+1

∞

冦

= i:c ≥

冱 (j − i)P 冧 j

j=i+1

= {i : c ≥ E[(Χ − i)+]}, where Χ is a r.v. representing the offer on a given day, i.e., P(Χ = j) = Pj, and x+ = max(x,0). Because (X − i)+ decreases in i we see that as the state cannot decrease (the maximum offer ever received cannot decrease in time), B is a closed set. Hence assuming stability (holds when E(Χ2) < ∞) the optimal policy accepts the first offer that is at least i*, the reservation offer, where

冦

i* = min i : c ≥

∞

冱 (j − i)P 冧. j

j=i+1

Because optimal policy never recalls a past offer, it is also a legitimate policy for the problem in which such recall is not allowed. Hence, the policy must be optimal for that problem as well. (Maximal return when no recall permitted cannot be larger than when recall is permitted.)

13.7 THE NATURAL RATE OF UNEMPLOYMENT Milton Friedman introduced the natural rate of unemployment in his 1968 Presidential address to the American Economic Association.21 In his Nobel lecture, Friedman (1977: 453) re-evaluates the relation between inflation and unemployment and claims that it illustrates the positive scientific character of economics—it is an admirable illustration because it has been a controversial political issue . . . yet the drastic change that has occurred in accepted professional views was produced primarily by the scientific response to experience that contradicted a tentatively accepted hypothesis—precisely the classical process for the revision of a scientific hypothesis. Friedman described three stages in the analysis of the relation between inflation and unemployment. In the first stage, the Phillips curve (1958) hypothesis was widely accepted, that is, there is a stable negative relation between the rate of change of wages and the unemployment level. This stable relation was regarded as causal and, therefore, one that policy-makers could employ in their decisions regarding the appropriate levels of inflation and unemployment.

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415

The second stage began with the recognition that the tendency for an acceleration of inflation to be accompanied by a drop in unemployment is explicable by unanticipated changes in nominal demands on markets characterized by long-term commitments of both capital and labor. Friedman presents a search-match-theoretic analysis of the explanation. Long-term labor commitments can be explained by the cost of acquiring information by employers about employees and by employees about alternative employment opportunities plus the specific human capital that makes an employee’s value to a particular employer grow over time and exceed his value to other potential employees. (Freidman 1977: 456; emphasis added) The second stage is characterized by the introduction of the Friedman– Phelps natural rate hypothesis. The upshot of this hypothesis was summarized by Friedman: What matters is not inflation per se but unanticipated inflation; there is no stable tradeoff between inflation and unemployment; there is a “natural rate of unemployment” . . . unemployment can be kept below that level only by accelerating inflation; or above it only by accelerating deflation. Only surprises matter. (Freidman 1977: 458) Friedman noted that the natural rate has been increasing in the U.S. for two reasons: (1) women, teenagers, and part-time workers are a growing fraction of the labor force, and (2) unemployment insurance is more generous and has larger coverage. Friedman, finally, identified a third stage, where the Phillips curve has a positive slope. Friedman conjectured that the explanation of this phenomenon will be based on political analysis, whereas the stage 2 phenomena were driven by search theory and human capital considerations. It may be useful at this point to show the relation between equilibrium unemployment and the key ingredients of the classical search model. Each period there is a probability γ that a worker survives into the next period. The Bellman equation for the classic search is now given by ∞

冮

w V(w) = max ,c + βγ V(y)dF(y) . 1 − βγ 0

冦

冧

(13.32)

Let ut be the unemployment rate at t. Its law of motion satisfies ut = (1 − γ) + ut − 1 γF(ξ). Thus, the stationary level of unemployment is denoted by

(13.33)

416

The Economics of Search u=

1−γ . 1 − γF(ξ)

(13.34)

Ljungqvist and Sargent call the model the associated “lakes model.” 13.8 A MODERN APPROACH TO THE NATURAL RATE OF UNEMPLOYMENT The 1997 issue of Economic Perspectives contained a symposium on the natural rate of unemployment. The article by Rogerson is almost a perfect fit for our purposes since it presents the natural rate in terms of the matching methods developed by Mortensen and Pissarides (1994). Rogerson correctly maintains that this model’s focus on the dynamics of the formation and dissolution of employment matches must be central to any theory of unemployment.22 There are two technologies associated with the Mortensen–Pissarides model: the first shows how output is derived from a matched worker–entrepreneur pair; the second depicts the matching process for workers and employers. Output of every pair is composed of two parts: an aggregate part which is identical for all pairs and an idiosyncratic part peculiar to each match. Aggregate shocks are good, bad, or null. Workers and entrepreneurs join one another by means of a matching function. This function is similar to a production function mapping inputs into outputs. In any given time period the sequence of events is as follows. At the start of each period, each extant match that produced in the previous period is struck by its idiosyncratic shock and all new matches obtain values for their idiosyncratic productivity. Then, everyone sees the aggregate shock and all matches decide whether to continue or quit. Search decisions follow. The equilibrium for the matching model depends on how wages are determined. Rogerson assumes that outcomes are the consequence of generalized Nash bargaining. This specifies how the surplus from a match is divided among its two members. Rogerson points to two key decisions which make the model understandable. From the workers perspective, the crucial ingredients of a separation decision are current and future prospects in the extant match, the probability of making a new match, and the future expected prospects at a match given that one is discovered. Clearly, the worker prefers the current match if its present discounted value is larger than the present discounted value of unemployment. A similar logic drives the entrepreneur’s decision-making. An entrepreneur will prefer a match that yields positive future expected returns. Note that it is always agreeable for both parties to continue a match when the surplus of the match is positive. With respect to advertising a vacancy, in equilibrium, entrepreneurs post

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vacancies provided the expected return from doing so net of the posting cost is greater than zero. Efficiency of the equilibrium depends on the share of the surplus captured by each member as a consequence of Nash bargaining. As we saw in Chapter 11, Hosios (1990) derived a simple condition for determining whether an equilibrium allocation is efficient. Rogerson gives the following example. Let the matching function be Cobb–Douglas: m(v,u) = Av1 − α uα, where v is the number of vacancies and u is the number of unemployed. Hosios showed that equilibrium allocations are efficient if and only if the surplus share obtained by the entrepreneur equals 1 − α. This matching model exemplifies dynamic stochastic general equilibrium modeling.23 It should be noted that the equilibrium in these models are stochastic processes. The behavior of such a model can be studied in the absence of aggregate shocks giving rise to a deterministic equilibrium. If this deterministic equilibrium exists and is unique, the actual equilibrium can be regarded, roughly, as fluctuations around the deterministic steady state equilibrium. In the matching model, efficient allocations maximize the expected present value of aggregate consumption subject to the feasibility constraints. Rogerson’s analysis of the Mortensen–Pissarides model shows that all unemployed workers are between jobs in the model’s equilibrium. In this sense, all unemployment is frictional. Still, employment is involuntary in that at any time, an unemployed worker prefers to change places with an employed worker. The message of Rogerson’s appraisal of the matching model is straightforward: unemployment cannot be separated into voluntary and involuntary components, or into frictional and cyclical components or into equilibrium and nonequilibrium components. All unemployment is partially voluntary; cyclical unemployment is frictional; and an equilibrium in one model may lose its equilibrium status in another model. With respect to the theoretical usefulness of Friedman’s natural rate, Rogerson conjectures that Friedman’s natural rate can be approximated by the deterministic steady-state equilibrium level of unemployment in certain classes of models. But why focus on the deterministic equilibrium when the matching model presents a clear definition of equilibrium? Rogerson surmises that the natural rate is an empirical concept. When so interpreted it may be useful, but there are other empirical concepts that are even more useful in capturing the empirical behavior of a particular employment series. Rogerson concludes that Friedman’s major contribution was extremely valuable. He found fault with the two major paradigms of macroeconomics—“the short-run Keynesian model was defective in that it did not distinguish permanent and temporary changes in monetary policy in terms of their effect on unemployment. And the ‘long-run’ growth model was at fault because it assumed no unemployment at all” (Rogerson 1997: 90; original emphasis).

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13.9 ADAM SMITH ON DYNAMIC STOCHASTIC MODELS In the Wealth of Nations, Smith (1776: 61) defines the “natural price of a commodity.” The actual price may be greater or smaller than this natural price. But, “the natural price, therefore, is, as it were, the central price, to which the prices of all commodities are continually gravitiating.” Smith also considers the impact of stochastic forces, namely, “accidents” on the market price. Just as several early economists had a literal command of the sequential job search process (see Chapter 4), so, too, Smith understands what a dynamic stochastic model of a perfectly competitive industry is, but he does not have the mathematics required to translate his insight into a testable hypothesis.

13.10 COORDINATION AND INFLATION24 Ball and Romer (2003) investigate the welfare effects of the relative price variability caused by inflation. The key concept in their study is that the formation of long-term relationships between customers and suppliers implies that prices have an informational role. In particular, customers use current prices as a signal of future prices. Inflation interferes with this informational process and causes customers to commit costly errors in choosing the “best” set of long-term relationships. Ball and Romer ask: why do firms allow relative prices to vary by seldomly occurring nominal price changes? Long-term relationships are the key to a correct answer. In their absence, relative price variability causes costly variation in firm sales and hence firms have strong motives to stabilize prices via frequent nominal adjustments. Deviations from this policy occur only if the menu costs of price adjustment are quite large. “With long term relationships, however, a firm’s sales remain steady as its price varies. Thus losses from infrequent change are small. So firms choose this alternative even if menu costs are small” (Ball and Romer 2003: 179). Earlier models studying the question of welfare loss due to inflation find the welfare effects to be ambiguous: “they do not capture the common intuition that inflation is harmful” (Ball and Romer 2003: 189). These early models do not consider repeat purchases—consumers purchase a single good only once so that long-term relationships are completely absent. Also absent is the informational role of prices, which Ball and Romer argue is the primary source of welfare loss. Friedman’s view is quite different from that of Ball and Romer. Building on Lucas (1973), Friedman believes that fluctuating inflation makes it difficult to distinguish relative prices from the observed nominal prices. This explanation assumes that the current price level is unobservable. Ball and Romer assume that it is observable so agents know current relative prices. Uncertainty enters the Ball and Romer model when the agent tries to forecast

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price movements. Another related difference between Friedman and Ball and Romer is the following. In Friedman’s model, the variance of inflation affects information, whereas Ball and Romer discover that even steady inflation erodes information by increasing microeconomic variability. It seems reasonable to conclude that the Ball–Romer model is applicable to moderate inflation countries like the U.S., where accurate information is available with a short lag. On the other hand, the Friedman–Lucas model seems germane for high inflation countries, where there usually is great uncertainty about the future price levels. Ball and Romer conduct a numerical analysis, which shows that welfare costs are significant at moderate inflation rates. Relative price variability appears to be important because of the central role of prices in a market economy. To the extent that inflation weakens the ability of prices to guide the economy to efficient allocations, the very soul of a competitive economy is damaged.25

13.11 ECONOMIC GROWTH AND COORDINATION Jones and Newman (1994, 1995) point out that the “new growth theory” models by Romer (1987, 1990), Lucas (1988), and Stokey (1988) do emphasize that informational considerations are decisive in our comprehension of long-term growth. However, Jones and Newman believe that these contributions “fail to provide a complete characterization of the informational dynamics of growth.” Their critique hinges on three features: (1) The “new growth theory” does not consider the potential coordination problems generated by persistent growth. Growth is perceived as an accumulation problem with almost no consideration for coordination problems; (2) The informational process guiding the coordination of consumption decisions with production opportunities is not considered; and (3) knowledge capital is viewed as a factor of production with increasing returns, but it is not subject to depreciation by the adjustments generated by the growth process. Jones and Newman offer the following prescription: A natural alternative to the above perspective—inspired by a different yet equally burgeoning literature on search and matching: Lippman and McCall (1976b), Diamond (1982, 1984), Mortensen (1986), Mortensen and Pissarides (1999a), and Hosios (1990)—is to give as much importance to the coordination aspects of growth as to its accumulation attributes. Their key assumption is: Continuous coordination depends upon a supplementary informational process—rational search—which allows agents to adapt to evolving

420

The Economics of Search production innovations by learning how best to exploit their benefits and how to optimally match consumption behavior with production opportunities. In this setting, the process by which agents generate search information becomes as important in understanding long-run growth as the process by which new technical information is generated.

Jones and Newman make an important distinction between “learning by doing” and “learning from experience.” Agents can obtain “coordination gains” increasing utility and output, even with a fixed state of knowledge capital (technology). These gains embody the proceeds of learning within a given technological environment reflecting the degree of adaptation to it. . . . They must be distinguished from benefits traditionally attributed to “learning by doing”. The coordination gains are indeed the consequence of “learning from experience”, but they are not a costless, automatic consequence of producing output. . . . They follow from a conscious choice of (in principle) all agents to reduce uncertainty about their prevailing technological environment through search so as to achieve a better matching of consumptive and productive activities. Jones and Newman characterize the most important idea underlying this perspective. Search information differs from the blueprints of the knowledge capital approach. It is fragile and it depreciates. The basic theme of a coordination problem perspective on growth is thus highlighted: the continuous augmentation of one type of information, technical knowledge, which promises a higher potential stream of benefits to agents puts a continuous strain on ongoing coordination, placing a permanent “tax” on another type of information—past search.

13.12 AUCTIONS VERSUS SEQUENTIAL SEARCH In the context of private information, Arnold and Lippman (1995) compare two fundamental selling institutions: search and auctions. Their comparison focuses on two related aspects of selling mechanisms: (1) timing in which offers are received and (2) seller’s ability to compare these offers. In search (sequential selling) one offer is received per period and it must be accepted or rejected before receipt of another offer. Thus, offers cannot be simultaneously compared in a period. The auction mechanism is quite different. Many potential buyers gather in a centralized marketplace and bids are announced in a single period. In this setting, bids can be compared simultaneously.26

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There are three major contributions of the Arnold and Lippman analysis. First, the analysis generalizes the basic sequential search model to encompass the sale of more than a single unit. They prove in Lemma 13.1 that Sn = ξn + Sn − 1, where Sn is the optimal return when n units remain for sale and ξn is the reservation price when there are n units to sell. Lemma 13.2 shows that the optimal policy comprises an increasing sequence of reservation prices.27 The expected return from adhering to such a policy is increasing in n, while the expected return per unit is decreasing in n. These two results are shown in Theorem 13.1. Their second basic contribution relates to the auction regime, where private bid solicitation costs are analyzed. Using several properties of the optimal return, they show in Theorem 13.2 that the expected return per unit is increasing when bidder valuations are random variables with either uniform or exponential distributions. The third major discovery flows from a comparison of the two selling mechanisms to determine which is preferable. It turns out that the preferred mechanism depends on the number of units offered for sale. In particular, Theorem 13.3 demonstrates that there is a crossing point, n*, of the two return functions such that sequential search is preferred when n < n*, whereas an auction is more desirable when n ≥ n*. In the auction environment with transactions costs, the seller determines the optimal number of bidders to solicit and then sells the n units to the n individuals submitting the highest bids. The solicitation and comparison of offers in a single period results in less refined sorting than that attained by sequential search. But the auction regime sells all n units in the initial period. Theorem 13.3 reveals that the auction not only avoids delays in selling, but also becomes an increasingly efficient means of price discrimination as n increases. In addition, Theorem 13.3 applies even when there are economies of scale in soliciting buyers. These scale economies cause Vn/n to increase more rapidly. The search methods used in this article are familiar to us. However, the auction regime may be obscure for some readers. Let us, therefore, spend a moment describing the auction regime as it is presented in Arnold and Lippman (1995).

13.12.1 The auction regime Securing bidder information is costly. The cost of soliciting m bids is md, where d > 0, and need not equal c of the search regime. The bids are i.i.d. random variables and Χ1, . . .,Χn ~ F and E(Χi) < ∞. At the beginning of the first period, a solicitation cost is paid and m bids are received. Payment from the winning bidder is received immediately thereafter. Consider the search cost as an advertising expenditure necessary to attract bidders; an increase in advertising increases the number of bidders. The analysis is limited to an open ascending-bid auction. With m > n

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bidders, all n units are sold at the n + 1st lowest bid. If the optimal sample size is positive then it is strictly greater than n. Revenue equivalence is extended by showing that the expected revenue generated is the same whether units are sold one at a time or simultaneously.28 Here they assume that the n units are sold simultaneously. Bidders bid up to their true valuations in an open ascending-bid auction. Hence seller’s expected revenue is the expected value of the (n + 1)st lowest bidders evaluation multiplied by the number of units sold. Define order statistics Y1 ≥ Y2, . . .,Ym as values of {Χi, . . .,Χm} ranked in descending order. Notice ∞

∞

冮

m

冮 冱 冢 j 冣F¯¯(y) F(y)

E(Yn + 1) = P(Yn + 1 > y)dy =

m

j

m−j

dy.

(13.35)

0 j=n+1

0

The cost of soliciting each bid is d. Therefore, the expected value Vn(m) of auctioning n items to m bidders is ∞

Vn(m) = n

m

冮 冱 冢 j 冣F¯¯(y) F(y) m

m−j

j

dy − md.

(13.36)

0 j=n+1

Analysis is limited to uniform and exponential distribution functions. If a sample of size m is drawn from a uniform distribution on [a,b] with a ≥ 0 (wlog set a = 0 and b = 1), then (13.36) gives 1

E(Yn) =

m

冮 冱 冢 j 冣(1 − y) (y) m

j

m−j

dy = 1 −

0 j=n+1

n . m+1

(13.37)

If offers are drawn from exp(θ) (wlog set θ = 1), then ∞

E(Yn) =

m

冮 冱冢 冣 0 j=n+1

m −jy e (1 − e−y)m − j = j

m

冱 j. 1

(13.38)

j=n

The return functions for the auction mechanism when F is uniform and exponential are:

冤

Vn(m) = n 1 − and

n+1 − md, m+1

冥

(13.39)

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m

Vn(m) = n

冱

1 − md, respectively. j=n+1 j

(13.40)

Using (13.39) and (13.40) Arnold and Lippman derive the optimal sample size mn as a function of the number of, n, of units being sold. They first establish that Vn(m) is concave. Lemma 13.1 Assume F is either uniform or exponential. For fixed n, Vn(m) is strictly concave in nonnegative integer values m. Proof

If F is uniform using (13.39) we obtain:

∆ = Vn(m) − [Vn(m − 1)/2 + Vn(m + 1)/2] = n(n + 1)/m(m + 1)(m + 2) > 0. If F is exponential, ∆=

n n n − + m m m+1

冤

冥冫2 = n/[2m(m + 1)] > 0.

Lemma 13.1 implies that a seller maximizes expected return from an auction by selecting the optimal sampling size mn equal to the smallest integer such that Vn(m) − Vn(m + 1) > 0. By strict concavity Vn(m) achieves a maximum at no more than two consecutive values of the integers. An explicit formula for mn in terms of the solicitation cost d is given by Lemma 13.2 Let d and n be given and denote the integer part of x by [x]. If mn is strictly positive, then for F uniform and F exponential mn satisfies, respectively, mn = [(1 − √(1 + 4n(n + 1)/d)/2)]

(13.41a)

mn = [n/d]

(13.41b)

Proof

If F is uniform,

Vn(m) − Vn(m + 1) = n(n + 1)/(m + 2) − n(n + 1)/(m + 1) = −n(n + 1)/{(m + 1)(m + 2)} + d Hence, Vn(m) − Vn(m + 1) > 0 > m2 + 3m + (2 − n(n + 1))/d > 0. From the quadratic formula (−3 + √1 + 4n(n + 1)/d)/2 is the positive root of this expression. Hence,

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The Economics of Search mn = (−3 + √1 + 4n(n + 1)/2)/2 + 1 = [(−1 + √1 + 4n(n + 1)/d)/2].

If F is exponential, then Vn(m) − Vn(m + 1) = d − n/(m + 1). Thus, n Vn(m) − Vn(m + 1) > 0 > m > − 1 and mn = [n/d]. d Treating mn as a continuous variable, it is easily seen from (13.41) that mn is a nonincreasing function. Hence the auction mechanism becomes more efficient as n increases. The optimal sample size mn maximizes Vn(m), but it does not guarantee a positive return from the auction! Let Vn ≡ Vn(mn). In general Vn will be negative for all values of d above some critical value d* which depends on F and n. If d > d*, the seller maximizes expected revenue by not soliciting any bids. (Presumably another selling mechanism than an auction should be selected.) d* is easily calculated as a function of n when F is uniform. When F is exponential, d* can only be bounded. Fix n. If F is uniform, d* = n/2(2n + 1). If F is exponential then 1 n/(ne + e) ≤ d* ≤ . e Lemma 13.3

Theorem 13.3 If F is either uniform or exponential, then the per unit return Vn/n is increasing. Proof

Fix n ≥ 1 and define Dn = Vn + 1/(n + 1) − Vn/n. Suppose F is uniform, fix

d ∈ (0,n/2(2n + 1)) and pick i > 0 such that d ∈ (4n(n + 1)/[(2(i + 1) + 1)]2 − 1), 4n(n + 1)/ [(2i + 1)2 − 1], whence 13.41a ⇒ mn = i. Use (13.40) to get Dn ≥ Vn + 1(i + 1)/(n + 1) − Vn(i)/n = {n[1 − (n + 1)/(i + 1)] − id/n} + 4(i − n)/[(2(i + 1) + 1)2 − 1] > 0. The first inequality follows from i + 1 being suboptimal sample size for n + i units, the next two equalities follows from (13.40) and the last inequality follows from facts that i > n and d > 4n(n + 1)/{(2(i + 1) + 1)2 − 1}. A similar argument can be made for the exponential.

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13.12.2 The preferred mechanism Given c, d, F and β, the number n of units to be sold determines which of the two mechanisms yields higher return. The auction is superior for all values of n sufficiently large. Theorem 13.4 in n.

The function Sn is concave; in particular, Sn/n is decreasing

Theorem 13.5 If F is either uniform or exponential, there is a critical number n*, 0 ≤ n* ≤ ∞, such that Vn − Sn < 0 for n < n* and Vn − Sn ≥ 0 for n ≥ n*. Proof Apply Theorems 13.3 and 13.4. As n↑ the auction mechanism becomes relatively more efficient for exerting market power and extracting consumer surplus. There may be no crossing point (n* = 0 or n* = ∞). If d > d*, then n* = ∞: the sequential mechanism is used for all values of n. If β, c and d are such that V1 > S1, then n* = 0: auction mechanism is preferred for all n. If d < d* and V1 < S1, n* is neither 0 or ∞. If there are economies of scale, Theorem 13.3 continues to apply. Economies of scale may result in more than one switch point. Sequential search is preferable when n is small and auctions are preferred when n is large. Arnold and Lippman also conduct a careful empirical analysis of the selling mechanisms in livestock markets. Their study revealed that technological advances in trucking, highways, and refrigeration, and changes in government shipping rate regulations reduced transportation costs giving rise to decentralized markets. Also telecommunication advances and the uniform livestock grading standards that were introduced, caused transactions costs to decline. Antitrust action limited oligopoly and led to the decentralization of markets and the creation of more cost-effective sequential selling institutions. These factors combined to increase rewards from sequential selling for all values of n which caused an increase in the critical number n*. Arnold and Lippman’s theoretical predictions were substantiated! Small cattle auctions virtually disappeared by 1985 and were replaced by larger auctions selling enough cattle (n > n*) to compete with the more efficient sequential selling institutions.29 13.13 AUCTIONS AND BIDDING BY MCAFEE AND McMILLAN McAfee and McMillan (1987a) begin their analysis with several important assertions. 1. 2.

Prevalence of asymmetric information in economic activity. Akerlof shows that markets need not be as efficient at transmitting information as Hayek argued.

426 3. 4. 5. 6. 7.

8.

The Economics of Search Inability of used car buyers to observe the quality of a single auto may cause the used car market to cease functioning. Similarly, for medical insurance markets (inability of insurance company to observe individual’s current health cause their collapse). Recent advances in microeconomic theory have been in modeling strategic behavior under asymmetric information. One component of this program is the theory of bidding mechanisms. Study of auctions is a way of approaching price formation. Arrow (1959) notes that the standard economic model of many small buyers and sellers that take as given the market price is deficient in that it does not reveal where prices are constructed or where they come from. Discarding Walrasian auctioneer (deus ex machina) as price setter, who replaces her? Who sets prices?

Even in a perfectly competitive market, Arrow claimed that much uncertainty is present during price formation so that each seller faces a downward sloping demand curve or each buyer faces an upward sloping supply curve. During adjustment to competitive equilibrium “the market consists of a number of monopolists facing a number of monopsonists.” Auction theory is one explicit model of price formation. Yet it lacks the bargaining features emphasized by Arrow (1959: 47). Auctions are of great practical value and the theory of auctions is closer to applications than most mathematical economics. Auction theory has both normative and positive aspects, that is, theoretical results explain the existence of certain trading institutions. Many results address the question: what is, from a monopolist’s perspective, the best form of selling mechanism to use under any particular set of circumstances? Should the seller impose a reserve price? At what level? Can the seller design an auction to price discriminate among bidders? Is it ever beneficial for the seller to require payment from an unsuccessful bidder? Whether it is feasible to make the payment depends not only on the bid, but also on something correlated with the true value of the item (as the case of royalties), should the seller do so? Should the seller distribute information he has concerning the item’s true value? What can the seller do to defend against collusion among buyers?

13.13.1 The types of auctions and their uses What is an auction? A market institution with an explicit set of rules determining resource allocation and prices on the basis of bids from market participants. The list of goods sold at auction range from complex military equipment to artwork to fish. Why use auctions? Cassady (1967) explains “Perhaps some products have no standard value—price of fish depends on demand and supply conditions which fluctuate over time.”

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For many government contracts, firms submit sealed bids and the contract is awarded to the lowest qualified bidder. In a double auction, several buyers and several sellers submit bids simultaneously. This takes place on organized exchanges: stock exchange and commodity market. There are four types of auctions when a unique item is bought or sold: 1 2 3 4

English auction (oral, open, or ascending-bid auction). Dutch auction (descending-bid auction). First price sealed-bid auction. Each bidder submits a single bid. Second price sealed-bid auction (Vickrey auction).

The Vickrey auction is seldom used! Auction theory avoids bargaining problems by assuming that the monopolist (or monopsonist) has all the bargaining power. Organizers of the auctions have the ability to commit themselves in advance to a given set of policies or procedures. These procedures cannot change after observing the bids—and all bidders know this in advance. The advantage of commitment is that procedures can be adopted that induce bidders to bid in desirable ways. Schelling (1956: 283) explains, “If the buyer can accept an irrevocable commitment in a way that is unambiguously visible to the seller, he can squeeze the range of indeterminacy down to the point most favorable to him.” This follows from “the paradox that the power to constrain an adversary may depend on the power to bind oneself” (1956: 282). There are many types of commitment. “Pledge of one’s reputation is a potent commitment” (Schelling 1960: 29). It does not follow that the party making the commitment is able to squeeze all of the gains from trade. The ability is limited by asymmetric information. The seller does not know any bidder’s valuation of the item for sale. If the seller observed a bidder’s valuation then he could offer the item to the highest valuator bidder at ε below the valuation. It is in the bidder’s interest to accept this “take-it-or-leave-it” offer. When information is asymmetric the seller’s ability to extract surplus is limited. The seller can exploit competition among bidders to drive up the price. But usually the seller cannot drive the price to equality of value to the bidder with the highest value because the seller does not know this valuation. Nature of uncertainty Asymmetry of information is a crucial element in the auction problem. Given the ability to make commitments, the auction organizer with perfect information extracts all gains from trades. However, the reason a monopolist chooses to sell by auction rather than by sorting price is that he does not know the bidders valuations! How the bidders respond to uncertainty depends on their risk attitude (risk averse–risk neutral–risk preferent). Hence one aspect of any bidding situation that the modeler must take into account is the bidders’ risk attitudes.

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Assume the seller is risk neutral. Differences among bidders’ valuations have two causes: at one extreme, suppose each bidder knows exactly how much he values the item—no doubt about the true value to him. He is ignorant of the valuations of other bidders. He knows their valuations as draws from some probability distribution. He also knows that other bidders regard his valuation as a draw from some probability distribution. The differences among valuations are caused by different tastes. So, for bidder i, i = 1, . . ., n, there is some probability distribution Fi from which he draws his valuation vi. Only the bidder observes his own valuation vi, while all other bidders and the seller know Fi. Any single bidder’s valuation is independent from any other bidder’s valuation. This is referred to as the independent-private-values model. It applies to an auction of an antique in which bidders are consumers buying for their own use and not for resale. It also applies to government contract bidding, when each bidder knows what his own production cost will be if he wins the contract. (Of course, in practice he does not even know what he is actually going to end up making and so does not have good information on production costs.) At the other extreme, consider the sale of an antique that is being bid for by dealers intending to resell it or the sale of mineral rights to a particular tract of land. The item being bid for has a single objective value, namely, the amount the antique is worth on the market or the amount of oil (and its selling price and extraction cost!) lying beneath the ground. However, no one knows the true value. Bidders may have different information for making guesses about worth. Let V be the unobserved true value. Then bidders perceived values vi, i = 1, . . ., n, are independent draws from some distribution H(·| V). All agents know H. This is the commonvalue model. The independent-private-values model and the common-value model are polar cases. Real world auctions are mixtures of the two! A general model allowing for correlation among bidders’ valuations also includes independent-private-value and common-value models as special cases, as were designed by Milgrom and Weber (1982). With n bidders, let xi denote a private signal about the item’s value observed by bidder i, let x = (x1, . . ., xn) and let (s1, . . ., sm) = S be a vector of variables measuring the quality of item for sale. Bidders cannot observe any of the components of s. The seller may observe some or all si’s. Let the ith bidder’s valuation of the item be vi(x,s). Thus any bidder’s valuation may depend not only on his signal, but also on what he does not observe, i.e., the other bidder’s private signals and the true quality of the item. This model reduces to the independent-private-value model when m = 0 and vi = xi∀i; and it reduces to the common-value model when m = 1 and vi = s1∀i. Bidders valuations may be correlated to some extent. This can be measured by affiliation30 a concept invented by Milgrom and Weber (1982).31 Auction models are easiest to analyze when based on four assumptions.

The ubiquity of search 1 2 3 4

429

Bidders are risk neutral. Independent-private-values assumption. The bidders are symmetric. Payment is a function of bids alone.

McAfee and McMillan call this the benchmark model (1987a: 706). McAfee and McMillan ask which of the four simple auction types are preferred by the seller. The answer is somewhat of a surprise. All four are equally desirable. On average, in the benchmark model, each type produces the same revenue for the seller! As an example let us look at the dynamics of the English auction. When do bidders stop increasing the price in this auction? When the price has been increased by the bidding process so that it is larger than the second to the last bidder’s valuation, he will exit from the auction. Hence, the highestvaluation person wins and pays a price equal to the valuation of his last remaining rival. This valuation is almost always less than his own valuation so that the “successful bidder earns some economic rent in spite of the monopoly power of the seller” (McAfee and McMillan 1987a: 707). The amount of this rent is subjective and known only by the winning bidder. On average however it can be calculated as follows: we just saw that the winning bidder receives a rent of v(1) − v(2), where v(i) is the ith order statistic. For the winning bidder, the n − 2 other bids are independent draws from distribution function F. The expected rent is the expected difference between v(1) − v(2), which is given by E[(1 − F(v1))/f(v1), where E is with respect to the distribution of v1. The expected amount received by the seller from the winning bidder in an English auction is the expectation of the random variable J(v(1)), defined by: J(v(1)) = v(1) −

(1 − F(v(1)) . f (v(1))

(13.42)

McAfee and McMillan assume that F is such that J is a strictly increasing function, that is, the expected receipts from the winning bid increases in v(1). They show that the expected payment from a second-price sealed-bid auction also equals the expected value of J(v(1)). The outcomes of the English and second-price auctions are dominant equilibria. This means that each bidder has a definite best bid independent of how high he believes the rival bids will reach. However, in a first-price sealed-bid auction there is no dominant equilibrium. The equilibrium in this case is weaker and given by a Nash equilibrium, that is, each bidder selects his maximum bid given his guess (which turns out to be correct in equilibrium) of the decision rules being pursued by his rival bidders.

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McAfee and McMillan show how to calculate this Nash equilibrium. Let bidder i have valuation vi. Bidder i conjectures that other bidders are following a rule given by bidding function B. This means that bidder i predicts (guesses) that any other bidder j has a bid of size B(vj) when his valuation is vj. Note that bidder i does not know vj! B is a monotone increasing function. Now what comprises i’s best bid? If the bid is bi, and it is a winning bid, he captures a surplus of vi − bi. The probability that bi is a winner equals the probability that all of the (n − 1) other bidders have valuations vj such that B(vj) < bi. This probability is given by [F(B−1(bi))]n−1. Bidder i selects bi so as to maximize his expected surplus, which is given by πi = (vi − bi)[F(B−1(bi))]n − 1.

(13.43)

Thus he selects bi so that ∂πi /∂bi = 0. Note that dπi dvi

=

∂πi dbi ∂πi + (∂πi /∂bi) = ∂vi dvi ∂vi

冢 冣

which is simply the Envelope Theorem. Hence, an optimal bid satisfies: dπi dvi

=

∂πi = [F (B − 1(bi))]n − 1. ∂vi

(13.44)

Thus, we have i’s best response to an arbitrary decision rule B which is being used by his/her (n − 1) rival bidders. Now assume the Nash (also known as rational expectations) criterion: “The rivals’ use of the decision rule B must be consistent with the rivals’ themselves acting rationally.” Coupled with a symmetry assumption, namely, any two bidders with the same valuation have identical bids, implies that i’s best bid bi (satisfies (13.44)) is the bid implied by B. That is, at a Nash equilibrium, bi = B(vi). Inserting this Nash condition in (13.44) yields an equation giving bidder i’s expected surplus at a Nash equilibrium, that is, dπ dvi

= [F(vi)n − 1].

(13.45)

Note that (13.45) holds for all n bidders, since at a Nash equilibrium all participants maximize simultaneously. Solving the differential equation (13.45) for πi is accomplished by integrating using the boundary condition that if vᐉ ≡ lowest possible valuation, then the bidder earns zero surplus which implies B(vᐉ) = vᐉ. Now invoking the definition of πi (13.43) plus the Nash condition bi = B(vi) we get each bidder’s decision rule.

The ubiquity of search 431 vi

冮[F(ξ)]

n−1

B(vi) = vi −

dξ

vᐉ

[F(vi)]n − 1

, i = 1, . . ., n.

(13.46)

Note that B is an increasing function and the second term on the righthand side reveals how much the bidder lowers his bid below his subjective valuation vi. McAfee and McMillan mention two crucial steps in this argument: 1

2

They discovered one bidder’s best function in response to a particular decision rule which the designated bidder conjectures all his (n − 1) competitive bidders are invoking. The optimization associated with (13.43) yielded a FOC which in turn produced equation (13.45). The Nash requirement that the conjectured decision rules were consistent with optimal behavior of the (n − 1) rival bidders was invoked. This gave rise to equation (13.45) derived from (13.44) and solved to give (13.46).

Suppose F is uniform with zero the lowest value. Substituting into (13.46) reveals that in a first-price sealed-bid auction, a bidder with valuation v n−1 tenders a bid of size B(v) = (n − 1)v/n. That is, he bids of his valuation. n The winning player is that bidder with the highest valuation v(1). In arriving at their bids, each player assumes his submission is based on the highest valuation v(1). Note that this assumption is costless if wrong, since losers pay nothing. McAfee and McMillan note that it can be proven that B(v(1)) given by (13.46) equals the expected second-highest valuation given the bidder’s information, namely, his own valuation v(1). “The bidder estimates how far below his own valuation the next highest valuation is, on average, and then submits a bid that is this amount below his own valuation” (McAfee and McMillan 1987a: 710). From the seller’s perspective, who is unaware of the winner’s valuation, the expected price is the expected value of B(v(1)), which in turn equals the expected value of J(v(1)) given by (13.42). But it was shown that the expected value of J(v(1)) equals the expected price in an English or second-price auction. Thus, the expected winning price in a first-price sealedbid auction equals that of an English or a second-price auction. By this argument McAfee and McMillan (1987a: 710; original emphasis) demonstrate the truth of the “Revenue Equivalence Theorem: For the benchmark model, each of the English auction, the Dutch auction, the 1st price sealed-bid auction, and the 2nd price sealed-bit auction yields the same price on average.” McAfee and McMillan (1987a: 660–1) note that the Revenue-Equivalence Theorem “is devoid of empirical predictions about which type of auction will be chosen by the seller in any particular set of circumstances.” If,

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however, the benchmark assumptions are relaxed, particular auctions can be ranked. Even though the auction problem is one of monopoly–oligopsony, the solutions are Pareto efficient. The highest valuation bidder obtains the item— assuming J is a strictly monotone function and that the seller values the item less than any bidder. It has been shown that an increase in competition (via an increase in the number of bidders) on average increases the seller’s revenue. Indeed, if there is perfect competition among the bidders (these numbers approaching infinity), all of the gains from trade go to the seller. McAfee and McMillan (1986) have shown that for special distributions like the normal and uniform, an increase in the variance of valuations increases both the average revenue of the seller and the rents of the winning bidders. What is the best-selling mechanism for the seller? McAfee and McMillan invoke the Revelation Principle to answer this query. The auction mechanism is like a production function. It is a production in which inputs are bids and outputs reveal the winning bidder and the payments incurred by the bidders. In a direct mechanism, each bidder reports his valuation of the item. A mechanism is incentive compatible if it is constructed so that each bidder finds that he/she maximizes his/her welfare by telling the truth. Also assume that bidders do not collude. McAfee and McMillan (1987a: 712) define the Revelation Principle as follows: “For any mechanism, there is a direct, incentive-compatible mechanism with the same outcome.” Consider the incentive-compatible mechanism identical with the first-price sealed-bid auction. Recall that the bidder with the largest valuation v wins and pays B(v), the size of his bid. If bidders were asked by the seller to submit their valuations, some will lie because it is optimal to do so. Instead the seller proclaims: “the bidder reporting the maximum valuation vˆ wins and pays B(vˆ ).” This direct mechanism is the same as the first-place sealed-bid auction. Recall that in this type of auction, bidders behave optimally when they announce that v = vˆ . They tell the truth and the mechanism is incentive compatible. The Revelation Principle reveals to the modeler that his search for the best mechanism can be performed on the set of direct, incentive-compatible mechanisms. The number in this set is very large revealing that the Revelation Principle is a theoretical tool. But the Revelation Principle shows that the auction maximizing expected price has the following properties: (a) if J(vi) < v0 for all i, the seller withdraws from the auction; (b) if the inequality in (a) does not hold, the seller offers the item to the bidder whose v is the maximum at a price equal to B(v). The price set optimally in (a) is called a reserve price. The optimal direct mechanism is the solution to a mathematical programming problem with two types of constraints: first, incentive-compatibility (self-selection) constraints, which entail telling the truth, and second,

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individual-rationality (free exit) constraints, which imply that bidders cannot improve their welfare by leaving the auction. Returning to the benchmark model and assuming that the seller assigns a value v0 to the auction item, McAfee and McMillan (1987a: 712) remark that an application of the revelation principle gives the following: “For the benchmark model, the auction that maximizes the expected price has the following characteristics: (a) if J(vi) < v0 for all bidders’ valuations vi, then the seller refuses to sell the item; (b) otherwise he offers it to the bidder whose valuation V is highest at a price equal to B(v).” Note that the (a) characteristic means that the seller optimally establishes a reserve price, and refuses to engage in the auction if all bidders’ valuations fall short. Now J(v) < v so that the seller’s withdrawal from the auction may be inefficient, that is, it is possible that the seller withdraws even though there may be a bidder with a valuation higher than the seller’s valuation. McAfee and McMillan remark that the seller resembles the textbook monopolist in that his best interest is a departure from Pareto optimality. McAfee and McMillan state a “powerful result”: given the four auction types comprising the benchmark model, any one of the four is the optimal selling mechanism given supplementation by the optimal reserve price. For any of these auctions, the optimal reserve price equals J − 1(v0). The reserve price is analogous to the reservation wage in the job search model. Before turning to the analytics of search with posted prices there are several transitions which we highlight. The first is based on a comparison of sequential search and auctions. This shows how a seller moves from one regime, say auction, to another, search. The Arnold–Lippman article is ideal for displaying these transitions. We are also interested in studying processes with asymmetric information. McAfee and McMillan provide an excellent evaluation of auctions which emphasizes the role of asymmetric information. They see information asymmetry at the heart of auction processes. We began with the Arnold–Lippman model and studied the interplay between search and auctions. We then moved to an investigation of the auction process by following the analysis of McAfee and McMillan. Thus we moved from sequential search to auctions and then from auctions to asymmetric information processes. Let us be more specific. When we observe these three phenomena—search, auctions, and asymmetric information—carefully, we notice connections among them. Search is characterized by a single critical number such that the process of search continues if the sampled number is less than critical, and the process terminates if the sampled number exceeds the critical number. The search process is transformed into an auction if a batch of sampled numbers appears almost simultaneously. Termination of search ends with the purchase of an item (in consumer search) and with the commencement of a job (in labor market search). Suppose a house is being sold. Then the transition from search to auction is clear when numerous potential buyers arrive almost simultaneously. The seller or the broker establishes an auction and the highest bidder buys the house.

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Akerlof (1970) considered the sale of used cars. Suppose the seller values used cars at .75 as much as buyers, where true quality is uniformly distributed over the unit interval (0,1) which also measures the amount buyers are willing to pay. Sellers know the exact quality for each of their cars. If all used cars are for sale, buyers are willing to pay the average value 1/2. If the products’ quality exceeds 2/3 the seller will withhold it from sale. Thus only cars of quality below 2/3 are for sale. The buyers respond by being willing to pay an average value of 1/3. Then the seller excludes cars whose quality exceeds 4/9. This process continues without limit until an equilibrium is reached where no trade takes place. The asymmetric information between buyer and sellers has eliminated the used car market. 13.14 THE ANALYTICS OF SEARCH WITH POSTED PRICES In an excellent article Arnold and Lippman (2001) studied the properties of the optimal posted price in the sequential search framework. They note that a posted price is used in most retail sales. This prominence occurred rather quickly after the posted price was introduced less than 200 years ago. Let us follow Arnold and Lippman as they describe the posted price regime and compare it with the reservation price regime.

13.14.1 The posted price regime The posted price regime entails the standard infinite horizon, continuous time search model with recall. The monopolist posts a price p for a single unit of produce. Customers are attracted when the seller spends c per unit time on advertising. The discount rate is δ ≥ 0 and each customer desires one unit of produce. The arrival rate, λ ≥ 0, of customers is Poisson. The seller considers the value of every customer as a random variable V with c.d.f. F and F ′ = f. They assume that E(V) < ∞. Consider a customer with valuation V ≥ p. This customer buys the good at the posted price p. The firm’s expected return R(p) is given by R(p) = − c/(δ + λ) + λ[R(p)F(p) + pF¯¯ (p)]/(δ + λ), where F¯¯ = 1 − F. This simplifies to R(p) = [− c + pλF¯¯ (p)]/[δ + λF¯¯ (p)].

(13.47)

By assuming a Poisson arrival process with parameter λF¯¯ (p), the discounted expected waiting time till a buyer32 arrives is λF¯¯ (p)/(δ + λF¯¯ (p)). Arnold and Lippman note that (13.47) states that the seller’s expected return when a price p is posted is equal to the expected discounted revenue pλF¯¯ (p)/(δ + λF¯¯ (p)) obtained from a sale minus the expected discounted cost c/(δ + λλF¯¯ (p)) of searching for a buyer. The numerator of (13.47) means that profitability requires

The ubiquity of search p*F¯¯ (p) > c/λ,

435

(13.48)

where p* is the optimal posted price. This condition always obtains in the subsequent analysis. Arnold and Lippman observe that E(V) < ∞ and pF¯¯ (p) → 0 as p → ∞ imply that there is a bound B < ∞ such that R(p) < 0 for p > B. The continuity of F¯¯ implies that R is continuous on [0,B]. It follows that there is an optimal posted price and a largest (and a smallest) optimal posted price exists. The first-order condition that presents the optimal posted price p* is given by 1 = h(p)(c + δp) /[δ + λF¯¯ (p)],

(13.49)

where h(x) = f(x)/F¯¯ (x) is the hazard rate function. Equation (13.49) equates the marginal revenue and marginal cost flowing from a small change in the posted price. The seller captures the entire increase in p when the item is sold. Hence, the marginal revenue from increasing p is 1. The marginal cost, MC(p) = h(p)(c + δp)/[δ + λF¯¯ (p)],

(13.50)

which is the right-hand side of (13.49).

13.14.2 Comparison with the reservation price regime Recall that letting V(x) be the seller’s expected return when he/she uses the reservation price x gives x

V(x) = c/(δ + λF¯¯ (x)) +

∞

冢冮V(x)dF(v) + 冮VdF(v)冣/δ + λF¯¯(x). 0

(13.51)

x

Now the optimal reservation price ξ must satisfy V(ξ) = ξ: the seller accepts an offer v if it is larger than the expected return V(ξ) from continued search when following an optimal reservation price policy ξ, and he rejects v if V(ξ) > v. Substitute V(x) = x in (13.51), and it is clear that ξ must satisfy the first-order condition c + δx = λH(x),

(13.52)

∞

冮

where H(x) = F¯¯ (y)dy. Divide both sides of (13.52) by δ + λ and we extract x

the following interpretation of (13.52). The left-hand side denotes the marginal cost of continued search which equals the discounted cost c/(δ + λ) of finding another customer and the drop x − λx/(δ + λ) = δx/(δ + λ) in the

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discounted value of x while searching. The right-hand side is the expected marginal gain H(x) in the selling price discounted by Ee − δτ, the expected discounted time till another arrival. Arnold and Lippman note two immediate differences between the two regimes. First, substituting (13.49) into (13.47) gives R(p*) = p* − 1/h(p*), implying R(p*) < p*. In the reservation price regime V(ξ) = ξ. The second difference is that in the reservation price regime a reservation price of ξ < c/λ may be optimal, whereas in the posted price regime p* > c/λ is essential for profitable search. These differences are direct consequences of the fact that the buyer pays exactly p* in the posted price regime, whereas in the reservation price regime the buyer pays V ≥ ξ. Arnold and Lippman also report subtle differences in the two regimes by comparing (13.52) and (13.49). In the reservation price regime, marginal revenue from search is continuous and strictly decreasing in the seller’s reservation price and the marginal cost of search is constant. Hence, there is a unique solution to (13.52). Things are less pleasant in the posted price regime. The marginal revenue is constant, but MC can be either increasing or nonmonotone. In addition, if h is not continuous, neither is MC. The authors demonstrate that when MC is not continuous, a solution to (13.49) may not exist. See Example 1 on p. 451 of Arnold and Lippman (2001). We draw the remaining remarks from the summary and conclusions of Arnold and Lippman. Note that there are 15 theorems and six examples that explain differences between the two regions. We select a few of the most interesting. Whereas the seller’s return is a unimodal function in the reservation price regime, R(p) is unimodal and p* ≥ ξ if h′ ≥ 0 and also if δ = 0 and h′ < 0, in which case p* < ξ (see Arnold and Lippman’s Theorem 3). The authors regard the following result as the most surprising difference between the two regimes. An increase in demand, that is, a stochastic increase in customer valuations from V to Y leads to an increase in ξ. Yet an increase in demand which benefits the seller, can induce a decrease in p*. (See their Example 6.) The hazard rate function h is unimportant in the analysis of the reservation price regime whereas the hazard rate plays a critical rule in the analytics of the posted price regime. The authors give several examples. The outcomes flowing from the analysis of the elasticity of demand or the unimodality of the sellers return as a function of posted price hinge on the hazard rate. That is, results are different if the hazard rate is increasing instead of decreasing. Arnold and Lippman also show that a mean preserving increase in risk causes the optimal reservation price to increase. The reason for this is that expected gain H(ξ) from an additional observation increases with a riskier distribution. On the other hand, in the posted price regime a mean

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preserving increase in risk may harm the seller and cause him to lower his posted price. This is a rich and original study and readers who find our account interesting should consult the article.

13.15 SALE OF A DETERIORATING ASSET VIA SEQUENTIAL SEARCH As far as we know, of the thousands of papers comprising the economic search literature none have modeled the sale of a deteriorating asset using sequential search. A very recent paper by Lippman and Mamer (2006) fills this void in an elegant manner. Lippman and Mamer show that as the value of a searcher’s asset diminishes with the passage of time, the optimal fraction pn of the asset’s value decreases. That is, the posted price (ξn in the reservation price regime) declines as time passes. This diminution holds under the fairly general requirement that the asset’s intrinsic value gn is a nondecreasing, log-concave function of n. Furthermore, this discovery is true in both the posted price and reservation price regimes. Lippman and Mamer note that this finding is by no means intuitively obvious. Two forces are in play. First, it would seem reasonable that with many periods remaining, the searcher must accept a lower portion of its value (to the buyer) to avoid the risk that the asset remains idle, and deteriorates so that it is of less value in the subsequent period. Lippman and Mamer identify a countervailing force: the seller has much more to lose if they sell all of their units for a small percentage of their value. The rate of deterioration determines which of these forces dominates. Theorems 1 and 2 of the Lippman and Mamer paper show that it is best for the seller to hold out for a larger fraction of the intrinsic value when more units are awaiting sales. Therefore, search is nonstationary.

13.16 MIDDLEMEN, MARKET-MAKERS, AND ELECTRONIC SEARCH Rubenstein and Wolinsky (1987) investigated a market with three kinds of agents: sellers, buyers, and middlemen. The focus of the paper is on the role of middlemen [sic] and a framework for studying their activity. The extent of the activity of middlemen is endogenously determined. We should note that the concept of a middleman is intimately tied to the matching models of Diamond, Mortensen and Pissarides.

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13.16.1 Middlemen In his 2004 article on middlemen, Shevchenko begins by quoting Rubenstein and Wolinsky (1987): “Despite the important role played by intermediation in most markets, it is largely ignored by the standard theoretical literature . . . because the study of intermediation requires a basic model that describes explicitly the trade frictions that give rise to the function of intermediation” (Shevchenko 2004: 1). Rubenstein and Wolinsky show that the function performed by middlemen is stimulated by search-theoretic models. Middlemen have an ability to exogenously purchase a unit of a homogeneous output from firms and sell it to consumers. Middlemen are economical if and only if the demand for their activity exists in equilibrium, that is, the middlemen can satisfy consumers better than if producers tried to do it directly. Shevchenko’s model of intermediation is different, but remains within a search setting. Shevchenko’s basic notion is that in an economy with a host of goods and a variety of tastes, there is a function for agents who maintain inventories of many goods. The key tradeoff confronting intermediaries is the same as that facing individual firms when they determine inventory levels, namely, larger inventories are more expensive to hold, but increase the probability that a random customer will discover a satisfactory item. Shevchenko assumes that a middleman chooses the number of shelves in the sellers store which he rents. Shevchenko also endogenizes the decision to become a middleman. Middlemen start with an inventory which fluctuates over time as customers arrive and try to exchange different goods. There is no money in this model! Also, there are no posted prices. Instead, prices are arrived at via bargaining. The steady-state equilibrium number of middlemen is determined along with their size, the distributions of inventories and prices. He obtains closed form solutions for these entities. Shevchenko observes that the equilibrium can be compared to the social planner’s solution. Conditions can be stated which produce too many or too few middlemen. These sizes can also be compared to the social planner’s decisions. His model predicts that an increase in the complexity of exchange causes both the size of the intermediaries and the diversity of goods to decline. The model’s fundamental structure is comparable to the search-theoretic model of the exchange process especially those monetary search models of Kiyotaki and Wright (1993), Shi (1995), and Trejos and Wright (1995). There is a diversity of goods and agents with heterogeneous tastes and this generates the double coincidence problem. Nevertheless, money is not included in the model. Shevchenko obtains an illustrative equilibrium solution when there are two goods. His analysis shows that the best configuration of goods in a store has the greatest variety. An interesting result on the behavior of prices is discovered: if the good purchased by a customer improves the store’s inventory composition, then the middleman lowers his price. In addition, price changes get smaller as the configuration of goods approaches the best one.

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13.16.2 Middlemen versus market-makers: competitive exchange Spulber’s dealer model In this model, producers and consumers must obtain price quotes by contacting middlemen. Middlemen are infinitely lived and calculate bid and ask prices to maximize expected discounted profits. The continuum of middlemen is indexed by k, the marginal cost of performing an exchange between a producer and a consumer. These costs k are uniformly distributed over [k, k¯ ]. The lower bound k is the marginal cost of the most efficient middleman. The marginal cost of the least efficient middleman who remain in the dealer market in equilibrium are given by k¯ . Both producers and consumers search sequentially. Each period a searcher receives one price quote from a middleman drawn randomly from U[k,k¯ ]. The future is discounted by ρ(1 − λ), where ρ ∈ (0,1) is the rate of time preference and (1 − λ) is called the “survival probability,” where λ is the rate of departure from the market before a trade takes place. Let Fα(a) denote the c.d.f. (of ask prices) confronting consumers and Fβ(b) the c.d.f. of bid prices confronting producers. Price quotes are i.i.d. draws from these c.d.f.s. Let V(a,v) be the expected discounted value of optimal search for type v consumer who has obtained an ask price of a from a randomly selected dealer. The consumer must select one of three options: (a) do nothing (don’t buy at price a and don’t search), (b) accept asking price a, or (c) reject a and continue search. The Bellman equation for the consumer’s problem is given by: a¯

冮

冦

冧

Vc(a,v) = max 0,v − a,ρ(1 − λ) V(a′,v)Fα(da′) a ¯

(13.53)

where (a,a¯ ) is the support of the ask price c.d.f. The first term in the braces corresponds to the option of no search, no trade, and no consumption. The reservation price rc(v) is the unique solution of the following equation: rc(v)

1 v = rc(v) + δ

冮 F (a)da,

(13.54)

1 −1 ρ(1 − λ)

(13.55)

α

a ¯

where δ = δ(p,λ) =

is called the “composite” exit adjusted discount rate per period. From (13.54) rc(v) is strictly increasing on (v c,1), where v c = rc(v c) ≡ r c = a.

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A similar argument yields the Bellman equation for the producer, p. b¯

冦

冮

冧

Vp(b,v) = max 0,b − v,p(1 − λ) Vp(b′,v)Fβ(db′) b ¯

(13.56)

where [b,b¯ ] is the support of Fβ. The reservation price for a type v producer is the solution to b¯

冮

1 (1 − Fβ(b))db. v = rp(v) − δ r (v) p

The function rp(v) is monotone increasing over (0,v¯ p), where ¯vp = rp(v¯ p) ≡ r¯ p = b¯ , is the marginal producer whose expected gain from search is zero. Extension of Spulber (1996a, 1996b) Rust and Hall (2003) distinguish between two types of intermediaries to facilitate the purchase and sale of an asset. The dealer or broker is the first type and are usually called “middlemen.” “Market-makers” or specialists comprise the second type. The market-makers post bid and ask prices, while middlemen conceal their prices which are unveiled by conducting costly search. Rust and Hall consider two enigmas: the first is what explains why market-makers are very successful in entering certain markets, like the bond market, and have strong impacts on the resulting trade, whereas these same specialists do not participate in other markets like steel. The second conundrum is to comprehend the process whereby a specialist can enter and thrive in, for example, the bond market even though the middlemen (in bonds) can respond by lowering their prices below the posted bid and ask prices of the specialist. Rust and Hall construct a model where the share of trade intermediated by both middlemen and market-makers is determined endogenously. Their model also addresses the effects of the enormous decline in search and transactions costs flowing from the World Wide Web. Their model extends Spulber (1996a). Four agents are considered: sellers (producers), buyers (consumers), price-setting middlemen, and an agent not considered by Spulber, marketmakers. Spulber’s model has every transaction mediated by middlemen. Exchange in Spulber’s dealer market takes place over individually negotiated prices accompanied by a costly sequential search process. Here we present the Bellman equations in this richer environment for consumers and producers. If a consumer has not yet decided to search, she has three options: (a) do nothing, (b) purchase a unit of the commodity in the exchange for price am, or

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(c) search in the dealer market for a lower price. The consumer’s value function is given by a¯

冮

冦

冧

Vc(am,v) = max 0,v − am,p(1 − λ) Vc(a′,am,v)Fα(da′) , a ¯

where Vc(a,am,v) is the value function for a consumer who chooses to search, and obtained an ask price a from a middleman—a random draw from Fα. When the consumer receives an offer, a fourth option occurs, namely, accepting the middleman’s offer. The Bellman equation is given by a¯

冦

冮

冧

Vc(a,am,v) = max 0,v − a,v − am,p(1 − λ) Vc(a′,am,v)Fα(da′) . a ¯

Theorem 13.6 Suppose there is an equilibrium such that both market-maker and middlemen live in the dealer market. Let a be the smallest ask price in the dealer market after the market-maker’s entry. Let vc(a,am) be the value of the marginal consumer with reservation value am. This marginal consumer is indifferent between trading with the market-maker and trading in the dealer market: an

冮

1 vc(a,am) = am + Fα(a)da. δa If vc(a,am) < 1, three different optimal search-purchase strategies exist, with each depending on the type of consumer. If v ∈ (0,a), it is not optimal for the consumer to trade with the market-maker or search for a middleman in the dealer market. If v ∈ [a,vc(a,am)], it is optimal to trade in the dealer market. If v ∈ (vc(a,am),1], it pays the consumer to ignore the dealer market and purchase the good from the market-maker at ask price am. Rust and Hall now study the market-maker’s entry and pricing decision and they derive the middlemen’s reaction to entry by the market-maker. They identify three possible equilibrium regimes: an unconstrained monopoly environment, a limit-pricing environment, and a competitive setting. They note that in the first two regimes, the market-maker forces all middlemen to exit. In the competitive setting, the market-maker lives with a group of middlemen. In both the competitive and limit pricing environments, the presence or possible entry of middlemen constrain the pricing decisions of the market-maker. If the monopolist market-maker sets an ask price of am and offers a bid price bm, Theorem 13.6 implies that the quantity demanded by the highestvaluation consumers in the interval (vc(a,am),1) is

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The Economics of Search am −1

Q (am,bm) = 1 − am − δ d m

冮F (a)da. α

a ¯

The equilibrium are obtained with and without a monopolist market-maker (Rust and Hall 2003: 376–81). They conclude (Rust and Hall 2003: 389) by summarizing the limiting properties of post-equilibrium equilibria as δ ↓ 0, km ↓ 0, and k ↓ 0 in Theorem 13.7 As δ ↓ 0, km ↓ 0, and k ↓ 0, the equilibrium prices, quantities and surpluses (consumer and producer) tend to Walrasian equilibrium values. In this limiting case the market-maker is involved in 1/2 of the market transactions. The most efficient middlemen (with k = k = 0) is essential in the remaining transactions. A limiting bid-ask spread of zero is charged at the common Walrasian equilibrium price of p* = 1/2. The authors consider Theorem 13.7 as a characterization of “efficient markets:” when search and transactions costs are small, bid–ask spreads are narrow. Thus there is little difference between trading with a market-maker and conducting exchanges in the dealer market. The authors realize that a more realistic analysis of competition between market-makers must take account of large fixed costs and frictions incurred during the process of becoming the “established” market-maker. They observe that “market segmentation” is possible and the competing marketmakers are similar to competing middlemen. Under these circumstances, producers and consumers will not know which market-maker offers the best bid–ask spread without engaging in sequential search for the best price quotes. Most of the benefits of a single market are eroded. In addition, if there are increasing returns to scale, they have the ingredients of “natural monopoly.” After presenting four applications of great interest (Rust and Hall 2003: 392–8), the authors conclude with explanations of the two puzzles stated in the article’s introduction. How could entry be profitable when middlemen uniformly undercut the market-maker’s posted bid and ask prices? The other perplex was to explain why market-makers intermediate most of the trade in financial assets like stocks and bonds but intermediate almost none of the trade in steel (see Rust and Hall 2003: 401 for answers).

13.16.3 Electronic search In a recent and illuminating article, Ellison and Ellison (2005) present some important lessons about markets which have been learned from the internet. They discuss two questions with answers comprising the core of our learning: how was the Internet to construct online marketplaces and how would it generate “frictionless commerce?” They first explain how the Internet presented new insights into the operation of markets where search costs are almost zero. In empirical analyses, it

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has been nearly impossible to study two similar markets where search costs are the only major difference. Now this is possible. In Diamond’s famous paradox, that there can be a large discontinuity between markets with miniscule search costs and those with zero search costs, they can be studied as search costs go to zero! Second, the Ellisons point out that several natural experiments can be performed. For example, comparing consumer response to traditional and online stores enables researchers to study the importance of location. Third, the Internet has improved dramatically the researchers ability to perform field experiments. The differences in auction rules’ impact on the revenue produced by many items. See Lucking-Reiley (1999) where the items were trading cards. Finally, the Internet has permitted economists to utilize new data sources. For example, Scott-Morton et al. (2001, 2002, 2003) have customer data on every car purchased from a large sample of dealers in a specific year that includes buyer’s name, address, make and model and price of auto purchased, etc. We now focus on search relevant aspects of the Internet. The Ellison’s show that the onset of frictionless commerce would not generate perfectly informed consumers, an absence of price-dispersion for identical items: the Law of One Price, and substantially lower online prices relative to offline prices. All three were anticipated by the popular press. Economic research has contradicted these popular predictions. This demonstration was based on search articles by Pratt et al. (1979), Varian (1980), Burdett and Judd (1983), Stahl (1989), and others. And, indeed, the early e-research showed that there is price dispersion online and online prices are not always lower than offline prices. However, there are observations flowing from economic analysis of the Internet which challenge the basic economics of search costs and product differentiation. For example, the size of price dispersion in markets with “branded” websites is large. Price dispersion is present even in markets where consumers find sellers by using search engines, the intertemporal price dispersion appears to be incompatible with basic search theory (see Ellison and Ellison 2005: 150–1), and price–cost margins on the Internet are not very small. In their section on “Lessons: Search Costs,” the Ellisons observe that because of the Internet’s promise to discover substantially smaller search costs, search models became more prominent in the industrial organization literature. At the same time information from the Internet raised suspicions about the validity of the search enterprise. The Ellisons propose an obfuscation theory as a lifesaver for search. They note that the Internet makes it easy for e-retailers to offer complicated menus of prices . . ., to make price offers that search engines will misinterpret, to personalize prices, and to make the process of examining an offer sufficiently tie-consumer so that customers will not want to do it many

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A second reaction to these findings contra basic search is to build a new search theory. Of several possibilities, the Ellisons find Baye and Morgan (2004) “most intriguing.” Baye and Morgan notice that results quite different from the traditional could be possible by assuming that firms conduct approximate, rather than exact, profit maximization.

13.16.4 An empirical analysis of Internet job search and unemployment durations Using the December 1998 and August 2000 Current Population Surveys (CPS) Computer and Internet Supplements matched with subsequent CPS files, Kuhn and Skuterud (2004) conduct a careful study of internet job search and unemployment durations. They find that Internet job-searchers do not find employment more rapidly than observationally similar searchers who make no use of the Internet. They note that their results may be a consequence of the sheer ineffectiveness of internet job search and negative selection on unobservables. They also observe that their findings are not compatible with a story where internet searchers are positively selected (on difficult-to-observe attributes) and in which Internet search facilitates re-employment. Kuhn and Skuterud note that Internet search companies often tell this story.

13.16.5 Middlemen: the emergence of trust The trust essential for exchange emerged from a brutal environment that has been characterized by Hobbes and Vico. The latter sees large dolts roaming the forests in search of food and mates. Mating was naturally undertaken and the dolts reproduced. There was virtually no trust in this Hobbesian landscape. How did trust begin? It no doubt had its origins in the early family. Two brothers would observe that they had a better chance at hunting if they cooperated in the hunt with the implicit agreement that the prey would be shared between the two. Each brother would then take his share and divide it among his family. Women harvesters confronted the same gains from trust and cooperation, and behaved accordingly. Tribes were formed and the genesis of exchange took place when two tribes realized that because of their land and weather they had a limited array of food. This realization probably occurred during a war between the two tribes. Eventually, adjacent tribes would initiate trade via gift giving which grew into a regular exchange between the two. Adventurous members of the tribe would envisage a richer menu of commodities consumed by more distant tribes. Thus a regular pattern of trade evolved, no doubt with many gruesome

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setbacks. As trade grew, so too did the institutions necessary for its smooth operation. These include institutions which stabilized and protected the key institution: property rights. Trust, promise-making and promise-keeping, and their legal enforcement were and remain the conditions for the contractual arrangements associated with exchange. This ensuing voluntary exchange is usually a positive sum game. Gray (1993: 53) observes that “without these background institutions, which are the matrix of spontaneous order, there is no reason to suppose that competition, or exchange, should be other than zero-sum, or even negative sums.” Two examples illustrate the presence of these institutions in modern exchange behavior. The first is drawn from Landa (1994). We first quote from James Buchanan’s Foreword to this volume by Landa. In developed Western economies . . . individuals, firms, and associations engage, one with another, in sometimes highly sophisticated and complex contractual exchanges without concerning themselves directly about the ultimate trustworthiness of those with whom they engage. That is to say, participants proceed “as if ” their exchange partners may be trusted to abide by the explicit and implicit rules of the economic game. . . . Only since the 1960s have modern economists generally come to recognize and to acknowledge the importance of the legal structure in promoting the overall efficiency of the complex exchange network. [Janet] Landa has demonstrated through a plethora of historical examples how ties of kinship and ethnicity have offered substitutes for the “as if ” trust in trading partners that a legal order facilitates. . . . Landa’s enterprise succeeds in offering an increment to our understanding of how market economies function, even those economies that allow observer-analysts to take for granted the existence of the institutions that participants in developing economies find so difficult to establish. (Landa 1994: vii) Buchanan makes a final and significant observation: The fragility and vulnerability of these institutions are not sufficiently appreciated by legal scholars, judges, or economists. An analyticalevaluative detour through a few of those settings where the “as if ” trust relationship does not exist is sure good for our social health. (Landa 1994: viii) In Part 1 of her monograph, Landa presents a framework for the social order problem from the viewpoints of several social science disciplines. She demonstrates the prowess of the Property Rights-Public Choice framework in explaining diverse exchange institutions including contract law, ethnic trading networks, and gift-giving exchanges. All three are explained using methods from law, sociology, and anthropology. She also shows how her

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theory applies to such diverse phenomena ranging from “Kula Ring system of gift-exchange to swarming in honeybees.” The second part of her monograph studies the “role of contract law in developed capitalist economies.” She also emphasizes the central position of division of labor in complicated exchange systems. In Part 2 Landa devises a highly interesting graphical analysis demonstrating how contract law gives rise to an exchange economy where expectations of dependent agents are coordinated by punishing those who breach contracts. Chapter 5 of Part 3 is the centerpiece of Landa’s research. It develops a theory of ethnically homogeneous middlemen among the Chinese in Southeast Asia. She views them as a clublike network functioning in place of a definitive contract law, which is not present in Southeast Asia. Landa (1994: xi) postulates that “in less developed economies where the legal framework is not well developed, personalized exchange relations along kinship or ethnic lines economizes on transactions costs of protecting contracts.” Part 4 specializes on gift-exchange. Part 5 studies altruism and cooperation among honeybees. Part 6 concludes with an analysis of the emergence of exchange institutions. 13.17 REAL OPTIONS

13.17.1 Prelude There are several key economic concepts that cradle the real option phenomenon. These concepts tend to be overlooked in the immense literature on real options. While these concepts are, for the most part, well known, it strikes us as important to recite them before we begin our survey of real options. The topics include: exchange and division of labor; learning in a sequential setting that is similar to the secretary problem and the adaptive version of the BSM in that it is Bayesian; pertinent comments by Arrow on the division of labor, agendas, and learning codes of a job are similar to acquiring human capital specific to the worker’s firm; the important distinction between sampling (exploration) and action (exploitation) which is common to processes ranging from acceptance sampling through multi-armed bandit problems; and specific knowledge and the corresponding learning experiences “can develop different perceptions of the laws that govern the world”—the “Rashomon effect.” Division of labor Real options and division of labor are inextricably bound. The division of labor, Adam Smith’s basic principle, supplies the power and energy to the real option concept. Smith’s famous pin factory, where ten specialized workers produce a daily output of approximately 40,000 units, is compared to the

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output of the 200 pins produced by the same ten workers operating independently.33 By specializing, the output of 10 workers increases by a factor of 200. Smith realized that machines were designed to coordinate with labor and yielded substantial increments to output. This nexus was also a consequence of the division of labor. Every body must be sensible how labor is facilitated and abridged by the application of proper machinery. It is unnecessary to give any example. I shall only observe, therefore, that the invention of all those machines by which labor is so much facilitated and abridged, seems to have been originally owing to the division of labor.34 Exchange, the second of Smith’s two great principles, is the driving force behind the division of labor. The two basic principles interact to yield increasing trade and finer specialization. Smith feared that the limits on exchange are probably not reached before specialization produces laborers who are stupified by the meniality of their work, the dire outcome of the division of labor. It would seem that Smith’s fear has been mitigated by exchange itself. This is beautifully described by Griswold (1999: 297; emphasis added): Life in a market society is an ongoing exercise in rhetoric. The necessity of developing our rhetorical skills is great in “civilized society,” where, for at least two reasons, we are highly interdependent. First, with the progress of the division of labor, each person is less and less capable of providing basic necessities for himself; everything depends on exchange. Second, as Smith also points out here, in a society of “great multitudes,” one’s “whole life is scarce sufficient to gain the friendship of a few persons” . . . in a “civilized” society, the arts of persuasion, communication, and noncoercive speech are essential . . . . Not only is our mutual dependence binding, but its accomplishment through language is civilizing. It is education that transmits rhetoric. Thus education is the means by which the evils of specialization can be defeated. It also seems that Smith was too literal in his interpretation of the division of labor. While some jobs fit this description, most do not. Work is itself an educational process. Arrow (1974: 55–6; emphasis added) has an original description: “Learning how a particular organization (firm) operates requires that workers become familiar with the firm’s code, which is always changing. Each firm has its own peculiar code and learning it is an irreversible investment for the worker.” Arrow also notes that the best code depends on the prior distribution of future signals and the coding costs. Coding “weakens . . . the tendency to increasing costs with the scale of operation and it creates an intrinsic irreversible capital commitment of the organization.” The firm’s “gains from increasing scale are

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derived by having its members make different experiments, that is, by specialization” (Arrow 1974: 55). The worker engages in a process that is similar to one which first accumulates information about alternative activities available at the firm and then chooses that activity for which he is best suited. The first step in this process is one of exploration, while the second is usually called exploitation. In the end, the firm has all of its employees engaged in a host of alternative experiments described previously. Specialization from this perspective is less menacing. The firm operates in a dynamic economy and the experiments required by employees are ever-changing. This can vivify the worker’s job, but it also imposes a risk of discharge. The worker also always has an option of quitting. These quits and layoffs certainly make jobs less stupifying and indeed challenge individual workers to enhance their knowledge of the skills required in currently available jobs. This knowledge is akin to an insurance policy. By paying his premiums, the worker’s fear of layoff is diminished and his education is enhanced. Bayesian sequential decisions It is curious that in the enormous literature on real options that was surveyed, there was hardly any mention of Bayes policies.35 It is well known that Bayes policies are the companions of the secretary problem. Learning, of course, is a vital component of real option decision-making. The similarity of real option decision-making to the BSM and the secretary problem makes the absence of Bayes learning, and the exchangeability which accompanies this learning, an odd conundrum. Specialized equipment When specialized equipment is sold, it usually requires a substantial discount. In terms of reversibility, an equipment that can perform only a small set of tasks which are not employed by other firms, is best sold as scrap. Thus the entire specialization process must begin anew. The same set of depreciation of outside skills confronts the firm-specific worker. Individual workers form Arrow groups where communication occurs within, but not among groups. History is like a ship that specializes its sailors by carrying each through a unique voyage. Only those seamen who participate in similar voyages can communicate effectively. According to Arrow (1974), one crucial aspect of specialization is specialized knowledge. This specialized knowledge leads to different perception of the laws that govern the world across individuals. Two individuals, both of whom are trying to be objective, may nevertheless disagree because of these different experiences. This is an extremely important observation revealing similarities between Arrow and deFinetti. In the quest for an equilibrium search distribution many economists attempted to accomplish this feat with identical individuals.

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This is a mathematical goal without contact with Arrow–deFinetti economics. Arrow concludes: But the very act of specialization produces a world in which different individuals have different experiences and therefore different perceptions of the world. . . . These differences are themselves a source of gain in increasing the realization of human potential . . . the division of labor leads to a division of society into smaller social groups among which communication is limited and both economic and social intercourse is restricted. Ultimately, different moral views on the nature of society can never be completely reconciled. (Arrow 1974: 163–4)

13.17.2 Introduction A real option, as opposed to the well-known financial option, is a choice among real investments whose consequences are uncertain. In dynamic sequential decision-making, almost all options are irreversible in the Heraclitean sense that the passage of time continuously changes the circumstances surrounding the choice of an option. The proper introduction of time usually means that today’s investment competes with the same investment tomorrow. Indeed, in many ways the theory of sequential decision-making is analogous to a theory of the sequential selection of options. Some real options are strikingly irreversible. The standard example is the decision to cut a forest of sequoias.36 Once cut, it is impossible to reverse the decision in a reasonable time period. Over the past twenty-five years there has been a large research effort by economists to construct a theory of real options. The theory has proven to be invaluable in rectifying some basic errors in capital theory and using financial options, optimal stopping, and search theory to enhance our comprehension of stochastic investment theory. Subsection 13.17.3 begins our survey of this rich area with answers to the query: what is a real option? In this subsection we elaborate on what we have said about options and show their linkage to phenomena studied in a host of economic subdisciplines ranging from environmental economics to business cycles. There is a sturdy nexus joining real options and search theory which is studied in Subsection 13.17.4. The multi-armed-bandit (MAB) and the Gittins index are included in this survey. Next in Subsection 13.17.5 we give a brief account of McDonald and Siegel’s model of waiting to invest. Kenneth Arrow was the first to recognize the economic importance of irreversible options as indicated by a host of articles. In Subsection 13.18.6 we focus on Arrow’s Limits of Organization (1974), a relatively neglected pamphlet, which contains the seeds of option theory and much else. The following section is a brief overview of several important contributions of

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option theory to environmental economics. The key issue is how to manage resources in a manner that preserves efficiency while obeying sensible environmental constraints. We conclude this introduction by noting the widespread use of real options which flows from the martingale, the kingpin of financial options that is also present, in nonfinancial environments. We observe that real options have many applications in biology. These are quite natural accompaniments to the music of search and the Gittins index, which synchronize the biological behavior of animals from the beaver to the honeybee to the neuron itself. The martingale, which is of utmost importance in financial options, also manifests itself in most of these real option applications. Of course, the martingale is a no-arbitrage condition and signifies a fair game. We conclude the introduction with four pertinent quotes, first from a neuroscientist, then from a physicist, and finally from Kenneth Arrow and Robert E. Lucas. Paul W. Glimcher: “In order to understand how the brain solves sensorimotor problems, we have to understand, we have to have a theory of, the sensorimotor problem posed by the environment to the organism. Bayesian economic mathematics is that theory” (2003: 203; emphasis added). Recall that in Bayesian methods according to deFinetti, coherence is the definitive ingredient. But coherence is a no-arbitrage condition and therefore corresponds to a martingale. This position is beautifully described in Lad (1996). We next quote from a famous physicist, K. R. Parthasarathy (1988) who espouses the following “naive philosophy”: In the language of classical probability a martingale is a sequence of random variables which “remains constant in conditional mean” and for a gambler, corresponds to a game which is on average fair. While investigating laws of nature, one adheres to the belief that nature plays a fair game. Putting these two intuitive ideas together one may ask whether some interesting physical law can be deduced from the principles of martingale theory. From this we might conclude that the no-arbitrage principle, also known as deFinetti’s coherence principle, is a natural law. Arrow begins his book The Limits of Organization (1974) with the following two paragraphs: The intricacies and paradoxes in relations between the individual and his actions in the social context have been put very well by the great sage, Rabbi Hillel: “If I am not for myself, then who is for me? And if I am not for others, then who am I? And if not now, when?” Here we have in three successive sentences, the essence of a tension that we all feel between the claims of individual self-fulfillment and those of social conscience and action. It is the necessity of every individual to express in some manner his intrinsic values. But the demands of society and the needs of the

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individual, expressed indeed only within that society, require that he be for others as well as for himself, that the others appear as ends to him as well as means. With two such questions with such different implications, it is no wonder we got the third question: How can I behave urgently and with conviction when there are so many doubtful variables to contend with? The tension between society and the individual is inevitable. . . . All I try to insist here is that some sense of rational balancing of ends and means be understood to play a major role in our understanding of ourselves and our social role. Lucas (2002) observes that: In a successfully developing society, new options continually present themselves and everyone sees examples of people who have responded creatively to them. Within a generation, those who are bound by tradition can come to seem quaint, even ridiculous, and they lose their ability to influence their children by example or to constrain them economically. The people who respond to the new possibilities that development creates are also the ones who make sustained development possible. Their decisions to take new risks and obtain new skills make new possibilities available for those around them. Their decisions to have fewer children and to try to prepare those children to exploit the opportunities of the model world increase the fraction of people in the next generation who can contribute to the invention of new ways of doing things. In economically successful societies, today, these are all familiar features of the lives of ordinary people. In pre-industrial societies, all of these features are rare, confined if present at all to small elites. If these observations are central to an understanding of economic growth, as I believe they are, then we want to work toward aggregative models of growth that focus on them.

13.17.3 What is a real option: a concise survey Introduction Over the past quarter century several outstanding articles have been written addressing the nature of real options and their applications. It is remarkable that the firm which was previously regarded as a collection of contracts has undergone a marked change. The firm is now an adaptive, stochastic ensemble where its players belong to one or more of the firm’s probabilistic, sequential investments. In the real option perspective contracts are stochastic and evolve over time in a Bayesian manner as new information is revealed. The basic technical change is that firm behavior is determined by stochastic dynamic programming as manifested by optimal stopping rules. The entities

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controlled by the BSM and those driven by the logic of the secretary problem are similar to the firms that use real option methods. Kandel and Pearson (2002) identify several important contributions that initiated the use of option-based models. The options studied are: operating options (McDonald and Siegel 1985; Brennan and Schwartz 1985), abandonment options (Myers and Majd 1990), the option to pursue or halt a series of investments (Lippman and Rumelt 1987; Pindyck 1987), the option to postpone an investment (McDonald and Siegel 1986), and the option to wait and choose a low interest rate as they fluctuate over time (Ingersoll and Ross 1992).37 Earlier contributions include: Abel and Eberly (1996), Abel et al. (1996) and Bar-Ilan and Strange (1998). This literature concentrates on the effects of uncertainty on the investment decision. Kandel and Pearson (2002) observe that the traditional method for selecting investments is to require that the ratio (option value multiple) of the present value of the cash flow associated with an investment to the out-of-pocket cash is greater than unity. The key error in this ratio analysis is its neglect of the option value, which expires when investment occurs. Kandel and Pearson identify some of the central findings of this recent literature: (1) frequently the best policy chooses to invest only when the option value multiple hits a threshold considerably larger than one; and (2) the threshold value of the option value multiple usually increases in measures of uncertainty for basic random variables. Condition (2) is a consequence of the payoff from taking the investment and is similar to a call option’s payoff. Thus, it is a convex function of the basic random variable. Convexity implies that the option value increases in uncertainty. This is a standard result in search theory. See Lippman and McCall (1976a) and Ljungqvist and Sargent (2004b). Kandel and Pearson model a firm which has access to a second technology requiring no investment, that is, the investment is fully reversible, with higher marginal costs. They then allow firms to reverse their capital investments at a cost. This is called partial reversibility. Finally, they study a simple competitive equilibrium. Some of their findings are: when the investments in irreversible capital are used to lower costs by replacing the reversible technology, the value of the foregone option to delay is lower and decreasing in demand uncertainty. Consequently the threshold value of the option value multiple also decreases in demand uncertainty. Absent the reversible technology, the threshold value of the option value multiple increases in demand uncertainty. When capital is employed to replace the reversible technology, the threshold value of the option value is unaffected by capital’s degree of reversibility. Finally, replacing monopoly with a competitive equilibrium has no influence on these findings. There are several more technical contributions to the real option literature which are especially lucid. These include: several articles in Lund and Øksendal (1991), the most noteworthy being: Kobila, Brekke and Øksendal, Olsen

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and Stensland and Flam, who present an extension of the stochastic programming literature initiated by Rockafellar and Wets (1976). The papers by Zervos and his coauthors: Duckworth and Zervos (2000), Lumley and Zervos (2001), Knudsen et al. (1998, 1999), and Zervos (2003) are also very important, as are the papers by McDonald and Siegel (1986), Brennan and Schwartz (1985) and Lippman and Rumelt (1982). The nature of a real option In their excellent monograph, Dixit and Pindyck (1994) survey the large literature on real option models. These models are closely related to the Black– Scholes option model in financial economics. Our plan is to describe the real option model following Dixit and Pindyck, and mention several early applications of this optimal stopping paradigm as well as some sophisticated examples. The option approach is nicely summarized in Dixit and Pindyck (1994: 6; emphasis added): The net present value rule . . . assumes that either the investment is reversible, that is, it can somehow be undone and the expenditures recovered should market conditions turn out to be worse than anticipated, or if the investment is irreversible, it is a now or never proposition, that is, if the firm does not undertake the investment now, it will not be able to in the future. Most investments do not satisfy these two conditions. The firm with an investment opportunity is similar to a firm which holds a stock option. When an irreversible investment is made, the option disappears. The value of the option at time t is the expected increase in the option value as new information arrives revealing that the option is more valuable than was thought at time t. Many investments have the option and irreversibility properties, which clearly invalidate the standard net present value criterion for investment selection. Dixit and Pindyck (1994: 352) mention that search models are relatives of the real options. “All of these models of learning and cost uncertainty belong to a broad class of optimal search problems analyzed by Weitzman (1979).” The decision-maker must determine which investment to accept and the order of acceptance. Dixit and Pindyck maintain that the option model is more general because the expected outcomes evolve stochastically even when no investment is undertaken. However, the option model is more restrictive in that the ordering of investments is “predetermined.” Note that the Whittle’s restless bandit problem allows unused options to change stochastically. It strikes us that a generalized MAB framework, such as Whittle’s restless bandit problem is sufficiently general so that many problems of investment opportunity can be evaluated within this setting. However, in this framework

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the optimal solution need not involve an index policy. Nevertheless, the classic MAB problem whose optimal solution involves an index policy (see, for instance, Cairoli and Dalang 1996) contains, modulo minor adjustments, the Black–Scholes option model. Indeed, Schöl (2002) shows how one can use a discrete Markov decision model to derive the Black–Scholes results. In particular, Schöl obtains the Fundamental Theorem of Asset Pricing: there is an equivalent martingale measure if and only if the no-arbitrage condition obtains. These results suggest that the solution to any optimal control problem must obey the no-arbitrage condition. This is a far-reaching consequence. All rational decision-making which is based on some form of optimal control is characterized by “fairness” in the no-arbitrage or martingale sense. It has been observed (see Lippman and McCall 1984) that the biological behavior associated with mating, foraging, and other activities of animals can be described by the optimal stopping rules associated with sequential search. Thus, the “fairness” principle extends to the animal kingdom! Dixit and Pindyck characterized the real options approach as “a new view of investment.” In this view, the three essential features of investment are that the investment is partially or completely irreversible; future rewards from an investment are stochastic; and, the decision-maker has some control over the timing of the investment. These three characteristics interact to yield a new theory of investment. The net present value (NPV) theory of investment is flawed in that it ignores the real option, which accompanies investment. Dixit and Pindyck (1994: 6) have a nice statement linking investment and options: the “firm with an opportunity to invest is holding an option analogous to a financial call option . . . when a firm makes an irreversible investment expenditure, it exercises or ‘kills’ its option to invest. It gives up the possibility of waiting for new information to arrive that might affect the desirability or timing of the expenditure.” Dixit and Pindyck construct a theory of stochastic irreversible investment in which options play a key role. An equivalent approach is based on optimal stopping or stochastic dynamic programming. They use the term real options to proclaim the strong analogy with financial options. In brief, a new theory of stochastic investment is built in which the concept of options is decisive and optimal stopping is the underlying link between real and financial decisionmaking. We must also reference the excellent article by Ingersoll and Ross (1992). The conclusion of this solid economic study is Our results have nothing to do with the usual addition of option-like features to the investment products themselves: rather, we have shown that, in an uncertain economy, nearly all investment projects have option rights values. . . . With uncertain interest rates, an investment should not be undertaken until its projected rate of return is substantially in excess of this breakeven rate. (p. 6)

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13.17.4. The origin of real options and their nexus with search theory In his ingenious analysis of business cycles, Bernanke (1983) presents a brief review of the original contributions to real option theory. Marschak (1949) was the first to expose a cogent analysis of the irreversibility or illiquidity problem. Environmental economics has generated important research on its prototypical version of the irreversible problem: “When can an effectively irreversible action, such as the cutting down of a sequoia forest, be economically justified?” (Bernanke 1983: 87). In their responses to this query, Weisbrod (1964), Arrow and Fisher (1974), Henry (1974a,1974b) and others observe that under conditions of uncertainty an option value can be assigned to avoiding irreversible actions. By not cutting down the forest, society keeps the option of choosing to cut or not to cut. On the other hand, cutting destroys the option of preserving the sequoias when new information reveals that keeping them is the optimal action. Bernanke generalizes this argument from one to k irreversible alternatives. An investor may either select one of the k irreversible projects or select none and collect additional information. One must either explore, i.e., accumulate more information, or exploit, i.e., select one of the k options. The optimal policy has a form similar to that for MAB problems, namely, exploit the option that has the largest reservation index. The decision-maker does nothing when the return from such a choice is greater than the k returns from exploitation. This is the essential message of Bernanke’s Proposition 1: the expected return to an accepted investment must be the best of the k alternatives and must be greater than a reservation value associated with the “expected value of deferring commitment.”38 Bernanke identifies, via his Proposition 1, three essential components of irreversible investment theory: 1 2 3

One should not use myopic rules when choosing among irreversible investments (Arrow 1968). It is more difficult to verify irreversible investments than their reversible counterparts (Arrow-Fisher 1974, and Henry 1974a). The option value is positive whenever disinvestment in the future may be desirable (Arrow 1968, deterministic case; Henry 1974b, stochastic case).

Bernanke also elaborates a bad news principle and observes that one-tailed decision-making is also present here as in search theory. The “bad news” theory is exposed as follows: let Χi,t + 1 be a random variable that is equal, for each state in t + 1, to the maximum value obtainable other than investing in i, less the value of investing in i. Note that positive values of Χi,t + 1 is a “bad news” outcome for i which implies that an investment in i made in t would be regretted in t + 1. Bernanke notes that one-tail decision rules occur frequently in search theory thereby showing another common feature of real option theory and search.

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Having discovered that it is precisely the expected value of bad news that determines the option value, gives Bernanke an engine for a comparative statics analysis of irreversible investment. It is clear that a mean preserving spread of the density of Χi,t + 1 around zero increases the option value, thereby reducing current investment. Notice that the effect of downside uncertainty (the bad news principle shows that a mean preserving spread can influence option value only through the bad news side of the density (Χi,t + 1 ≥ 0)) on investment bears no relation to risk preferences. Risk neutrality is the assumption Bernanke makes throughout his analysis. This result resembles the outcome in search theory: a greater dispersion increases the value of information and lengthens the average optimal search time. Bernanke (1983) is one of the few papers that uses Bayesian methods in designing his dynamic model of irreversible investment. Indeed, he mentions that in an earlier version of his article a Dirichlet distribution was employed to show that “more diffuse investor priors led to higher option values. Thus, in early stages of the learning process, waiting for information is a valuable activity, and investment tends to be deferred. As knowledge accumulates and priors concentrate, the propensity to invest increases” (Bernanke 1983: 95). In an environment where uncertainty is periodically renewed, Bernanke (1983: 95) notes that “Howard showed how correct inference could be made by repeated application of Bayes law.” Many of Bernanke’s insights are also present in the paper by Roberts and Weitzman (1981). They observe that: we require a sequential decision rule which indicates whether or not to continue at each stage as a function of the information then available. The optimal stopping rule maximizes expected benefits minus costs, taking account of the fact that at all future stages we will also be following an optimal stopping rule.39 (Roberts and Weitzman 1981: 1263; emphasis added) Their Theorem 1: The Optimal Stopping Rule, states: at any stage, let the expected cost to completion of the project be C and the perceived benefit be distributed N(µ,σ2). It is optimal to proceed with the Sequential Development Project 40 if and only if C ≤ E[Y |Y ≥ µ], where Y ~ N(0,σ2). The authors solve for the cutoff value of µ, µˆ, which satisfies ∞

冮(y − C)f(y)dy = 0 µˆ

or ∞

∞

C

µˆ

冮(y − C)f(y)dy = 冮(C − y)f(y)dy,

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which we recognize as the key equation defining the reservation quantity in search models. Another key paper in relating real option theory to search theory is Venezia and Brenner (1979). They show that the determination of the optimal duration of a growth investment corresponds to a search problem. All that is required to get a search interpretation is replacement of the phrase “selling the investment” by “accept the offer and terminate search.” Let Χt be a random variable denoting the proceeds from investment at age t, and let (x1, . . .,x2, . . .,xt, . . .) be an infinite vector denoting the proceeds from the investment over an infinite time span. The utility function is given by: ∞

U(x1, . . ., xt, . . .) =

冱e

−it

u(xt ),

t=1

with u a von Neumann-Morgenstern (VNM) utility function and i the interest rate at the time of termination T, the investor gets ΧT, which has value U(0,0, . . ., ΧT,0,0) = e − iTu(ΧT ). Hence the investor chooses T to max e − iTE[u(ΧT )]. For a risk neutral investor the objective function is: max βT − 1E[ΧT], where βT − 1 replaces e − iT. There are two sources of uncertainty in future proceeds: (1) uncertainty concerning general economic conditions and the physical properties of the investment, and (2) given these, uncertainty concerning the investment’s future prospects. At any period of time the economy occupies one of K states: S1,S2, . . ., SK which are referred to as states of the investment. The motion among these K states is governed by a Markov chain.41 Conditional on Si and t, the age of the investment, the value of the investment (proceeds) is a random variable Χ ti with C.D.F., F i(x,t). Assume that for all i and t the expectations µti = E[Χ ti] are finite and satisfy µti ≤ µit + 1 ≤ . . . ≤ µi < ∞, i = 1, . . ., K, and the Χ tis converge in distribution to Χi ~ F i(Χ). The µtis are monotone and bounded so they converge. Let µi = lim µti. On average the investment t→∞

appreciates with time. The investor must decide at each t whether to stop and receive an income of x or wait till t + 1. Let Vit(x) be expected discounted proceeds from an

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optimal decision at age t given si and a current offer x; let Wit = Eit[Vit(x)], where Eit is the expectation operator with respect to F i(x,t). The expected discounted proceeds Vit(x) from the optimal decision must be the maximum of the proceeds x from stopping the investment and the discounted expected income, K

ηit = β

冱π W ik

k,t + 1

,

k=1

from continuing. Hence, Vit(x) = max{x,ηit}. The optimal strategy is to inspect, at each t, the state si and the current value of x. If x ≥ ηit, the process should be stopped. If x < ηit, retain it till t + 1 when the decision is reconsidered. Search interpretation: Χ tis are the price offers for a seller of a product or wage offers to a worker in the job market. In job search, the worker stays at the same job once it is accepted. There are no quits or layoffs. His expected utility is βT (1 − β)−1E[u(ΧT )] and the problem is the same as the optimal duration of investment. We now present the proof by Mortensen and Pissarides (1999a) showing that the continuous time elementary sequential search model is equivalent to its option pricing representation, that is, rU = (b − a) + λ冮[max{W,U } − U ]dF(W ),

(13.57)

where the stationary value of search is U=

λ b−a + 冮Max{W,U }dF(W ), r+λ r+λ

(13.58)

and λ is the constant hazard (in this exponential case 1/λ is the expected duration of the waiting period between offers), r is the risk free interest rate, b is the income flow received when unemployed less search costs, and a is the per period cost of search. The optimal strategy in the discrete model compares the current outcome of the sampling process Wt with the values of continued search Ut + 1 in the next period. In the continuous model (13.51) Wt and Ut + 1 are replaced by W and U. With an option interpretation, (13.56) prices the option by equating the opportunity cost of holding it, the left-hand side, with the current income flow, b − a, plus the expected flow of capital gains which is the product of λ and the expected capital gain conditional on an offer’s arrival. Mortensen and Pissarides (1999a: 2573) conclude that: “The general fact that the option

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value of search solves a general asset pricing equation and transversality condition [lim U(t)e−rt = 0], provides a quick and powerful characterization of t→∞

optimality conditions in equilibrium models in continuous time.” There is, of course, a martingale lurking in this analysis42 so that (13.53) is a no-arbitrage condition. It is also worth noting that there is no fundamental difference between the discrete and continuous dynamic programs characterizing this Markov decision process. Mortensen and Pissarides show that the reservation wage property is preserved when the discrete model is replaced by its continuous version. Zuckerman (1984) gives a rigorous proof of this result. The result is not surprising (see Serfozo 1982). It is worthwhile to observe that the first dynamic program is presented in Wald and the first real option model is Wald’s sequential probability ratio test (SPRT). In their presentation of the elementary sequential search model, Ljungqvist and Sargent (2004b) observe with some puzzlement that a meanpreserving increase in risk raises the reservation wage, thereby improving the situation of unemployed workers. Their resolution of this puzzle is based on a result from option pricing theory: [The] value of an option is an increasing function of the variance in the price of the underlying asset. This is so because the option holder receives payoffs only from the tail of the distribution. In our context, the unemployed worker has the option to accept a job and the asset value of a job offering wage rate w is equal to w/(1 − B). Under a measure preserving increasing risk, the higher incidence of very good wage offers increases the value of searching for a job, while the higher incidence of a very bad wage offer is less detrimental because the option to work will in any case not be exercised at such low wages.

13.17.5 A glimpse of ecological search There have been several resource management models which explicitly consider irreversibility. Four articles are Brock et al. (1983, 1988), McDonald and Siegel (1986), and Brennan and Schwartz (1985). The first two articles are closely connected to our dynamic economy model in Chapter 7. McDonald and Siegel (1986: 709) investigate the investment decision of a firm that is contemplating the following opportunity: “At any time t (up to a possible expiration date T), the firm can pay Ft to install an investment project, for which expected future net cash flows conditional on undertaking the project have a present value Vt . . . (which) represents the appropriately discounted expected cash flows, given the information available at time t.” Both Vt and Ft are stochastic and the installation capacity is irreversible, that is, the added capacity is specialized to accommodate the idiosyncrasies of this particular firm.

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The present value Vt follows a geometric Brownian process given by dV V

= αvdt + σvdzv,

(13.59)

where zv is a standard Wiener process. Ft, the cost of installation also follows a similar process: dF = α f dt + σf dzf. F

(13.60)

McDonald and Siegel observe that the assumption of geometric Brownian motion for V is as sensible as the assumption that stock prices follow geometric Brownian motion. However, they show that this assumption may imply unrealistic behavior. Suppose that the investment project is irreversible. How does this affect the optimal timing of installation? To answer this query the authors consider the case of V and F following (13.59) and (13.60) and demonstrate the calculation of the correct discount rate when investors are risk averse. Let both Vt and Ft be random. The problem can be formulated as a first passage problem. It is easy to show that investment should take place when V/F hits a boundary. The boundary B is chosen to max{Eo[(Vt − Ft)e−µt]} subject to (13.60) and (13.61). Let V ′ = kV and F ′ = kF, with k an arbitrary positive number. Now consider the problem: choose a boundary B′ to

冦 冤冢

冣 冥冧

Max Eo V ′t − F′t e −µt subject to

dV ′ dF ′ = αvdt + σv dzv and = α f dt + σf dzf. V′ F′ It is clear that B = B ′ The authors show that the correct rule is to invest when V/F hits a fixed boundary which is represented by the barrier C*. The expected present value of the payoff is therefore Eo{Ft′[C* − 1]e−µt′} = [C* − 1]Eo{Ft′e−µt′},

(13.61)

where the expectation is with respect to Ft and the first-passage times for Vt/Ft′. We summarize the results we have just presented. McDonald and Siegel investigate the optimal timing of an irreversible project where both the product’s benefits and the investment cost behave according to continuous-time

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stochastic processes. They derive the optimal investment rule and an explicit formula for the option value, assuming the option is valued by risk averse investors who are well diversified.

13.17.6. On the limits of organization In classical maximizing theory it is implicit that the values of all relevant variables are at all moments under consideration. All variables are therefore agenda of the organization, that is, these values have always to be chosen. On the other hand, it is a commonplace . . . that a potential decision variable be recognized as such may be much greater than that of choosing a value for it. (Arrow 1974: 47) Arrow (1974: 48; emphasis added) outlines those factors which determine agenda. “The point of view is that an optimizing model but in a rich framework of uncertainty and information channels.” He assumes information to be costly. His primary theme is that the uncertainty, indivisibility, and capital intensity affiliated with information channels have two implications: (a) that the behavior of an organization frequently depends on random events, i.e., history, and (b) by striving for efficiency the organization may stultify. Arrow (1974: 49; emphasis added) emphasizes the necessity for distinguishing between “decisions to act in some concrete sense, and to collect information.” He notes that this distinction is the central feature of statistical decision theory where these two classes of decision are called “terminal acts” and “experiments,” respectively. Acceptance sampling is the classic example studied by Raiffa and Schlaifer (1961). Raiffa and Schlaifer distinguish between terminal acts and experiments. The classic example is a real option model, namely, acceptance sampling. (A firm receives a batch of inputs and wishes to determine their quality. Rather than test the entire batch, a sample is scrutinized and the entire lot is accepted or rejected based on the information contained in this sample scrutiny. Of course, the sophisticated version of this procedure is Wald’s SPRT test.) Arrow observes that an experiment with one goal in mind may produce additional information pertinent to quite different terminal acts. For example, Jorgenson et al. (1967) present a model of opportunistic replacement such that when a complex mechanism, such as a missile, is examined to check for possible malfunctioning of one subunit, it is then optimal to examine or replace other subunits. The logic of this procedure was first discovered by Friedman and Wallis and gave rise to Wald’s sequential probability ratio test (SPRT) and much else. The SPRT was probably the first real option and was discussed thirty years before the Black–Scholes financial option. Their similarity resides in

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their common reliance on sequential decision-making as manifested by optimal stopping rules. The role of codes in information transfer by organizations is also analyzed by Arrow. Coding has two economic consequences: “(a) it weakens, but does not eliminate the tendency to increasing costs with scale of operation; (b) it creates an intrinsic irreversible capital commitment of the organization.” Learning a code is an irreversible action by the relevant employee, and also an irreversible accumulation of capital by the organization. Once again we have the condition producing real options. Arrow elaborates on this using Becker’s study of on-the-job training and its firm-specific component. Real options pervade the acquisition of human capital. Optimal stopping is the basic method for solving real and financial option problems and, of course, Arrow was one of the first to exploit the fact that sequential decision-making is a problem of optimal stopping and to extend Wald’s basic result (Arrow et al. 1949).

13.18 RESOURCE ALLOCATION IN A RANDOM ENVIRONMENT In this section we present several important variations of BSM which testify to both its significance and omnipresence. They are presented in the house hunting format, and also in the format of Chapter 7 where the “dynamic” economy is one that changes according to a Markov chain. These versions of BSM are based on the classic article by Derman et al. (DLR) (1972b), “A Sequential Stochastic Assignment Problem,” which we consider first. This is followed by the Bayesian approach to a generalized house-selling problem, an important article by Albright (1977). We then summarize the piece by Righter (1989), a resource allocation problem in a random environment. It should be noted that the BSM problem in its house-selling version is embedded in a more general setting by DLR (1972b). They show that if the decision-maker must pair n known quantities, p1 ≤ p2 ≤ . . . ≤ pn with n random variables Χ1, . . ., Χn, that are i.i.d. from a known distribution, in order to maximize the total expected return, given that a pairing of pi and Χj yields a reward piΧj, then the identical critical numbers policy is optimal for any values of p1, . . ., pn. As a special case, this policy is best for the problem of selecting the k best Χs when the reward is the sum of the k values obtained: simply let p1 = . . . = pn − k = 0 and pn − k + 1 = . . . = pn = 1. The policy in DLR solves the problem of choosing the k best Χs simultaneously for every k ≥ 1. The DLR model has been generalized by Albright (1974), Albright and Derman (1972), and Sakaguchi (1984). In Albright (1977), it is assumed that the Χs are produced from a distribution with known family, but unknown parameters. The unknown parameters have prior distributions which are revised in Bayesian fashion as successive

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Χs are observed. Similar problems are discussed in DeGroot (1968), Diamond (1976), Rothschild (1974) and Sakaguchi (21).43 Albright notes that DeGroot is the one that most resembles his.

13.18.1 The Derman, Lieberman, and Ross article (1972b) The sequential stochastic assignment problem is described as follows. Suppose n men are available to perform n jobs. The jobs arrive in sequential order: (1st job 1 appears, then job 2, etc.). Associated with the jth job j = 1, . . .,n, is a random variable Χj taking on value xj. The Χs are i.i.d. random variables. The jth job is called a “type xj” job. Derman, Lieberman, and Ross note that if a “perfect” man is assigned to type xj job, a reward of xj is garnered. However, perfection is often an illusion and when the ith man is assigned to any type j job, the expected return is pixj, where 0 ≤ pi ≤ 1, i = 1, . . .,n are known constants. Anyone assigned to a job is not free for any future assignment. The problem then is to assign the n men to the n jobs to maximize the total expected return. Note that an assignment of workers is equivalent to a sequential assignment of the ps to the Χs. A policy is any rule assigning workers to jobs. If the random variable ij is defined to be the worker (with a given number) assigned to the jth incoming job, then the total expected return is given by: n

E

冤冱p Χ 冥, ij

j

(13.62)

j=0

and the preferred policy is that which maximizes (13.61). Note that (i1,i2, . . .,in) is a random permutation of the integers 1,2, . . .,n. There are two clever interpretations of this model. Let there be n cards with a probability pi associated with the ith card. A sequence of i.i.d. random variables is observed in sequential fashion. If a random variable Χj appears, a card must be chosen and played on that random variable. If the ith card is played when Χj = xj is realized, the expected return is pixj. As an example, this process occurs when xj is received with probability pj and zero is obtained with probability (1 − pj). The problem is to choose n plays of the cards to maximize (13.61), the total expected return. Derman, Lieberman, and Ross also note that a special case of their model generalizes the house-hunting problem (Karlin 1962). Suppose there are k ≤ n identical houses to be sold. Offers arrive sequentially. The offers are a sequence of i.i.d. random variables, Χ1,Χ2, . . .,Χn. The seller can accept or reject the offers, but must sell all houses before the nth offer. In the previous “card version” let k of the cards have ps equal to 1 and let (n − k) cards have ps equal to zero. If the seller accepts the jth offer he assigns a card to it with a p equal to 1 and receives xj, with card and house becoming “sold” and no longer available. If the seller rejects the jth offer he assigns a card with p equal

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to 0 and obtains 0. This process continues until all houses (and cards) are sold. The problem is to decide which offers to accept so that the total expected return, (13.62), is a maximum. The critical result required to obtain an optimal policy is to show that the policy has the following form. If there are n stages to go (n workers to assign) and probabilities p1 ≤ p2 ≤ . . . ≤ pn, then the optimal choice in the first stage is to use pi (i.e., the ith worker) if the random variable Χ occupies the ith nonoverlapping interval comprising the real line. These intervals depend on n and the cumulative distribution function of Χ, but are independent of the ps. Lemma (Hardy’s Theorem) If x1 ≤ x2 ≤ . . . ≤ xn and y1 ≤ y2 ≤ . . . ≤ yn are sequences of numbers, then n

max

n

冱x y = 冱x y ,

(i1,i2, . . .,in ∈ ℘ j = 1

ij

j

j

j

(13.63)

j=1

where ℘ is the set of all permutations of the integers (1,2, . . .,n). This lemma implies that the largest sum is obtained when the smallest of the xs and ys are paired, the next smallest of the xs and ys are paired and continues until the largest of the xs and ys are joined. Let f(p1, . . .,pn) = Total expected reward when an optimal policy is followed and the probabilities are p1,p2, . . .,pn, f(p1,p2, . . .,pn|x) = Total conditional expected return given Χ = x under an optimal policy when the probabilities are p1, . . .,pn. An induction argument proves that optimal policies exist. Let GΧ(z) be the cumulative distribution function of the random variable Χ. Χ1,Χ2, . . .,Χn are assumed to be i.i.d. random variables with C.D.F. GΧ(z) and that ∞

µ = E(Χ) =

冮 zdG (z) < ∞. Χ

−∞

The following theorem contains the optimal policy. Theorem 13.8

For each n ≥ 1, there exist numbers

−∞ = a0,n ≤ a1,n ≤ . . . ≤ an,n = + ∞, such that whenever there are n stages to go and probabilities p1 ≤ p2 ≤ . . . ≤ pn then the optimal choice in the initial stage is to use pi if the random variable Χ1 occupies the interval (ai − n, ai,n). The ai,n depend on GΧ, but are independent of the ps. In addition, ai,n is the expected value in an (n − 1) stage problem, of the quantity to which the ith smallest p is assigned (assuming an optimal policy is being pursued), and

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n−1

f(p1,p2, . . .,pn − 1) =

冱 pa

for all p1 ≤ p2 ≤ . . . ≤ pn − 1.

i i,n

(13.64)

i=1

Proof Suppose there are numbers {aj,m}mj =− 11, m = 1,2, . . .,n − 1, such that the optimal policy in an m stage problem is to initially use the ith smallest p if the initial value is contained in the interval (ai − 1,m,ai,m), where a0m = − ∞ and am,m = ∞. Then in the n stage problem where pk is selected first f(p1,p2, . . .,pn|x) = max[xpk + f(p1,p2, . . .,pk − 1,pk + 1, . . .,pn)].

(13.65)

k

By the induction hypothesis, it follows that the optimal policy for an (n − 1) stage problem is independent of the (n − 1) values of p. Hence, defining ai,n as the expected value (under optimality) of the quantity to which the ith smallest p is assigned in the (n − 1) stage problem, the total expected reward of that problem is given by n−1

f(p¯ 1,p¯ 2, . . .,p¯ n − 1) =

冱 p¯ a

,

(13.66)

i i,n

i=1

for every p¯ 1 ≤ p¯ 2 ≤ . . . ≤ p¯ n − 1 (the p¯ 1,p¯ 2, . . .,p ¯ n − 1 represent the remaining (n − 1) ps of the original nps after the first, i.e., pk is chosen in the n-stage problem). In addition, since ai,n is independent of the ps and other policies are n−1

obtained by permuting the ps, any sum of the form

冱p a

ji i,n

(where

i=1

j1, j2, . . ., jn − 1 is a permutation of the integers) can be obtained for the total expected reward of the (n − 1) stage problem. Thus, using Hardy’s result (Lemma 1) it follows that a1,n ≤ a2,n ≤ . . . ≤ an − 1,n

(13.67)

since by the induction assumption, f(p¯ 1,p¯ 2, . . .,p¯ n − 1) must be maximized. Using (13.66) and (13.67), equation (13.65) can be written k−1

冱 pa

冤

f(p1,p2, . . .,pn|x) = max xpk + k

n

i i,n

+

i=1

冱

Appealing to Lemma 1 once more gives k* − 1

f(p1,p2, . . .,pn|x) = xpk* +

冱 i=1

n

piai,n +

冱

piai − 1,n,

i = k* + 1

冥

piai − 1,n .

i=k+1

(13.68)

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where k* is such that (with a0,n = − ∞, an,n = + ∞) ak* − 1,n < x ≤ ak*,n. This result follows because the ps and as are ordered so that if x is greater than or equal to the (k* − 1) smallest a, then the corresponding p (i.e., pk*) must be greater than or equal to the (k* − 1) smallest p. Thus, the first choice in an n stage problem is to choose pi if x ∈ (ai − 1,n,ai,n). This result is trivial for n = 1 which completes the induction. Equation (13.63) follows at once from equation (13.66), and the proof is complete. Intuitively, the optimal solution first partitions the real line using the expected values of the Χs. If the first chosen Χ is observed to fall in the interval (ai − 1,n,ai,n) it is then paired with pi. Using an alternative p for pi can’t improve the situation because if a lower pj is switched for pi then you gain Χ1pj + ajpi and give up Χ1pi + ajpj. Since Χ1 > aj and pi > pj by Lemma 1 your expected value declines. A similar argument holds for choosing a pk > pi. This matching of ps and Χs then continues with a1,n removed from the partition. Theorem 13.8 does not show how to obtain the ai,n. For that DLR use Corollary 1

Let a0, = − ∞ and an,n = + ∞. Then ai,n

ai,n + 1 =

冮

zdGΧ(z) + ai − 1,nG(ai − 1,n) + ai,n(1 − G(ai,n))

(13.69)

ai − 1,n

for i = 1,2, . . .,n, where − ∞ · 0 = ∞ · 0 ≡ 0. Proof Remember that ai,n + 1 is the expected value in an n stage problem of the quantity to which the ith smallest p is assigned. The result follows by conditioning on the initial x and recalling that pi is used if and only if this value lies in (ai − 1,n,ai,n). Derman, Lieberman, and Ross present a stochastic generalization of Hardy’s Theorem. Theorem 13.9 that is

Let successive values of Χ1,Χ2, . . .,Χn be a submartingale,

E[Χj|Χ1, . . .,Χj − 1] ≥ Χj − 1, for all j ≥ 2. The corresponding optimal policy is to use p1, then p2, etc., until final use is made of pn. The proof is by induction. See DLR. Till now it was assumed that the ps were a fixed set of numbers. Derman, Lieberman, and Ross provide an extension which allows the ps to be determined in optimal fashion. In the setting provided by the stochastic sequential assignment problem, a firm usually can attract skilled workers (those with high ps), by paying larger salaries. Suppose that c(p) denotes the cost to retain a worker with productivity index p. Let a1,a2, . . .,an denote the expected values of the quantity to which the smallest p is assigned (these as are the

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ai,n + 1s considered previously). The appropriate total expected return for a specific allocation p1,p2, . . .,pn, p1 ≤ p2 ≤ . . . ≤ pn, is given by n

Q(p1,p2, . . .,pn) =

n

n

冱a p − 冱c(p ) = 冱[a p − c(p )]. i i

i=1

i

i=1

i i

i

(13.70)

i=1

Hence the problem is: maximize Q(p1,p2, . . .,pn) subject to pi ∈ Π, i = 1, . . ., n,

(13.71)

p1 ≤ p2 ≤ . . . ≤ pn,

(13.72)

and

where π is a nonempty subset of [0,1]. Note that p = (pi) maximizes  subject to (13.66) if and only if pi maximizes aipi − c(pi), i = 1,2, . . .,n. If c is lower semicontinuous and π is compact, these maxima are achieved. Note also that because a1 ≤ . . ., ≤ an, the maximizing pi automatically satisfies (13.70) provided pi is equated to pj whenever ai = aj. Equation (13.72) becomes redundant and no assumption is required on c. In DLR’s final section, a general assignment model is studied. Here r(p,x), the expected return if a p worker is assigned to an x job, replaces px. This is a more realistic model which readers may wish to consult. The importance of the work by DLR is indicated by the literature it generated. These include Albright (1974, 1977), Sakaguchi (1984a, 1984b), Nakai (1986), and Righter (1989), to name but a few. We now turn to an adaptive version designed in Albright (1977), which begins by observing that the following model has received much attention. Suppose n random variables are observed sequentially. The goal is to select the best given that recall is not permitted. This problem has several versions requiring distinct mathematical analyses. If only relative ranks of the n variables are observed, one obtains the secretary problem. Here the problem begins by noting that there are n! equally likely orderings. The goal is either to maximize the probability of selecting the k highest rankings, k ≥ 1 or to minimize the expected rank actually selected. Solutions of these and other variations are in Chow et al. (1964), Gilbert and Mosteller (1966), and Ferguson (1989). An alternative approach assumes the random variables Χ1,Χ2, . . .,Χn are i.i.d. from a given distribution. Here the goal is to maximize the values of the Χs actually chosen. This is a BSM and usually called the house-selling problem. Albright notes that it has been analyzed by Karlin (1962), Lippman and McCall (1976a, 1976b), Sakaguchi (1961), and Telser (1973). MacQueen and Miller (1960) should be added to the list. As noted above, Albright envisions this problem as being embedded in a

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more general setting by DLR (1972). As we just saw they proved that if the decision-maker must pair n known quantities p1 ≤ p2 ≤ . . . ≤ pn with the n Xs to maximize the total expected return, where a return of piΧi is received when pi and Χi are paired, then the same critical numbers policy is best for any values of p1,p2, . . .,pn. Albright studies the following Bayesian model. Let Χ1,Χ2, . . .,Χn be i.i.d. random variables observed sequentially and assume they must be paired with known values p1 ≤ p2 ≤ . . . ≤ pn. If a p is jointed to an x a return of px is received. The decision-maker’s goal is to maximize the sum of the n returns. Note that the most likely version of this model is when each of the ps is 0 or 1, and therefore the problem collapses to either the house-selling problem where the best is selected or a generalized house-selling problem where the k best, k ≥ 2 are chosen. Albright discusses the following application with ps other than 0 and 1. Suppose a firm wishes to purchase N items of a given type in the next n days and the prices on these days are Χ1,Χ2, . . .,Χn. If p¯ i is the amount purchased on the ith day and (p¯ 1,p¯ 2, . . .,p¯ n) must be a permutation of fixed numbers (p1,p2, . . .,pn). Then the problem is the same as Derman et al. except that one minimizes rather than maximizes. See Albright (1977) for other examples. Turning to the Bayesian problem, assume there are one or more unknown parameters in the known distribution from which the Χs are drawn. These parameters have a prior distribution which is revised using Bayes rule as successive Χs are presented. Assume the order of operations is that an Χ is observed, the prior is updated, and then a decision is made. Let q be the prior distribution and qx the posterior given that an x is observed. The following theorem reveals that the major result in Derman et al. is obtained without any distributional assumptions. Theorem 13.10 (Albright) Suppose n Xs remain to be seen and the current prior is q. Then there are numbers − ∞ ≡ a0,n ≤ a1,n ≤ . . . ≤ an − 1,n(qx) ≤ an,n ≡ + ∞ such that if the next Χ has value x, it is best to assign the ith smallest p to this x if and only if ai − 1,n(qx) < x ≤ ai,n(qx). These critical numbers are independent of the ps. In addition, before this first Χ is observed, ai,n + 1(q) is the expected value of the Χ which, under the optimal policy, will eventually be paired with pi. Finally, the critical numbers satisfy the following recursive equation

冮

冮

冮

ai,n + 1(q) = xdH(x) + ai − 1,n(qx)dH(x) + ai,n(qx)dH(x). A

A ¯¯

A¯

where H is the current marginal distribution of the Χs A = {x:x ≤ ai − 1,n(qx)}, A = {x: ai − 1,n(qx) < x ≤ ai,n(qx)} and A¯ = {x:x > ai,n(qx)}.

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469

Proof Similar to that in DLR. Theorem 13.8 almost solves the problem, but the actual calculations of the ai,ns from (13.63) is difficult. There is no guarantee that A, A, and A¯ are intervals. Albright performs calculations for three families of distributions, the Normal, the Uniform, and the Gamma. Albright also conducts a sensitivity analysis comparing Bayesian and non-Bayesian policies. These are interesting and the reader may wish to obtain Albright’s article. It is useful to summarize what has been discovered in these models. First, optimal stopping manifests itself as a family of secretary problems, a family of BSM problems, and each family is “easily” converted to a Bayesian learning process when the parameters of the distribution are unknown, but have specified forms. The secretary problem and Bayes methods will be discussed rather intensively in the companion volume. We conclude this section with a summary of Righter (1989) which continues research on resource allocation in a random environment. Her model is close to the search model in a dynamic economy presented in Chapter 7. Suppose there are n activities each requiring one resource. Resources arrive according to a Poisson process with rate λ. Each arriving resource has a value Χ, a nonnegative random variable with distribution F(·), independent of other arriving resources. The expected value of Χ is bounded as follows: 0 < E[Χ] < ∞. An arriving resource is known on arrival and may be rejected or assigned to one of the firm’s activities. Each activity has a value ri and these are ordered as follows: r1 ≥ r2 ≥ . . . ≥ rn ≥ 0. Assignments are irreversible. The return obtained when a resource of value x is assigned to the ith activity is the product rix. The goal is to assign arriving resources to available activities to maximize the firm’s total expected return. Righter considers two models, A and B. In Model A, returns are discounted with an exponential discount function with rate α. This is equivalent to assuming no discounting but a termination of the problem after an exponential distributed time with rate α. Righter notes that model A is identical to having a single exponentially distributed deadline for all activities and returns are garnered only for activities that receive assignments prior to the deadline. In Model B, each activity has its own deadline which is exponential with rate αi and is independent of other deadlines. Activity i terminates when the deadline is hit. The resource arrival rate, the distribution of resource values, and the activity values and deadline rates (for B) change according to independent continuous time Markov chains. This is the link with Chapter 7, “Job Search in a Dynamic Economy.” When all activities have identical values, the model specializes to the BSM, i.e., the house-selling problem. Activities correspond to houses and resources are offers for the houses. The goal is maximize total discounted value of accepted offers (A), or maximize the total return for houses sold before removed from the market (B). Righter recapitulates the findings of DLR and the discoveries by others who have modified and extended the DLR model.

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The analysis begins with two lemmas. The first states that when all deadline rates are the same, αi = α, for all i, then the structure of the optimal policy for model B is identical to A’s. In this circumstance, the optimal policy is determined by thresholds which are independent of activity levels except through their order, and do not depend on the total number of activities. Furthermore, the thresholds of the optimal policy are equal to the expected resource values assigned to the activities by the optimal policy. The second lemma shows that for model B, if all deadline rates are identical, the expected resource value allocated to activity i by the optimal policy is identical to the optimal expected value of the resource it would receive if it were a single with deadline rate αi. If there is but one activity, A and B are the same. A discovery of Lemma 2 is that the optimal policy for model B can be obtained from A’s policy. Righter achieves results similar to Lippman and McCall (1976c) by allowing her model’s parameters to fluctuate according to continuous time Markov chains. While different calculations are performed, the ideas are similar. In order to get the thresholds to be monotone (increasing in the state) Righter employs the following three assumptions: ∞

1

For all states m,

冱P

j−k

is increasing in j.

k=m ∞

冮

2

For all c, dGj(ᐉ) is increasing in j.

3

µj = µ, for all j.

c

In words, (1) maintains that the next state of the Markov chain given the current state is stochastically increasing in the current state. This is equivalent ∞

to the condition that

冱P f(k) is increasing in j for all increasing functions f. jk

k=0

On the other hand, (2) is equivalent to the assertion that Gj is stochastically ∞

冮

increasing in j, which is equivalent to f(ᐉ)dGk(ᐉ) is increasing in j for all 0

increasing functions f. Theorem 13.11 Under assumptions (1)–(3), for all activities i, V Ai(j,λ) and V Bi(j,λ) are increasing in j where V Ai(j,λ) is the expected discounted resource value assigned to activity i under the optimal policy when environment is in state j and the arrival rate is λ and Model A obtains. V Bi(j,λ) is the same for Model B. These assumptions guarantee that the Markov chains are monotone in the sense of Conlisk and Daley. The analysis by Righter demonstrates the power of monotone Markov chains.

14 Topics for further inquiry

There are several topics in the economics of search which have been neglected almost completely or have not been treated in the detail they demand. In this concluding chapter we will outline some of the topics that we consider in our companion volume. These include: 1 2 3 4 5

Exchangeability and the Polya urn The secretary problem Sequential search in biology and ecology Markov Chains Monte Carlo (MCMC) techniques Equilibrium models revisited.

14.1 EXCHANGEABILITY DeFinetti’s greatest and most creative discovery was exchangeability, which replaced independence in both his philosophical and mathematical theory of learning. Kyburg and Smokler (1980) note that until exchangeability was discovered by deFinetti in 1931, the subjectivist probability theory was a quaint “philosophical curiosity.” By showing how exchangeability connects with classical statistics and modifies the practice of statistics, deFinetti altered this attitude once and for all. Consider the following example of an exchangeable sequence of cointosses. Instead of assuming that the probabilistic behavior must be in accordance with a Bernoulli construct, deFinetti sought and found the individual’s personal probability distribution for the sequences (Χ1,Χ2, . . .) of heads and tails. To say that this sequence is exchangeable means that the distribution of any n-sequence of outcomes is invariant to their ordering. The probability of (Χ1,Χ2,Χ3) = the probability of (Χπ(1),Χπ(2),Χπ(3)), where π is a permutation operator. Let us present several key examples of exchangeable sequences presented by deFinetti himself. He observes that in detecting the flaws of independence he has responded by replacing it with the simplest, cogent choice—the order

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in which the Bernoulli trials occurs is irrelevant. If there are n tosses of a coin with h heads and (n − h) tails, n and h are all that is required to calculate our personal probability. In another terminology they are called sufficient statistics, where the order of occurrence has no influence on the joint probability. Example 14.1 Drawings from an urn with unknown color composition (with replacement) are exchangeable when the urn contains an unknown number of red and black balls. Example 14.2 Drawings from the same urn without replacement also generates exchangeable sequences of red and white balls. Example 14.3 In the case of double replacement (if a red ball is chosen, two red balls are placed in the urn and the same for white balls). The sequence of colored balls is exchangeable. this is Polya’s urn!1 Example 14.4 Suppose we toss a coin 50 times and obtain h = 26 heads. The sequence is exchangeable and an equivalent description is the statement that n = 50 and h = 26 are sufficient statistics. DeFinetti (1975: 212; original emphasis) notes: “In other words, so far as ‘learning from experience’ is concerned, it does not matter whether we observe the complete sequence, or whether we simply observe that n = 50 and h = 26; this is a consequence of exchangeability.” It should be noticed that in all these examples the distribution of outcomes are fully determined. There are no “unknown parameters”! Example 14.5 Suppose there are n urns containing red and white balls. Let pi be the probability of drawing a white ball from urn i. Let the prior probability of choosing urn i be denoted by ai. Consider the following sequence: an urn is selected according to ai. A ball is then sampled from the selected urn. This sampling is done n times with the replacement. From the total probability formula, the probability of r white balls in a sample of size n is: n

冢 r 冣Σa p (1 − p ) r i i

i

n−r

.

(14.1)

This mixture of Bernoulli processes yields a probability that is exchangeable. Example 14.6 Finally, consider drawing a number p ∈ [0,1] according to the prior density f. The total probability is now an integral: 1

冢 冣冮

n r p (1 − p)n − rf (p)dp, r 0

(14.2)

which is also exchangeable. We now come to deFinetti’s representation

Topics for further inquiry 473 theorem: consider a probability law that is exchangeable for all n. Then there exists a unique distribution F (called the prior or the deFinetti measure) such that 1

冢 冣冮

n r w(n)r = p (1 − p)n − rdF(p), r 0

(14.3)

where w (n)r is probability of r successes in n trials. Von Plato (1994) observes that deFinetti’s philosophical interpretation of (14.3) is that it demonstrates how to remove “unknown objective probabilities” of independent events, and replace them with their subjective counterparts. The mixture of urns was, for deFinetti, a clear path to the discovery of exchangeability. DeFinetti was asked how he arrived at his representation theorem. He responded that it was by “contemplating a set of several urns with varying compositions” (Von Plato 1994: 251). 14.1.1 Extensions of exchangeability 1 2

3

In 1932, Khintchine presented a simpler proof of the representation theory and also the strong law of large numbers for exchangeable events. Von Plato (1994: 253; emphasis added) notes that the subjectivist approach leads deFinetti to conclude that “probabilistic independence is not only a lack of causal connection, but a lack of any influence on our judgment of probability” (1934). Exchangeability was devised to reconstruct this crucial influence. Partial exchangeability is the next generalization. (We return to this in our discussion of insurance in Volume II.) For now, we observe that perfect symmetry is replaced by k classes in which exchangeability is present in each class. The main result is the representation theory for a k-fold partial exchange sequence of events:

冢 r 冣 . . . 冢 r 冣 冮 p 1 (1 − p )

.,n ) w(nr ,, .. .. .,r = 1

r

1

k

k

k

(n1, . . .,nr) r1, . . .,rk

4

nk

1

k

r1

1

pr (1 − pk)n k

n1

− rk

n1 − r1

...

(14.4)

dF(p1, . . .,pk),

where w is the joint probability of a sequence of r1,. . .,rk successes in a sequence of n1,. . .,nk trials. This is but one example of partial exchangeability. We will see that this concept includes: hierarchical models, an introduction to stochastic processes via partially exchangeable Markov chains, and Aldous’s row x column exchangeability which connects with the U-statistic. Some consider this connection deeper than deFinetti’s theorem. In the 1930s deFinetti had resolved the problem of predictive probability

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The Economics of Search (n) , r

p

5

the conditional probability of success on the (n + 1)th trial. In a 1952 paper, he presented the geometry associated with this notion—a twodimensional random walk on a lattice. Success coincides with a step to the right and up. Failure has a step to the left and down. In 1930, deFinetti considered finite exchangeability. A basic example is a sequence of drawings from a finite urn without replacement. He presents an algorithm with which one can check whether a finite sequence extends to an infinite sequence.

14.2 POLYA URNS It is surprising at first that many fundamental, dynamic models in economics and related sciences can be portrayed as simple urn models.2 What seems even more remarkable is the frequency with which complicated economic processes are simplified by urn applications, and, in some important cases, inherit their symmetry or exchangeability.3 It should also be noted at the outset that the presence of these exchangeable models in diverse applications suggests a statistical methodology quite different in principle from the Neyman–Pearson–Fisher (NPF) methods employed by many economists. We believe that economists will find the Bayesian methodology more illuminating, in many cases, than the NPF alternative. There is a partial explanation for the usefulness of urn models in economics. Motion is the essence of economics. The movement accompanying the allocation of scarce resources among competing alternatives can be visualized as an urn model. Balls, representing resources, are thrown at a box containing many urns, each signifying one of society’s alternative uses of resources. These throws are not completely haphazard. They are guided by a price system, which determines approximately how many balls occupy each urn. The entire system is subject to shocks which alter the prices assigned to each urn. Balls leave urns that have reduced prices and flow on to higher priced urns. An equilibrium set of prices is affiliated with each shock and determines an equilibrium allocation of balls among urns. These equilibria are altered as new shocks occur. This simple metaphor captures a good deal of the activity (motion) each of us experiences every day as we interact with the economy. Our understanding of dynamic economic problems often is enhanced when the problem is formulated within an urn model. Learning by participants in the economy is perhaps the fundamental problem. We view this learning from a Bayesian perspective and show how the Polya urn sometimes captures the essence of the problem. From the earliest manifestations of economic behavior, those seeking an understanding of it have also thought that symmetry would aid their quest.

Topics for further inquiry 475 We will see that these economic urn models are characterized by a symmetry which flows from the Bayesian perspective and is called exchangeability. DeFinetti discovered exchangeability in 1929 and is responsible for much of the analysis that may help us understand the complex dynamic economy. His philosophy is individually oriented in that it is a subjective edifice. Probability is basically an opinion held by each individual and based on her personal experience which is codified by Bayes theorem. This subjective philosophy seems at odds with the objectivity sought by most economists. We attempt to show that deFinetti’s subjectivity has important advantages which frequently augment the ability of economics to understand the “real world.”4 Search and learning are two pivotal activities. Search accompanies learning. The mind must identify information which is relevant for its welfare. This identification process entails search of the enormous information available to the mind and the discovery of that small information set pertinent to this idiosyncratic individual. We see that search is an important component of induction. It collaborates with learning in accumulating information that is essential for personal decision-making.

14.3 URN METHODS Any problem of probability appears comparable to a suitable problem about bags containing balls, and any random mass phenomenon appears as similar in certain essential respects to successive drawings of balls from a system of suitably combined bags. (Polya 1990: 61; emphasis added) Every Markov chain is equivalent to an urn model as follows. Each occurring subscript is represented by an urn, and each urn contains balls marked E1,E2, . . . . The composition of the urns remains fixed, but it varies from urn to urn, in the jth urn the probability to draw a ball marked Ek is Pjk. (Feller 1957: 339) 14.3.1 Introduction It is difficult to identify historically the initial use of urn models. Johnson and Kotz (1977) observe that the method of urns is described in the Old Testament and in the early Jewish theological literature. Apparently the method of urns was first employed in probabilistic problems by Huygens (1629–95). Urn models5 are closely related to combinatoric methods,6 which were the foundation for the initial research in probability theory in the seventeenth and eighteenth centuries. The family of urn models known as Polya urns was first presented by Markov (1856–1922). These urns and their close relatives are our major interest. The particular version studied herein was presented in

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Eggenberger and Polya (1923, 1928). It has a simple structure: a ball is drawn from an urn containing a red balls and b black balls. The selected ball is returned to the urn together with s balls of the same color. The first applications of this model were in studies of contagion. Various versions of the Polya model have been successful in improving our comprehension of contagious diseases and also analogous information processes like advertising and insurance. The most important theoretical insights of the Polya urn flow from its close relation to the two fundamental theorems of Bayesian analysis: Bayes theorem and deFinetti’s theorem. As we saw, the Polya urn is similar to Bayes theorem and is linked to the deFinetti theorem via the exchangeable sequences it generates. The Polya urn and its variations are applicable to several subdisciplines in Economics. These include: learning, search, income mobility and inequality, insurance, evolutionary economics, and, especially, econometrics. We show that the Polya distribution associated with the Polya urn is a beta mixture of a binomial distribution. Lad (1996: 308) notes that: The twentieth-century tradition of objectivist statistical practice has largely honored the characterization of events that we regard exchangeability, as events “generated by an unknown but constant probability, θ, for success in each particular instance.” Thus, the paradigmatic statistical program has been to “estimate the true probability of occurrence of an event” by means of observations from a sequence of independent events that occur with the same probability. . . . It is interesting in this context that the equivalence of the Polya distribution with the beta-binomial mixture identifies a representation of the problem that does not involve any parameter θ at all! As an introduction to both urn analysis and Bayes theory, we present an urn version of Bayes theorem. 14.3.2 An urn version of Bayes theorem Assume there are k urns U1, . . .,Uk and that the proportion of black balls in the ith urn is given by πi, i = 1, . . ., k. One of these k urns is selected by drawing from another urn, the k + 1st. The probability of selecting the ith urn is Pi, i = 1, . . ., k. Suppose a ball is drawn from the selected urn and is black. What is the probability that the selected urn is Uj? First note that the probability of choosing a black ball given that it is drawn from urn Uj is: P[B|Uj] = πj. Thus, P[B∩Uj] = P[Uj]P[B|Uj] = Pjπj. We wish to calculate P[Uj|B] = P[B∩Uj]/P[B]. Now k

P[B] = P[U1]P[B|U1] + . . . + P[Uk]P[B|Uk] =

冱P π . i i

i=1

(14.5)

Topics for further inquiry 477 It follows that P[Uj|B] =

Pjπj k

.

冱P π

i i

i=1

This relation is called Bayes theorem and is commonly given as: P[Hj|E] =

P[Hj]P[E|Hj]

,

k

冱P[H ]P[E|H ] i

i

i=1

where E is an event and Hi, i = 1, . . ., k, the hypotheses regarding the conditions under which the event is observed to occur. P[Hj] is the prior probability of Hj, j = 1, . . ., k. P[Hj|E] is the posterior probability of Hj given that E is observed, j = 1, . . ., k. Equation (14.5) is called the formula of total probability. Johnson and Kotz presents a host of urn models that represent basic probabilistic notions like the inclusion–exclusion principle, the birthday problem, etc. We seek those urns which generate exchangeable sequence. Perhaps the best way to begin is with D. Freedman’s analysis (1965) of Bernard Friedman’s general urn model (1949). 14.3.3 Bernard Friedman’s urn model An urn initially contains w white balls and b black balls. Balls are drawn at random, one at a time. Each drawn ball is replaced in the urn together with α balls of the same color and β of the opposite color. This procedure gives rise to four important special cases (models). First, if α = β = 0, a binomial model emerges. In the second case, let α = −1 and β = 0. This gives a model of sampling without replacement. The third case is the most important for our purposes. If α = 1 and β = 0, the resulting sequence is drawn from a Polya urn. The fourth model sets α = −1 and β = 1. It is called the Ehrenfest model of heat exchange. If α = c, which may be negative, the third case is still called the Polya model. Note that if P(n,x) denotes the probability of x successes in n trials, then P(n + 1,x + 1) =

w + cx w + b + cn

P(n,x) +

b + c(n − x) P(n,x + 1). w + b + cn

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In Freedman’s analysis of Friedman’s urn, let wn and bn be the number of white and black balls at time n. The parameters α and β have the same meaning as before. Some of the Polya urn properties derived by Freedman are: (a) The process (wn + bn)−1 wn ; n ≥ 0 is a martingale and converges with probability 1 to a limiting random variable Z. This limiting random wo bo variable has a beta distribution with parameters , . 7 Given Z, the α α differences wn + 1 − wn, n ≥ 0 are conditionally independent and identically distributed, that is, they are exchangeable, given Z. They take on values 1 and 0 with probabilities Z and 1 − Z, respectively. n−1

(b) Freedman notes that the second part of his Lemma 2.1 is: n

冱Χ con-

−1

j

j=0

verges with probability 1. This holds for any ᏸ′ exchangeable process by Birkoff ’s ergodic theorem or by a martingale argument. This observation coupled with his proof of Theorem 2.2 yields a proof of deFinetti’s theorem. (c) In Remark 2.3 Freedman observes that in (Wn + Bn)−1 Wn is equal to P(Χn + 1 = 1|Χ1, . . .,Χn), where Χn = Wn + 1 − Wn for n ≥ 0. Let Χn be any n−1

−1

exchangeable process of 0s and 1s such that Z = lim n n→∞

冱Χ , j

then

j=0

P(Χn + 1 = 1 | Χ0, . . .,Χn) = E(Z|Χ0, . . .,Χn) is a martingale converging to Z.8 14.3.4 On Bayesian learning Bayes theorem is a method for adjusting our prior information as we acquire experience. This revision process is the prototype of learning. Indeed, deFinetti maintains that Bayes theorem is the learning mechanism that solves the induction problem posed by David Hume. Zellner (1988, 1991, 2002) studies an information processing rule (IPR) which has two inputs: π(θ|Io), a prior density over θ given the initial information Io and f (y|θ,Io), a data density for observation y given θ and Io. Zellner (1988) describes the outputs of an IPR as g(θ|y,Io), a “post-data” density for θ and h(y|Io), a marginal density for y defined by h(y|Io) ≡ 冮 f (y|θ,Io)π(θ|Io)dθ. Zellner used the following functional as his criterion ∆[g(θ|y,Io)] = Output information − Input information He then chose g to solve the following problem

Topics for further inquiry

479

min ∆[g(θ|y,Io] subject to 冮g(θ|y,Io)dθ = 1. g

In this way he minimized loss of information, measured by Output information − Input information. The solution to this problem is g*(θ|y,Io) =

π(θ|Io)f (y|θ,Io) , h(y)

which is Bayes theorem. He also showed that ∆[g*(·|·, . . .,)] = 0, that is, the Bayesian IPR is 100 percent efficient! He concluded by proving that when one sets g(θ|y,Io) = g*(θ|y,Io) a maximum entropy solution is obtained. Thus Bayesian learning is similar to a learning process that maximizes entropy. An excellent summary of maximum entropy is presented in Golan (2002), which also summarizes Zellner’s discoveries.9 Zellner (2002) also derives an important link between Bayes theorem and Bellman’s dynamic programming. He shows that formulating dynamic information processing entails the following problem. Choose an optimal sequence of functions π(θ|Dt,t) ≡ π(θ,t), where Dt = (y1, . . .,yt) is the information available at t, to minimize T

冱 冮 π(θ,t) log[π(θ,t)p(t)/π(θ,t − 1)f (θ,t)]dθ, t=1

subject to 冮π(θ,t)dθ = 1, for all t and π(θ,0) a given prior at t = 0, where p(t) ≡ p(Dt|t), f (θ,t) ≡ f (θt|P,t) and π(θ,t) ≡ π(θ|Dt,t), the post-data density. Zellner observes that this optimization problem is a Bellman dynamic programming problem. Furthermore, the optimal sequential solution is a Bayesian revision, i.e., the solution for period t, t = 1, . . .,T, is π*(θ,t) = π*(θ,t − 1)f (θ,t)/p(t). 14.3.5 The thumbtack experiment (Lindley and Phillips 1976) Interaction and dependence characterize the everyday economics in which all of us participate. Yet many econometric and statistical methods assume that the actors in the economic drama behave independently. It is surprising that a simple thumbtack, when properly interrogated, shows us how to inject interdependence into frequentist models of inference.10 The thumbtack model was used skillfully by Lindley and Phillips (1976) to contrast Bayesian methods of inference with their frequentist counterparts.11 A similar example was used in Kreps’s edifying and witty conversations describing the essence of deFinetti’s theorem, which Kreps (1988: 145)

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regards as “the fundamental theorem of statistical inference—the theorem that from a subjectivist point of view makes sense of most statistical procedures.” Choose a thumbtack12 at random from a newly purchased box of tacks. After an inspection to insure the chosen tack is not defective, it is tossed 12 times. If a toss lands with the pin facing upward(downward) the toss is labeled with a U(D). Of the 12 tosses 9 are Us and 3 are Ds. The actual order of the 12 outcomes is: UUUDUDUUUUUD. For a Bernoulli experiment like this, the sequence of outcomes is exchangeable. This means that the order of the outcomes is irrelevant (it conveys no additional information once I know there are 9 Us and 3 Ds). Recall that in a sequence of independent trials with constant probability θ of success, my sequence of r successes and s failures has probability θr (1 − θ)s regardless of the order of the successes and failures. This shows that Bernoulli trials are exchangeable. DeFinetti’s representation theorem for Bernoulli sequences is extremely important. Let p(r,s) be the probability of obtaining the particular sequence of r Us and s Ds. DeFinetti showed that 1

p(r,s) =

冮 θ (1 − θ) p(θ)dθ r

s

0

1

for some p(θ) ≥ 0 with

冮 p(θ)dθ = 1. The theorem can be stated quite simply: 0

exchangeable sequences are mixtures of Bernoulli sequences. DeFinetti also showed that the law of large numbers is a special case of the representation theorem. Let n = r + s be the number of tosses. He demonstrated that for exchangeable sequences lim r/n, call it θ, exists with probn→∞

ability one. Furthermore this θ has distribution p(θ). In our example this means that the proportion of Us has a limit with distribution p(θ). For Bernoulli trials the limiting distribution is degenerate, concentrating on a single value, which is exactly what the law of large numbers requires. Based on the information revealed so far about the behavior of the thumbtack on twelve tosses, Lindley and Phillips note that it would be impossible for a classical statistician to perform a significance test (or calculate a confidence interval). One must also specify the sample space13 (or equivalently, a stopping rule) before a test for fairness (P(U) = P(D) = 1/2) could be performed. The probability of the observed result, or one more extreme, is 12 12 12 1 12 + + ⬵ 7.5 percent. Thus, the null hypothesis of fairness 3 1 0 2 cannot be rejected at the 5 percent level. The Bayesian need only assume exchangeability to calculate the probability that the tack will be a U at the 13th toss given 9 Us and 3 Ds on 1st 12 tosses. That is, we seek

冦冢 冣 冢 冣 冢 冣冧 冢 冣

Topics for further inquiry p(U13|9,3) =

481

p(10,3) . p(9,3)

DeFinetti’s theorem shows us how to calculate the numerator and denominator on the right-hand side. This gives the same result as Bayes theorem with prior p(θ) and likelihood θr(1 − θ)s. To see this, write out the numerator giving 1

p(U13|9,3) =

冮 θ{θ (1 − θ) p(θ)dθ/p(9,3)}. 9

3

0

The term enclosed by the braces is the posterior distribution when p(θ) is the prior and equals p(θ|9,3). Therefore, 1

p(U13|9,3) =

冮 θp(θ|9,3)dθ. 0

Exchangeability which is weaker than the assumption of Bernoulli random variables gives p(θ), the prior, and demonstrates the validity of the calculations associated with Bayes theorem. Lindley and Phillips (1976: 115) observe: Thus from a single assumption of exchangeability the Bayesian argument follows. This is one of the most beautiful and important results in modern statistics. Beautiful, because it is so general and yet so simple. Important, because exchangeable sequences arise so often in practice. The thumbtack experiment is also used in the definitive monograph Theory of Statistics by Schervish (1995). Let us briefly describe some of Schervish’s remarks surrounding this example. Throw the tack onto a relatively soft surface and note whether it stops point up or down position. It is reasonable to assume that the tosses are symmetrical. The classical model proceeds as follows. The outcomes are IID random variables with Χi = 1, if the ith outcome is point up and Χi = 0, if the point is down. The frequentist approach “invents” a parameter θ which is of fixed and unknown value. Thus, the experiment is Bernoulli with the probability of an up(down) equal to θ(1 − θ). The Bayesian constructs a probability distribution µ for an unknown θ such that

冮

P(Χi = xi, i = 1, . . ., n) = θΣx (1 − θ)n − Σx dµ(θ). i

i

(14.6)

Instead of assuming IID and fixed but unknown θ′, we appeal to the natural symmetry of the problem and only assume exchangeability every permutation of the n outcomes on the left-hand side of (14.6) gives the same value for the right-hand side of (14.6). By deFinetti’s theorem, there is a µ which satisfies

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(14.6) for sufficiently large n. The fixed and unknown parameter θ is replaced by a random variable Θ. Finally, and decisively the Χis are not assumed to be IID, but instead are assumed to be conditionally IID given Θ. (Clearly, IID implies exchangeability.) The only other form of exchangeability besides conditionally IID random quantities is illustrated by a problem of drawing balls from an urn without replacement.14 The frequency interpretation of probability used by Kolmogorov in his 1933 foundations requires that an infinite amount of data be collected. DeFinetti resolves this problem by interpreting the limit of frequencies Θ as conditional on information not yet acquired. These probabilities are based on subjective judgments (opinions). Individuals with different opinions assign different probabilities to the same sequence of Bernoulli trials. Nevertheless, if these individuals assume exchangeability of the observed sequence, they must believe that there is a Θ such that given Θ = θ, the random variables are IID with parameter θ.15 It’s probably just as well that Kolmogorov did not seek guidance from deFinetti for then we would not have as rich a theory of complexity as now exists.16 Where is the Polya distribution? 14.3.6 The Polya distribution is equivalent to the beta-binomial distribution We concluded the last section with the query: where is the Polya distribution in the thumbtack experiment? We answer by first recalling the structure of a Polya urn. The urn contains r white balls and s black balls and n random draws are made from the urn. When a black(white) ball is drawn it is replaced in the urn with another black(white) ball. Let Χi = 0(1) if a black(white) ball is sampled on the ith draw. Letting θ be the probability of drawing a white ball, the sequence Χ1,Χ2, . . .,Χi, . . . is composed of exchangeable random variables.17 By deFinetti’s theorem an exchangeable sequence of r white balls and s black balls with r + s = n is given by 1

冮

p(r,s) = θr(1 − θ)sp(θ)dθ,

(14.7)

0

1

冮

with p(θ) ≥ 0 and p(θ)dθ = 1. 0

In the most important part of the thumbtack experiment, an inference (forecast) is made and p(θ) is assumed to be a beta distribution with parameters a,b > 0. Thus, deFinetti’s theorem specializes to 1

p(r,s) =

Γ(a + b)

冮冢Γ(a)Γ(b)冣θ 0

r+a−1

(1 − θ)s + b − 1dθ.

(14.8)

Topics for further inquiry

483

The integrand in (14.8) is the beta-binomial density. It is straightforward18 to show that the beta-binomial distribution is equivalent to the Polya distribution. In calculating the probability p(Un + 1 |r,s), we calculate the expected value of the Polya density in the integrand of (14.8) and obtain 1

Γ(a + b)

冮 冤Γ(a)Γ(b) θ

p(Un + 1|r + s) = θ

r+a−1

0

=

冥

(1 − θ)s + b − 1 dθ

a+r . a+b+n

14.3.7 The Polya urn is Bayesian (a) Consider the following version of the Polya urn. At time 0, the urn is occupied by two balls, one black, and the other white. At each time t = 1,2,3, . . . a single ball is selected at random and replaced in the urn with another ball of the same color. After n draws the urn has n + 2 balls. If Bn is the number of black balls chosen by n, then Bn + 1 of the n + 2 balls are black. (b) Now, envision the following Bayes-like process. A random number θ is selected from the uniform distribution on (0,1). A coin is then produced such that the probability of a head is equal to θ. The coin is tossed n times and the number of heads in the n tosses is Bn. The probabilistic structure of Bn is identical to that of the Bn in (a). (Bn + 1) (c) Let bn = in (a). Show that b is a martingale with respect to B. We (n + 2) have a good start since bn is a function of Bn and 0 ≤ bn ≤ 1. Proceeding, note that P{Bn + 1 = k + 1|B1 = i1, . . .,Bn − 1 = in − 1, Bn = k} = P{Bn + 1 = k|B1 = i1, . . .,Bn − 1 = in − 1, Bn = k} =

k and n+2

n+2−k . n+2

Hence E [bn + 1|B0, . . ..,Bn] =

Bn + 1 Bn bn + (1 − bn) = bn. n+3 n+3

14.4 THE SECRETARY PROBLEM The secretary problem, also known as the marriage problem, was introduced in Martin Gardner’s Mathematical Games column of Scientific American in February 1960. Since then it has attracted the interest of many distinguished

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mathematicians who have composed hundreds of articles. Why has this seemingly elementary problem become a cynosure for mathematicians? While it is easy to pose the secretary problem and versions of it, the solutions are not easily obtained, and usually require ingenious use of probability, combinatorics, and number theory. It is precisely these attributes that have proved irresistible to talented mathematicians. In the past twenty-five years, the practical aspects of the secretary problem have been manifested in economics, operations research, biology, etc. As versions of the secretary problem have become important in a variety of disciplines, it has generated great interest among applied mathematicians in these areas. Indeed, it is present in elementary economics textbooks! In his superb article, Ferguson (1989) traces the origin of the problem to Cayley and Kepler. Ferguson’s version of the secretary problem has six components: (1) a single secretary position is vacant; (2) there are n contenders for the position; (3) interviews of the n candidates are sequential and the order is random; (4) the secretaries can be ranked from best to worst by the interviewers. There are no ties and acceptance or rejection of a candidate is based on the relative ranks of candidates examined so far; (5) a rejected candidate can not be recalled at any later time; and (6) the utility derived from hiring the ith secretary is 0 unless the secretary is the best of the n candidates, in which case the utility is 1. Briefly stated, Ferguson defines the fundamental secretary problem as “a sequential observation and selection problem in which the payoff depends on the observations only through their relative ranks and not otherwise on their actual values” (1989: 284). Compatible with this definition, Ferguson prefers the googol version of the secretary problem. That is, he asserts (1989: 285) that “we should take as the secretary problem, the problem as it first appeared in print, in Martin Gardner’s February 1960 column in Scientific American.” Gardner called it the game of googol with the following description: Ask someone to take as many slips of paper as he pleases, and on each slip write a different positive number. The numbers may range from small fractions of 1 to a number the size of a googol (1 followed by a hundred 0’s) or even larger. These slips are turned face down and shuffled over the top of a table. One at a time you turn the slips face up. The aim is to stop turning when you come to the number that you guess to be the largest of the series. You cannot go back and pick up a previously turned slip. If you turn over all the slips, then you must pick the last one turned. The reader should stop and ponder the following question. What is the probability, for large n, of choosing the best secretary? The secretary problem as posed by Ferguson has a simple solution. It can be shown that one can concentrate on the class of rules such that for an integer r, the first r − 1 applicants are rejected and then the next candidate who is best in the relative ranking of those candidates interviewed thus far is

Topics for further inquiry 485 selected. Ferguson notes that for this rule, the probability n(r), of choosing the best candidate is 1/n for r = 1, and for r > 1, n

n(r) =

冱P(jth applicant is best and is chosen) j=r n

=

冱冢 冣 冢 j=r

n

r−1 r−1 1 = . j−1 n j=r j − 1

1 n

冱冢

冣

冣

The best value of r maximizes this probability. Now let n tend to infinity and let x be the limit of r/n. Substitute t for j/n and dt for 12/n. The sum is a Riemann approximation to an integral, that is r−1 n(r) = n

冢

n

1

冣冱冢 j=r

n j−1

冮冢 冣

1 1 →x dx = −x log x. n t x

冣冢 冣

Now we simply set the derivative with respect to x equal to 0 and solve for x. 1 This gives x = ≈ .3678, the optimal probability. This means that the best e 1 selection strategy is to do nothing19 until candidates have been interviewed, e and then choose the next candidate who is the best we have seen so far. Then 1 the optimal x = ≈ 37 percent.20 e

14.4.1 Cayley’s problem This appeared in the Educational Times in 1875 (p. 18): 4528 (Proposed by Professor Cayley)—A lottery is arranged as follows: there are n tickets representing a,b,c, . . . pounds respectively. A person draws once; looks at his ticket; and if he pleases, draws again (out of the remaining n − 1 tickets); and so on, drawing in all not more than k times; and he receives the value of the last ticket drawn. Supposing that he regulates his drawings in the manner most advantageous to him according to the theory of probabilities, what is the value of his expectation? Cayley assumes that k, n, and the a,b,c, . . . are known numbers and solves the problem using backward induction of dynamic programming. Ferguson notes that Cayley’s problem and the secretary problem are similar, but have a striking difference. The payoff is not 1 or 0 depending on selection of the best of the less-than-best; it is a numerical quantity which

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depends on the value of the chosen object. In a word, it is the job search problem.

14.4.2 The Dynkin (1963) solution to the secretary problem: the Markov connection The stages r(0) = 1,r(1),r(2), . . . at which candidates are observed comprises a Markov chain where a stage r(i) is defined as the time at which a new best candidate arrives. This follows from p(r(i + 1) = ᐉ|r(0) = 1,r(1) = a, . . .,r(i) = k), which is the probability that the (k + 1)th, (k + 2)th, . . . (ᐉ − 1)th candidates are less desirable than the kth and the ᐉth candidate is more desirable and, hence does not depend on r(0), . . ., r(i − 1). Actually, Pkᐉ = P(r(i + 1) = ᐉ|r(i) = k) =

P(r(i) = k and r(i + 1) = ᐉ) p(r(i) = k)

1

(ᐉ − 1) = (1 ≤ k ≤ ᐉ ≤ n), 1 k ᐉ

since the numerator is the probability that the kth and ᐉth candidates are the 2nd best and the best of the 1st ᐉ candidates. At r(i) = k, the probability of getting the best candidate, if selection stops, is k/n, while if one continues evaluation until r(i + 1) and then stops, the wining probability is n

冱P

kᐉ

k+1

ᐉ k k 1 1 1 = ak = + +L+ . n n n k k+1 n−1

冢

冣

The myopic rule compares these probabilities. This rule is optimal since the conditions for the Chow, Robbins, Siegmund monotone case are satisfied. This is only true if one looks ahead from one relatively best to the next. The one-item-ahead policy is suboptimal.

14.4.3 The dynamic programming solution of the secretary problem (Lindley, 1961) Let r be the number of candidates inspected by the employer and let s be the relative rank of the candidate who was inspected last. If s ≠ 1, the employer continues his search and state variables transit from (r,s) to (r + 1,s′). The relative rank s′ takes on the values 1,2, . . ., r + 1 with equal probability

Topics for further inquiry 487 assigned to each. If s = 1, then this candidate may be accepted, with r/n the probability that he is the best of the n. Let V(r,s) be the optimal expected probability of selecting the best candidate when the state of the system is (r,s). Applying dynamic programming gives two equations: r+1

冱

r 1 V(r,1) = max , V(r + 1,s′) , n r + 1s′ = 1

冦

冧

(14.9)

r+1

冱

1 V(r + 1,s′), s = 2,3, . . ., r V(r,s) = r + 1s′ = 1 where V(n,s) = 1 when s = 1 and 0 otherwise. Lindley obtained the solution by backward recursion over r = n,n–1, . . .,1. Let ar be given by 1 1 1 ar = + +L+ . r r+1 n−1 The optimal procedure in state (r,1) is: stop, if ar < 1 and continue, if ar > 1. This implies that if r* is the r such that ar − 1 ≥ 1 > ar, the best policy is to reject the 1st r* − 1 candidates and then hire the first candidate who is better than all of those interviewed. (r* − 1)ar* − 1 This policy wins with probability . As n → ∞, this probability n (which is approximated by r*/n) converges to e−1 = 0.368. We now see that the secretary problem is both the prototypical real option model and the archetype sequential search model!

14.5 THE ECONOMICS OF INFORMATION, BEHAVIORAL BIOLOGY, AND NEURO-ECONOMICS In our companion volume we will introduce several of the outstanding theoretical and empirical studies in behavioral biology and its closely related disciplines: cultural anthropology, neurobiology, and the cognitive sciences. The three major components of behavioral biology are foraging, mate selection, and the choice of a settlement or nest. From our perspective search theory is a unifying ingredient of this research with game theory also assuming a conspicuous role. We believe there is a fundamental connection between biology and economics where search is merely a single manifestation. Another way of recognizing this linkage is to show the close connections among Adam Smith, Charles Darwin, and other prominent biologists and economists. We

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are always aware of egregious mistakes attached to the supposition that economics is a universal life-saver for any of the problems endogenous to another discipline.21 Context must never be overlooked either within or, especially, across disciplines. Given this basic restriction, we agree wholeheartedly with Frank (1998: 3): The theory of natural selection has always had a close affinity with economic principles.22 Darwin’s masterwork is about scarcity of resource, struggle for existence, efficiency of form, and measure of value. If offspring tend to be like their parents, then natural selection produces a degree of economic efficiency measured by reproductive success. The reason is simple: the relatively inefficient have failed to reproduce and have disappeared. Exchange, division of labor, the “invisible hand,” and decisionmaking under uncertainty are hallmarks of Smithian economics. Frank considers three exchange rates connected with natural selection: 1 2 3

Fisher’s (1958) formulation of reproductive value by analogy with the discounted value of money.23 The second factor is marginal value. Frank maintains that this gives the proper scale for comparing costs and benefits. The third exchange is also a scaling factor: the coefficient of relatedness from kin selection theory (Hamilton, 1964). This appears to be peculiar to evolutionary theory, but Frank shows that kin selection is tightly connected with the notion of correlated equilibrium in game theory and economics. He also presents the nexus with the basic ideas of statistical information and prediction. Thus, he shows that not only are Smith and Darwin joined by exchange, but so too are Bayes, Fisher, and Hamilton, i.e., there exists a “logical unity of social evolution, statistical analysis of cause, aspects of Bayesian rationality, and economic measures of value.”

Our discussion concentrates on the similarities among Smith, Darwin, and Hamilton and the close nexus with Bayesian methods. Biological and economic decision-making are similar in that the search and exchange of information comprise their essence. Search lies at the foundations of foraging, mate selection, and nest(house) hunting problems— the three basic problems in behavioral biology. An organism’s life is a learning process and it is exactly this learning which constructs and shapes the unique individual self. Adjustment to the environment is central to learning. Environmental fluctuations affect all organisms, where the precise impact on individual behavior depends on its life history and the unique cognitive(neural) structure assembled in response to this history. In economic terms, the environment confronting each individual in a particular society is characterized by the marketplace. Exchange is the decisive

Topics for further inquiry

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activity of each individual as he responds to changes and the repeated exchanges by each of society’s members yield a spontaneous order. It is precisely this unplanned order which enables each member to exploit all the pertinent information dispersed throughout society. The prices in a pure economy carry this information, enabling each and every exchange among individuals to be based on all the relevant information available throughout the society. It is exchange which gives rise to the division of labor, the mechanism which enhances both individual productivity and society’s welfare. The recognition that a man’s effort will benefit more people, and on the whole satisfy greater needs, when he lets himself be guided by the abstract signals of prices rather than by perceived needs, and that by this method we can best overcome our constitutional ignorance of most of the particular facts, and can make the fullest use of the knowledge of concrete circumstances widely dispersed among millions of individuals, is the great achievement of Adam Smith. (Hayek 1978b: 269) The crucial point yoking these familiar Smithian principles is that every economic response is a reply to unforeseen changes. Uncertainty lives in both the socialeconomic “order” and is the source of the idiosyncratic cognitive architecture of each of society’s members. One might say that at the level of society there is an unfolding “objective” order, whereas each individual is busy constructing his own “subjective” self. In our discussion of biological behavioral systems we rely heavily on the intensive research of the behavior of honeybees by Seeley and his colleagues. The bee attracted the interest of Bernard Mandeville and led to the development of both evolution and spontaneous order together with division of labor and the central position of exchange. His original poem, composed in 1705, had as its major thesis the paradoxical assertion that private vices combine to create public virtue. We also plan to discuss the foraging problem and its link with search theory. We outline the exchangeable approach to foraging, extending the Bayesian formulation by Krebs and Davies (1986) and provide a fairly detailed discussion of adaptive sequential search. We present problems of patch use and prey choice and an important new approach to foraging via game theory: social foraging theory. We explore the foraging and nesting behavior of bumblebees and its relation to the cognitive architecture of bumblebees. We also consider mate choice, both one-sided and two-sided models are investigated, and we discuss the handicap (or signaling) model and note its renewal by Grafen (1990). Two-sided mating models can now be designed with search on one side and signaling on the other. As we see, almost all of the economic analysis that is covered in a complete course in microeconomics and that is presented in a course in the economics of

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information have applications and extensions in the marvelous research by Seeley and his colleagues on honeybees. The topics range from division of labor as it is sequentially practiced by the clever honeybees to the notion of a superorganism which addresses current controversies over the level of selection. We also plan to give an overview of the important and recent research in neuroscience and its economic implications. We rely heavily on the research by Bingham, Edelman, LaCerra, Quartz, and Senjowski. Their articles combine basic market economics with complex neuronal structures. We summarize their work emphasizing the important interchange among economics and neuroscience. We argue that Bayesian analysis may be useful in analyzing neuroscientific structures. This is compatible with recent recommendations by Shephard, Tennenbaum, and others and can be translated into an exchangeability approach similar to that flourishing in population genetics and econometrics.

14.6 EQUILIBRIUM SEARCH MODELS REVISITED The construction of novel equilibrium search models that are closer and closer to real problems continues unabated. These models are ingenious and are now having significant practical implications. Although we have devoted two chapters in this book to equilibrium search we have merely skimmed the surface of this rapidly growing area of research. In the next volume, we will consider additional topics in equilibrium search including topics on matching.

Notes

Preface 1 Eckstein and Wolpin (1990) is a structural estimation of an equilibrium search model, Albrecht and Axell (1984), which we discuss in Section 12.3. 2 The house-selling problem was first solved by MacQueen and Miller (1960) and Karlin (1962). 3 See Alchian (1969). 4 An excellent treatment of inventory theory is unveiled in Porteus (2002). 1 Introduction 1 The evolution of money as a medium of exchange is presented in Clower (1967, 1973), Jones (1976), Oh (1989), Ostroy and Starr (1990), Kiyotaki and Wright (1989, 1993), Tommasi (1994) and Ball and Romer (2003). The argument by Jones hinges on search costs. These costs are higher for barter because of heterogeneous tastes and the trading process is essentially sequential in that individuals need not specify a fixed sequence of trade before entering the market (Jones’s assumption). Instead, the sequence of trades depends on the realization of encounters. This is a complex combinatorial problem. Oh devises an ingenious recursion for its resolution. It is related to the algorithmic research that is beginning to influence economics. We contend that economics could have a powerful effect on all algorithmic research, and of course, vice versa. 2 A stopping time is a random variable T satisfying {T ≤ t} ∈ Ft,

for all t ≥ 0 (+).

Roughly speaking, Ft is the history of the process from 0 till t. The event {T ≤ t0} means that the process is stopped at or before t0. Clearly, we want this event to belong to the process history and this is implied by (+). 3 Stochastic dominance is central to the development and implementation of the economics of search. The purpose of stochastic dominance (mean preserving spread) is to measure the information content of alternative “experiments.” The “experiments” are searching for new information. Hence, the mean preserving spread is a criterion for choosing the most economical method of search. 4 These stages can be collapsed into a single stage by generalizing the sufficiency concept. This is valuable in that it forces the decision-maker(s) to concentrate on the value of the action and not be diverted by intermediate evaluations. Of course, this corresponds to the Savage–Wald–deFinetti theory of decision-making under uncertainty.

492

Notes

5 This has been studied by Jones and Newman (1995) in relation to technological change. As the worker ages, he/she learns from experience and a tight fit is achieved between job and worker. If technology changes suddenly, the job may require new skills to which the older worker cannot adapt. 6 A discussion of these principal–agent problems and their proposed resolution are reviewed with many references in Lippman and McCall (2002). 7 Two excellent volumes on game theory and the economics of information are Binmore (1992) and Hirshleifer and Riley (1992), respectively. 8 A detailed discussion of the history and evolution of search is contained in Chapter 3. 9 For a penetrating analysis of Walrasian theory and search see Diamond (1984). 10 For a comprehensive discussion, see Hirshleifer (1973). 11 For excellent analyses of intermediaries see Rubinstein and Wolinsky (1987) and Shevchenko (2004). 12 We must be careful in comparing these surveys, but certainly the surveys by Mortensen (1986) and Mortensen and Pissarides (1999a, 1999b) are best for those interested in labor economics. The same can be said for Sargent (1987) and Ljungqvist and Sargent (2004b) for readers interested in macroeconomics. The McMillan–Rothschild survey is best for game theorists, and Ferguson (1989) surveys the secretary problem. Levine and Lippman cut across many economic problems that display the ubiquity of search spelled out in Chapter 8. The two Phelps’s volumes also cover all bases. See Phelps et al. (1970), Aghion et al. (2002), and the splendid review by Leijonhufvud (2004). 13 See Tweedie (2001). 3 The history and evolution of sequential analysis 1 George Polya (1945: 64) has observed that interesting problems occur in batches like a patch of mushrooms: “If you find one look nearby for others.” 2 See Lai (2001) and Siegmund (2003) for superb surveys of the sequential literature. 3 The title of Mortensen’s seminal contribution is: “The Matching Process as a Noncooperative Bargaining Game” (1982). 4 This historical sketch is taken from the papers by Feinberg (1978) and Lippman and McCall (1993). 5 Because the price system functions so well even in the absence of the centralization of such information, Hayek acclaims it (1945: 527) a “marvel.” 6 In his concluding section Stigler (1961: 224) extends this definition of search to include problems “in the detection of profitable fields for investment, and in the workers’ choice of industry, location, and job.” 7 Tommasi (1993) provides an example in which an increase in one parameter of the model (how many units of the good each consumer demands) induces more search but the variance of the equilibrium price distribution does not increase. 8 The notion that by rejecting a low offer the worker is electing voluntary unemployment is due to Lucas. The term “voluntary unemployment” may have originated with Lucas (1976) in his critique of the Phillips curve. 9 Stigler’s search theory has been extended in several excellent papers by Manning and Morgan. See Morgan and Manning (1985), Manning and Morgan (1982), Morgan (1983), and Manning and Manning (1997). 10 Shortly after 1970, dynamic programming became the basic analytical tool used in the sequential search literature. 11 See Howard (1960), Blackwell (1962), Denardo (1967), Veinott (1969), and Ross (1983a). 12 The Statistical Research Group (SRG) at Columbia University was organized in July 1942 to advise and assist the U.S. military on various statistical problems that

Notes

13

14

15

16 17

18 19

20 21

22

23 24 25 26

493

arose in the conduct of their wartime activities. The group members included: A.H. Bowker, Milton Friedman, M.A. Girshick, Harold Hotelling, L.J. Savage, G.J. Stigler, A. Wald, W.A. Wallis and J. Wolfowitz. Other World War II research groups in Britain and the U.S. contributing to the technical foundations of sequential analysis and optimal stopping include the Enigma Project, the Anti-Submarine Warfare Operations Research Group (ASWORG) and the Quality Control Group in the Ministry of Supply (QCMS). In his book Sequential Analysis (1947), Wald credits Milton Friedman and W.A. Wallis with proposing the problem of sequential analysis. During the war, the cryptographic research conducted in England, headed by Alan Turing, recorded many contributions to statistics and sequential analysis. See Good (1979). In Dodge and Romig (1959) a population is sampled to ascertain whether the proportion of defectives exceeds some specified level. After drawing a sample of size n1 and counting the number x of defectives, the population is: (a) accepted if x is less than a fixed value α, (b) rejected if x exceeds a fixed value β > α, and (c) a second sample size n2 is taken if α ≤ x ≤ β and the accept–reject decision is based on the number of defectives in the sample size of n1 + n2. In sequential analysis the decision to accept or reject the null hypothesis is reconsidered after each observation: the actual sample size depends on the ongoing history of the sampling process. If a random variable which tells the decision-maker whether to stop or continue at time t is based only on knowledge of the history of the stochastic process up to time t and not upon the future history of the stochastic process, then the random variable is a stopping time (alternatively stopping rule). A very nice sequential formulation of the secretary problem is exposited by Ross (1983b). See Ross’s Chapter 3. See Chow and Robbins (1961), Breiman (1964), Dubins and Savage (1965), Robbins (1970), DeGroot (1970), and Chow et al. (1971), and the recent reviews of sequential analysis Lai (2001) and Robbins’s seminal contributions, Siegmund (2003). See Bather (1970) and Harrison (1985) for controlling Brownian motions and Merton (1973) for an introduction to option pricing. Lippman (1975b) introduced the case of unbounded rewards. The extension to competitive settings was introduced by Sobel (1971) for stochastic dynamic programming and Mamer (1987) for Chow and Robbins’s (1961) monotone stopping problems. Mortensen (1970) also developed a sequential search model. Mortensen links his novel model with the contributions of Friedman and Phelps on the “new” Phillips curve. We believe Rothschild (1973) was the first to use the term reservation price/wage in the search literature. In the pre-search labor literature, the reservation wage was the intercept of the labor supply curve, the largest wage at which no labor is supplied. Lucas states (1987: 54–5; original emphasis) that the “analysis of unemployment as an activity was initiated by John McCall.” In particular (1987: 66), “it is exactly this ‘voluntary’ aspect of McCall’s formulation that leads it immediately into the first coherent analysis of employment-related risks.” In most states unemployment insurance requires active search for employment. See Theorem 4 of Lippman and McCall (1986). McCall and others did not consider the alternative response of firms to the minimum wage regime: increase the offer distribution by converting all offers less than m to an offer of m. The name is apt, but offensive to some. It is apt in that bandit problems can be viewed as sequential policies for playing slot machines possessing unknown probabilistic characteristics, where each machine is a one-armed bandit.

494

Notes

27 The Gittins index is discussed at length in Chapter 9. 28 A more complete discussion of optimal stopping is contained in Chapter 4. 29 We note in passing that in 1933 deFinetti extended his representation theorem to exchangeable random quantities. This theorem resolves the frequency problem which tortured Kolmogorov for many years. 30 We return to this observation in the concluding chapter. 31 Note that optimal stopping rules are inherently Bayes. They are also closely linked to martingale theory and they can be formulated as dynamic programs and Markov decision processes. 32 If a keen observer of economics were asked to identify three of the most important economists of the twentieth century, it is very likely that the list would include Arrow and Friedman. It is interesting that both men made substantial contributions to the foundations of sequential and Bayesian decision-making early in their careers. 4 The basic sequential search model and its ramifications 1 McAfee and McMillan (1987a) is a fine survey. 2 Recently, many investment problems have been identified as “real” versions of the famous stock option model. Dixit and Pindyck (1994) claim that irreversibility and the possibility of delay are significant properties of most “real” investments. This converts them into “real options.” The essence of the stock-option model is optimal stopping. Several early versions of real options are identified in the next section. 3 Two excellent collections of important articles together with superb commentaries are contained in Diamond and Rothschild (1978) and Levine and Lippman (1995). 4 The problem of showing the conditions for which myopic policies are optimal was posed in Breiman’s seminal article in 1964. This article shows the tight link between the basic job search model and optimal replacement policies. Both can be viewed as real asset problems in the Dixit–Pindyck sense. As we will see, the BSM is a special case of a multi-armed bandit. 5 An excellent discussion with illuminating exercises is contained in Diamond and Rothschild (1978). 6 P(min(Χ1, Χ2, . . ., Χn) < x) = P(Χ1 < x,Χ2 < x, . . ., Χn < x) = F(x)n. 7 An example is a job-seeker who mails off a number of applications to prospective employers who make the hire/no-hire decision purely on the basis of the applications they receive and for whom the time required to hear back from any of the employers so approached is large relative to the time required to complete a single application. 8 A fine and abundant source of exercises are in Sargent (1987) and Ljungqvist and Sargent (2001). Less difficult, but rewarding exercises are in de Meza and Osborne (1980). 9 See Table 4.1 for the standard assumptions of the classic search model. 10 This calculation is an application of Leibniz’s rule to equation (4.5). Let β (t)

ψ(t) =

冮 f (x,t) dx for t ∈ [a,b].

α (t)

Assume f and ft are continuous and that α and β are differentiable on [a,b]. Leibnitz’s rule claims that ψ(t) is differentiable on [a,b] and β(t)

ψ′(t) = f(β(t),t)α′(t) +

冮 f (x,t)dx. t

α (t)

Notes

495

11 Unlike the purchase of goods in deterministic neoclassical theory, the value of what the purchaser of information receives in return is a random variable. In the sequential search model, it is the difference between the searcher’s income while unemployed and the eventual wage at which he begins work. Likewise, TC, the total cost of the information in this model is also random. Specifically, TC = cT where T is the duration of search. Thus, instead of speaking as we do in neoclassical theory of what happens to total expenditures as the price of the good goes up, we focus here on expected total expenditures. 12 If a random variable Y is stochastically smaller than the random variable Χ, then E(Y) ≤ E(Χ). The fact that E(T) decreases with c clearly follows from equation (4.10) and the fact that a rise in c lowers ξ, the reservation wage. 13 The hazard rate function associated with a distribution function F, λ(x), is given by λ(x) = f(x)/[1 − F(x)], where f(x) is the density at x. 14 It is worth noting that while in the general case dynamic programming involves at each stage of the problem a choice between many (sometimes an infinite number of) distinct actions, in this application the agent must, after receiving each new piece of price information, take one of only two actions: (1) purchase the good at the price p and terminate the search or (2) pay a cost of c and search another vendor. 15 Attempts to solve functional equations often begin with some kind of educated guess regarding the solution. 16 Functional equations such as equation (4.18) often have many solutions. If there were more than one solution to equation (4.18), the optimal strategy might not be well defined (i.e., there might be several very different optimal strategies for given values of c and F). 17 The reader may wish to review Markov decision processes as presented in Chapter 2 or in the recent monographs by Kallenberg (2002) and Putterman (1994). 18 Serfozo (1979) shows that there is an equivalence between continuous and discrete time Markov decision processes. He notes that Lippman (1975a) was “the first to recognize the usefulness of this equivalence in establishing the existence of optimal control policies with certain monotonicity properties” (Serfozo 1979: 617). 19 For a full discussion of replacement policies see Jorgenson et al. (1967). 20 We will use this property in our discussion of search in a dynamic economy. 21 We return to monotone Markov chains in Chapter 7. 22 A nonnegative integer r.v. N is a stopping rule for the sequence Χ1,Χ2, . . . if the event N = k depends only on the observed values of Χ1,Χ2, . . .,Χk and not on the as yet unobserved values Χk + 1,Xk + 2, . . . In other words, the decision to stop is not based on knowledge of the future. See Karlin and Taylor (1975: 254–256) for examples of Markov times. 23 Borel–Cantelli lemma: let A1,A2, . . . be an infinite sequence of independent events. Then the event {Ai infinitely often (i.o.)} which is the occurrence of an infinite number of the Ai, is given by ∞ ∞ A¯ ∞ = {Ai,i.o.} = ∩ ∪ Ai. j=1i=1

The Borel–Cantelli lemma states that the probability of A¯ ∞ is zero or one, according to whether ∞

∞

冱P{A } < ∞, or 冱P{A } = ∞. i

i=1

i

i=1

See Billingsley (1979), DeGroot (1970), and Chung (1974).

496

Notes

24 A sequence of r.v.s Y1, Y2, . . . is called uniformly integrable if lim |Yn|dP = 0 uniformly in n.

Z=∞

{|Yn| ≥ Z}. DeGroot 1970: 359. 25 Recall that E(N) = 1/F(p¯ ) and that lowering c decreases p¯ . 26 Morgan and Manning (1985; and Morgan 1983) in a series of articles argue that Stigler and sequential search are special cases of a general search strategy in which the searcher obtains a number of quotations and must decide how many more to solicit. We consider these qualifications later. 27 Lai (2001) is essential reading for those interested in the sequential design of experiments and its recent advances. 5 Estimation methods for duration models 1 For a good discussion of maximum likelihood estimation see Wooldridge (2003). 2 Controls for gender, race, marital status, age, education, immigrant status, region of country, industry of lost job, tenure in lost job, blue-collar–white-collar status, reason for displacement, weekly wage in lost job, and year of displacement were also included. The sample includes only those who are imputed to be eligible for UI benefits. 3 Heckman and Singer (1984) have shown that by increasing the number of mass points with the sample size, as the sample size approaches infinity the mass point distribution consistently estimates the distribution of the unobserved variable. 4 For different approaches to specifying a discrete-time competing risks model see Han and Hausman (1990) and Sueyoshi (1992). 5 For additional specification diagnostic methods for duration models see Schoenfeld (1980), Wei (1984), and Lancaster (1990). 6 Unemployment, unemployment insurance and sequential job search 1 For an illuminating and erudite discussion of the microeconomics of unemployment see Phelps (1972). 2 The mismatch problem has stimulated several studies. See Becker et al. (1977), Diamond and Maskin (1979), Jovanovic (1979a), Kormendi (1979), Mortensen (1977), and Wilde (1979). 3 See Lippman and McCall (1980) and references therein. 4 Also, Var(Np) = [p + 1 − qp]/(1 − p)2 q2p. 5 Of course, this is not surprising in view of that fact that ENp → ∞ as p → 1. 6 Because M is independent of (θi), (6.11) follows by applying Wald’s equation to M

Up =

冱θ . i

i=1

7 The economic rationale for the existence of temporary layoffs is spelled out in the papers by Baily (1977), Feldstein (1976), and Topel (1982). 8 Notice that this model does not include the possibility of subsequent layoffs. 9 See Danforth (1978) and Hall et al. (1979). 10 See Classen (1979). 11 For an overview of unemployment insurance systems in other countries see Atkinson and Micklewright (1991). 12 More recently see the paper by Fallick and Ryu (2003).

Notes

497

7 Job search in a dynamic economy 1 A similarly structured model has been used by Derman (1963) in his study of optimal maintenance policies (see Appendix). Jorgenson et al. (1967) extend Derman’s model to include an environmental state which has an opportunistic effect on policy. 2 The state space can be the entire real line provided lim µx < ∞, where x→∞

∞

冮

µx = ydFx(y). 0

3 Note this includes as a special case the economy moving according to a nonsymmetric random walk. In the continuous case, this would correspond to Brownian motion with a positive drift. 4 For example, see Parsons (1973) who argues that firms reduce search costs “by increasing advertising, arranging convenient interview times, paying moving expenses, etc.” k

5 Let α =

冱 p(1 − p)

j−2

< 1, so for k > 1,

j=2

P(Yi + 1 ≤ k) − P(Yi ≤ k) = pi + 1 + (1 − pi + 1)α − [pi + (1 − pi)α] = (1 − α)(pi + 1 − pj) ≥ 0. 6 See McCall (1973) for details. 7 E(Yn + 1|Yn, . . .,Y1) = E(Yn(q/p)x |Yn, . . .,Y1) = Yn[(q/p)p + (q/p)−1q] = Yn. 8 Those interested in a full treatment of this problem should consult Salop (1973). The article by Barron (1975) is also pertinent. 9 Conlisk (1985) is a working paper which was revised and published in Conlisk (1990). 10 A classic reference on monotonicity is Stoyan (1983) which has undergone basic transformations and has reappeared as the splendid monograph: Muller and Stoyan (2002). We refer to this monograph in our next application. 11 Studies of harvesting are contained in Reed and Clarke (1990) and Dupuis and Wang (2002). These studies are closely related to Brock and Rothschild (1984), Brock et al. (1988), and to the excellent text by Malliaris and Brock (1982). Apparently, the first article on this important subject was Miller and Voltaire (1983). 12 Real option models are presented in our next search book. 13 For a clear description of Brock, Rothschild, and Stiglitz (1989), see Malliaris and Brock (1982). 14 Conlisk also finds Markov monotonicity in the classic study by Nelson and Winter (1985). 15 See Lehoczky (1980) and Brumelle and Gerchak (1980). 16 For a Bayesian approach, which measures the robustness of Bayes, see Muller (1996). 17 This theory is the subject of two remarkable articles: Muller (1997b, 1997c). In the Preface to their monograph, Muller and Stoyen (2002), Stoyen observes that the unified study of integral stochastic orders of Chapter 2 yields an analysis of the order generators “which D.S. had dreamt of in the 1970s.” n+1

498

Notes

8 Expected utility maximizing job search 1 The early sections of this chapter are based on the doctoral dissertation of Jeffrey Hall. Jeffrey died in a mountaineering accident and the profession lost a first-rate economist. While the thesis was laid out by Jeffrey, many of the important technical points connected with the basic theorems were developed by Steve Lippman. Some of the later sections of the chapter are drawn from Scarsini (1994). 2 Most of this section is drawn from the excellent survey by Scarsini (1994), and Lippman and McCall (1981). 3 Early references include Arrow (1971), Diamond and Rothschild (1978), McCall (1971), and Hirshleifer and Riley (1979). 4 See Scarsini for the four conditions of Landsberger and Meilijson corresponding to the four RS conditions. 5 A concave function has decreasing marginal slope; an anti-starshaped function has a decreasing average slope from the point v at which it is anti-starshaped. See Scarsini for a complete discussion of anti-starshaped functions. 6 See Karni and Schwartz (1977) and Karni (1999). 7 Kohn and Shavell (1974) specifically consider the problem of utility maximizing search, but because search costs are assumed constant when measured in utility and possible wealth effects are neglected, their model is equivalent to the basic Stigler model; the problem is the same, the only difference is that the term “utility” has been substituted for the term “wages.” It should be noted that Danforth in two essays (1974a, 1974b) has introduced a utility-based objective into a search model in a very meaningful and innovative way. Danforth makes utility a function of consumption over time, and hence draws an essential link between the consumption decision and the job acceptance decision. In addition, the search environment he constructs is very general; for example, there is the possibility of partial recall in which any of the last k offers may be accepted, variable search costs, variable interest rates, uncertainty as to the actual distribution of wages, and unemployment benefits. Because of the generality of his model, it is difficult to isolate the effect of the introduction of expected utility maximization; nevertheless, his independent work contains a number of the results presented in this chapter. 8 In the analysis that follows, it will be convenient to have the utility function defined for all real numbers. Thus, if limz → b U ′(z) = − ∞, we define U(z) = −∞ for all z < b and U(b) = limz → b U(z). This says that infinite disutility results from having net worth less than b. Thus, an individual will not search if the search results in a net worth less than or equal to b with positive probability. While wealth (possibly negative) is being explicitly introduced into the model, there is no explicit use of a budget constraint. This corresponds to the Stigler model where there is no limit to the number of offers one may receive, even though there is a positive cost for each offer. Also, we shall use the symbol M to stand for the searcher’s current wealth. 9 Robbins (1970) has given a very clever proof to demonstrate that the optimal return is finite (and equal to ξ) if U is linear and E(Χi ) < ∞. By the concavity of U, there are integers n and m such that U(x) ≤ n + mx. Hence: (i) V(M,w) ≤ n + m(M + w + ξ) by Robbins’ result. Now by E(Χ1) < ∞ and (i) we can conclude that (ii) V(M − c,w ∨ Χi) is integrable. Coupling (8.10) and (i) yields (8.11), whereas (8.12) follows from (8.8), (8.11), and (ii). 10 Again we abbreviate the notation both for convenience and to highlight the parameter of interest. More formally, (8.17) reads

Notes

499

π(x,Z,U *) > π(x,Z,U ) for all x and Z whenever rU * ≥ rU. 11 Under recall, the reservation wage ξn when n periods (observations) remain and the individual is maximizing a linear utility function satisfies ξn ≡ ξ (see Lippman and McCall 1976: 170). 12 Theorems 3 and 5 have appeared in Lippman and McCall (1976). 13 As we saw in Chapter 7 there are variants of the standard model which yield decreasing reservation wages. For example, the reservation wage drops if there is systematic search (see Salop 1973). 14 In the finite horizon model with recall and linear utility function, the recall option is, on occasion, exercised; however, it is exercised only after receipt of all offers. 15 A similar result is contained in Rothschild (1974). 16 To do so, note that the continuity of Y yields that of Vi, and a simple induction argument establishes that Vi is continuous for each i. But now the finiteness of the search process implies that V(M,w) = VB (M,w) which, in view of the fact that the integer BM jumps by at most one on any (M−) interval of length c, allows us to assert the continuity of V. 17 Because W (M) ≠ (0,∞), there is a w0 such that E(M + YM,w ) > U(M + w0). On the other hand, we know from Theorem 2 that U(M + ξ) > EU(M + YM,ξ ). Thus, the continuous function g(w) ≡ EU(M + YM,w) − U(M + w) is positive at w0 and nonpositive at ξ, yielding existence of a wage satisfying g(w) = 0 as desired. 18 Recall that if ξg and ξf be the reservation wages corresponding to the random variables G and F having densities g and f, where g = f + s and s is a mean preserving spread then ξf ≤ ξg. An alternative proof could be based on second order stochastic dominance as follows. Define Cx by Cx(y) = 0 for y ≤ x and Cx(y) = (y − x) for y > x. Hence Cx is convex. Consequently, M

0

Hg(x) = ECx(G) ≥ ECx(F) = Hf (x), each x. 19 Note that a random variable with density g has the same mean as one with density ∞

冮

∞

冮

f and that its variance is larger since s(x)dx = 0 and x2s(x)dx > 0. 0

0

20 A simpler argument than the one based on (8.33) and (8.34) relies on the fact that if Χ is riskier than Y in the sense of second order stochastic dominance, then Eu(Χ ) ≤ Eu(Y ) for all concave increasing functions u. 21 It is somewhat surprising that (8.57) obtains in spite of the fact that V2(·,w) is not concave even if we assume U ″′ ≥ 0. The assumption that U has a convex derivative implies that (8.57) holds when i = 1 as well as that rU is decreasing. 9 Multi-armed bandits and their economic applications 1 In most cases, the state of the bandit will be denoted simply by xi or x. 2 This result was reported in McCall and McCall (1981). If all job opportunities are the same so Z* = Zi(0,0), then this result reduces to equation (8) of Lippman and McCall (1981) with p = 1. 3 Note that the feasible values of di may be state dependent. 4 The precision of a density is simply the inverse of its variance. 5 The i subscript will be dropped for the remainder of this section. 6 Note that by limiting the risk of layoff to the first good job match in these examples, the benefits of occupation-specific information are attenuated. 7 For example, when u = .01, α = 200, and δ = 1.5, ∆Z(0|c1 = 1000)/ ∆c1 = [Z(0|c1 = 1500) − Z(0|c1 = 1000)]/500 = − 1.94.

500

Notes

8 Neal actually looks at occupation/industry changes rather than simply occupational changes. 9 Also see Glazebrook et al. (2002). 10 A sample of early responses to Diamond’s paradox and Rothschild’s complaint 1 One might think that by labeling his critique a complaint we are emphasizing its trenchancy to the neglect of its constructive content. Most of the articles surveyed in this chapter, and many others, are responses to Rothschild’s constructive critique! 2 This chapter is based on an unpublished paper by Lippman, McCall and Berninghaus. 3 Recall that the reservation price associated with any of our optimal stopping rules is a fixed point. Furthermore, the contraction mapping used to prove this result is constructive, i.e., we can easily calculate the reservation price. Unfortunately, this is not in general true for F. A more delicate fixed point argument is required, and it only yields existence. See Berninghaus (1984). 4 This heterogeneity can be caused by differences in the value of time, differences in taste for shopping, and differences in the dollar cost of shopping. 5 See Butters, Diamond, and Stiglitz. 6 This incongruity would probably vanish if i were a random variable. 7 Note that δF(p) is a decreasing in p when firm sector is atomless so that a change in p for a single firm does not cause F to change. 8 The definition of δF(p) corresponds to MacMinn’s with R replaced by Re. 9 See Burdett and Judd (1983: 965). 10 See Berninghaus (1984) for details. A generalized Kakatani fixed point theorem is used. This is a nonconstructive theorem so no guide is given as to how the fixed point is calculated. 11 Equilibrium search after the Diamond–Mortensen–Pissarides breakthrough 1 2 3 4

5

6 7 8 9 10

This is the model of Butters (1977). This quote is from Petrongolo and Pissarides (2001: 425). Also see the recent book by Mortensen (2003). An interesting statistical model has been devised by Nagaraja and Barlevy (2003), which was stimulated by Burdett and Mortensen (1998). See Nagaraja and Barlevy (2003) for a description of the geometric random record model and the application to Burdett and Mortensen (1998); also see Barlevy (2005). Note, however, that such an equilibrium outcome is knife-edged with respect to the employer costs of raiding (which are assumed to be zero). If the costs were positive then rational firms would never raid and the two-mass point equilibrium would collapse to one mass point at the opportunity cost of working. This knifeedged result, however, does not persist in the general model with productivity differences across firms. In another study, Alvarez and Veracierto (1998) found that insurance features in a model with incomplete markets had “very small” effects. Alvarez and Veracierto believe that these lotteries are unrealistic. But they do render tractability to the problem of γ > 0. Panel data from the 1987–88 Current Population Surveys. Ljungqvist and Sargent (1998, 2003, 2005a, 2005c). Ljungqvist and Sargent (2004b). Sargent (1987) is a harbinger of Sargent’s future research in this area.

Notes

501

11 See quote by Robert J. Myers (1964: 172–3). 12 This subsection is based on Bloch and Ryder (2000), and others. 13 Smith (1997) calls this partitioning of the type space [0,1] perfect segregation. Perfect segregation was also discovered by Burdett and Coles (1997), Chade (2001), and Eeckhout (1999). 14 Bloch and Ryder (2000) also obtain the partitioning result since their assumed utility function is a limiting case of multiplicative separability. 15 The seminal articles are those of Diamond, Mortensen, and Pissarides. The key model is Mortensen and Pissarides (1994), which has as its monetary analog in the Rocheteau–Wright bargaining model. 16 The price taking monetary model by Rocheteau and Wright is analogous to the Lucas and Prescott (1974) search model and its expansion by Alvarez and Veracierto (1999). 17 The Rocheteau–Wright price posting model with directed search is the monetary theoretic analog of Moen (1997) and Shimer (1996). 18 Even when the size of implicit impulses is not criticized, Mortensen and Pissarides (1994) obtain a correlation of −0.26 between vacancies and unemployment. This differs considerably from the empirical value of −0.88. 12 Structural estimation methods 1 Eckstein and Wolpin (1990) is a structural estimation of an equilibrium search model, Albrecht and Axell (1984), which we discuss in Section 12.3. 2 For other examples see Kiefer and Neumann (1981) and Lancaster and Chesher (1983). 3 Other early applications include Rust (1987) and Wolpin (1984). 4 Some of this research is also reported in Gotz and McCall (1980, 1983). 5 Interestingly, an updated version of this model is still used by the Air Force to help craft compensation policies. 6 For an application of this method see Rust and Phelan (1997). 7 The authoritative study of complexity analysis in economics is Velupillai (2000). Also see Hernandez-Lerma and Lasserre (2002). 8 See Keane and Wolpin (1994) and Rust (1997). 9 See Glazebrook and Nino-Mora (1999) for a complete discussion of this adaptive greedy algorithm. 10 See Weiss (1988). 13 The ubiquity of search 1 Skyrms (2004: xii) in his Preface states: “But in resolving the complex into the simple we will follow Hobbes—for Hobbes was really the grandfather of game theory—in focusing on simple interactions modeled as games” (emphasis added). 2 It is commonly believed that Pascal was the father of probability. But see Ore (1953). 3 Cited in G. Mazzotta (1999: 170–1). 4 The significance of Hume’s economics in modern theory is clear in Lucas (1996), his Nobel lecture. 5 The house-selling problem was first solved by MacQueen and Miller (1960) and Karlin (1962). 6 See Alchian (1969). 7 An excellent treatment of inventory theory is unveiled in Porteus (2002). 8 Ludvig von Mises has stated: “It is impossible to assign any function to indirect exchange, media of exchange, and money within an imaginary construction, the characteristic mark of which is in unchangeability and rigidity of conditions.

502

9 10

11 12

13

14 15 16

17

18 19 20

21 22

23

Notes

Where there is no uncertainty concerning the future, there is no need for any cash holding” (von Mises 1949: 414). His reference to the time dimension is particularly relevant in regard to a premature sale (see Section II, Part C). Hirshleifer was the first author to explicitly note the importance of uncertainty and search in determining an asset’s liquidity. He forcefully observed: “It is immediately evident that uncertainty is of the essence here”; and “limitations of information may prevent buyers and sellers from finding one another, at least without incurring the costs and uncertainties of a search process” (Hirshleifer 1968b: 1–2). See Boulding (1955: 310–11) for a lucid discussion of liquidity and the role of money. A cogent analysis of the role of money in economic theory is contained in the work of Robert Clower (1977). Armen Alchian (1977) suggests that the transactions costs in trading i for j will be minimized via trading i for money and money for j with the first trade being effected by a specialist in commodity i and the second trade by a specialist in commodity j. An expanded discussion of this point that includes the importance of search and information is presented in Karl Brunner and Allan Meltzer (1971). Milton Friedman’s demand for money function (1957) contains a variable u that represent uncertainty, among other things. However, uncertainty is a minor actor in his theory of money. Instead of being the tail, Friedman’s u variable is now the dog (see p. 9). In a heuristic formulation, Jack Carr and Michael Darby (1981) have based a short-run money demand function upon the effects of money supply shocks on money holdings given reservation prices (presumably the result of optimal search behavior) set by asset sellers and buyers. Most portfolio analyses (for example, Tobin 1958) assume that the appropriate measure of risk is that associated with immediate sale of the assets held in the portfolio. But immediate sale may not be optimal. A structural characteristic leading to this situation arises when the asset is a business and the current owner’s managerial talents in running this business substantially exceed his talents (and implicit wages) in any other employment. One might say that such an asset is perfectly marketable. Not only is the owner capable of effecting a quick sale, there is nothing to be gained (on average) from waiting for a better price; i.e., a quick sale can be effected at the market price. This raises the question of whether the concept we have provided measures liquidity or marketability. In our view, this question is largely semantic. The discussion in this section is in the spirit of Albert Hart (1942), John Hicks (1974), and Robert Jones and Joseph Ostroy (1984). The articles by Bernanke (1983), McDonald and Siegel (1986), and Roberts and Weitzman (1981) are also pertinent. Jones and Ostroy (1984: 26–7). The (gross) return on an asset is the value of its payoff (dividend or service stream plus the following period asset value) divided by current asset value. B represents a set of states for which stopping is at least as good as continuing for one period and then stopping. The policy that stops the first time the process enters a state in B is called a one-step-look-ahead policy and the policy is said to be myopic. At the same time Ned Phelps also discovered the natural rate. See Phelps (1967). In passing, Rogerson claims that Ljungqvist and Sargent’s (1995) model is related to that of Mortensen and Pissarides. We will see that the relation is weak. Ljungqvist and Sargent use a generalization of BSM without bargaining and the special matching technology. A more detailed analysis of matching model is presented in Chapter 11.

Notes

503

24 Leijonhufvud (2000) studies the costs of inflation in a nonsearch setting. Essentially, this chapter has imbedded Leijonhufvud’s insights into a search milieu. 25 The Ball and Romer analysis is fortified by the work of Carlton (1982) and Tommasi (1994). 26 Arnold and Lippman note that their study is close in spirit to that of DeVany (1987), who considers a seller with one unit of a commodity choosing among three mechanisms. Customers arrive according to a Poisson process. The first mechanism initiates an auction after a certain length of time T has elapsed. In the second mechanism, an auction is held after the arrival of a fixed number of customers. Both these mechanisms affect transaction costs and the expected winning bid. Posted prices is the third mechanism considered. Here each customer decides whether to buy at once at the posted price. DeVany (1987) observes that the lower transaction costs associated with posted prices explains their widespread use. 27 Lemma 13.2 extends Theorem 4 of Lippman and McCall (1986), and provides a simple algorithm for computing reservation prices. 28 Private values such that no individual’s valuation is affected by knowledge of others valuations. Buyers are risk neutral and F is strictly increasing and continuously differentiable. Maskin and Riley (1989) demonstrate that these assumptions insure revenue equivalence for multiple unit auctions. Recall that Vickrey’s (1961) revenue equivalence result shows (assuming risk neutral customers) that bidders’ strategic behavior yields an identical expected revenue as that captured by other auction formats like sealed-bid, second price and Dutch auctions. Vickrey (1962) showed that revenue equivalence applied to three auction formats (with multiple units). Maskin and Riley (1989) generalized this theorem to encompass all possible auction formats. 29 The reader is encouraged to study the splendid article “Search Mechanisms,” by McAfee and McMillan (1988). Among other results, they show that with costly communication, the monopsonist’s optimal procurement mechanism is a combination of reservation-price search and auction. 30 Recall that a random variable Χ (or its distribution Px) is associated if Cov(f(x),g(Χ)) ≥ 0 for all increasing f and g. A probability measure P on Rn is said to be affiliated if P(A∩B|L) ≥ P(A|L)P(B|L) for all increasing subsets A and B and for all sublattices L ⊂ Rn. Affiliation ⇒ Association. 31 An excellent discussion of dependence is presented in Muller and Stoyan (2002: s. 3.10). 32 A buyer is a customer who will pay at least p for the unit. 33 The classic treatment of the pin factory is Leijonhufvud (1986). 34 Smith (1937: 9). 35 Hu and Lee (2003), and Bernanke (1983) are notable exceptions. 36 For those more spiritually (aesthetically) inclined, Henry (1974) considers the following startling irreversible investment. He supposes that we must choose whether Notre Dame Cathedral should be preserved or demolished and replaced by a parking lot. Preserving Notre Dame is not an irreversible decision, but constructing a parking lot after demolishing Notre Dame is clearly an irreversible decision. 37 The book edited by Schwartz and Trigeorgis (2001) contains many important articles on real options. 38 The reader is encouraged to read Bernanke’s proof. 39 The significance of optimal stopping in economics, finance, and engineering is emphasized in DuPuis and Wang (2002). 40 An excellent discussion of the importance of sequential decisions in stochastic investment analysis is Bar-Ilan and Strange (1998). 41 This model is very close to Lippman and McCall (1976c). 42 Optimal stopping problems, which can be solved using Markov decision processes,

504

Notes

continue search until the process becomes a martingale at which point the no arbitrage condition is satisfied. 43 Kennedy (1986) gives a martingale analysis of Optimal Sequential Assignment. Glazebrook (1990) presents a Gittins analysis of allocation in a stochastic environment. 14 Topics for further inquiry 1 Actually, this is a special case of Polya’s urn. In general, the chosen ball is replaced in the urn with s ≥ 0 balls of the same color. 2 This has been demonstrated in the seminal work of Aoki, Arthur and DeVany and Walls. 3 The urn models which produce exchangeable sequences are a special family of all possible urn processes. Yet, it is precisely this special class which has widespread applicability in economics. Two of the most prominent members of this class are the Polya–Eggenberger urn (hereafter referred to as the Polya urn) and the ballot urn associated with the ballot theory. 4 Recent developments in simulation, the Markov Chain Monte Carlo procedures, have yielded approximations required for the implementation of Bayesian methods, and account for the sudden increase in Bayesian applications. Many of these approximations are founded on Polya urns. It should be noted that several economists were leaders in developing MCMC methods. They include Chib, Geweke, Kiefer, Lancaster, and, of course, Zellner. 5 Also known as occupancy models and allocation models. 6 The link between urns and combinatorics is explored in the classic works of Feller (1970) and Takacs (1967). 7 This is Freedman’s Theorem 2.2, which is a special case of deFinetti’s theorem. Freedman shows that (Wn + Bn)−1 Wn is a martingale. 8 This is useful in the thumbtack experiment, Section 4. 9 Jaynes (1988: 28) observes: “There is something fundamental in these principles (Bayes Theorem and Maxent), although we think that more unified ways of presenting them are still needed and will be found . . .” 10 The problem of inference as viewed by a scientist (or philosopher of science) involves testing a hypothesis. In frequentist statistical inference, the hypothesis is replaced by a fixed parameter θ and calculating p(xn + 1 | x1, . . ., xn) is the basic statistical problem. Solving this problem involves the introduction of independence. Lindley (1990) observes that even in the autoregressive process, xn + 1 = axn + εn, the dependent xs are expressed in terms of IID random errors, the εs. DeFinetti solves this inference problem by assuming the xs are exchangeable and introduces Θ such that given Θ the xis are IID. Thus, deFinetti himself relies on IID random variables. But Θ is not a fixed, unknown parameter, but a random variable with distribution µ. The probabilistic prediction of xn + 1 given x1, . . ., xn requires the calculation of p(x1, . . ., xn) for all n. This calculation follows from deFinetti’s theorem for Bernoulli random variables. 11 Lindley and Phillips (1976: 113) note that Karl Pearson (1920), a founder of modern statistics, described the following version of the thumbtack experiment as “one of the more important problems in applied statistics”: “If on n1 occasions an event has occurred r1 times, what is the chance that on n2 further occasions it will occur r2 times?” 12 A thumbtack is useful relative to a coin because most of us do not have strong beliefs about how the tack will land when it is tossed several times. If the coin were flipped, most would feel that head and tail would be equally likely. 13 That is, all of the outcomes that could have occurred in an experiment of n = 12 tosses is required before testing or estimation can commence.

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14 See Example 1.18 in Schervish where sampling from an urn without replacement generates exchangeable outcomes which are not conditionally IID. 15 See Theorem 1.47 in Schervish (1995) (deFinetti’s theorem for Bernoulli random variables). 16 See Velupillai (2000) for an excellent discussion of Kolmogorov-complexity and its economic applications. 17 For a proof, see Schervish (1995: 9). 18 For example see Lad (1996). 19 Note we are observing the quality of each rejected candidate. This information gives rise to a Bayes solution! 20 The clearest derivation of this result is Ross (1983b). 21 See Dawkins (1996) for a lucid discussion of this error. 22 This is clear from the flourishing activity of the “new” discipline: evolutionary economics. 23 This exchange rate is meaningful in that the value of tomorrow’s offspring is discounted by the population growth rate, thus measuring the population’s future.

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Index

Aalen, O. 124, 126–9 Abel, A. 452, 506 Abowd, J. 313, 506 Abreu, D. 506 acceptance sets 231, 242–4 Acemoglu, D. 184, 185, 186, 189, 506 achievable region approach 383–5 adaptive information xix, 390 Adelman, I. 506 admissible decisions 60 adverse selection 6, 7, 18, 162, 197, 281 Aghion, P. 492, 509, 525 Akerlof, G. xx, 4, 7, 277, 390, 425, 434, 506 Albrecht, J. 369, 491, 501, 506 Albright, S. 462–3, 467–9, 506 Alchian, A. 10, 14, 49, 51–2, 491, 501–2, 506 Aldous, D. 473 Alvarez, F. xviii, 16, 304, 325–31, 337, 500–1, 506–7 Andersen, P. 116, 121, 507 Ansell, P. 197, 386, 500, 517 Aoki 504 argmax 310 Arjas, E. 116, 159, 507 Arnold, M. xix, 17, 390, 420–1, 423, 425, 433–6, 503, 507 Arrow, K. 61–2, 219–20, 223, 226, 230, 426, 446–50, 455, 461–2, 494, 498, 507, 521, 534 Arrow-Pratt measure of relative risk aversion 220 Arthur, W. 55, 504, 507 assignment theorem xxi, 392 asymmetric information xx, 5, 6, 7, 8, 17, 18, 281, 390, 425–7, 433–4 Athreya, K. 507 Atkinson, A. 496, 508

auction xix, xx, 17, 67, 390, 420–9, 431–3, 443, 503 auction regime xix, 390, 421 Autor, D. xx, 333, 391, 521 Aven, T. 113, 508 Avrachenkov, K. 63, 508 Axell, B. 282, 294–6, 369, 491, 501, 506, 508 Azariadis, C. 508 Baily, M. 496, 508 Ball, L. xix, 390, 418–9, 491, 503, 508 Baltes, P. 525 Banks, J. 273, 508 Bar-Ilan, A. 452, 503, 508 Barlevy, G. 500, 508, 530 Barron, J. 497, 508 barter economy xviii, 389 baseline hazard function 136–7, 141 baseline survivor function 141 Basic Sequential Search Model 11, 14, 65, 72, 97 Bather 493 Bawa, V. 508 Baye, M. 444 Bayes, T. 6, 18, 61–2, 65–6, 70, 106, 109–10, 308, 359–60, 393, 448, 456, 468–9, 475–9, 481, 483, 488, 494, 497, 504–5 Bayes adaptive learning 393 Bayesian methods xxi, 450, 456, 479, 488, 504 Bayes rule 70, 106, 468 Beckenbach, E. 508–9 Becker, G. xvi, xviii, 10, 217, 304, 349–50, 462, 496, 508 belated information 68, 94–6, 163–4, 275 Bellman, R. xv, 14, 55, 61–3, 66, 82, 84,

Index 90, 97–8, 162, 215, 310, 313, 320, 328, 340, 367, 381, 415, 439, 440–1, 479, 508, 513 benchmark model 429, 431, 433 Benhabib, J. 509 Bernanke, B. 455–6, 502–3, 509 Berninghaus, S. 301, 500, 509 Bernoulli trials 472, 480, 482 Berry, D. 56 Bertrand, J. 317–8 Bertrand solution 317 Beta distribution 266 Bewley, T. 355, 509, 528 bidding xx, 390, 411, 426–30 Bingham, R. 490 biology 67, 281, 450, 484, 487–8 Birge, J. 509 Black, F. 453, 454, 461 Blackwell, D. xv, 61–3, 492, 507, 509, 522, 525 Blanchard, O. 509 Bloch, F. xviii, 304, 345–9, 501, 509 Bloemen, H. 206, 509 Blundell, R. 509 Borch 498 Borel (-algebra) 111 Borel-Cantelli Lemmas 495 Borgan, O. 116, 507 Boulding, K. 397, 502 Bowker, A. 493 branching bandits 385 Braverman, A. 509 Breiman, L. xv, 68, 100, 493–4, 509 Brekke, K. 452 Bremaud, P. 48, 113, 121 Brennan, M. 452–3, 459, 510 Brenner, M. 457, 510 Breslow, N. 140–1, 510 Breslow approximation 140–1 Brock, W. xviii, 213, 304, 459, 497, 510, 526 Brumelle 497 Brunner 502 Bruss, F. 17 Buchanan, J. 445 Bull, C. 509 Burda, S. 510 Burdett, K. xvii, 282, 298–9, 302, 304, 313, 317, 324, 369, 373, 443, 500–1, 510 Butters, G. xvii, 277, 286–7, 298, 302, 304, 308, 500, 510 Cagan, P. 510 Cahuc, P. 369, 377–8, 380, 510

541

Cairoli, R. 454 Cannan, E. 4, 50, 510 Card, D. 176, 329–30, 511, 521, 529 Carlson, J. 297–8, 301, 511 Carlton 503 Carr 502 Cassady, R. 426, 511 categorical variable 142 Cayley, A. 14, 49, 55, 484–5, 511 censoring time 122, 131 Chade, H. 305, 350, 352–3, 501, 511 Champernowne, D. 211, 511 Chernoff, H. 110 Chesher, A. 501, 523 Chetty, R. 217, 244–5, 511 Chib, S. 13–14, 504, 511 Chow, Y. 62, 102, 467, 486, 493, 511 Christensen, B. 357, 362–4, 511 Clark, W. 511 Clarke, H. 497, 532 Classen, K. 496, 511 Clower, R. 397, 491, 502, 511 coalition model 330 Coe, D. 512 Coffman, I. 384, 512 Cole, H. 512 Coles 501 Collins, E. xviii, 305, 345–8, 528 commodity money 394–5 comparative statics 86, 292, 294, 298, 456 competitive equilibrium (posting) 354 complete information 6, 163, 286, 377 complexity xviii, 9, 91, 140, 214, 267, 356, 381–3, 438, 482, 501 complexity problems 381, 383 computer science 67 Conlisk, J. xv, 15, 208, 210–13, 470, 497, 512, 536 consistent estimator 125, 128, 145–6 constructive proof xxi, 391 Continuous Search Model 98 continuous time duration models with covariates 116 convex polyhedron 197, 384–5 coordination xix, 9, 306, 390, 396, 419–20 counting process 45, 46, 47, 111, 116, 121, 122, 125, 126, 138 counting processes 46, 121, 126, 138 Cox, D. 116, 136–7, 140–3, 150, 507, 512, 514 Cox, J. 512 Cox Regression Model 136

542

The Economics of Search

creative destruction xix, 390 cumulative hazard function 129 Cyert, R. 65, 512 Dacre, M. 197, 383–6, 512 Dalang, R. 454 Daley, D. 15, 208–9, 470, 512 Danforth, J. 170, 206, 229, 496, 498, 512 Darby 502 Dardia, M. 535 Darwin, C. 487–8 Davies, N. 489, 522 Davis, S. 311–2, 333, 513 Dawkins, R. 505, 513 dealer/broker xx, 391 decision-making under uncertainty 3, 59, 69 decreasing absolute risk aversion 170, 189, 206, 231, 237, 243 deFinetti, B. 65–6, 220, 448–50, 471–82, 491, 494, 504, 505, 513 DeGhellnick 63 DeGroot, M. 65, 101, 263, 266, 463, 493, 495–6, 512–3 de la Desha, G. 528 Dellacherie, C. 110 deMere 501 De Meza, D. 494, 513 Demsetz, H. 10, 513 Denardo 492 Den Haan, W. 513 Derman, C. xxi, 68, 100, 209–10, 392, 462–3, 468, 497, 506, 513 DeVany, A. 10, 503–4, 513 Devine, T. xviii, 12–13, 116, 356, 513 Dewey, J. 1, 514 Diamond, P. xv, xvii, xix, 9, 12, 15, 16, 49, 225, 277, 282, 302–4, 308, 313, 317, 389–90, 419, 437, 443, 463, 492, 494, 496, 498, 500–1, 514 Diamond paradox xvii, 304 Dickens, W. 333, 514 Dirac measure 111 discounted present value 73, 75, 200–1, 208 discrete-time Kaplan-Meier estimator 146 discrete-time life history methods 155 division of labor 392, 395–6, 446–7, 449, 488–90 Diwekar, V. 514 Dixit, A. 212, 452–4, 494, 506, 514 Dodge, H. 55, 493, 514 dominant equilibria 429

Doob, J. 122, 125 Doob-Meyer Decomposition Theorem 122, 125 Dornbusch, R. 509 double auction 427 Dubins, L. 62, 514 Duckworth, K. 453, 514 Dupuis, P. 497, 503, 514 duration analysis xv, 15 duration data 116, 121, 129, 137, 143, 152, 382 Durlauf, S. 510 Dutch auction 427, 431, 503 dynamic programming xviii, 7, 55, 56, 61–2, 64, 67, 82, 89, 229, 273, 356, 361, 366, 369, 382, 383, 479, 485, 487, 492, 495 Dynkin, E. 486, 514 Eberly, J. 452, 506 Eckstein, Z. xviii, 12, 356, 365, 369–70, 372–3, 376, 491, 501, 514 ecology of search xxi e-commerce xx, 391 economic growth 202, 451 economic search 3, 5, 9, 10, 14, 18, 56, 437 economics of search xv, xxi, 10, 11, 51, 391, 443, 471 Edelman, G. 490 Edling, J. 514 Eeckhout, J. 305, 349, 501, 514 Efron, B. 141, 514 Eggenberger, F. 476, 504, 514 Ehlich 217 eigenvalue 26 electronic search xx, 391 Ellison, G. xx, 391, 442, 515 Ellison, S. xx, 391, 442, 515 employed-no benefits (ENB) 155 Engelbrecht-Wiggans, R. 515, 534–5 Engels, F. 392 English auction 427, 429, 431 equilibrium xvii, xix, 6, 7, 8, 9, 10, 14–18, 22, 26–7, 50, 66, 97–8, 161, 184–9, 193–6, 198, 216–7, 244, 277–9, 281–99, 300–5, 309–13, 315–18, 321–6, 328–9, 331–2, 334–7, 340, 344–56, 369, 371–3, 375–7, 389, 394, 407–12, 415–17, 426, 429–30, 434, 438–9, 441–2, 448, 452, 459, 474, 488, 490–2, 500–1 equilibrium search model 15–16, 66, 304, 331, 334, 354, 356, 490–1, 501 ergodic 25, 27, 478

Index Erickson, G. 520 Esary, J. 209, 515 evolutionary speed 383 exchange 2, 3, 9, 10–11, 67, 219, 388, 392–4, 396, 399, 427, 438–40, 444–7, 473, 477, 488, 489, 491, 501, 505 exchangeability xxi, 17, 448, 471, 472–6, 480–2, 490 expectation 30, 33, 43, 107, 213, 219, 263, 335, 358, 385, 399, 408, 412, 429, 458, 460, 485 expected utility maximizing model xvi, 217 Fallick, B. 496, 515 Farber, H. 515 Feinberg, E. 89, 90, 492, 508, 515, 520 Feldman, A. 393, 515 Feldstein, M. 496, 515 Feller, W. 26, 28–9, 31, 48, 287, 475, 504 Ferguson, T. 12, 17–18, 55, 467, 484–5, 492, 515 Filar, J. 508 firing taxes 17, 325, 329–30 Fisher, A. 455, 507 Fisher, R. 474, 488, 515, 517 Flam 453 Fleming, T. 116, 121, 515 Flinn, C. 362, 515 flow sampling 130, 143 Frank, S. 219, 405, 488, 515 Fredriksson, P. 194–5, 515 Freedman, D. 477, 478, 504, 516 French, K. 110, 113, 380, 516 frictional unemployment 57–8, 202–3, 332 frictionless commerce xx, 391, 442–3 Friedman, B. 477–8, 516 Friedman, M. xix, 60, 354, 390, 414–5, 417–9, 461, 493–4, 502, 510, 516 Friedman, T. 516 Fristedt, B. 56 Frydman, R. 492, 509, 525 Fundamental Convergence Theorem 27 Gaarder, S. 113, 508 Gale, D. xviii, 304, 345, 349–50, 516 Galiani, F. 388 Gardner, M. 483–4 Gaver, K. 516 Geman, D. 14, 516 Geman, S. 14, 516

543

generalized Nash bargain 310, 416 geometric random variable 57, 75–6, 86, 166, 253, 401, 405 Gerchak, Y. 497, 516 Geweke, J. 13–14, 504, 516 Ghosh, B. 60, 516 Gibbons, R. 8, 516 Gilbert, J. 467, 516 Gill, R. 116 Girshick, M. 61–2, 493, 507 Gittins, J. xv, xvi, xviii, 15, 18, 56, 59, 64, 246, 252, 255, 263, 273–5, 302, 361, 383, 385, 449–50, 494, 504, 508, 516, 527, 538 Glazebrook, K. 197, 268, 273–5, 383–4, 386, 500–1, 504, 512, 516–7 Glimcher, P. 450 Golan, A. 479, 517 Good 493 Gottschalk, P. 333, 517 Gotz, G. 17, 357–9, 501, 517 Grafen, A. 489, 517 Graham, D. 517 Grandy, W. 539 graph-theoretic xxi, 391 Gray 445 Green, J. 5, 517 Greenwood, J. 517 Griswold, C. 447, 517 group sequential designs 110 Groves, T. 517 Gruber, J. 517 Haefke, C. 513 Hall, G. xx, 391, 440–1, 533 Hall, J. 496, 498, 517 Hall, R. xviii, xxi, 305, 355, 517 Haltiwanger, J. 513 Ham, J. 179, 517 Hamermesh, D. 517 Hamilton, W. 18, 488, 517 Hammond, N. 517 Han, A. 496, 518 Hansen, G. 193, 337, 518 Hansen, R. 518 Hardy, G. 464–6, 518 Harrington, D. 116, 121, 515 Harris, M. 518 Harrison 493 Harsanyi, J. 6, 7 Hart, O. 219, 502 Hausman, J. 496, 518 Haviv, M. 508 Hayek, F. 2, 5, 54, 425, 489, 492, 518

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hazard function 80, 116–20, 130, 132, 136–7, 141, 143–5, 167, 361 He, S. 110, 518 Heckman, J. 12, 15, 55, 152, 356, 362, 496, 515, 518 Henry, C. 455, 503, 518 Hernandez-Lerma 501 Hey, J. 519 Heymann, D. 396, 519 H-function 78 Hicks, J. 14, 50, 52, 219, 397, 407, 502, 519 Hinderer, K. 519 Hirano, K. 13, 519 Hirshleifer, J. xvi, 18, 217, 389, 397–8, 401, 492, 498, 502, 519 Hobbes, T. 387, 444, 501 Holmlund, B. 194–5, 515 Holt, C. 519 Honoré, B. 152, 518 Hopenhayn, H. 192–3, 329, 519 Hosios, A. xvii, 304, 312, 324, 417, 419, 519 Hotelling, H. 493 Hougaard, P. 148, 519 house selling problem 462, 467–9 housing market xix, 389 Howard, R. 63, 456, 492, 520 Howitt, P. 355, 520 Hu, I. 503, 520 human capital xvii, xviii, 4, 303–4, 331, 333–7, 340, 415, 446, 462 human capital marriage model xviii, 304 Hume, D. 388, 389, 478, 501, 520, 535 Hurwicz, L. 6 Hutt, W. 14, 49, 50 Huygens 475 Imrohoroglu, A. 193, 518 incentive-compatible mechanism 432 incentives xvii, 2, 4, 5, 8, 16, 192, 196, 303, 318, 337 independent increments 45–7 industrial organization 6, 7, 281, 443 inflation xix, 353–4, 390, 395–6, 414–5, 418–9, 503 information set 6, 43, 366, 475 Ingersoll, J. 452, 454, 520 institutional dynamics 9 insurance policies 162 integrated hazard function 123–4, 128 irreversible actions 455 island model xviii, 304, 328, 337–8, 340, 345

Jacobson, L. 334, 520 Jaynes, E. 504, 520 Jevons, W. 392, 394 job matching xvi, 194, 196, 246, 262, 272 job search xv, xx, 3, 14, 18, 20, 23, 50–1, 57, 68–9, 73, 92, 101–2, 118, 133, 156, 162, 170–1, 176–7, 180, 184–5, 191, 194–6, 203, 206–8, 214–6, 227, 235, 309, 313, 317, 328, 345, 357, 366, 369, 377, 380, 391, 399, 418, 433, 444, 458, 486, 494 Johnson, N. 475, 477, 516, 520, 538 Jones, R. xviii, xix, xxi, 59, 64, 218, 389–94, 397, 407, 419–20, 491–2, 502, 520 Jorgenson, D. 461, 495, 497, 520 Jovanovic, B. 9, 50, 496, 520 Judd, K. 298–9, 443, 500, 510 Kakatani 500 Kallenberg, L. 63, 90, 495, 520 Kamae, T. 100, 520 Kambourov, G. 520 Kandel, E. 452, 520 Kaplan, E. 128, 146–7 Karlin, S. xv, 463, 467, 491, 495, 501, 521, 534 Karni, E. 171, 191–3, 195, 229, 393, 498, 521 Kasper, H. 206, 521 Katsuaki, T. 532 Katz, L. 184, 333, 521 Keane, M. 501, 521 Keiding, N. 116 Keilson, J. 521 Kemeny, J. 212–3, 521 Kennedy 504 Kepler, J. 484 Kester, A. 521 Keynes, J. 397, 399, 401, 521 Kiefer, N. xviii, 12–13, 15, 55, 116, 356–7, 362–4, 501, 504, 511, 513, 521, 522 Kiyotaki, N. xviii, xix, 67, 305, 389, 395–6, 438, 491, 522 Klein, B. 116, 522 Klein, J. 522 Kletzer, L. 522 Knight, F. 49, 51–2, 219, 286, 405 Knudsen, T. 453, 522 Kobila, T. 452, 522 Kocherlakota, N. 512 Kohn, M. 12, 237, 498, 522 Kolmogorov, A. 60, 482, 494, 505 Kormendi, R. 496, 522

Index Kotz, S. 475, 477, 516, 520, 538 Krainer, J. 389, 407–13, 522 Kramarz, F. 313, 506 Kramer xix Krebs, J. 489, 522 Krengel, T. 100, 520 Kreps, D. 7, 479, 522 Krueger, A. 313, 329, 511 Krugman, P. 523 Kuhn, P. 444, 523 Kyburg, H. 471, 523 Kydland, F. 523 labor economics 55, 67, 492 labor market internet xx, 391 labor market policies 17, 325–6, 329 LaCerra, P. 490 Lad, F. 450, 476, 505, 523 Laffont, J. 5, 517, 523 Lagos, R. 353, 523 Lai, T. xv, 105, 110, 492–3, 496, 523 LaLonde, R. 520 Lancaster, T. 12–13, 55, 116, 132–3, 496, 501, 504, 523 Landa, J. 445–6, 523 Landes 496 Landsberger 224–6, 498 Lasserre 501 law of large numbers 30, 123, 219, 473, 480 Lawrence, R. 523 Layard, R. 509, 523, 529 layoffs 27, 58, 68, 72, 161, 163, 168, 172, 271, 334, 448, 458, 496 Lazear, E. 523 learning process xix, 390, 394, 456, 469, 479, 488 Ledyard, J. 517 Lee, C. 503, 520 Leffler, K. 522 Lehoczky, J. 497, 523 Leijonhufvud 503 Leijonhufvud, A. 396, 492, 519 Leonardz, B. 524 LeRoy, S. xix, 389, 407–13, 522, 524 less risky 77, 221 Levin, B. 105, 523 Levine, D. 12, 277, 492, 494, 524 Levine, P. 176, 511 Lieberman, G. xxi, 392, 463, 513 likelihood ratio 60, 108 Lindley, D. 110, 479–81, 486–7, 504, 512, 524 Lippman, S. xvi, xix, 12, 17, 57, 66–8,

545

162, 213–4, 217, 227, 229, 243, 277, 390, 397–9, 400, 407, 419–21, 423, 425, 433–7, 452–4, 467, 470, 492–6, 498–9, 500, 503, 507, 511–2, 517, 522, 524–5, 538 liquidity xix, 193, 219, 389, 395–9, 400, 401, 402–7, 411, 502 Livingston, D. 388, 525 Ljungqvist, L. xxi, 12, 13, 16, 50, 78, 304, 312, 331–4, 337, 395, 416, 459, 492, 494, 500, 502, 509, 525 local risk aversion 223, 245 log-likelihood function 148, 150, 155, 160, 364 log-logistic distribution 119 log-logistic hazard function 120 Loveaux, F. 509 Lucas, R. xviii, 12, 16, 50, 53, 58, 97, 161–2, 304, 325, 395, 397, 418–9, 450–1, 492–3, 501, 525–6, 529, 536 Lucking-Reiley, D. 443, 526 Lumley, R. 453, 526 Lund, D. 452, 522 Luton, R. 526 MAB problem xvi, 64, 246, 256, 265, 267, 272–3, 454–5 Machin, S. 526 Macho-Stadler, I. 7, 8, 526 MacMinn, R. 289–94, 297–8, 301, 500, 526 MacQueen, H. xv, 55–6, 113, 413, 467, 491, 501, 526 macroeconomics 12, 67, 308, 331, 417, 492 Majd, M. 452, 530 Makower, H. 218, 398, 526 Malliaris, A. 526 Malueg 302 Mamer, J. 437, 493, 524 Mandeville, B. 489 Manne, A. 63, 526 Manning, A. 526 Manning, J. 492, 496, 526, 528 Manning, R. 68, 106, 110, 492, 526 Manovskii, I. 520 Margolis, D. 313, 506 market makers xx, 391, 440, 442 Markov, A. 198–9, 202–4, 208–15, 325, 327, 337, 347, 383, 454, 457, 459, 462, 469–71, 473, 475, 486, 497, 503–4 Markov chains 15, 18–19, 20, 23–4, 26, 63, 210, 212–3, 469–70, 473, 508

546

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Markov decision processes 55, 62–3, 89, 90, 214–5, 383, 494–95, 503 marriage 13, 53, 67, 280, 305, 345–6, 348, 483 Marschak, J. 5, 218, 398, 403, 455 Marshall, R. 277, 517 Martin, J. 483, 526 Martingale 33, 48, 101, 139, 507, 524 Martingale Central Limit Theory 139 martingale difference sequence 159 Marx, K. 392 Maskin, E. 9, 302, 496, 503, 514, 523, 527 mass point distribution 496 matching xv, xvi, xvii, xix, xxi, 7, 9, 11, 17, 50, 53, 185, 194, 217, 244, 246, 266, 271–2, 302–4, 306–9, 312, 318, 331, 337–8, 345–50, 352–5, 357, 382, 389–91, 395, 407, 416–7, 419–20, 437, 466, 490, 502 matching equilibrium xvi, xvii, 303–4, 312, 350, 352 matching models xix, 17, 50, 302, 304, 308, 389, 395, 437 matching technology xvii, 303, 306, 308, 347, 353, 502 matchmakers 304, 345 match surplus 310, 339, 377 Matras, J. 527 Matthews, S. 527 maximum likelihood estimator 131, 363 Mazzotta, G. 501, 527 McAfee, R. xx, 17, 297–8, 301, 390, 425, 429–33, 494, 503, 511, 526–7 McCall, B. xvi, 152, 159, 179, 182, 246, 254, 257, 266, 268–9, 271, 275–6, 332, 499, 527 McCall, J. xvi, 12, 16–17, 57, 162, 213–4, 217, 227, 229, 243, 254, 257, 275–6, 357–9, 397–9, 400, 407, 419, 452, 454, 467, 470, 492–3, 495–9, 500–1, 503, 511–2, 517, 520, 522, 524–5, 527–9, 538 McCormick, R. 516 McDonald, R. 449, 452–3, 459–60, 502, 528 McFadden 498 McMillan, J. xx, 12, 16, 17, 390, 425, 429–33, 492, 494, 503, 519, 527–8 McNamara, J. xviii, 305, 345–8, 528 Meier, P. 110, 128, 146–7 Meilijson 224–6, 498 Meister, B. 522 Meltzer 502 Menger, C. 392, 397, 528

Mertens, A. 510 Merton 493 Meyer, B. 113, 179, 184, 521, 528 Meyer, P. 122, 125 Meyn, S. 528 Michael 496 Micklewright, J. 496, 508 middlemen xx, 5, 11, 53, 391, 437–42, 446, 492 migration 67, 254–5, 275–6 Milgrom, P. 428, 528 Millard, S. 528 Miller, B. 528 Miller, R.A. xvi, 17, 246, 262–4, 357, 361, 497, 528 Miller, R.G. 55–6, 113, 413, 467, 491, 501, 526 minimal expected total cost 88 minimum wages 17, 325, 329 Mirman, L. 302, 528 mismatch unemployment xv, 163–4 Mitrani, I. 384, 512 Moen, E. xviii, 304, 324–5, 501, 528 Moeschberger, M. 116, 522 Moffitt, R. 333, 517 money xviii, 2, 67, 90, 196, 222, 353, 389, 392–9, 401–2, 404, 407, 438, 488, 491, 501–2 monotone Markov chain 100–1, 208–9, 470, 495 monotonicity xv, 100, 198–9, 200, 210–13, 234, 241, 243, 351, 495, 497 moral hazard problem xv, 50, 163, 171, 196 Morgan, P. 68, 106, 110, 444, 492, 496, 508, 526, 528 Morgenstern, O. 3, 61, 457 Mortensen, D. xv, xvii, xix, 9, 12–13, 15–16, 49, 68, 171, 176, 192–3, 302–6, 309, 311, 313, 317, 324, 331, 337, 354, 369, 373, 389–90, 395, 416–7, 419, 437, 458–9, 492–3, 496, 500–2, 510, 528–9 Mosteller, F. 467, 516 Muller, A. xxi, 214–6, 391, 497, 503, 529 multi-armed bandit (MAB) xvi, 246 Muth, J. xxi, 391, 529 Myers, R. 501, 530 Myers, S. 452, 530 Myerson, R. 530 myopic (one-step-look-ahead OSLA) policy 69 myopic property 83 Nachman, D. 229, 231, 530

Index Nagaraja, H. 500, 530 Nakai, T. 467, 530 Nash, J. xxi, 6, 8, 11–12, 15, 18, 50, 194, 283, 285, 310, 317, 331, 339, 349, 354, 370, 372, 377, 380, 409, 411, 416–7, 429–31 Nash equilibrium 6, 8, 18, 283, 370, 372, 409, 411, 429–30 natural filtration 111–2 natural rate of unemployment xix, 390, 414–6 Neal, D. 271–2, 500, 530 negative dynamic programming 87 Nelson, P. 530 Nelson, R. 51, 218, 497, 530 Nelson, W. 124, 126–9 Nelson-Aalen estimator 124, 126–8 Neumann, G. 12, 15, 55, 356, 501, 521–2 neuroscience 67, 490 Newman, G xix, xxi, 390–1, 419–20, 492, 520 Ney, P. 507 Neyman, J. 60, 474 Neyman-Pearson theory 60 Nickell, S. 523 Nicolini, J. 192–3, 519 Niehans, J. 393, 530 Nino-Mora, J. 197, 383–4, 386, 500–1, 517 non-anticipative 384–5 non-cooperative game 6–8 non-transferable utility 347 no recall case 84 Norris, J. 48 Novick, M. 524 null recurrent state 26 null state 26 Nyarko, Y. 520 O’Brien 100 Oakeshott, M. 387, 530 objective function 188, 457 occupational choice xvi, 246, 262, 266–7 Oh, S. 397, 491, 530 Øksendal, B. 452, 522 Olsen, T. 452 on-the-job search 67, 92, 308, 313, 317–8, 369 operations research xv, 56, 345, 356, 383, 484 optimality equation 36–7, 39, 40, 247–8, 250 optimal replacement policy 68

547

optimal stopping rules xv, 15, 18, 61–2, 279, 399, 451, 454, 462, 494, 500 optimal stopping theory 58, 61–2, 351 optimal strategy 82, 84–5, 104, 212, 228–9, 232, 234, 236, 240, 351, 361, 458, 495 optimal unemployment insurance 195 Ore 501 Ore, O. 530 organizational theory 4 Osborne, M. 494, 513 oscillating heterogeneity 4 Ostroy, J. 10, 218, 393, 397, 407, 491, 502, 520, 530 Palfrey, T. 530–1 Pareto optimality 11, 433 Parsons, D. 497, 531 Parthasarathy, K. 450 partial exchangeability 473 partial likelihood function 137, 143 partial order 77, 220 partition 4, 212, 213, 303, 323–24, 348, 350, 352, 466 Patinkin, D. 13 Pavan, R. 272, 531 Pavoni, N. 531 Pearce, D. 506 Pearson, K. 60, 474, 504 Pearson, N. 452, 520 Perez-Castrillo, D. 7, 8, 526 perfect Bayesian equilibrium 8 perfect foresight allocations 10 perfect information 6, 163, 219, 283, 319, 427 perfect segregation theorem 348 performance space 197, 384–6 permutation 463, 465, 468, 471, 481 persistent state 25–6 Pescatrice, D. 511 Petrongolo, B. 306–8, 500, 531 Phelan 501 Phelps, E. xix, 12, 191, 390, 415, 492–3, 496, 502, 506, 509, 525, 531 Phillips, A. 217, 414–5, 492–3, 528, 531 Phillips, L. 479–81, 504, 524 Pindyck 494 Pindyck, R. 212, 452–4, 506, 514 Pissarides, C. xv, xvii, xix, 9, 12–3, 16, 49, 194, 303–9, 311, 324, 331, 337, 353–4, 389–90, 416–7, 419, 437, 458–9, 492, 500–2, 529, 531 Plott, C. 531

548

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Poisson process 45–7, 98, 113, 177, 403, 469, 503 Poisson Random Measure 111, 518 Pollard, D. 48 Pollicott, H. 213, 531 Polya, G. 17, 23, 308, 471–2, 474–8, 482–3, 492, 504, 514, 531 Polya urns 475, 504 polyhedron 383 Porter, W. 302, 528 Porteus 501 positive externalities 195 posted prices xx, 390, 433, 438, 503 Postel-Vinay, F. xvii, 304, 317–9, 324, 369, 377–8, 380, 510, 531 Pratt, J. 220, 223, 226, 229, 230–1, 282, 296, 443, 507, 532 predictable (-field 112 predictable covariation process 125 predictable process 112, 114, 122, 125–6 predictable quadratic variation of M(t) 125 Prescott, E. xviii, 16, 50, 304, 325, 337, 501, 523, 526, 532 Preston, I. 509 principal-agent problems 492 probability space 41, 43, 62, 111, 224 property rights 51, 97, 445 proportional hazards model 129, 137 Proschan, F. 209, 515 psychology 67, 244, 389 Putterman, M. 495, 532 quality control example 108 Quartz, S. 490 Quirk, J. 532 quits 58, 68, 72, 161, 163–4, 359, 374, 448, 458 Radner, R. 5, 281, 495, 497, 520, 532 Raiffa, H. 461, 532 Ramsey, J. 532 random walk 25, 27–9, 30–1, 60, 89, 209, 212, 474, 497 Rao, C. 537 Rasmussen, E. 6, 532 Raviv, A. 518 Rea, S. 179, 517 real options xxi, 18, 392, 446, 448–51, 453–4, 462, 494, 503 recall xvi, 56, 67–8, 81, 84–6, 90–1, 104, 163–4, 168–70, 177, 183–4, 208, 214, 216–7, 228–9, 233, 236, 241–3, 257,

260–1, 313, 374, 400–1, 413–4, 434, 467, 498–9 recall case 85–6, 90 recurrent state 26 Reed, W. 497, 532 reflecting barriers 25 regular conditional distribution 112 Reinganum, J. 288–9, 300–1, 532 replacement rate 193–4, 339, 345 reservation price property 82–3, 85 reservation wage 14–15, 50, 57–8, 64, 66–8, 73, 76, 78, 80, 90–2, 96–7, 100–1, 106, 118, 164–6, 169–71, 173–5, 177, 179, 181–2, 191–2, 196, 198–9, 200–4, 206–8, 228–9, 232–4, 236, 241–3, 253–5, 257, 259–62, 276, 299, 314–5, 317–9, 336, 356–7, 362–3, 368, 370, 374–6, 380, 433, 459, 493, 495, 499 revelation principle 433 Ridder, G. 369, 373, 376–7, 537 right censoring 131 right-continuous martingale 122, 125 right-continuous predictable process 122, 125 Righter, R. 462, 467, 469–70, 532 Riley, J. 492, 498, 503, 519, 527, 532 risk aversion xvi, 66, 184, 193, 217–8, 220, 222–3, 225–7, 229–30, 235–6, 244–5, 534 riskier 78–9, 103, 237, 436, 499 Robbins, H. 14, 58–9, 62, 102, 105, 486, 493, 498, 511, 523, 532, 535 Roberts, K. 456, 502, 532 Robin, J. 304, 317–9, 324, 369, 377, 378, 380, 531 Rocheteau, G. xviii, 305, 353–4, 501, 532 Rockafellar, R. 453, 532 Rogerson, R. xix, 305, 329, 337, 390, 416–7, 502, 519, 532–3 Romer, D. 491, 503, 508 Romer, P. xix, 390, 418–9, 533 Romig, H. 55, 493, 514 Ross, A. 530 Ross, S. xxi, 34, 64, 87–9, 90, 392, 413, 462–3, 492–3, 505, 533 Ross, S.A. 452, 454, 520 Rossi, P. 13, 539 Roth, A. 350, 533 Rothschild, M. xvi, xxi, 3, 7, 9, 12, 16, 49, 56, 213, 217, 223–4, 226, 237, 277, 279, 308, 313, 391, 394, 459, 463, 492–4, 497–9, 500, 507, 510, 514, 528, 533 Rubenstein, A. xx, 377, 391, 437–8, 492, 533

Index ruin 207 Ruiz-Hernandez, D. 273–5, 516 Rumelt, R. 67, 452–3, 525 Ruschendorf, L. xxi, 216, 391, 529 Rust, J. xviii, xx, 17, 356, 381–2, 391, 440–1, 501, 533 Ryder, H. xviii, 305, 345–9, 501, 509 Ryu, K. 496, 515 Saario, V. 534 Sakaguchi, M. 462–3, 467, 534 Salanie, B. 7, 534 Salop, S. 207, 282–4, 497, 499, 534 sampling with replacement 140 Samuelson, L. 534 Samuelson, W. 532, 534 Sandmo, A. 227, 534 Sargent, T. xxi, 12–13, 16, 50, 78, 304, 312, 331–4, 337, 395, 416, 452, 459, 492, 494, 500, 502, 509, 525, 529, 534 Savage, L. 62, 491, 493, 514 Scarf, H. 214, 521, 534 Scarsini, M. xvi, 217, 223–6, 498, 534 Schöl 454 Schelling, T. 213, 427, 531, 534 Schervish, M. 481, 505, 534 Schick, L. 539 Schlaifer, R. 461, 532 Schmitz, N. 68, 106–10, 534 Schoenfeld, D. 496, 535 Schoeni, R. 535 Scholes, M. 453–4, 461 Schultze, C. 523 Schumpeter, J. xix, 51–2, 388, 390, 535 Schwartz, A. 286, 498, 538 Schwartz, E. 452–3, 459, 503, 510 score function 138, 363 Scott-Morton, F. 443, 535 sealed-bid auction 427, 429, 431–2 search/matching approach xix, 390 search econometrics 14 search equilibrium (bargaining) 353 secretary problem xxi, 17, 55–6, 62, 446, 448, 452, 467, 469, 483–5, 492–3 Selten, R. 6, 7 Senjowski, T. 490 sequential analysis xv, 49, 55–6, 58, 110, 413, 493 sequential model 9, 55, 57, 68, 80, 86, 105, 294 sequential probability ratio test (SPRT) 60, 66, 459, 461 Serfozo, R. 459, 495, 535 Shanbhag, D. 537

549

Shapley, L. xv, xviii, 14, 63, 66, 162, 304, 345, 349–50, 516, 535 Shavell, S. 12, 191–2, 237, 498, 522, 535 Shephard, R. 490 Shevchenko, A. xx, 391, 438, 492, 535 Shi 438 Shimer, R. xviii, xxi, 184–6, 189, 305, 347, 354–5, 501, 506, 533, 535 Shorrocks, A. 535 Shubik, M. 515, 534–5 Shwartz, A. 89, 90, 508, 515, 520 Siegel, D. 449, 452–3, 459–60, 502, 528 Siegmund, D. xv, 58–9, 61–2, 105, 486, 492–3, 523, 535 signaling 7, 8, 18, 312, 489 Silva-Risso, J. 535 Simon, H. 49, 52, 395, 535 Singer, B. 12, 15, 55, 496, 518 Skinner, A. 388, 535 Skuterud, M. 444, 523 Skyrms 501 Smallwood, D. 536 Smelser, N. 525 Smith, A. xix, 388–90, 392, 418, 446–7, 487–9, 503, 517–8 Smith, C. 520 Smith, L. 345, 347, 501, 536 Smith, V. 512, 536 Smokler, H. 471, 523 Snell, J. 62, 212–3, 521 Snower, D. 512, 524, 528 Sobel 493 sociology 67, 389, 445 Sommers, P. 536 Sotomayor, O. 350, 533 specialist 440, 502 Spence, M. 4, 7 spot trading 97 Spulber, D. 439, 440, 536 Stahl, D. 443, 536 Stancenelli, E. 206, 509 Starr, R. 393, 397, 530 stationary increments 46 steady state equilibrium xx, 329, 391, 417 Stensland, G. 453 Stidham 216 Stigler, G. xvi, 4, 8, 9, 11, 49, 50, 51–7, 66, 68, 70–2, 105–6, 217–8, 277–9, 286, 292, 492–3, 496, 498, 536 Stiglitz, J. xvi, 3, 4, 7, 213, 217, 223–6, 237, 283–4, 459, 492, 497, 500, 509, 514, 525, 533–4 stochastically larger 40, 77, 86, 100, 203, 220–1

550

The Economics of Search

stochastically monotone Markov chains 100, 209 stochastic dynamic programming xvi, 19, 34, 89, 246, 358, 451, 454, 493 Stokey, N. 12, 419, 536 Stoyan, D. 214–6, 497, 503, 529 Strange, W. 452, 503, 508 Strassen 391 Stratified Cox Regression 142 structural estimation xviii, 18, 262, 356–7, 365, 369, 380–1, 491, 501 submartingale 28, 121–2, 466 Sueyoshi, G. 496, 536 Summers 313 Sundaram, R. 273, 508 supermartingale 28, 62, 101–2, 121 superprocesses xvi, 246, 265–7 survival analysis 117 symmetric 8, 25, 77, 89, 409, 411, 429 Takacs 504 Takacs, L. 536 Telser, L. 467, 537 temporary layoff 168, 170, 496 Tennenbaum, J. 490 Terkel, S. 537 time-varying covariates 117, 130, 150 Tobin 502 Tommasi, M. 491–2, 503, 537 Topel, R. 496, 537 Townsend, R. 518 transient state 26 Trejos 438 Trigeorgis 503 trust 355, 387, 444–5 Turing, A. 493 Tweedie, R. 492, 528, 537 two-sided search xvii, 303–4, 345 unbalanced panel 143 unemployed-benefits (UB) 155 unemployed-no benefits (UNB) 155 unemployment xv, xvii, 4, 10, 15–17, 20–3, 50, 54–5, 58, 67, 69, 116–7, 143, 149, 155–6, 161–4, 166–71, 174–7, 179–80, 182, 184–6, 189, 191–7, 203, 245, 260–2, 303–5, 307–9, 311–5, 317–8, 321–2, 324–8, 330–9, 343, 345, 354–5, 362, 366, 372–4, 376–8, 382, 414–7, 444, 492–3, 496, 498, 501 union boss 330 unions 17, 41, 162, 325, 330 unobserved heterogeneity 132–3, 148 urban economics 67

urn model xvii, 17, 304, 306, 308, 474–5, 477, 504 utility functions xvi, 217, 232, 237 Van Damme, E. 6, 537 Van Den Berg, G. xviii, 176–7, 356, 369, 373, 376–7, 514, 537 Varaiya, P. 269, 537 variable intensity of search 68 Varian, H. 282, 284–5, 443, 537 variance-covariance matrix 139 Veinott, A. 66, 69, 70, 492, 537 Vellupillai, K. 501, 505, 537 Venezia, I. 457, 510 Veracierto, M. xviii, 16, 304, 325–31, 337, 355, 500–1, 506–7, 537 vertical heterogeneity 349 Vesterlund, V. 217, 244, 537 Vickrey, W. 427, 503, 523, 537 Vico, G. 388, 444 Volcker, P. 404 Voltaire, K. 497, 528 von Mises 501 von Neumann, J. 3, 457 vonNeumann-Morgenstern game theory 61 Von Plato, J. 473, 538 Von Zur Meulea, P. 538 wage gap 244 Wald, A. xv, 14, 50, 52, 55, 59, 60–2, 65–6, 109–10, 162, 413, 459, 461–2, 491, 493, 496, 516, 524, 538 Walkup, D. 209, 515 Wallis, W. 60, 493, 538 Walls, D. 504 Walrand, J. 537 Walrasian auctioneer 10, 426 Walrasian equilibrium xx, 391, 442 Wang, H. 497, 503, 514 Weber, R. 216, 428, 528 Wei, L. 496, 538 Weibull, J. 6, 537 Weibull, W. 119, 131–3 Weibull distribution 119, 131 Weibull-Gamma models 133 Weil, S. 162, 538 Weisbrod, B. 455, 538 Weiss, G. 501, 538 Weiss, H. 213, 531 Weiss, L. xv, 60–1, 191–2, 535, 538 Weitzman, M. 453, 456, 502, 532 Werning, I. 538 Wets, R. 453, 532

Index Whittle, P. xvi, 56, 59, 246, 266, 272–3, 275, 302, 453, 516, 538 Wilde, L. 286, 298, 496, 538 Williams, D. 538 Williams, J. 538 Wilson, R. 7, 522 Winter, S. 51, 497, 530 Wise, D. 296, 517, 532 Wolfers, J. 509 Wolfowitz, J. 61, 110, 493 Wolinsky, A. xx, 391, 437–8, 492, 533 Wolpin, K. xviii, 12, 17, 356, 362, 365, 369–70, 372–3, 376, 380–2, 491, 501, 514, 521, 538 Woodford, M. 492, 509, 525 Wooldridge, J. 143, 496, 539

551

Wright, R. xviii, xix, 67, 305, 353–4, 389, 395–6, 438, 491, 501, 522–3, 529, 532, 539 Wu 498 Yavas, A, 539 Yee 106 Yorukoglu, M. 517 Young, J. 13 youth unemployment xv, 163 Zeckhauser, R. 296, 532 Zellner, A. 13–14, 478–9, 504, 539 Zervos, M. 453, 514, 526 Zimmerman, J. 516 Zuckerman 459