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FUNCTIONAL STRUCTURE INFERENCE

International Symposia in Economic Theory and Econometrics Series Editor: William A. Barnett Volume 14:

Economic Complexity Edited by W.A. Barnett, C. Deissenberg & G. Feichtinger

Volume 15:

Modelling Our Future: Population Ageing, Social Security and Taxation Edited by Ann Harding & Anil Gupta

Volume 16:

Modelling Our Future: Population Ageing, Health and Aged Care Edited by Anil Gupta & Ann Harding

Volume 17:

Topics in Analytical Political Economy Edited by Melvin Hinich & William A. Barnett

International Symposia in Economic Theory and Econometrics Volume 18

FUNCTIONAL STRUCTURE INFERENCE

EDITED BY

William A. Barnett University of Kansas, Lawrence, USA

Apostolos Serletis University of Calgary, Alberta, Canada

Amsterdam – London – New York – Oxford – Paris – Shannon – Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier website at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53061-5 ISSN: 1571-0386 For information on all Elsevier publications visit our website at books.elsevier.com

Printed and bound in The United Kingdom 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

Introduction to the Series The series International Symposia in Economic Theory and Econometrics publishes quality proceedings of conferences and symposia. Since all articles published in these volumes are refereed relative to the standards of the best journals, not all papers presented at the symposia are published in these proceedings volumes. Occasionally these volumes include articles that were not presented at a symposium or conference, but are of high quality and are relevant to the focus of the volume. The topics chosen for these volumes are those of particular research importance at the time of the selection of the topic. Each volume has different coeditors, chosen to have particular expertise relevant to the focus of that particular volume. William A. Barnett Series Editor

v

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Contents Introduction to the Series . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by William A. Barnett and Apostolos Serletis

ix

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Part I:

Nonparametric Approaches to Separability Testing

1

1 Testing the Significance of the Departures from Weak Separability Philippe de Peretti

3

2 Quantity Constraints and Violations of Revealed Preference Adrian R. Fleissig and Gerald A. Whitney

23

3 Nonparametric Tests of the Necessary and Sufficient Conditions for Separability Barry E. Jones, Nadine McCloud and David L. Edgerton

33

Part II:

Flexible Functional Forms and Theoretical Regularity

57

4 Bayesian Estimation of Flexible Functional Forms, Curvature Conditions and the Demand for Assets Apostolos Serletis and Asghar Shahmoradi

59

5 Productivity and Convergence Trends in the OECD: Evidence from a Normalized Quadratic Variable Profit Function Guohua Feng and Apostolos Serletis

85

6 The Theoretical Regularity Properties of the Normalized Quadratic Consumer Demand Model William A. Barnett and Ikuyasu Usui

107

Part III: Areas

Functional Structure and the Theory of Optimal Currency 129

7 On Canada’s Exchange Rate Regime Apostolos Serletis and Karl Pinno vii

131

viii

Contents

8 Tests of Microeconomic Foundations of an Australia/New Zealand Common Currency Area James L. Swofford, Lena Birkelöf, Joshua R. Caldwell and John E. Filer Part IV:

Functional Structure and Asymmetric Demand Responses

9 Food Demand in Sweden: A Nonparametric Approach Per Hjertstrand

143 155 157

10 A Systems Approach to Modelling Asymmetric Demand Responses to Energy Price Changes David L. Ryan and André Plourde

183

Part V: Seasonality, Liquidity Constraints and Hopf Bifurcations

225

11 Rational Seasonality Travis D. Nesmith

227

12 Hopf Bifurcation within New Keynesian Functional Structure William A. Barnett and Evgeniya Aleksandrovna Duzhak

257

13 Insurance and Asset Prices in Constrained Markets Pamela Labadie

277

Part VI:

293

Fiscal Policy and Real Business Cycles in Open Economies

14 Labor Adjustment Costs, Shocks and the Real Business Cycle Zuzana Janko

295

15 Optimal Fiscal Policy in an Asymmetric Small Open Economy Constantine Angyridis

307

Part VII:

333

Monetary Policy and Capital Accumulation

16 Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect Eric Kam

335

17 On the Implications of Different Cash-in-Advance Constraints with Endogenous Labor Arman Mansoorian and Mohammed Mohsin

349

Introduction Most of the chapters in this volume were delivered as papers in two sessions that William Barnett organized at a conference held in Vigo (Spain) in July 2005, sponsored by the Society for the Advancement of Economic Theory. The volume brings together fundamental new research in economics, including relevant significant innovations in microeconometrics, macroeconomics, and monetary and financial economics, thereby making substantive contributions to the literature. The volume consists of seven parts. Part I deals with nonparametric approaches to separability testing and consists of three papers. The paper by Philippe de Peretti (Chapter 1) extends his previous work in this area and introduces a new nonparametric test for weak separability within an explicit stochastic framework. The paper by Adrian Fleissig and Gerald Whitney (Chapter 2) evaluates UK meat consumption (over the period from 1900 to 1919), by applying a revealed preference test that allows for rationing. It shows that revealed preference tests that fail to allow for rationing reject optimal consumer choices with this data set. The third paper by Barry Jones, Nadine McCloud and David Edgerton (Chapter 3) surveys the current state of the art in the use of nonparametric methods to separability testing. Part II consists of three papers regarding the use of flexible functional forms in economics and finance. The first paper (Chapter 4), by Apostolos Serletis and Asghar Shahmoradi, uses Bayesian inference to revisit the demand for money in the United States in the context of five popular locally flexible functional forms – the generalized Leontief, the basic translog, the almost ideal demand system, the minflex Laurent and the normalized quadratic reciprocal indirect utility function. The second paper by Serletis and Guohua Feng (Chapter 5) investigates aggregate productivity convergence among 15 European Union countries plus Canada and the United States, during the 1960–2002 period, using a normalized quadratic variable profit function (within a multiple-output and multiple-input framework). The third paper by William Barnett and Ikuyasu Usui (Chapter 6) conducts a Monte Carlo study of the global regularity properties of the normalized quadratic model. In particular, it investigates monotonicity violations, as well as the performance of methods of locally and globally imposing curvature. It finds that monotonicity violations are especially likely to occur when elasticities of substitution are greater than unity. It also finds that imposing curvature locally produces difficulty in the estimation, smaller regular regions, and poor elasticity estimates in many cases considered in the paper. Part III of the book is about functional structure and the theory of optimal currency areas. The first paper by Serletis and Karl Pinno (Chapter 7) uses a dynamic ix

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W.A. Barnett and A. Serletis

equilibrium model of the Canadian economy to investigate the degree of currency substitution between the Canadian dollar and the US dollar which potentially has implications for the theory of optimum currency areas and can be used to evaluate the desirability of a monetary union between Canada and the United States. The second paper by James Swofford and his co-authors (Chapter 8) presents estimates of transactions costs savings and results from revealed preference tests of microeconomic foundations of an Australian/New Zealand common currency area. Their results can be viewed as favorable toward the formation of an Australian/New Zealand common currency area. Part IV of the book has two papers regarding functional structure and demand responses. The first paper by Per Hjertstrand (Chapter 9) is a study of separability structures and the demand for food in the Swedish market, over the period from 1963 to 2002, using newly developed computationally intensive nonparametric tests for weak separability. The second paper by David Ryan and André Plourde (Chapter 10) takes a systems approach to the demand for energy in the residential sector of Ontario (Canada). In particular, it investigates asymmetric responses to price changes by extending the single-equation framework to a framework with multiple inter-related demands. The consequences on some standard properties of demand systems (homogeneity and symmetry) are also investigated. Their results suggest that demands for energy sources (electricity, natural gas and oil products) are characterized by asymmetric responses to price changes, even after allowing for inter-fuel substitution. Part V has three papers about functional structure and seasonality, liquidity constraints and Hopf bifurcations. The first paper by Travis Nesmith (Chapter 11) incorporates seasonal behavior into aggregation theory. Using duality theory, it extends earlier work by Erwin Diewert to a larger class of decision problems. It also relaxes Diewert’s assumption of homotheticity and provides support for Diewert’s preferred seasonally-adjusted economic index using weak separability assumptions that are shown to be sufficient. The second paper by Barnett and Evgeniya Duzhak (Chapter 12) pursues a bifurcation analysis of new Keynesian functional structure that has become increasingly popular in recent years. In doing so, they build on a series of recent papers by Barnett and He (1999, 2002, 2006) and provide background theory relevant to locating bifurcation boundaries in loglinearized New-Keynesian models with Taylor policy rules or inflation-targeting policy rules. The third paper by Pamela Labadie (Chapter 13) examines the effects of liquidity and borrowing constraints on intermediation and consumption insurance in the context of a stochastic version of Kehoe and Levine (2001). It shows that the impact of the constraints on asset prices and consumption allocations generally depends on whether the aggregate shock is i.i.d. or Markov. Part VI of the book is about fiscal policy and real business cycles. The first paper (Chapter 14), by Zuzana Janko, analyzes a real business cycle model with labor adjustment costs and finds labor adjustment costs to be an important propagation mechanism for both technology and government spending shocks. The

Introduction

xi

second paper (Chapter 15), by Constantine Angyridis, is a study of optimal fiscal policy in a dynamic stochastic small open economy with a financial asymmetry between the government and the representative household. It solves numerically for the optimal fiscal policy and discusses its predictions in relation to the existing literature. Finally, the last part of the volume is about monetary policy and capital accumulation. The paper by Eric Kam (Chapter 16) describes the effects of monetary growth on real sector variables using optimizing models with an endogenous rate of time preference. Results depend on whether money is introduced in a utility function or a cash-in-advance constraint. The last paper by Arman Mansoorian and Mohammed Moshin (Chapter 17) compares the effects of inflations on cash in advance models with endogenous labor. The steady state effects of an increase in the inflation rate on welfare, consumption and capital are significantly different, depending on whether there are cash in advance constraints on consumption alone, or on both consumption and investment. The steady state effects on employment are not significantly different for the two models. William A. Barnett Apostolos Serletis

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List of Contributors Angyridis, C., Department of Economics, Ryerson University, 350 Victoria Street, Toronto, ON, Canada M5B 2K3 (Ch. 15) Barnett, W.A., Department of Economics, University of Kansas, Lawrence, KS 66045, USA (Chs 6, 12) Birkelöf, L., Department of Economics, Umeå University, Umeå, Sweden (Ch. 8) Caldwell, J.R., Department of Economics and Finance, University of South Alabama, Mobile, AL 36688, USA (Ch. 8) de Peretti, P., Université Paris 1 Panthéon-Sorbonne, France (Ch. 1) Duzhak, E.A., Department of Economics, University of Kansas, Lawrence, KS 66045, USA (Ch. 12) Edgerton, D., Department of Economics, Lund University, P.O. Box 7082, S-22007 Lund, Sweden (Ch. 3) Feng, G., Department of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4 (Ch. 5) Filer, J.E., Department of Economics, Arizona State University, Tempe, AZ 85281, USA (Ch. 8) Fleissig, A.R., Department of Economics, California State University, Fullerton, Fullerton, CA 92834, USA (Ch. 2) Hjertstrand, P., Department of Economics, Lund University, P.O. Box 7082, S-220 07 Lund, Sweden (Ch. 9) Janko, Z., Department of Economics, University of Calgary, 2500 University Drive, N.W., Calgary, AB, Canada T2N 1N4 (Ch. 14) Jones, B.E., Department of Economics, Binghamton University, P.O. Box 6000, Binghamton, NY 13902-6000, USA (Ch. 3) Kam, E., Department of Economics, Ryerson University, Toronto, ON, Canada M5B 2K3 (Ch. 16) Labadie, P., Department of Economics, George Washington University, 300 Academic Building, Mount Vernon Campus, Washington, DC 20052, USA (Ch. 13) Mansoorian, A., Department of Economics, York University, North York, ON, Canada M3J 1P3 (Ch. 17) McCloud, N., Department of Economics, Binghamton University, P.O. Box 6000, Binghamton, NY 13902-6000, USA (Ch. 3) Mohsin, M., Department of Economics, University of Tennessee, Knoxville, TN 37996, USA (Ch. 17) xiii

xiv

List of Contributors

Nesmith, T.D., Board of Governors of the Federal Reserve System, 20th & C Sts., NW, Mail Stop 188, Washington, DC 20551, USA (Ch. 11) Pinno, K., Department of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4 (Ch. 7) Plourde, A., Department of Economics, University of Alberta, Edmonton, AB, Canada T6G 2H4 (Ch. 10) Ryan, D.L., Department of Economics, University of Alberta, Edmonton, AB, Canada T6G 2H4 (Ch. 10) Serletis, A., Department of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4 (Chs 4, 5, 7) Shahmoradi, A., Faculty of Economics, University of Tehran, Iran (Ch. 4) Swofford, J.L., Department of Economics and Finance, University of South Alabama, Mobile, AL 36688, USA (Ch. 8) Usui, I., Department of Economics, University of Kansas, Lawrence, KS 66045, USA (Ch. 6) Whitney, G.A., Department of Economics and Finance, University of New Orleans, LA 70148, USA (Ch. 2)

Part I Nonparametric Approaches to Separability Testing

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Chapter 1

Testing the Significance of the Departures from Weak Separability Philippe de Peretti Université Paris 1 Panthéon-Sorbonne, France

Abstract This chapter introduces a new nonparametric test for weak separability within an explicit stochastic framework. The procedure is a development of de Peretti (2005). It is therefore based on both (i) An adjustment procedure that searches for the minimal perturbation in the data in order to satisfy weak separability, (ii) A test procedure that checks the significance of the discrepancy between the observed data and the adjusted ones. Two theoretical frameworks are used. The standard only sufficient Varian’s (1983) framework and a necessary and sufficient one. Running Monte Carlo simulations using the latter, the test turns out to be extremely powerful.

Keywords: nonparametric test, weak separability JEL: D12, C10, C63

1. Introduction The weak separability of preferences is a concept of central importance in microeconomics and in micro-econometrics. As emphasized by Barnett (1980), weak separability is the key existence condition for an aggregate to exist. Indeed, if preferences are rationalized by a weakly separable utility function, it is possible to build cost and distance functions, and thus price and quantity indexes for a subset of goods. As a corollary, all the substitution effects being internalized, it becomes possible to estimate demand systems only over those goods, dramatically simplifying the analysis. In addition, weak separability returns a crucial information about the structure of preferences, and then about some natural structuring of goods. This latter has been widely used in monetary economics in order to select assets entering an aggregate of money (see Serletis, 1987; Swofford and Whitney, 1987; Belongia and Chrystal, 1991; or Fisher and Fleissig, 1997). To test a finite set of quantities and prices for compliance with weak separability, most researchers have focused on nonparametric tests as defined by Varian International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18001-7

4

P. de Peretti

(1983), since parametric ones have low power (Barnett and Choi, 1989). The procedure amounts to sequentially using the General Axiom of Revealed Preference (GARP) to test three different subsets for compliance with utility maximization. Nevertheless, this three-step procedure being based on GARP is nonstochastic, and a single violation of the axiom leads to reject the null of weak separability, even if the violation is due to purely stochastic causes, as measurement error. Empirically, the procedure appears to be often inconclusive, especially if few violations are found. Moreover, apart from the nonstochastic nature of GARP, Fleissig and Whitney (2003) have questioned the efficiency of an algorithm used in step three of the Varian’s procedure. They showed that using such an algorithm led to an increased type I error. They, instead, developed an astute program, and showed that at least under nonstochastic considerations, it allowed to restore the power of the procedure. The goal of this chapter is to develop a new nonparametric test of weak separability in which the recent contribution of Fleissig and Whitney (2003) is also considered. The test is a development of de Peretti (2005) and allows to check the significance of the violations of GARP, when the violations occur when testing for weak separability. It therefore inherits a similar structure being based on both (i) An adjustment procedure that computes the minimal perturbation in the data in order to satisfy weak separability, (ii) A test procedure that checks the significance of the necessary adjustment. With regard to the adjustment procedure, two theoretical frameworks are used: The standard only sufficient Varian’s one, in which the recent contribution of Fleissig and Whitney (2003) is considered, and a second, based on a necessary and sufficient condition for homogeneous weak separability. Major advantages of such a test are as follows. Concerning the adjustment procedure, it uses the information content in three different closure matrices. The algorithm is therefore extremely fast, even for large dataset and/or if a large number of violations is found. It is also easy to implement, based on the minimization of quadratic functions subject to linear constraints. At last, it is general enough to deal with Afriat inequalities based tests, or necessary and sufficient conditions. With regard to the test procedure, the significance of the adjustment is tested by using discrepancy tests belonging to the power divergence family (Basu et al., 2002). It appears to be extremely powerful, as shown by Monte Carlo simulations. This chapter is structured as follows. Section 2 focuses on the General Axiom of Revealed Preference and associated nonstochastic weak separability tests within the Varian’s (1983) framework. Section 3 deals with the significance of the violations of GARP in weak separability tests, introducing the adjustment procedure and the test procedure. Section 4 details how the adjustment procedure is solved. Section 5 suggests a necessary and sufficient test for homogeneous weak separability. Section 6 presents an empirical application using the LP program of Fleissig and Whitney (2003). At last Section 7 investigates the power of the procedure test by running Monte Carlo simulations.

Testing for Weak Separability

5

2. Nonparametric Tests for Weak Separability Let x be a vector of observed real commodities, let x1 and x2 be two partitions of x such that x1 ∪ x2 = ∅, and that x1 ∩ x2 = x. There exists a weakly separable utility function if there exists a utility function U (x), and if this one can be re-written as (1) for a sub-set x1 . U (x) = V Ux1 x1 , x2 , (1) where: U (·) is the overall utility, V (·) is a strictly increasing function, known as the macro-function, Ux1 (·) is the sub-utility function, or the micro-function. Testing for such a preference structure therefore amounts to sequentially testing if U (x), Ux1 (x1 ) and V (Ux1 (x1 ), x2 ) exist. It then reduces to a three-step test of utility maximization introduced now. Let X be a (T ×k) matrix of observed real quantities, where T denotes the number of observations and k the number of goods, and let xi = (xi1 , xi2 , . . . , xik ) be the ith row of the matrix. Similarly, define P as a (T × k) matrix of corresponding prices, and let pi = (pi1 , pi2 , . . . , pik ) be the ith row of the matrix P. In order to check an observed set of quantities and prices {xi , pi }Ti=1 for consistency with utility maximization, first define the three following binary relations: xi is said to be strictly directly revealed preferred to xj if pi · xi > pi · xj , written xi P 0 xj ; xi is said to be directly revealed preferred to xj if pi · xi pi · xj , written xi R 0 xj ; at last, xi is said to be revealed preferred to xj if xi R 0 xm , xm R 0 xk , . . . , xp R 0 xj , written xi Rxj , where R can be easily computed by using the transitive closure of R 0 , R, computed for instance according to the Warshall algorithm. Using the above definitions and following Varian (1982) define the General Axiom of Revealed Preference (GARP) as, for a couple of observations (i, j ), i ∈ {1, . . . , T }, j ∈ {1, . . . , T }: xi Rxj ⇒ pj · xj pj · xi . Using GARP, Varian (1982) proved that: Theorem 1 (Varian). For a set {xi , pi }Ti=1 , the three following conditions are equivalent: • There exists a locally nonsatiated utility function U (·) that rationalizes the data. • There exist strictly positive utility indices Ui and marginal income indices λi that satisfy ∀i ∈ {1, . . . , T }, ∀j ∈ {1, . . . , T } the Afriat inequalities (2), Ui Uj + λj (pj · xi − pj · xj ).

(2)

• The data satisfy GARP. Using the above theorem, testing for weak separability is now straightforward. Define the (T × a) X1 matrix as a partition of X, with x1i = (xi1 , xi2 , . . . , xia ),

6

P. de Peretti

and let P1 be a (T × a) matrix of associated prices with p1i = (pi1 , pi2 , . . . , pia ), where a ∈ [1, k − 1]. Let X2 be a (T × (k − a)) matrix with x2i = (xi(a+1) , xi(a+2) , . . . , xik ), and let P2 be a (T × (k − a)) matrix of associated prices with p2i = (pi(a+1) , pi(a+2) , . . . , pik ). Then, testing the weak separability of the first a columns of X, amounts to running the following three-step procedure: Step 1: Test for the overall utility U (·) running GARP on {xi , pi }Ti=1 . Let nvio be the number of violations: nvio = 0, then go to step 2, if: otherwise, reject the null of maximization and weak separability. Step 2: Test for a sub-utility Ux1 (·) running GARP on {x1i , p1i }Ti=1 . Let nvio1 be the number of violations: nvio1 = 0, then go to step 3, if: otherwise, reject the null of maximization and weak separability. Step 3: Solve the Afriat inequalities (2) for {x1i , p1i }Ti=1 using, for example, the Fleissig and Whitney (2003) LP program. Use Ui as utility indices and 2 2 −1 T λ−1 i as price indices. Run GARP on {(xi , Ui ), (pi , λi )}i=1 . Let nvio2 be the number of violations: nvio2 = 0, then accept weak separability, if: otherwise, reject the null of maximization and weak separability. Hence, if and only if nvio = nvio1 = nvio2 = 0 preferences are said to be weakly separable. The above test is extremely stringent since a single violation of GARP in only one of the three steps leads to reject the null of maximization and thus weak separability. It totally ignores that some violations may be caused by purely stochastic elements as measurement errors or small optimization errors. Extending the above decision rule, consider that under the null, data behave as if they were generated by a weakly separable utility function, but are unobservable. In particular, assume that observed quantities incorporate an error term whose form (multiplicative) is given by (3). xij = xij∗ (1 + εij ),

(3)

where xij is the observed quantity, xij∗ is the unobserved one, consistent with weak separability, εij is an i.i.d. error term with zero mean and variance σ 2 . Equation (3) leads to take into account that some violations of GARP at one or several steps, might not be significant, conveying no significant information about the rejection of weak separability. Hence, the need to discriminate between significant and non significant violations, when violations occur when testing for weak separability. We next introduce such a procedure.

Testing for Weak Separability

7

3. A Stochastic Nonparametric Extension The procedure we want to consider here has the same logical structure than in de Peretti (2005). Given a dataset not consistent with weak separability, it consists in: (i) Finding the minimal perturbation in the quantities in order to satisfy weak separability. (ii) Testing the necessary adjustment for its significance. We first focus on the adjustment procedure. In empirical work, under the null, the perturbations εij are generally unknown. We therefore search for adjusted quantities that are consistent with weak separability, such that the distance with the observed ones is minimal. Given the form of the error term, this amount to solving the following quadratic program (4) over zij . obj = min zij

k T zij i=1 j =1

xij

2 −1

(4)

subject to: ∀i ∈ {1, . . . , T }, ∀j ∈ {1, . . . , T }: zi Rzj ⇒ pj · zj pj · zi ,

(C.1)

∀i ∈ {1, . . . , T }, ∀j ∈ {1, . . . , T }: z1i Rz1j ⇒ p1j · z1j p1j · z1i , where: z1i = (zi1 , zi2 , . . . , zia ) and

p1i = (pi1 , pi2 , . . . , pia ),

(C.2)

∀i ∈ {1, . . . , T }, ∀j ∈ {1, . . . , T }: z3i Rz3j ⇒ p3j · z3j p3j · z3i , where: z3i = (zi(a+1) , zi(a+2) , . . . , zik , Ui ), p3i = pi(a+1) , pi(a+2) , . . . , pik , λ−1 , i for some {Ui , λi }Ti=1 satisfying the Afriat inequalities or consistent with the economic theory for {z1i , p1i }Ti=1 .

(C.3)

In the above program, the (C.1) constraint is relative to the overall maximization and the constraints (C.2) and (C.3) are respectively relative to the sub-utility and to the weak separability. Once the minimal adjustment computed, arises the question of its significance, that is whether or not data are consistent with weak separability. A natural way to test for the null is to base the decision rule on the discrepancy between the observed and the adjusted quantities. If the series are near, then one will be tempted

8

P. de Peretti

not to reject the null, the departures from weak separability being caused by stochastic elements, supporting the assumption (3). Conversely, a large distance would tend to reject the null. To give a statistical sense to the word discrepancy, we use four different disparity measures. Let oi = (oi1 , oi2 , . . . , oiT ) be a vector of observed data, and let ti = (ti1 , ti2 , . . . , tiT ) be associated theoretical or expected data. According to the distribution of ti , create c classes, and let f t = (f1t , f2t , . . . , fct ) be the theoretical ft

ft

ft

frequencies and π t = (π1t , π2t , . . . , πct ) = ( T1 , T2 , . . . , Tc ) be the theoretical pro portions, with ci=1 πit = 1. For those categories,1 let f o = (f1o , f2o , . . . , fco ) be fo

fo

fo

the observed frequencies and let π o = (π1o , π2o , . . . , πco ) = ( T1 , T2 , . . . , Tc ) be the observed proportions. Following Cressie and Read (1984), define the power divergence statistic I λ (π o , π t ) by (5), and following Basu et al. (2002) the disparity measure ρG (π o , π t ) between π o and π t by (6), where the function G(·) will be defined after. λ

o

I π ,π

t

=

c i=1

o

ρG π , π

t

πio λ(λ + 1)

πio πit

λ −1

,

λ ∈ R,

o c c πi t = G − 1 πi = G(δi )πit . πit i=1

(5)

(6)

i=1

As emphasized by Basu and Sarkar (1994), under the null of equalities of proportions, 2T I λ (π o , π t ) and 2T ρG (π o , π t ) have the same asymptotic distribution, 2 that is are asymptotically distributed as a χ(c−1) . Interestingly, the power divergence statistic can generate 5 of the most used goodness-of-fit tests. For instance, if λ = 1 it returns the classical Pearson’s test, and if λ = −2 it is equivalent to the Neyman’s modified test. If λ = − 12 it is also equivalent to the Freeman– Tukey statistic (Hellinger distance). If λ converges to −1 and 0 then it approaches respectively the log-likelihood and the modified log-likelihood ratio statistic. The power divergence statistic is itself generated by the power divergence family when in particular G(δ) is defined as (7). G(δ) =

(δ + 1)λ+1 − (δ + 1) δ − . λ(λ + 1) λ+1

(7)

Since such statistics may not have the same power compared to outliers and inliers, additional statistics can also be used: The Pearson–Blended Hellinger Mixture (PBHM) where the appropriate function G(·) for the disparity measure

categories are built to ensure equiprobability in c − 1 classes. The last category is used as a buffer category. In each class the frequency is more than 4.

1 The

Testing for Weak Separability

9

is given by (8), and a combined divergence where G(·) is defined as (9). δ2 δ2 G(δ) = τ + (1 − τ ) , √ 2 2(τ δ + 1 + (1 − τ ))2 G1 (δ) with λ = −0.5 if δ 0, G(δ) = G2 (δ) with λ = 1 if δ > 0,

with τ = 0.5,

(8) (9)

where the corresponding functions are given by (7). Define now the (T × k) Z matrix containing all the adjusted quantities consistent with weak separability solution of the above program, let zˆ ·j = (ˆz1j , zˆ 2j , . . . , zˆ Tj ) be the j th column of the matrix, and let x·j = (x1j , x2j , . . . , xTj ) be the corresponding observed quantities. Identifying the zˆ ·j to theoretical data, let π tj = (πjt 1 , πjt 2 , . . . , πjt c ) be the theoretical proportions for good j , and let π oj = (πjo1 , πjo2 , . . . , πjoc ) be the corresponding observed ones. Then, the null can be easily tested, for each good j by using (5) or (6). Let pvj be the p-value of the test for the good j , the decision rule is therefore at a threshold α: min(pv1 , pv2 , . . . , pvk ) α, accept the null of separability, if: min(pv1 , pv2 , . . . , pvk ) < α, reject the null. The decision rule is therefore directly related to the objective function (4): A large objective function leading to reject the null, and a small one supporting the null. In this chapter, we use four different disparity measures: (5) for λ = −1/2 and λ = 2/3, the PBHM and the combined divergence.

4. Solving the Adjustment Procedure This section focuses on the algorithm used to solve the quadratic program (4). Conditional to a particular method to solve the Afriat inequalities, we want to produce a dataset that is consistent with: (i) An overall utility function. (ii) A sub-utility. (iii) The weak separability. Since our method is a development of de Peretti (2005), we first recall this procedure. For a set {xi , pi }Ti=1 violating GARP, define the binary relation xi VRxj if xi Rxj and xj P 0 xi or if there exists a sequence between xi and xj such that xi Rxk and xk P 0 xi , xk Rxl and xl P 0 xk , . . . , xm Rxj and xj P 0 xm . Similarly de fine the binary relation xi SRxj if S(i) = S(j ), where S(i) = ( Tj=1 rij ) − 1 and rij is the element being at the ith row and j th column of the transitive closure matrix R. S(i) therefore returns how many bundles xi is revealed preferred

10

P. de Peretti

to, excluding itself. Since xi VRxj implies xi SRxj , de Peretti (2005) has suggested, for the bundles violating GARP, to build Bl set(s), l = 1, . . . , n, such that B1 ∪ B2 ∪ · · · ∪ Bn = V , B1 ∩ B2 ∩ · · · ∩ Bn = ∅ and such that every couple (xi , pi ) ∈ Bl , (xj , pj ) ∈ Bl ∀l ∈ {1, . . . , n} satisfy xi SRxj . In other words, each set contains bundles violating GARP that are candidates to be at the same place in the preference chain. Each set corresponds to a particular rupture of the preference chain. Define the set B1 as the set containing bundles violating GARP, which are all candidates to be at the same highest place in the preference chain, such that for (xi , pi ) ∈ B1 and (xj , pj ) ∈ / B1 : S(i) > S(j ). Searching for adjusted quantities that are consistent with GARP can now be achieved for {(xi , pi ), (xj , pj )} (xi , pi ) ∈ B1 , (xj , pj ) ∈ B1 such that xi Rxj and xj P 0 xi , by solving the quadratic program (10) over zij . obji = min zij

k zij j =1

xij

2 −1

(10)

subject to: pi · x i = pi · z i , p j · x j pj · z i

and pm · xm pm · zi

for all xm related to xi by xi VRxm , m = j, pq · x q p q · z i

(C.1)

for all (xq , pq ) ∈ / B1

such that riq = 1, where riq is an element of the transitive closure matrix (ith row, qth column).

(C.2)

The (C.1) constraint in the above program, ensures that zi will not violate GARP anymore with the other bundles of the set, and that zi will not violate GARP with bundles located above it in the preference chain. The (C.2) constraint ensures that zi will remain revealed preferred to bundles at a lower place in the preference chain, hence causing no new violation. Computing the minimal adjustment in the data in order for the whole dataset to satisfy GARP, can be now achieved by using the following four-step iterative procedure: Step 1: Test D = {xi , pi }Ti=1 for consistency with GARP, let nvio be the number of violations (0 nvio T (T − 1)): nvio = 0 then stop the iterative procedure, if: otherwise go to step 2. Step 2: Build n set(s) Bl , l = 1, . . . , n. Go to step 3.

Testing for Weak Separability

11

Step 3: Among the sets Bl , search for the one written B1 , containing the bundles being potentially at the same highest place in the preference chain, such / B1 : S(i) > S(j ). Go to step 4. that for (xi , pi ) ∈ B1 and (xj , pj ) ∈ Step 4: In the set B1 , compute (10) subject to (C.1) and (C.2) for each bundle related by xi Rxj and xj P 0 xi , and select the one, such that among all objective functions, its one is minimal. That bundle will be revealed preferred to the others. Let (ˆzi , pi ) be the bundle solution of this procedure. Replace, in D, (xi , pi ) by (ˆzi , pi ) and go to step 1. We now turn to the weak separability adjustment procedure. Define D = T {xi , pi }Ti=1 , D 1 = {x1i , p1i }Ti=1 and D 2 = {x3i , p3i }Ti=1 = {(x2i , Ui ), (p2i , λ−1 i )}i=1 , T where as previously {Ui , λi }i=1 are indices satisfying the Afriat inequalities for D 1 . Under the null, there exists an overall utility for D, a sub-utility for D 1 and a macro-function for D 2 . If violations appear, at least two sequences can be used to adjust the data, i.e. to produce a dataset consistent with weak separability. The one we want to consider here is as follows: (i) Adjust D 1 to produce data consistent with the sub-utility. (ii) Adjust D to produce data consistent with both the overall utility and the sub-utility. (iii) Adjust D 2 to produce data consistent with both weak separability and the overall utility. We now detail the sequence. We first consider adjusting the data in order for D 1 to be consistent with the subutility. This can be done by simply implementing the above iterative procedure. The four-step sequence is as follows: Step 1.1: Test D 1 for consistency with GARP, let R1 be the transitive closure and rij1 an element at the ith row and j th column. Let nvio be the number of violations, (0 nvio T (T − 1)): nvio = 0 then stop the iterative procedure, if: otherwise go to step 1.2. Step 1.2: Build n set(s) Bl , l = 1, . . . , n. Go to step 1.3. Step 1.3: Among the sets Bl , search for the one written B1 , containing the bundles being potentially at the same highest place in the preference chain, such / B1 : S(i) > S(j ). Go to step 1.4. that for (x1i , p1i ) ∈ B1 and (x1j , p1j ) ∈ Step 1.4: In the set B1 , search, for all bundles related by x1i Rx1j and x1j P 0 x1i by solving (11) subject to (C.1) and (C.2), for the bundle that will be revealed preferred to the others, such that for this bundle, among all objective functions, its objective function is minimal. Let (ˆz1i , p1i ) be 1 , zˆ 1 , . . . , zˆ 1 ). the bundle solution of this procedure, where zˆ 1i = (ˆzi1 ia i2 1 1 1 1 1 Replace, in D , {(xi , pi )} by {(ˆzi , pi )} and go to step 1.1.

12

P. de Peretti

obji = min

a z1 ij

1 zij j =1

xij1

2 −1

(11)

subject to: p1i · x1i = p1i · z1i

and

p1m · x1m p1m · z1i

p1j · x1j p1j · z1i ,

for all x1m related to x1i

by x1i VRx1m , m = j, p1q · x1q p1q · z1i 1 such that riq = 1.

(C.1)

/ B1 for all x1q , p1q ∈ (C.2)

1 = {(ˆz1 , p1 )}T consistent with the subThe above sequence returns a set D i i i=1 utility maximization, given sub-budgets {(p1i · x1i )}Ti=1 . 1 = {(ξ i , p1 )}T . To simplify, define ξ i = (ξi1 , ξi2 , . . . , ξia ) = zˆ 1i , and thus D i i=1 1 1 1 and rˆ be an element of it (ith row, j th colLet R be the transitive closure of D ij umn). At last, redefine D as {(ξ i , x2i ), (p1i , p2i )}Ti=1 . Thus D is formed by possibly adjusted quantities that are rationalized by a sub-utility function and unadjusted quantities x2i . 1 and Given D R1 , the second step is to find the minimal adjustment in D, in order to satisfy GARP (overall utility), under the additional constraint that adjusted 1 must still be consistent with the sub-utility. This is achieved by quantities for D using a similar iterative procedure and by also taking advantage upon the information content in the transitive closure matrix R1 . In other words, we adjust the data D in order to define a coherent transitive closure matrix for the overall utility, and also constraint the adjusted quantities for the sub-utility to be still consistent with R1 . The sequence is as follows: Step 2.1: Test D for consistency with GARP, let R be the transitive closure and rij an element at the ith row and j th column. Let nvio be the number of violations (0 nvio T (T − 1)): nvio = 0 then stop the iterative procedure, if: otherwise go to step 2.2. Step 2.2: Build n set(s) Bl , l = 1, . . . , n. Go to step 2.3. Step 2.3: Among the sets Bl , search for the one written B1 , containing the bundles being potentially at the same highest place in the preference chain, such that for {(ξ i , x2i ), (p1i , p2i )} ∈ B1 and {(ξ j , x2j ), (p1j , p2j )} ∈ / B1 : S(i) > S(j ). Go to step 2.4. Step 2.4: In the set B1 , search, for all bundles related by (ξ i , x2i )R(ξ j , x2j ) and (ξ j , x2j )P 0 (ξ i , x2i ), by solving (12) subject to (C.1) to (C.4), for the bundle that will be revealed preferred to the others, such that for this bundle, among all objective functions, its objective function is minimal.

Testing for Weak Separability

13

Let {(ˆz1i , zˆ 2i ), (p1i , p2i )} be the bundle solution of this procedure. Replace in D {(ξ i , x2i ), (p1i , p2i )} by {(ˆz1i , zˆ 2i ), (p1i , p2i )}, and go to step 2.1.

obji = min

a z1 ij

1 ,z2 zij ij j =1

xij

k z2 i(j −a)

2 −1

+

j =a+1

xij

2 −1

(12)

subject to: p1i · ξ i + p2i · x2i = p1i · z1i + p2i · z2i , p1j · ξ j + p2j · x2j p1j · z1i + p2j · z2i and p1m · ξ m + p2m · x2m p1m · z1i + p2m · z2i for all observations ξ m , x2m related to ξ i , x2i by ξ i , x2i VR ξ m , x2m , m = j, 1 1 2 2 1 2 where z1i = zi1 and z2i = zi1 , (C.1) , zi2 , . . . , zia , zi2 , . . . , zi(k−a) p1q · ξ q + p2q · x2q p1q · z1i + p2q · z2i

for all (xq , pq ) ∈ / B1

such that riq = 1,

(C.2)

p1i · ξ i = p1i · z1i ,

(C.3)

p1r · ξ r p1r · z1i such that rˆir1 = 1.

for all ξ r , p1r (C.4)

In the above sequence, the (C.1) and (C.2) constraints, as previously ensure that the data will be consistent with the overall utility. The (C.3) and (C.4) constraints in step 2.4 ensure that {(ˆz1i , zˆ 2i ), (p1i , p2i )} will be consistent with both the overall utility function and the sub-utility function, since it forces data to be consistent 1 . Whereas (C.4) is crucial, (C.3) can be omitted. It with the transitive closure of D has nevertheless an appealing economic interpretation, with regard to the allocation of the sub-budget. Note at last that uncorrected quantities enter the objective function. Define ς i = {ςi1 , ςi2 , . . . , ςia ) = zˆ 1i , ωi = {ωi1 , ωi2 , . . . , ωi(k−a) ) = zˆ 2i = {(ς i , ωi ), (p1 , p2 )}T . Let and thus D R be the associated transitive closure, i i i=1 T R. At last, let D 2 = {(ωi , Ui ), (p2i , λ−1 and let rˆij be an element of i )}i=1 , where 1 = {ς , p1 }T , or alternatively {Ui , λi }Ti=1 satisfy the Afriat inequalities for D i i i=1 let Ui be a measure of the sub-utility level for period i, and λi the inverse of

14

P. de Peretti

the corresponding price index (see next section). The problem is now to produce weak separability, under the additional constraint that nonseparable data must remain consistent with the overall utility. This is achieved by running the following sequence, in which we force weak separability and constraint data to be still consistent with R. Step 3.1: Test D 2 for consistency with GARP, let R2 be the transitive closure and rij2 an element at the ith row and j th column. Let nvio be the number of violations (0 nvio T (T − 1)): nvio = 0 then stop the iterative procedure, if: otherwise go to step 3.2. Step 3.2: Build n set(s) Bl , l = 1, . . . , n. Go to step 3.3. Step 3.3: Among the sets Bl , search for the one written B1 , containing the bundles being potentially at the same highest place in the preference chain, 2 −1 such that for {(ωi , Ui ), (p2i , λ−1 / i )} ∈ B1 and {(ωj , Uj ), (pj , λj )} ∈ B1 : S(i) > S(j ). Go to step 3.4. Step 3.4: In the set B1 , search, for all bundles related by (ωi , Ui )R(ωj , Uj ) and (ωj , Uj )P 0 (ωi , Ui ) by using (13), for the bundle that will be revealed preferred to the others, such that for this bundle, among all objective functions, its objective function is minimal. Let {(ˆz2i , Ui ), (p2i , λ−1 i )} be the bundle solution of this procedure. Replace in D 2 {(ωi , Ui ), (p2i , λ−1 z2i , Ui ), (p2i , λ−1 i )} by {(ˆ i )}, and go to step 3.1.

obji = min

k z2 i(j −a)

2 zij j =a+1

xij

2 −1

(13)

subject to: p2i · ωi = p2i · z2i

and

2 2 p2j · ωj + λ−1 j (Uj − Ui ) pj · zi

2 2 and p2m · ωm + λ−1 m (Um − Ui ) pm · zi

for all (ωm , Um ) related to (ωi , Ui ) by (ωi , Ui )VR(ωm , Um ), m = j, 2 2 p2q · ωq + λ−1 q (Uq − Ui ) pq · zi 2 such that riq = 1,

p2r · ωr p2r · z2i such that rˆir = 1.

(C.1) for all (ωq , Uq ), p2q , λ−1 ∈ / B1 q (C.2)

for all ωr , p2r (C.3)

Testing for Weak Separability

15

In the above sequence, the (C.1) and (C.2) constraints are relative to weak separability. Note the quantity indices Ui are seen as being at their “true values”, and are therefore not adjusted, the adjustment taking place only over zˆ 2i . The (C.3) constraint is relative to the overall utility, forcing the zˆ 2i to be consistent with the corrected transitive closure matrix of the overall utility (sequence 2). Hence, at this step, the zˆ 2i will be consistent with both overall utility maximization and separability. Note that here again, the overall budget is kept unchanged. The overall sequence, thus appears to be a generalization of the de Peretti’s = {(ς , zˆ 2 ), (p1 , p2 )}, (2005) adjustment procedure and produces a dataset D i i i i which given prices and a method to compute utility and price indices, is consistent with weak separability. An alternative sequence would have been to first adjust D to be consistent with the overall utility, second to adjust the data for a sub-utility, constraining the data to be also consistent with the overall utility, and at last to force weak separability, still in constraining the data to be consistent with overall utility. Empirically, those two procedures return similar results.

5. A Necessary and Sufficient Test for Homogeneous Weak Separability T In the Varian’s approach, a set {Ui , λ−1 i }i=1 satisfying the Afriat inequalities is used. Based on first-order conditions, and property of concave functions, it avoids using measures of the sub-utility function. This dramatically simplifies the analysis, but leads to an only sufficient condition. Nevertheless, this is not the only way to test for weak separability. Focusing on (1), and assuming linear homogeneity of the sub-utility function one may wish to run nonparametric tests with “true” measures Ui of the sub-utility. From a theoretical point of view, the output of the sub-utility function is the quantity aggregate. To estimate it, one has therefore two possibilities. The first one, is to define an appropriate flexible functional form for the sub-utility, and estimate the unknown parameters by using first-order conditions. The second one, emphasized by Diewert (1976, 1978) is to use a superlative index, as the Törnqvist–Theil index. Indeed, the Törnqvist–Theil index QV (p11 , p12 , x11 , x12 ) is exact for the homogeneous translog utility function, which means that it is able to track the

variations of the sub-utility: QV (p11 , p12 , x11 , x12 ) =

Ux1 (x12 ) . Ux1 (x11 )

Hence, in this case,

one can measure the sub-utility without any econometric estimation. The price index is deduced by using the weak reversal property (in index form or not). The nonparametric tests can then be implemented using QV (·) and the deduced price T index instead of {Ui , λ−1 i }i=1 , in step 3 of the Varian’s procedure, and in step 3.1 of the adjustment procedure. One will note that the test is necessary and sufficient.

16

P. de Peretti

6. An Empirical Application We now illustrate both the adjustment procedure and the test procedure by an application. We first introduce the data generating process (DGP).2 Let {x∗i , pi }20 i=1 ∗ , x∗ , . . . , x∗ ) be a set of quantities and associated prices, where x∗i = (xi1 i2 i5 and pi = (pi1 , pi2 , . . . , pi5 ). Also define an other set of quantities and prices ∗ ∗ ∗ ∗ {y∗i , qi }20 i=1 , where yi = (yi1 , yi2 , yi3 ) and qi = (qi1 , qi2 , qi3 ). Define at period ∗ be i the overall simple Cobb–Douglas maximization program (14), and let pi1 xi1 in fact a sub-budget. This allows us to develop a utility-tree, and hence introduce a weakly (strongly) separable utility function. The maximization program for the sub-utility being given by (15). ∗1/5 ∗1/5 ∗1/5 ∗1/5 ∗1/5

max U = xi1 xi2 xi3 xi4 xi5 s.t. pi · x∗i = Ii ,

(14)

∗1/3 ∗1/3 ∗1/3 max Uy = yi1 yi2 yi3 ∗ s.t. qi · y∗i = pi1 xi1 .

{x∗i , pi }20 i=1

(15)

{y∗i , qi }20 i=1

Generating and consistent with such a preference structure is achieved as follows. We generate an income I, as an I(1) series, such that an element Ii is defined as: 1000 if i = 1, Ii = Ii−1 + εi if 2 i 20, εi being distributed as N (0, 1). Similarly, we define the prices, also as I(1) processes, such that an observation pij is defined as: 100 if i = 1, pij = p(i−1)j + εij if 2 i 20, εij being distributed as N (0, 1). And qij is defined as: 100 qij = q(i−1)j + εij

if i = 1, if 2 i 20, εij being distributed as N (0, 1).

We then solve the maximization program for the overall utility, and then given ∗ for the sub-utility. To introduce a stochastic structure, quantities are divided pi1 xi1 by a random error term. For instance, we have: xij =

xij∗ (1 + ij )

,

2 The data used in this section are available at the author’s e-mail address: [email protected]

Testing for Weak Separability

yij =

yij∗ (1 + υij )

17

,

where xij and yij are the observed series and υij and ij are two normally distributed error terms, with zero mean and variance 0.052 . Let D = {(yi1 , yi2 , yi3 , xi2 , xi3 , xi4 , xi5 ), (qi1 , qi2 , qi3 , pi2 , pi3 , pi4 , pi5 )}20 i=1 , and suppose we want to test the separability of yi . To compute utility indices, we use here the recently introduced LP program of Fleissig and Whitney (2003). For the sub-utility function, for period i define QVi as a quantity superlative index, and especially as the Törnqvist index. The idea is to compute the minimal perturbation in the quantity index and in the price index, defined according to the weak reversal property, in order to produce utility and price indices satisfying the Afriat inequalities. First define Ui = QVi + Q+ i − QVi + − + − + − and λ = + λ − λ , where Q , Q , λ and λ are small positive Q− i i i i i i i i qi ·yi perturbations, solution of the minimization program (16): min Z =

T

Q+ i

+

i=1

T

Q− i

+

i=1

T i=1

λ+ i

+

T

λ− i

(16)

i=1

subject to: + − − + − Q+ i − Qj − Qi + Qj − λj qj (yi − yj ) + λj qj (yi − yj )

−QVi + QVj +

QVj qj (yi − yj ), qj · y j

(C.1)

QVi − iλ , q i · yi

(C.2)

− −Q+ i + Qi QVi − i ,

(C.3)

− −λ+ i + λi

Q

Q i

where iλ and are small errors. Given the above program, the solutions to the Afriat inequalities are given by Q QVi − + − λ Ui = QVi + Q+ i − Qi − i and λi = qi ·yi + λi − λi − i . Table 1 returns the minimal adjustment in order to produce weak separability, given the above method to built utility and price indices. Concerning the Table 1: Results of the Adjustment Procedure

# of observations # of goods # of violations Adjustment Overall adjustment

Sub-utility

Overall utility

Separability

20 3 0 0

20 7 0 0

20 5 4 0.0100073 0.0100073

18

P. de Peretti

Table 2: Results of the Test Procedure x·2

x·3

x·4

x·5

Freeman–Tuckey p-value

0.586 0.989

1.201 0.945

0 1

0.691 0.983

Cressie and Read, λ = 2/3 p-value

0.543 0.990

1.196 0.945

0 1

0.669 0.985

PBHM p-value

0.560 0.990

1.201 0.945

0 1

0.679 0.984

Combined, λ = −0.5 and 1 p-value

0.615 0.987

1.308 0.934

0 1

0.749 0.980

adjustment, recall that we first adjust data to be consistent with the sub-utility, second data to be consistent with the overall utility, and then data to be consistent with weak separability. Note that neither the tests for the sub-utility nor the tests for overall utility produced violations of GARP. Conversely, GARP is violated when testing for the existence of the macro-function. Four violations appear giving an adjustment of 0.0100073. In this latter step, only two bundles are adjusted: x9 and x7 with corresponding adjustments of 0.0092399 and 0.0007674. We now turn to the test procedure that consists in testing the discrepancy between the observed quantities, and adjusted ones, seen as theoretical values. Since only the four last goods of xi are adjusted, we only focus on those latter. As presented by Table 2, the null of weak separability can not be rejected at usual threshold, which is coherent with our data generating process.

7. Monte Carlo Simulations We now turn to Monte Carlo simulations in order to estimate the power of the procedure. We use a similar DGP as in the previous section but focus on the necessary and sufficient test for weak separability, since the DGP involves a first degree homogeneous sub-utility function. As mentioned above, this is done by T replacing {Ui , λ−1 i }i=1 satisfying the Afriat inequalities by the quantity index and a price index computed thanks to the weak reversal condition. This is clearly a special case of the Fleissig and Whitney approach. In order to estimate the type I error we use the previously defined preference structure for T = 40 and use two different measurement errors σ = 0.05 and σ = 0.10. We use 10000 replications. At each iteration, we collect the p-values for all tests. For a given test, the type I error at α is defined as the probability to reject weak separability whereas there is weak separability divided by the number of times GARP is violated in at least one of the three steps. Table 3 presents some key statistics about the adjustments as well as about the

Testing for Weak Separability

19

Table 3: Descriptive Statistics, H0 True σ = 0.05 # of replications

σ = 0.10

10000

Nonstochastic type I error1 Adjustment: Mean Std Max Median Min

10000

0.808

0.983

0.0016 0.0039 0.0645 0.0007 4.89E−10

0.0125 0.0145 0.1089 0.0079 1.55E−07

1 Number of times at least one violation of GARP is found divided by 10000.

Table 4: Estimation of the Type I Error Threshold α = 0.01

α = 0.05

α = 0.10

α = 0.15

Freeman–Tuckey Cressie and Read, λ = 2/3 PBHM Combined, λ = −0.5 and 1

Measurement error: σ = 0.05 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

Freeman–Tuckey Cressie and Read, λ = 2/3 PBHM Combined, λ = −0.5 and 1

Measurement error: σ = 0.10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

nonstochastic test for weak separability. It should be noted first that, in our framework, measurement error often produces violations of GARP, since for σ = 0.05 and σ = 0.10, in only 19.2% and 1.70% of the cases the data are consistent with separability. Thus, measurement error produces a high type I error. Concerning the adjustments, as expected, it appears to be larger for large errors. The maximal adjustment found is 0.0645 for σ = 0.05 and 0.1089 for σ = 0.10. Note that median adjustment appears to be very low: 0.0007 for σ = 0.05 and 0.0079 for σ = 0.10. We now turn to the estimation of the type I error, returned by Table 4. Clearly the procedure introduced is particularly well suited since the type I error is not different from zero for all considered statistics, and at all considered thresholds. We now turn to the estimation of the type II error. In order to estimate the probability to accept weak separability whereas there is not weak separability,

20

P. de Peretti

we first produce a dataset that is not weakly separable. Especially, we produce a dataset that is not inter-temporally weakly separable. To achieve such a goal, we use the same data generating process, but force the weights of the sub-utility to change every 10 periods. The data are not multiplied by a measurement error. Table 5 presents some descriptive statistics associated with the necessary adjustment. Clearly, adjusting data to be consistent with weak separability, whereas data are nonseparable requires a much larger adjustment with a median value of 9.451. Nevertheless, in few cases, ruptures in the utility function can also produce a low number of violations, and then a small adjustment (min = 3.19E−07). This implies a very small type II error for the nonstochastic weak separability tests (0.001). We now focus on the type II error of the procedure, presented by Table 6. For the Freeman–Tuckey, PBHM and combined divergence the procedure appears to be quite powerful, with low type II errors, respectively of 0.065, 0.069 and 0.062 at 5%. For λ = 2/3, the type II error is larger, 0.118 at 5%, suggesting not using this statistic. The type II error may be explained as follows. In some cases, ruptures in the utility function produces few violations, and hence a low adjustment, causing the test to accept the nonsignificance of it. The test appears therefore to be quite powerful. Table 5: Descriptive Statistics, H0 False # of replications

10000

Nonstochastic type II error1

0.001

Adjustment: Mean Std Max Median Min

11.004 7.937 38.554 9.451 3.19E−07

1 Number of times no violation of GARP is found divided by 10000.

Table 6: Estimation of the Type II Error Threshold

Freeman–Tuckey Cressie and Read, λ = 2/3 PBHM Combined, λ = −0.5 and 1

α = 0.01

α = 0.05

α = 0.10

α = 0.15

0.068 0.157 0.108 0.067

0.065 0.118 0.069 0.062

0.062 0.102 0.065 0.056

0.057 0.087 0.062 0.054

Testing for Weak Separability

21

8. Conclusion and Discussion The goal of this chapter was to introduce a general framework to test for weak separability. Clearly, it consists in first finding the minimal perturbation in the data to ensure compliance with weak separability. This is achieved by using a procedure closely related to de Peretti (2005), that takes advantage upon the information content in different closure matrices. Second, goodness-of-fit tests are used to check the significance between the observed series and the adjusted ones satisfying weak separability by construction. It is very important to point out that the performance of such a procedure strongly depends on the method used to compute utility and price indices, if the standard Varian’s framework is used. In this chapter, we have used the recent contribution of Fleissig and Whitney (2003). Nevertheless, this is not the only way to test for weak separability. In particular, one may wish to consider necessary and sufficient tests. This can be done by using proxies of utility levels. In this chapter, with an a priori knowledge of the sub-utility, we have used a quantity Divisia index, known to track the variations of the sub-utility, if homogeneous. When the form of the sub-utility is unknown, utility levels (as well as the price index) can be approximated by estimating demand systems, or by using first-order conditions. One can therefore build a necessary and sufficient test in our framework, which is general enough to deal with both methods. This is left for future research. A second direction would be considering testing for additive separability.

References Barnett, W.A. (1980). Economic monetary aggregates: An application of index number and aggregation theory. Journal of Econometrics 14, 11–48; Reprinted in Barnett, W.A. and Serletis, A., The Theory of Monetary Aggregation. Amsterdam: North Holland, 2000, Chapter 2. Barnett, W.A. and Choi, S.A. (1989). A Monte Carlo study of tests of blockwise weak separability. Journal of Business and Economics Statistics 7, 363–377; Reprinted in Barnett, W.A. and Binner, J., Functional Structure and Approximation in Econometrics. Amsterdam: North Holland, 2004, Chapter 12. Basu, A. and Sarkar, S. (1994). On disparity based goodness-of-fit tests for multinomial models. Statistic and Probability Letter 19, 307–312. Basu, A., Surajit, R., Chanseok, P. and Basu, S. (2002). Improved power in multinomial goodness-of-fit tests. Journal of the Royal Statistical Society 51, 381–393. Belongia, M.T. and Chrystal, K.A. (1991). An admissible monetary aggregate for the United Kingdom. The Review of Economics and Statistics 73, 497–503. Cressie, N. and Read, T.R.C. (1984). Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society 46, 440–464. de Peretti, P. (2005). Testing the significance of the departures from utility maximization. Macroeconomic Dynamics 9, 372–397.

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Diewert, W.E. (1976). Exact and superlatives index numbers. Journal of Econometrics 4, 115–145. Diewert, W.E. (1978). Superlatives index and consistency in aggregation. Econometrica 46, 883–900. Fisher, D. and Fleissig, A.R. (1997). Monetary aggregation and the demand for assets. Journal of Money, Credit and Banking 29, 458–475. Fleissig, D. and Whitney, G. (2003). A new PC-based test for Varian’s weak separability condition. Journal of Business and Economic Statistics 21, 133–144. Serletis, A. (1987). Monetary asset separability tests. In Barnett, W.A. and Singleton, K.J. (Eds), New Approaches to Monetary Economics. Cambridge: Cambridge University Press. Swofford, J. and Whitney, G. (1987). Nonparametric tests of utility maximization and weak separability for consumption, leisure and money. Review of Economics and Statistics 69, 458–464. Varian, H.R. (1982). The nonparametric approach to demand analysis. Econometrica 50, 945–973. Varian, H.R. (1983). Nonparametric test of consumer behavior. Review of Economic Studies 50, 99–110.

Chapter 2

Quantity Constraints and Violations of Revealed Preference Adrian R. Fleissiga and Gerald A. Whitneyb,∗ a Department

of Economics, California State University, Fullerton, Fullerton, CA 92834, USA b Department of Economics and Finance, University of New Orleans, LA 70148, USA

Abstract A revealed preference test that allows for rationing is applied to evaluate UK meat consumption from 1900–1919. We find the data consistent with effective rationing for all meat products over the period 1914–1918. In contrast, tests without allowing for rationing reject optimal consumer behavior. In addition, estimated virtual prices for the corresponding rationed goods give a data set that satisfies revealed preferences. Using the virtual prices, the results suggest that from 1913 to 1918 meat costs rose by approximately 15 percentage points more than the observed price index. Keywords: Afriat inequalities, rationing, virtual prices JEL: C19, D12, D45

1. Introduction In this chapter we use an approach similar to Fleissig and Whitney (2006) to evaluate for rationing of multiple goods over multiple periods using UK data for the period 1900–1919. Our study differs from their approach as we examine a subset of goods, not the entire consumption set. Rationing of all the goods in the subset is allowed over the war years of 1914–1918. The procedure also accommodates for periods when there was no rationing. Hence we can determine if a demand theoretic framework is consistent with observed consumer choices during the prewar period 1900–1919. It is important to note that binding rationing constraints is neither a necessary nor a sufficient condition for there to be any detectable violations of revealed preference. ∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18002-9

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The approach also allows us to recover the revealed preference version of virtual prices for the rationed goods as in Neary and Roberts (1980). Their virtual prices are those prices at which the rationed quantities would be freely demanded. In the revealed preference context, the virtual prices are prices that make choices (made with and without rationing constraints) consistent with the Generalized Axiom of Revealed Preference. In our application to wartime UK data, the virtual prices of effectively rationed goods give an estimate of how much prices would have to change to avoid shortages in the short-run when rationing was ended. The revealed preference approach also avoids the drawback of having to select a functional form for utility. Moreover, this procedure does not have to impose a priori rationing constraints for which goods are binding. This is important because the desired consumption set is not reduced when rationing on a good is not binding. For example, voluntary export constraints may result simply in prices rising for the domestic product to clear markets, or as argued by Brenton and Winters (1993), if competition keeps supplier prices down rationing will result. Over the period 1900–1919, we find that the observed consumption pattern of meat products over the entire period is consistent with optimizing behavior if we allow for rationing during the war years. We also find that inflation where virtual prices are substituted for war years (giving a better measure of the welfare cost of inflation) was approximately 15 percentage points higher than recorded inflation.

2. Data The UK data cover the period 1900–1919. This is an interesting period to analyze because of the various constraints on consumption. The data are obtained from Prest (1954), Consumers’ Expenditure and Behavior in the United Kingdom 1900–1919. All of the quantity data are converted into consumption per adult equivalent population. Adult equivalent population is obtained by weighing UK population figures according to age and gender. The weights used are: 1 for males 15 years and older; 0.9 for females 15 years and older; 0.675 for those between 5 to 14 years old; and 0.28 for those under 5 years. The weights are from Stone and Rowe (1954), Appendix 1 to Measurement of Consumers’ Expenditure and Behavior in the United Kingdom 1920–1938, Volume I. While official rationing of food did not begin until 1918, the movement towards quantity controls began in 1914 when a system of returns for retail prices was established as was a plan for publishing a weekly list of approved retail prices (Beveridge, 1928). During this period, much attention was directed to sugar prices and the Royal Commission on Sugar Supplies was formed on August 20, 1914. However, inspection of imports shows that the effects of the war already had a serious impact by 1916. For example, beef and veal imports were 9,952,000 cwt in 1913 and by 1916 had dropped to 5,034,000. Total consumption of beef and veal fell from 22,004,000 cwt in 1913 to 18,085,000 cwt in 1916. It has been

Quantity Constraints and Violations of Revealed Preference

25

Table 1: Meat Products 1900–1919a Beef veal Mutton Pork Offal Bacon and ham a Prices are in pounds per thousands. Quantities are measured

in hundred weights thousand per adult equivalent.

estimated that by October of 1916, two million tons of shipping destined for the United Kingdom were destroyed by the German navy as discussed in Beveridge (1928). Nonetheless, meat rationing only began in London and the surrounding districts on 25 February 1918 and national rationing was established on 17 April 1918, see Beveridge (1928). We focus on the demand for the five types of meats shown in Table 1 for which both price and quantity data are available. A similar break down for poultry is not available. If this group is weakly separable from the rest of the consumption basket, it should conform to the axioms of revealed preference. However, rationing limits consumer choices and may lead to choices that appear to violate optimal consumer behavior. Consequently tests that fail to account for quantity constraints may reject optimal consumer choices even though the consumer may be acting rationally. Thus we begin our analysis by first examining if consumer choices are consistent with optimal behavior without imposing any quantity constraints. Prior to performing these tests, we briefly discuss the revealed preference approach to evaluating consumer demand.

3. Testing Revealed Preference The revealed preference approach is frequently applied to data to evaluate consumer demand. In particular, a utility function U (·) rationalizes a set of n-observations of k-element vectors of prices and goods (p i , x i ), if U (x i ) U (x) for all x such that p i x i p i x. While the simplest utility function that rationalizes the data is a constant utility function u(x) = 1, Afriat (1967) and Varian (1982) formally produce conditions under which a well-behaved utility function rationalizes the data. Afriat’s theorem. The following conditions are equivalent (Varian, 1982). (A1) There exists a nonsatiated utility function that rationalizes the data. (A2) The data satisfy GARP.

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(A3) There exist numbers U i , λi > 0 for i = 1, . . . , n that satisfy the Afriat inequalities: U i − U j − λj p j x i − x j 0

for i, j = 1, . . . , n.

(1)

(A4) There exists a concave, monotonic, continuous, nonsatiated utility function that rationalizes the data. The Generalized Axiom of Revealed Preference (GARP) of Varian (1982) is equivalent to the “cyclical consistency” conditions of Afriat (1967) and is easier to empirically evaluate. GARP states that if observation j is revealed preferred (directly or indirectly through a sequence with other observations) to observation i, then observation i cannot be strictly directly revealed preferred to observation j . Varian (1982) refers to condition (A3) as the “Afriat inequalities”. A unique solution to the Afriat inequalities require some form of normalization. We begin the analysis by evaluating if there exists a well behaved utility function that can rationalize the data over the period 1900 through 1919. Using the transitive closure procedure of Varian (1982), we find 9 violations of GARP which implies the data are not consistent with utility maximization (see Table 2). Note that this test does not impose any quantity constraints and thus it is possible that the apparent violations of revealed preference may be caused by rationing. All but one pair of GARP violations involve the war years 1914–1918. We assume that the violations may have been caused by rationing or shortages of the goods. We now modify the Afriat inequalities and derive new restrictions to evaluate if consumer choices are consistent with optimal consumer choices in the presence of quantity constraints. Table 2: Testing GARP Period 1900–1919a Years of violations 1910 and 1919 1914 and 1910 1914 and 1918 1914 and 1919 1918 and 1910 1918 and 1914 1919 and 1910 1919 and 1914 1919 and 1918 Number of GARP violations = 9. a The GARP test using the transitive closure procedure of Varian (1982) is performed using Visual Basic.

Quantity Constraints and Violations of Revealed Preference

27

4. Rationing and Utility Maximization Consider the following optimization with consumer choices constrained by some type of rationing on goods or expenditure in period i: Max u(x) s.t. p i x mi , Ai x b i ,

(2)

where Ai is a h by k matrix with each row representing a rationing constraint and bi is a h vector of quantity limits. Hence, Ai x bi is a rationing constraint that can represent an upper bound on the purchase of one or more goods, or a period where expenditure is constrained directly or by ration coupons for various goods, or shortages are not met by increased prices. Under rationing, there are now Lagrange multipliers (μgj ) associated with the rationing constraints for good g in period j . These multipliers will be positive for binding constraints and zero otherwise. Some special cases of the rationing model from (2) were developed by Varian (1983) with the matrix Ai replaced by a vector a i . The vector a i can have a one and several zeros, which results in an upper bound on a single good. Alternatively, the vector a i can be prices in ration coupons needed to purchase a good. For example, it might take two coupons to buy a loaf of bread, five to buy a pound of meat, etc. Under these special rationing cases of Varian (1983), there exists a utility function that rationalizes an observed data set if u(x i ) u(x) for all x, such that mi p i x and bi a i x for all x, where mi is the ith period budget constraint. This system is extended to multiple expenditure rationing constraints by Fleissig and Whitney (2006). We now setup an alternative optimization that imposes quantity constraints. The model that we solve for to test for quantity constraints over the period 1900–1919 is as follows: Min Z = μij subject to

j U i U j + λj p j x i − x j + μ1j x1i − x1 + · · · j + μhj xhi − xh 0,

λi > 0, μgi 0, μgi = 0 in periods without rationing,

(3)

for j = 1 to n. Note that λj > 0 for the entire period and ugi = 0 for all goods if i represents a pre-war or post-war year with μgi 0 for all the war

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years. Normalization is implemented by requiring λi > 0.001. Hence for the war years, we allow for rationing of one or more goods. This system differs from the approach of Varian (1983) which only tests for a single ration constraint. If a solution exists then the data can be rationalized by a utility function assuming the consumer faces rationing constraints. If all μgj are set to zero, (3) becomes the familiar Afriat inequalities (A3). Some important information can be obtained from the constrained optimization (3). In particular, the virtual prices are obtained by rearranging the terms in the Afriat inequalities under rationing which gives: μj i i j j j U U + λ p + j x − xj , (4) λ where μj is k-element vector with elements that are zero or μgj for g = 1, . . . , h. Hence, (p j + μj /λj ) is a k-element vector of prices, including virtual prices for the rationed goods, that satisfy the Afriat inequalities. For the gth rationed good in period j , the virtual price is: μgj (5) . λj Equation (5) is used to derive the virtual prices that would make the choices under the quantity constraints consistent with optimal consumer choices. If a feasible solution exists for (3), then the corresponding solutions for λj can be used to obtain the virtual prices in (5). p ∗gj = p gj +

5. Quantity Constraints Results Under rationing and given the shortages of goods, the observed expenditure on the meat categories were inconsistent with the Afriat inequalities. In addition, the observed expenditure was below that desired at the prevailing prices and income. Given that the violations of GARP occurred over the period 1914 through 1918 we set μgj 0 for g = 1, . . . , 5 over the period j = 1914, . . . , 1918. For the remaining years, μgj = 0 for g = 1, . . . , 5 for j = 1900, . . . , 1913 and j = 1919. Recall that even under these quantity constraints, it is still possible that the data will not be consistent with rational consumer behavior. It turns out that there exists a feasible solution for the solution from (3) using the Excel add-in Large-Scale GRC Solver Engine version 3.5. Thus per capita meat consumption is consistent with utility maximization with effective rationing over the years 1914–1919. To analyze how much meat prices would need to change by to make preferences consistent with GARP, the virtual prices are calculated using formula (5). Recall that the virtual prices are those prices at which the rationed quantities would be freely demanded. The virtual prices are graphed along with the actual prices in Figures 1–5.

Quantity Constraints and Violations of Revealed Preference

29

Figure 1: Beef and Veal.

Figure 2: Mutton.

Given that the difference between the observed price of a rationed good and its virtual price is the amount the consumer would have paid for an extra unit of the rationed good, these differences are reported in Table 3. The results show how

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Figure 3: Pork.

Figure 4: Offal.

much prices would have to change to make the observed quantities for 1914–1918 consistent with revealed preference. For example, the price of beef and veal in 1914 should have been 11.4% higher to satisfy the Afriat inequalities.

Quantity Constraints and Violations of Revealed Preference

31

Figure 5: Bacon. Table 3: Difference Virtual and Observed Prices Year

Beef and veal

Mutton

Pork

Offal

Bacon and ham

1914 1915 1916 1917 1918

11.4% 21.4% 16.2% 3.6% 6.7%

12.1% 22.1% 16.8% 3.6% 6.8%

10.2% 19.1% 14.7% 3.3% 6.0%

15.6% 23.2% 16.6% 3.4% 7.6%

16.6% 17.8% 14.6% 2.9% 4.8%

The observed Laspeyres price index shows that prices for the five meat products increased by 133.8% from 1913 to 1918. Using virtual prices, the increase was 148.5% percent which is almost 15 percentage points above what was reported.

6. Conclusions A revealed preference test that allows for rationing quantity rationing finds that there exists a utility function that can rationalize UK data on meat consumption over the period 1900 through 1919. In contrast, given that quantity constraints were frequently imposed over this period, revealed preference tests that fail to allow for rationing reject optimal consumer choices. Our results suggest that there were shortages of all five meat items from 1914–1918. Using the calculated virtual prices, it appears as though the cost to consumers of the 1913 market basket of

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meat rose approximately 15 percentage points more than the cost calculated with observed prices.

References Afriat, S. (1967). The construction of a utility function from expenditure data. International Economic Review 8, 67–77. Beveridge, W.H. (1928). British Food Control. Oxford University Press. Brenton, P.A. and Winters, L.A. (1993). Voluntary export restraints and rationing UK leather footwear imports from Eastern Europe. Journal of International Economics 34(3–4), 289–308. Fleissig, A. and Whitney, G. (2006). A revealed preference test of rationing. Unpublished manuscript. Neary, J.P. and Roberts, K.W.S. (1980). The theory of household behavior under rationing. European Economic Review 13, 25–42. Prest, A.R. (1954). Consumers’ Expenditure in the United Kingdom 1900–1919. Cambridge: Cambridge University Press. Stone, R.A. and Rowe, D.A. (1954). The Measurement of Consumers’ Expenditure and Behaviour in the United Kingdom, 1920–1938, Vol. I. Cambridge: Cambridge University Press. Varian, H. (1982). The nonparametric approach to demand analysis. Econometrica 50, 945–973. Varian, H. (1983). Nonparametric tests of consumer behavior. Review of Economic Studies 50, 99–110.

Chapter 3

Nonparametric Tests of the Necessary and Sufficient Conditions for Separability Barry E. Jonesa,∗ , Nadine McClouda and David L. Edgertonb a Department

of Economics, Binghamton University, P.O. Box 6000, Binghamton, NY 13902-6000, USA b Department of Economics, Lund University, P.O. Box 7082, S-22007 Lund, Sweden

Abstract We survey the current state of the art in the use of nonparametric methods to test separability. We focus on three tests: Swofford and Whitney’s (1994) joint test of the necessary and sufficient conditions for weak separability; Fleissig and Whitney’s (2003) sequential test of those conditions; and Jones and Stracca’s (2006) test of the necessary and sufficient conditions for additive separability. We illustrate the latter two tests by applying them to data generated from Barnett and Choi’s (1989) WS-branch model. The empirical results show that these two tests are able to correctly identify separable structure in nearly all cases.

Keywords: nonparametric tests, revealed preferences, weak separability, additive separability JEL: C14, C63, D12

1. Introduction Nonparametric revealed preference methods can be used to test economic data for consistency with utility maximization and various types of separability of the utility function. Specifically, Varian (1982) proves that a dataset consisting of observed goods prices and quantities can be rationalized by a well-behaved utility function if and only if the dataset satisfies the generalized axiom of revealed preference (GARP). Varian (1983) provides necessary and sufficient nonparametric

∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18003-0

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conditions for various types of separability. These necessary and sufficient conditions can be stated in terms of the existence of indexes, which solve sets of inequality constraints given the observed price and quantity data. In this chapter, we survey the current state of the art in the use of nonparametric tests of these conditions. As Barnett and Choi (1989, p. 363) explain, the practical importance of assumptions about separability results from three facts: First, separability provides a fundamental linkage between aggregation over goods and the maximization principles in economic theory; Second, separability provides a basis for partitioning the economy’s structure into sectors; Finally, separability assumptions can produce powerful parameter restrictions permitting great simplification in the estimation of large demand systems. Weak separability, in particular, is a key property in economics, because of its connection to two-stage budgeting. If a block of goods is weakly separable from all other goods, then the marginal rates of substitution between any pair of goods in that separable block does not depend on the quantities consumed of any good that is not in the block. This in turn implies that the demand functions for goods in the separable block depend only upon the prices of those goods and total expenditure allocated to the block. Thus, weak separability allows for the estimation of the second stage of a two-stage demand system and, consequently, greatly reduces the number of parameters to be estimated. Without weak separability, it is possible for the prices of all other goods to enter into the demand functions for the block of goods being considered. As Blundell and Robin (2000, p. 54) discuss, “[t]he advantage of weak separability is in the reduction of the allocation decision problem to a recursive sequence of manageable choice problems. The drawback of such separability assumptions is well known and stems primarily from the strong restrictions placed on substitution possibilities between commodities occupying different groups . . .”.

See Blackorby et al. (1978), Deaton and Muellbauer (1980), Barnett and Choi (1989), Pollack and Wales (1992), Blundell and Robin (2000), Barnett and Serletis (2000) and Barnett and Binner (2004) for further discussion. Separability assumptions can be evaluated using either statistical tests of parameter restrictions on a parametric functional form or using nonparametric tests of the appropriate necessary and sufficient conditions provided by Varian (1983). Unlike parametric tests, nonparametric tests are not hypothesis tests in the statistical/econometric sense. Rather, they consist of nonstochastic checks to determine whether or not the price and quantity data are consistent with Varian’s separability conditions. Nonparametric methods have a number of well-known advantages relative to parametric tests. First, they do not require any functional form assumptions or estimation and, consequently, avoid problems associated with model misspecification (see Barnett and Choi, 1989). Second, nonparametric tests can be applied to any number of observations. In contrast, parametric tests require more observations than estimated parameters and, more importantly, are really

Nonparametric Tests for Separability

35

joint tests of the imposed separability structure and the functional form being estimated (see Swofford and Whitney, 1994). Third, nonparametric methods are used to test separability of the direct utility function. In contrast, parametric tests are often based on imposing separability on flexible functional form representations of the indirect utility function (or cost function), i.e. imposing separability in prices rather than in goods. Unless the direct utility function is homothetic, blockwise separability of the indirect utility function in prices does not correspond to block-wise weak separability in goods (see Barnett and Choi, 1989).1 As a consequence of these attractive features, nonparametric revealed preference methods have been applied in a variety of different contexts. See, for examples, Varian (1982), Swofford and Whitney (1987, 1994), Manser and McDonald (1988), Belongia and Chalfant (1989), Patterson (1991), Fisher and Fleissig (1997), Spencer (1997), Rickertsen (1998), Fleissig et al. (2000), Drake et al. (2003) and Jones et al. (2005a). The most significant criticism of nonparametric revealed preference methods is that they are nonstochastic and are, consequently, unable to account for random error from any source, see Varian (1985) and Swofford and Whitney (1994). With respect to revealed preference tests of optimizing behavior, Varian (1985, p. 445) states “[t]he data are assumed to be observed without error, so that the tests are ‘all or nothing’: either the data satisfy the optimization hypothesis or they do not. However, it seems that if some data fail the tests, but only by a small amount, we might well be tempted to attribute this failure to measurement error, left out variables, or other sorts of stochastic influences rather than reject the hypothesis outright.”.

As we discuss below, there are a number of promising approaches, which attempt to address this issue. See, for examples, Varian (1985), Epstein and Yatchew (1985), Gross (1995), Fleissig and Whitney (2003, 2005), de Peretti (2005), Jones and de Peretti (2005) and Jones et al. (2005b). At present, no consensus seems to exist as to what is the best approach and, consequently, this remains an area of active research. Moreover, much of that work focuses on testing optimization hypotheses, such as the utility maximization hypothesis, rather than on nonparametric separability tests. In contrast, statistical tests applied to parametric models have a clear advantage in this respect, since they can account for stochastic factors directly through the error term in the estimated model (Swofford and Whitney, 1994). Nonparametric weak separability tests have also been criticized for being biased towards rejecting separability. The bias in such tests results from testing the necessary and sufficient conditions for weak separability sequentially rather than jointly. For example, Barnett and Choi (1989) generated data from a three good

1 In addition, parametric tests are often formulated as tests of approximate local separability

rather than global separability. Again, see Barnett and Choi (1989) for further discussion.

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Cobb–Douglas utility function with no random disturbances and then checked for weak separability using the nonparametric test in the computer program, NONPAR, developed by Varian. That sequential test rejected weak separability in all cases, in spite of the fact that the utility function used to generate the data is weakly separable in all possible blockings of the goods. As noted by Barnett and Choi (1989, p. 364), “[t]hat approach checks sufficient but not necessary conditions for the data to satisfy a separable structure. Those sufficient conditions are so restrictive that the test is very strongly biased towards rejection.”.

Swofford and Whitney (1994) address this criticism by providing an unbiased nonparametric procedure, which jointly tests the necessary and sufficient conditions for weak separability. Their joint test is based on obtaining the solution to a nonlinear inequality constrained minimization problem, however, which they found to be computationally challenging to actually solve. Specifically, Swofford and Whitney needed to divide up their dataset, which had 62 observations, into two overlapping subsamples to avoid exceeding 40 observations in order to solve the minimization problem using a CRAY super-computer. Consequently, their test has not been widely used in empirical work. Subsequent advances in computing technology have reduced these difficulties, however, making it more practical to use Swofford and Whitney’s test. For example, Jones et al. (2005b) were able to run the test on the full 62 observation dataset from Swofford and Whitney (1994) on a standard PC. Other recent research has focused on making improvements to the more widely used sequential test approach, although that approach is intrinsically biased towards rejecting weak separability. Specifically, Fleissig and Whitney (2003) develop a method for computing certain indexes (referred to as Afriat indexes in this chapter), which are used in sequential weak separability tests. Using this new method, they obtain impressive results from their sequential weak separability test relative to the sequential weak separability test in NONPAR. In an empirical study, Jones et al. (2005a) found that Fleissig and Whitney’s test produced the same accept/reject results as Swofford and Whitney’s weak separability test in most cases, although they found one case where Fleissig and Whitney’s test incorrectly rejected weak separability. Given that Fleissig and Whitney’s test is less computationally burdensome and runs much more quickly, a sensible approach is to run Swofford and Whitney’s test only if Fleissig and Whitney’s test has already rejected weak separability.2 This is the approach adopted by Elger et al. (2007).

2 Fleissig and Whitney’s (2003) test is based on solving a linear programming problem, whereas Swofford and Whitney’s (1994) test is based on solving a nonlinear inequality constrained minimization problem. See Binner et al. (2006) for additional empirical work using Fleissig and Whitney’s weak separability test.

Nonparametric Tests for Separability

37

Jones and Stracca (2006) provide a test of the necessary and sufficient conditions for additive separability. Their new test procedure is based on obtaining the solution to a linear inequality constrained minimization problem. If the objective function for the minimization problem can be reduced to zero, then the test accepts additive separability. The new test can be integrated into an overall testing strategy in two possible ways. Logically speaking, testing for additive separability prior to running any tests for weak separability would make sense, since additive separability implies block-wise weak separability. Thus, if additive separability is accepted, then further testing is unnecessary. On the other hand, in practice it makes sense to run Fleissig and Whitney’s test prior to either the additive separability test or Swofford and Whitney’s weak separability test, since of these tests it is the least computationally burdensome and time consuming to run. We illustrate Fleissig and Whitney’s weak separability test and Jones and Stracca’s additive separability test on data generated from Barnett and Choi’s (1989) WS-branch utility tree model, under the assumption of additive separability, over a range of different elasticities of substitution. The simulated data consists of 60 observations of prices and quantities for three goods. We ran both tests in FORTRAN using standard IMSL subroutines. We found that Fleissig and Whitney’s test correctly identified weak separability in all 35 of the cases that we examined. We also found that Jones and Stracca’s test correctly identified additive separability in 33 of the 35 cases. Thus, both tests are fairly straightforward to implement and appear to perform well in practice. The remainder of the chapter is organized as follows: In Section 2, we discuss the necessary and sufficient conditions for the various separability hypotheses. In Section 3, we discuss nonparametric tests of these conditions. We also discuss stochastic generalizations of revealed preference methods. In Section 4, we describe the WS branch utility tree model and provide empirical results. Section 5 concludes the chapter.

2. Separability Hypotheses and Necessary and Sufficient Conditions In this section, we define the various separability hypotheses and briefly state and interpret the corresponding necessary and sufficient nonparametric conditions. 2.1. Notation and Definitions Let z = (z1 , . . . , zm ) and y = (y1 , . . . , yn ) denote two blocks of goods of size m and n respectively and let u(z, y) denote a utility function defined over all m + n of these goods. We can define various separability hypotheses in terms of these two blocks of goods. The utility function is said to be weakly separable in the y block if there exists a macro-function, u, ¯ and a sub-utility function, V , such that u(z, y) = u(z, ¯ V (y)).

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The utility function is said to be block-wise weakly separable in the two blocks if there exists a macro-function, u, ¯ and sub-utility functions, U and V , such that u(z, y) = u(U ¯ (z), V (y)). Finally, the utility function is said to be additively separable (or, more rigorously, preferences are said to be additively separable) if there is some monotonic transformation, f , such that f (u(z, y)) = U (z) + V (y), where U and V are utility functions defined over the two blocks of goods; see Varian (1983).3 Additive separability implies block-wise weak separability, which in turn implies that each block of goods is weakly separable from the other one.4 2.2. Rationalizing Observed Data with a Utility Function Varian (1982, 1983) provides the basic theory underpinning the use of nonparametric revealed preference methods. Specifically, nonparametric tests can be used to determine whether or not a set of observed data on prices and quantities can be rationalized by a well-behaved utility function. In this subsection, we briefly discuss Varian’s nonparametric test of the utility maximization hypothesis. We begin by setting notation. We assume that there are m + n goods divided into two blocks of size m and n. There are T observations of prices and quani ) and yi = (y i , . . . , y i ) tities for each block of goods. Let zi = (z1i , . . . , zm n 1 denote the ith observed quantities for the two blocks of goods, where i runs i ) and ri = (r i , . . . , r i ) denote the ith from 1 to T . Let pi = (p1i , . . . , pm n 1 observed prices corresponding to the z and y blocks respectively. Finally, let i , y i , . . . , y i ) and vi = (p i , . . . , p i , r i , . . . , r i ) denote the comxi = (z1i , . . . , zm n m 1 n 1 1 bined quantity and price data for all m + n goods. Varian (1982) provides several equivalent necessary and sufficient conditions for a dataset to be rationalized by a well-behaved utility function, which he refers to as Afriat’s theorem. We say that a utility function, w(x), rationalizes the observed price and quantity data (vi , xi ) if w(xi ) w(x) for all x such that vi x vi xi for all i. The theorem is as follows:

3 See

also Deaton and Muellbauer (1980) for further discussion. The latter two concepts can be easily generalized to include more than two blocks. 4 Blundell and Robin (2000) develop a generalization of weak separability called latent separability, which preserves some of the desirable properties of weak separability without the strong assumption of mutual exclusivity of commodity groupings. One interpretation of latent separability is that purchased goods are used in the production of more than one intermediate good. Latent separability is equivalent to weak separability in latent rather than purchased goods. Crawford (2004) provides necessary and sufficient nonparametric conditions for latent separability. As noted by Deaton and Muellbauer (1980, p. 126), there are many types of separability besides weak separability. Specifically, under implicit or quasi-separability the prices in the cost (expenditure) function are broken up into separable groups, as opposed to the quantities in the direct utility function being broken up into separable groups as under weak separability. See Deaton and Muellbauer (1980, pp. 133– 137) for further discussion.

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Theorem 1 (Afriat’s theorem). The following conditions are equivalent: (i) There exists a nonsatiated utility function, which rationalizes the data. (ii) The data satisfy the generalized axiom of revealed preference (GARP).5 (iii) There exist numbers W i , τ i > 0 such that ∀i, j = 1, . . . , T . W i W j + τ j v j x i − xj (1) (iv) There exists a nonsatiated, concave, monotonic, continuous utility function, which rationalizes the data. In addition to a formal proof of the theorem, Varian (1982, pp. 969–970) provides a heuristic argument to give more economic meaning to the inequalities in (iii), which we refer to as Afriat inequalities. To see why (iv) implies (iii), suppose that a nonsatiated, concave, and differentiable utility function, w(x) rationalizes the observed price and quantity data. Concavity of the utility function implies that w xi w xj + ∇w xj xi − xj (2) for any pair of observations i and j . Further, utility maximization implies that τ j vj = ∇w(xj ), where τ j is the Lagrange multiplier on the budget constraint for the j th observation. Letting W i = w(xi ) for all i, we obtain (1) by substituting the utility maximization conditions into (2). This shows that the Afriat conditions are a necessary condition for utility maximization assuming differentiability of the utility function (the formal proof of the theorem does not, however, involve differentiability). Varian (1982) motivates the corresponding sufficiency result as follows: Note that w(x) W j + τ j (x − xj ) for all j , which provides T overestimates of the utility function (one for each observation) at some point x. Thus, the lower envelope of these overestimates, minj {W j + τ j vj (x − xj )}, provides a reasonable measure of the utility of x, which has the required properties. It follows from these arguments that W i can be interpreted as a measure of the utility provided by each observed bundle, xi . Similarly, τ i (the Lagrange multiplier) is a measure of the marginal utility of income at each observation. 2.3. Necessary and Sufficient Conditions for Weak Separability Varian (1983) provides additional necessary and sufficient nonparametric conditions for a dataset to be consistent with weak and additive separability. The

5 Let xi Rxj mean that xi is revealed preferred to xj . GARP states that xi Rxj implies vj xj vj xi , i.e. if xi is revealed preferred to xj , then it cannot be the case that xj is strictly directly revealed preferred to xi .

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necessary and sufficient conditions for weak separability are described by the following theorem: Theorem 2 (Varian’s Weak Separability Conditions). The following three conditions are equivalent: (i) There exists a weakly separable (in the y block), nonsatiated, concave, monotonic, continuous utility function, which rationalizes the data. (ii) There exist numbers W i , V i , τ i , μi > 0 such that τ j (V i − V j ) W i W j + τ j p j z i − zj + μj ∀i, j = 1, . . . , T , i ∀i, j = 1, . . . , T . V V j + μ j r j y i − yj

(3) (4)

i , 1/μi ) and x ˜i = (iii) The data (ri , yi ) and (˜vi , x˜ i ), where v˜ i = (p1i , . . . , pm i i i i i (z1 , . . . , zm , V ), satisfy GARP, for some choice of (V , μ ) satisfying (4).

There exist numbers V i , μi > 0 satisfying (4) if and only if the observed price and quantity data for the y block of goods, (ri , yi ), satisfy GARP. This is due to the fact that (4) are the appropriate Afriat inequalities for the y block. Further, (3) and (4) together imply that the combined price and quantity data for all m + n goods, (vi , xi ), also satisfy GARP.6 Intuitively, if the combined data violate GARP, then it cannot be rationalized by any utility function – weakly separable or otherwise. Thus, two necessary conditions for weak separability of the y block of goods are that the data (ri , yi ) and (vi , xi ) must satisfy GARP.7 Condition (iii) can be interpreted as follows: Suppose that the data satisfy the two necessary conditions for weak separability and, consequently, can be rationalized by a well-behaved utility function. Further, suppose that we replace the quantity data for the y block of goods, yi , with V i and we replace the corresponding price data, ri , with 1/μi , where V i and μi satisfy (4). Correspondingly, Fleissig and Whitney (2003, p. 134) refer to V i as a “group quantity index” and 1/μi as a “group price index”. The price and quantity data for all goods resulting from these replacements is then given by (˜vi , x˜ i ). Weak separability implies that there exist some choice of indexes, (V i , μi ), satisfying (4), such that (˜vi , x˜ i ) satisfies GARP.8 In other words, assuming that the data are consistent with utility

substituting (4) into (3), it follows that the data (vi , xi ) satisfy (1). and Whitney (1994, pp. 238–239) provide a heuristic derivation of the inequalities in (ii), which is along the same lines as the one in Varian (1982), which we discussed above. 8 We note, however, that it is not generally correct to interpret V i and 1/μi as quantity and price indexes in the usual aggregation-theoretic sense. V i can be interpreted as a measure of 6 By

7 Swofford

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maximization, weak separability implies that there exist group quantity and price indexes such that if the quantities and prices for the separable block of goods are replaced by these group quantity and price indexes, then the data can still be rationalized by a well-behaved utility function. 2.4. Necessary and Sufficient Conditions for Additive Separability The necessary and sufficient conditions for weak separability are well known and have been tested in a variety of different contexts. Less well known are the necessary and sufficient conditions for additive separability contained in the following theorem: Theorem 3 (Varian’s Additive Separability Conditions). The following two conditions are equivalent: (i) There exist two nonsatiated, concave, monotonic, continuous utility functions, U and V (defined over the z and y blocks respectively), whose sum rationalizes the data. (ii) There exist numbers U i , V i , τ i > 0 such that ∀i, j = 1, . . . , T , U i U j + τ j p j z i − zj (5) V i V j + τ j r j y i − yj (6) ∀i, j = 1, . . . , T . To interpret these inequality conditions, suppose that U and V are nonsatiated, concave, and differentiable utility functions whose sum rationalizes the data. Since, U and V are concave in z and y respectively, it follows that U (zi ) U (zj ) + ∇U (zj )(zi − zj ) and V (yi ) V (yj ) + ∇V (yj )(yi − yj ) for any pair of observations i and j . Utility maximization implies that τ j pj = ∇U (zj ) and τ j rj = ∇V (yj ), where τ j is the Lagrange multiplier on the budget constraint for the j th observation. Letting U i = U (zi ) and V i = V (yi ) for all i, we obtain (5) and (6) by substituting the utility maximization conditions into the corresponding inequalities implied by concavity.9 The necessary and sufficient conditions for additive separability also represent sufficient conditions for block-wise weak separability. In particular, if there exist numbers, U i , V i , τ i > 0, satisfying (5) and (6), then it can easily be shown that

the subutility provided by the observed quantities of goods in the y block (see Swofford and Whitney, 1994, p. 238), but the subutility function is not a quantity aggregate in aggregation theory unless it is linearly homogeneous. See Barnett (1987) for detailed discussion of aggregation theory with and without homotheticity. In addition, the product of V i and 1/μi does not generally equal ri yi (expenditure on the y block) for all i, unless the data (ri , yi ) are consistent with homotheticity, see Varian (1983, p. 103). 9 This argument can be generalized to allow for more than two blocks in the obvious way.

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there also exist numbers, W i , U i , V i , τ i , λi , μi > 0, satisfying the following conditions: τ j (U i − U j ) τ j (V i − V j ) + ∀i, j = 1, . . . , T , λj μj U i U j + λj p j zi − zj ∀i, j = 1, . . . , T , i j j j i j ∀i, j = 1, . . . , T V V +μ r y −y Wi Wj +

(7) (8) (9)

which are the obvious generalizations of (3) and (4) for block-wise weak separability.10 Given U i , V i , τ i > 0, which satisfy (5) and (6), the additional numbers needed to satisfy (7)–(9) can be obtained by defining W i = U i + V i and λi = μi = τ i for all i. Similar arguments to those used previously imply that three necessary conditions for the data to be consistent with either block-wise weak separability or additive separability are that the price and quantity data for the two blocks of goods, (ri , yi ) and (pi , zi ), and the combined price and quantity data (vi , xi ) must all satisfy GARP.

3. Nonparametric Separability Tests In this section, we discuss nonparametric tests of weak and additive separability as well as stochastic generalizations of nonparametric revealed preference methods. 3.1. Sequential Weak Separability Tests The necessary and sufficient conditions for weak separability can be tested sequentially using GARP tests. As discussed previously, two necessary conditions for weak separability (of the y block) are that the data (ri , yi ) and (vi , xi ) satisfy GARP. The initial step in a sequential test is to determine if these necessary conditions are satisfied. If they are satisfied, then a sufficient condition can be easily tested in a subsequent step as follows: First, construct a particular choice of Afriat indexes, (V i , μi ), satisfying the appropriate Afriat inequalities: V i V j + μj rj (yi − yj ) for all i, j . Next, using this particular choice, test (˜vi , x˜ i ) for GARP. If this GARP condition is also satisfied, then the data are consistent with the necessary and sufficient conditions for weak separability. If any of the three GARP conditions are not satisfied, then weak separability is rejected by the sequential test.

we treat U i and V i as group quantity indexes for the z and y blocks of goods respectively, with corresponding group price indexes 1/λi and 1/μi , then (7) is equivalent to the condition that the resulting dataset (1/λi , 1/μi ; U i , V i ) satisfies GARP. 10 If

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Sequential tests are inherently biased towards rejecting separability. Varian’s weak separability theorem (condition (iii) of Theorem 2) merely states that there exist some choice of Afriat indexes, (V i , μi ), such that (˜vi , x˜ i ) satisfies GARP. The Afriat indexes are not unique, however, which implies that (˜vi , x˜ i ) could violate GARP for a particular choice of Afriat indexes even if the data are consistent with maximization of a weakly separable utility function. The data are only inconsistent with weak separability if (˜vi , x˜ i ) violates GARP for all possible choices of Afriat indexes. 3.2. Joint Test of the Necessary and Sufficient Conditions for Weak Separability Swofford and Whitney (1994) derive an unbiased joint test of the necessary and sufficient conditions for weak separability. The test procedure is based on minimizing the following objective function: F =

T

τ i − μi φ i

2

(10)

i=1

in strictly positive numbers, W i , V i , τ i , μi , φ i , subject to the following two sets of inequality constraints: ∀i, j = 1, . . . , T , W i W j + τ j p j zi − zj + φ j V i − V j (11) i j j j i j V V +μ r y −y (12) ∀i, j = 1, . . . , T . If a feasible solution can be found such that the objective function is minimized to exactly zero, then φ i = τ i /μi for all i and, therefore, (11) and (12) are equivalent to (3) and (4) respectively. In that case, the data are consistent with maximization of a weakly separable utility function. If there is no feasible solution or if the objective function cannot be minimized to zero, then weak separability is rejected. In practice, the minimization problem is solved using a numerical optimization procedure. Weak separability can be checked using a convenient post-test based on the solution. The post-test consists of testing (˜vi , x˜ i ) for GARP using the Afriat indexes, (V i , μi ), obtained from the feasible solution to the minimization problem. If GARP is satisfied, then weak separability is accepted. If GARP is not satisfied, or if no feasible solution exists, then weak separability is rejected. The post test just confirms that (11) is equivalent to (3), see Jones et al. (2005b) for further discussion. The procedure can be given an attractive economic interpretation by defining a new variable, θ i , such that φ i = (τ i + θ i )/μi . Viewed in terms of this new variable, the minimization attempts to find a feasible such that θ i = 0 for T solution i 2 all i, since the objective function becomes F = i=1 (θ ) . Swofford and Whitney (1994) show that if an agent maximizes a weakly separable utility function subject to a standard budget constraint and an additional expenditure constraint

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on the y block, then θ i can be interpreted as the Lagrange multiplier (or shadow price) associated with the additional constraint. Similarly, τ i is the Lagrange multiplier associated with the standard budget constraint. Thus, θ i /τ i represents the ratio of the shadow price of the expenditure constraint on the y block to the marginal utility of income at each observation. The necessary and sufficient conditions for weak separability are satisfied if there is a feasible solution to the constraints, such that the shadow price of the additional expenditure constraint, θ i , is zero for all i. This would mean that the constrained value of expenditure on the y block always corresponds to its optimal value from a standard utility maximization problem or, in other words, there is complete adjustment of expenditure in all periods. If there is a feasible solution to the constraints, but θ i is nonzero for some i, then there is incomplete adjustment of expenditure on the y block for some observations. Consequently, Swofford and Whitney (1994, p. 244) interpret the average value of |θ i /τ i | over all i as a measure of incomplete adjustment of expenditure on the y block.11 3.3. Weak Separability Tests in Practice Varian (1982) provides an iterative algorithm, which can be used to construct Afriat indexes for a dataset that satisfies GARP. The sequential test of weak separability in the NONPAR computer program is based on Afriat indexes constructed using that algorithm. As discussed in the Introduction, Barnett and Choi (1989) found that the weak separability test in NONPAR is extremely biased towards rejecting separability. Swofford and Whitney’s (1994) weak separability test is attractive, because it jointly tests the necessary and sufficient conditions for weak separability and is, consequently, not biased towards rejection. They found, however, the underlying minimization problem on which the test is based to be computationally challenging to actually solve. There are T (T − 1) nontrivial linear inequality constraints and, more significantly, there are T (T − 1) nontrivial nonlinear inequality constraints (in addition, there are 5T sign restrictions) in the problem. Thus, there are 7,564 linear and nonlinear inequality constraints for a problem with 62 observations. Swofford and Whitney had to divide up their sample of 62 observations into two overlapping sets of 40 observations each (reducing the number of inequality constraints to 3,120) in order to solve the minimization problem using a CRAY super-computer. Consequently, their test has not being widely used in empirical work. Fleissig and Whitney (2003) propose a new numerical method of constructing Afriat indexes, which they use in their sequential weak separability test. Their

11 Based

on this interpretation, Jones et al. (2005b) propose minimizing the following al = T (θ i /τ i )2 , which they find usually results in lower ternative objective function: F i=1 measures of incomplete adjustment relative to the original objective function.

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method is based on the idea that a natural starting point is to use a superlative quantity index, such as the Törnqvist index, to obtain estimates for V i and a corresponding range for μi satisfying the Afriat inequalities. By definition, a superlative quantity index can provide a second-order approximation to the true quantity aggregate (Diewert, 1976). A superlative index may, however, require small adjustments in order to actually satisfy the Afriat inequalities due to several factors including third and higher order approximation errors and measurement errors in the data (Fleissig and Whitney, 2003, p. 135). Fleissig and Whitney provide a linear programming (LP) problem, which seeks to minimize the adjustments (in absolute value terms) needed for the Törnqvist index to satisfy the Afriat inequalities. The problem can be easily solved using a sparse matrix LP solver such as the one in IMSL.12 Assuming that the two necessary conditions for weak separability are satisfied, Fleissig and Whitney’s weak separability test consists of testing (˜vi , x˜ i ) for GARP using Afriat indexes derived from the solution to their LP problem. They found that this sequential test performed well in a Monte Carlo experiment based on Cobb–Douglas utility functions, thereby addressing the problem encountered by Barnett and Choi (1989). Nevertheless, their test still has an inherent bias towards rejecting separability, because the necessary and sufficient conditions are not tested jointly. Recent advances in computing technology have made Swofford and Whitney’s test more practical to use, especially if the number of observations in the dataset is not too large. For example, Jones et al. (2005b) were able to run the test on a PC using the full 62 observation dataset from Swofford and Whitney (1994). They used an advanced commercial solver (FFSQP) to solve the underlying minimization problem.13 The test remains impractical, however, for datasets with very large sample sizes, since the number of nonlinear inequality constraints is a function of the square of the sample size. Jones et al. (2005a) ran both Fleissig and Whitney’s test and Swofford and Whitney’s test on quarterly US macroeconomic data from 1993 to 2001. They found that the two tests produced the same accept/reject results in 20 out of the 21 cases they examined. In the remaining case, however, Swofford and Whitney’s test identified a weakly separable structure, which was incorrectly rejected by Fleissig and Whitney’s test.14 In practice, therefore, Fleissig and Whitney’s weak separability test appears to produce fairly reliable results. Given that Fleissig and

12 See

Fleissig and Whitney (2003) or Jones et al. (2005a) for more detailed technical discussion of the LP problem. 13 See Zhou et al. (1997) for a description of the solver. 14 Jones et al. (2005a, p. 501) distinguish between “weak separability with complete adjustment” and “weak separability with incomplete adjustment” when interpreting Swofford and Whitney’s test. If the data are consistent with Varian’s necessary and sufficient conditions for weak separability, then θ i equals zero for all i, as discussed above. They refer

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Whitney’s test is also less computationally burdensome and runs more quickly, a sensible approach is run Swofford and Whitney’s test only if Fleissig and Whitney’s test has already rejected weak separability. This is the approach adopted by Elger et al. (2007). 3.4. Test of the Necessary and Sufficient Conditions for Additive Separability Jones and Stracca (2006) propose a test of Varian’s necessary and sufficient conditions for additive separability, which can also be interpreted as testing a sufficient condition for block-wise weak separability. The test is based on minimizing the following objective function: T i τ − μi 2 G= (13) μi i=1

in strictly positive numbers, U i , V i , τ i , μi , subject to the following two sets of inequality constraints: U i U j + τ j p j z i − zj (14) ∀i, j = 1, . . . , T , i j j j i j V V +μ r y −y (15) ∀i, j = 1, . . . , T . If a feasible solution is found such that the objective function is minimized to exactly zero, then τ i = μi for all i and, therefore, (14) and (15) are equivalent to (5) and (6) respectively. In that case, the data are consistent with maximization of an additively separable utility function. If there is no feasible solution or if the objective function cannot be minimized to zero, then additive separability is rejected. Thus, in theory, additive separability is accepted if the objective function can be minimized to exactly zero. In practice, however, due to the convergence properties

to this as weak separability with complete adjustment, whereas we simply refer to this as weak separability in this chapter. If a feasible solution is identified, but θ i is nonzero for some i, then they refer to this as weak separability with incomplete adjustment, since θ i can be interpreted as the shadow price of an additional expenditure constraint affecting the y block as discussed above; see also Jones et al. (2005b). In such cases, Jones et al. (2005a) report the average value of |θ i /τ i | following Swofford and Whitney (1994). When comparing the results from Fleissig and Whitney’s test to results from Swofford and Whitney’s test, only weak separability with complete adjustment is relevant. The reason is that only Swofford and Whitney’s test can be interpreted in terms of incomplete adjustment. (Stated differently, in this terminology, Fleissig and Whitney’s test is a test of weak separability with complete adjustment. It cannot be used to test for weak separability with incomplete adjustment.) As discussed by Jones et al. (2005b), the time profile of |θ i /τ i | may be useful in helping to point out particular observations that are associated with violations of weak separability (with complete adjustment).

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of any numerical optimization procedure, this condition can only be approximately satisfied in an application. Unlike with Swofford and Whitney’s test, there is no convenient post-test (in the form of a GARP condition) for the additive separability test, see Varian (1983, p. 108) for related discussion. In applications, therefore, the minimized value of G must be compared to an extremely small number, which the tester is willing to treat as being approximately zero. In contrast, block-wise weak separability can be easily checked by testing whether or not the data (1/τ i , 1/μi ; U i , V i ) satisfy GARP using the indexes obtained from the solution to the minimization problem (see footnote 10). 3.5. Stochastic Generalizations The discussion has so far focused on deterministic revealed preference test procedures. These procedures are not hypothesis tests in the statistical/econometric sense. If, for example, GARP is satisfied, then there exists a well-behaved (nonsatiated, concave, monotonic and continuous) utility function that rationalizes the data. In this case, the observed data can be explained without error by the utility maximization model. In contrast, parametric demand models are estimated under particular assumptions about the error structure and many commonly used parametric demand models can violate the theoretical regularity conditions from microeconomics. On the other hand, if a dataset violates GARP, then it cannot be rationalized by any nonsatiated utility function. GARP violations may result, however, from a variety of factors including small measurement errors in the data, small optimization errors, heterogeneity in preferences (either across consumers if applied to cross-sectional data or across time if applied to time series data), as well as other factors (see Gross, 1995).15 As Gross (1995, p. 701) notes, “. . . revealed preference methods to date have suffered from inadequate statistical procedures and goodness-of-fit metrics”. Several approaches have been advanced, which attempt to determine whether or not violations of revealed preference axioms are significant in a statistical sense and/or assess the severity of such violations. In this subsection, we focus on approaches that are related to GARP violations. One approach is based on the idea that if only small adjustments to the data are sufficient to eliminate all of the GARP violations, then intuitively we would probably not want to reject the utility maximization hypothesis. Building on this idea, Varian (1985) proposes calculating the minimal adjustments to the quantity data, which are needed in order to render it consistent with a specific revealed preference axiom, given the observed prices. These adjustments can be computed

15 Of course, a lack of GARP violations does not necessarily mean that the data are truly consistent with utility maximization. The lack of violations might instead result from either large income variation or low relative price variation, which would imply that budget lines infrequently intersect. Again, see Gross (1995).

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by minimizing the sum of squared differences (or proportional differences) between the adjusted quantities and the observed quantities, subject to the constraint that the adjusted data satisfy the specific axiom. For GARP, this takes the form of jointly solving for adjusted quantities and Afriat indexes, which together satisfy the Afriat inequalities. See Jones et al. (2005a, 2005b) and Jones and de Peretti (2005) for empirical results based on this approach. Varian (1985) formulates a test of the null hypothesis that the true data (i.e. the data measured without error) satisfy the revealed preference axiom, but the observed data violate it due to i.i.d. measurement errors in the quantity data. The minimized sum of squares is, by definition, less than the true sum of squared measurement errors under the null. If the true measurement errors are assumed to be normally distributed, this result can be used to produce a chi-square test statistic. The test requires, however, that the tester knows the true standard deviation of measurement errors, which would usually not be the case.16 Alternatively, when the true standard deviation is not known, the minimized sum of squares can be used to calculate a bound on the unknown standard deviation such that if the true standard deviation is greater than or equal to this bound then the null hypothesis cannot be rejected, see Varian (1985, p. 450). In empirical work, good judgment may often be more than adequate to make a decision and a formal test procedure may be unnecessary. If, for example, only extremely small adjustments of the data are needed in order for it to comply with GARP and the data are known to be very noisy, then one would probably not want to reject the null hypothesis of utility maximization. The minimal adjustments of the quantity data needed to render a dataset consistent with GARP can be difficult to actually compute, since the Afriat inequalities are nonlinear when viewed as being functions of both the Afriat indexes and the adjusted quantities (see Jones and de Peretti, 2005, p. 618; for related discussion).17 de Peretti (2005) proposes a much simpler iterative procedure, which also adjusts the quantity data to be consistent with GARP. One of the ways that de Peretti’s adjustment procedure differs from the one proposed by Varian is that it only adjusts bundles that are directly involved in GARP violations. Jones and de Peretti (2005) discuss the two adjustment procedures (and the associated test procedures) and compare and contrast them in an empirical application.18

16 de Peretti (2005) provides an alternative statistical test based on adjusted data, which does not have this unattractive feature. His test requires the assumption, however, that the errors computed by the adjustment procedure inherit the i.i.d. property of the true measurement errors under the null hypothesis. 17 Varian’s (1985) empirical application was based on adjusting data to satisfy the weak axiom of cost minimization (WACM), which is a less difficult problem to solve. 18 In addition to calculating the number and rate of GARP violations, Jones and de Peretti (2005) also gauge the severity of GARP violations in terms of inefficiency indexes. Intuitively, the inefficiency index can be thought of as measuring how close observed consumer

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An alternative to adjusting the data to render it consistent with GARP would be to determine the number of observations that are causing most of the violations. If, for example, deleting just a few observations in a large dataset is sufficient to eliminate all of the violations, then one might want to consider whether idiosyncratic factors associated with those specific observations are playing a role. Another alternative is to consider the total number of violations. In that regard, Fleissig and Whitney (2003, p. 141) found in their Monte Carlo experiment that “[m]easurement error generally produces relatively few violations of GARP and weak separability, and thus, surprisingly, measurement error may not be the likely source of the rejection when there are many violations.”.

Gross (1995) proposes a formal statistical test, which is based on partitioning the data into two subsets referred to as CS and VS. The violator subset (VS) contains the observations that are causing most of the violations, while the consistent subset (CS) is its complement. A test statistic can be formulated by estimating the fraction of expenditure that is wasted by the VS observations in maximizing utility consistent with CS. Gross recommends using bootstrap methods to estimate the distribution of the test statistic. Finally, Fleissig and Whitney (2005) propose two methods for assessing the statistical significance of GARP violations. Their methods are based on adding measurement errors to the observed data to create a large number of perturbed datasets. Suppose that a dataset violates GARP in an empirical application. For their least lower bound test, you add measurement errors to the quantity data while holding the expenditure and price data constant and test this perturbed dataset for GARP. This step is repeated a large number of times. The test statistic is the fraction of perturbed datasets that do not violate GARP. They propose the following test: If more than α% of the perturbed datasets have no GARP violations, then you accept the null of utility maximization and conclude that measurement errors are the cause of the violations (see Fleissig and Whitney, 2005, p. 359). In their simulations, Fleissig and Whitney (2005) use an α of 5% in the test. Their upper bound test is based on adding slack terms into the Afriat inequalities, which allow them to be violated. A test statistic can be constructed by minimizing the maximum slack term required for the data to satisfy all of the constraints, which can be evaluated by comparing its value to a simulated distribution. A promising direction for future research would be to extend these various approaches to assessing the significance of violations of weak separability. For example, Swofford and Whitney (1994, pp. 247–248) found violations of weak

choices are to the maximizing choices. If the inefficiency index equals zero, then the data satisfy GARP; see Varian (1990) for further details. Varian (1990) also emphasizes the relative importance of determining whether a consumer’s violation of the optimizing model is economically significant as opposed to being statistically significant. See also Chalfant and Alston (1988).

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separability for a particular preference structure using their weak separability test and explored the possibility that those violations could be attributed to measurement errors in the data. They concluded, however, that their analysis was just suggestive of how measurement errors may have affected the results from their test and go on to argue that a more satisfactory solution than the one they pursued “. . .would be to develop a stochastic test that explicitly accounts for measurement error along the lines suggested by Epstein and Yatchew (1985) and Varian (1985)”.19 Echoing this conclusion, Jones and de Peretti (2005, pp. 626–627) suggest that a promising direction for further research in this area would be to generalize the two measurement error approaches explored in their paper to test violations of weak separability for significance.20

4. Empirical Results In this section, we illustrate the nonparametric separability tests proposed by Fleissig and Whitney (2003) and Jones and Stracca (2006) on price and quantity data generated from the three good WS-branch utility tree model of Barnett and Choi (1989). 4.1. WS-Branch Utility Function In a Monte Carlo simulation study, Barnett and Choi (1989) generate data using the WS-branch utility tree model. In homothetic form, with three goods, the direct utility function is as follows: u = U q1 (x1 , x2 ), x3 ρ 2ρ 1/(2ρ) = A B11 q1 (x1 , x2 )2ρ + 2B12 q1 (x1 , x2 )ρ x3 + B22 x3 (16) , where

1/(2δ) q1 (x1 , x2 ) = A11 x12δ + 2A12 x1δ x2δ + A22 x22δ

(17)

and ρ < 1/2, δ < 1/2, Aij > 0 and Bij > 0 for i, j = 1, 2. In addition, A11 + 2A12 + A22 = 1 and B11 + 2B12 + B22 = 1. The parameter, A > 0, can be

19 Swofford

and Whitney (1994), following Manser and McDonald (1988) and Patterson (1991), also consider the sensitivity of their empirical results to measurement error when accepting weak separability. In particular, they identified a utility structure, which passed their weak separability test in two overlapping sample periods. They disturbed the data with measurement errors and tabulated the percentage of cases where the structure still passed. As the magnitude of measurement errors increased, they found that the percentage of cases where the structure continued to pass declined. The finding of separability also appeared to be more robust to measurement errors in the earlier period than in the latter one. 20 Jones and de Peretti (2005) cite a working paper by de Peretti along these lines.

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51

normalized to one. These parameter restrictions are used to impose monotonicity and quasi-concavity. The WS-branch utility function is block-wise weakly separable, with x1 and x2 representing the first block and x3 representing the second block. The model contains several interesting special cases: If B12 = 0, then the utility function is also additively separable in these two blocks. If A12 = 0, then the subutility function, q1 , is additively separable in x1 and x2 . If A12 = B12 = 0 and δ = ρ, 2ρ 2ρ 2ρ then u is a pure CES utility function: (B11 A11 x1 + B11 A22 x2 + B22 x3 )1/(2ρ) . In that case, the utility function is additively separable in all possible blockings of the three goods. Finally, if A12 = B12 = 0 and δ = ρ = 0, then u is Cobb– Douglas (in the limit). 4.2. Calibration Barnett and Choi (1989) simulate data based upon 60 pre-selected observations of quantities for three goods and total expenditure. These three quantities are each normalized to equal 20 at the 31st observation and expenditure is normalized to equal 60 at the 31st observation. The utility function is calibrated to produce prices that are equal to one for all three goods at the 31st observation and to achieve desired elasticities of substitution for both the macro function, U , and the subutility function, q1 . They set A12 and B12 equal to either 0 or 0.1. We closely follow their approach in calibrating our model. Barnett and Choi added white noise disturbances to their data when evaluating parametric separability tests, but used data generated from a Cobb–Douglas utility function without random disturbances when evaluating the nonparametric weak separability test in NONPAR. We do not incorporate random disturbances into our data, but we consider a wide range of elasticities of substitution for both functions. We obtained the pre-selected quantity and expenditure data used by Barnett and Choi (1989). We convert these data to per-capita terms using the corresponding population series and then carried out the required normalizations.21 We calibrate the model by assuming that A12 = B12 = 0, A11 = A22 = 1/2, B11 = 2/3 and B22 = 1/3. Under these assumptions, the values of δ and ρ can be chosen to achieve desired substitution elasticities at the 31st observation using the following formulas: 1 σ12 = (18) , 1 − 2δ 1 σ13 = (19) 1 − 2ρ 21 The data consist of three categories of US consumption: perishables, semi-durables and services. The data are annual data from 1890 to 1953 excluding 1942 to 1945. We use per-capita quantities and expenditure in our data generation to be consistent with the use of per-capita data in most empirical studies.

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which correspond to substitution elasticities for the subutility function, q1 (i.e. within the aggregator function), and for the macro-function, U (i.e. between aggregates), respectively.22 We calibrate the model for all 25 possible combinations of values for σ12 and σ13 drawn from the set {0.1, 0.6, 1.0, 3.0, 5.0}. These values are taken from Table 1 in Barnett and Choi (1989).23 Given the calibrated values of all parameters and the values of the pre-selected quantities and expenditures, prices can be generated from the inverse demand system for the WS-branch function.24 Under our assumptions, the utility function is additively separable in two blocks. One block consists of x1 and x2 , while the other consists of x3 . If σ12 = σ13 , then the utility function is a pure CES under our assumptions and is, consequently, additively separable in all possible blockings of the three goods. 4.3. Weak Separability Test Results In this subsection, we test for weak separability using Fleissig and Whitney’s (2003) sequential test. For that test, a dataset is found to be consistent with the necessary and sufficient conditions for weak separability if the data (ri , yi ) and (vi , xi ) both satisfy GARP and, in addition, the data (˜vi , x˜ i ) satisfy GARP for the Afriat indexes calculated from the solution to Fleissig and Whitney’s LP problem. We calculate these Afriat indexes in FORTRAN using the double precision IMSL subroutine DSLPRS, which solves sparse matrix LP problems.25 First, we test for weak separability of x1 and x2 (the y block) from x3 (the z block) for the 20 possible combinations of values where σ12 = σ13 . The test results indicate that weak separability is accepted in all 20 cases. Next, we test for weak separability for the 5 combinations of values where σ12 = σ13 . In these cases, the utility function is a pure CES and is weakly separable in all possible blockings. Specifically, x1 and x2 are weakly separable from x3 , x1 and x3 are weakly separable from x2 , and x2 and x3 are weakly separable from x1 . We test all three possibilities for all five elasticity values for a total of 15 cases. Again, the test results indicate that weak separability is accepted in all 15 cases. Thus, Fleissig and Whitney’s weak separability test performs extremely well, correctly identifying weakly separable structure in all of the 35 cases we considered.26 22 See

Barnett and Choi (1989) for the general formulas. (18) and (19) are produced by applying our assumptions to the general formulas at the 31st observation. 23 In practice, we used 0.99 instead of 1.0. Barnett and Choi (1989) also use values of 0.3, 0.8, 1.2 and 1.5 in some of their other tables. 24 The prices of all three goods are equal to one at the 31st observation for our parameter values. 25 See Appendix A.2 of Jones et al. (2005a) for technical details. 26 We also programmed a sequential weak separability test in FORTRAN, which uses Afriat indexes constructed from the iterative algorithm in Varian (1982). We found that

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53

4.4. Additive Separability Test Results In this subsection, we test for additive separability using the test proposed by Jones and Stracca (2006). A dataset is consistent with the necessary and sufficient conditions for additive separability if a feasible solution exists such that the objective function, G, can be minimized to exactly zero. As discussed previously, however, this condition can only be approximately satisfied in practice, due to the convergence properties of any numerical optimization procedure. We solve the minimization problems in FORTRAN using the double precision IMSL subroutine DLCONG, which solves linear inequality constrained problems with a general objective function having an analytic gradient. First, we ran the test for additive separability with x1 and x2 representing one block and x3 the other for the 20 possible combinations of values where σ12 = σ13 . In 19 of these 20 cases, the solver obtained a feasible solution with a minimized objective function of 1.23E−12 or less. The minimized value is less than 1.0E−14 in 17 of these 19 cases.27 In the remaining case, the solver also obtained a feasible solution, but the terminal value of the objective function is 0.00034. Next, we ran the test for the 5 combinations of values where σ12 = σ13 . In these cases, the utility function is a pure CES and is additively separable in all possible configurations of the two blocks: x1 and x2 representing one block and x3 the other; x1 and x3 representing one block and x2 the other; and x2 and x3 representing one block and x1 the other. We tested all three possible configurations for all five elasticity values for a total of 15 cases. In 14 of these 15 cases, the solver obtained a feasible solution with a minimized objective function of 1.23E−12 or less. The value is less than 1.0E−14 in 13 of these 14 cases.28 In the remaining case, the solver also obtained a feasible solution, but the terminal value of the objective function is 0.39. Thus, the IMSL solver is able to obtain feasible solutions with minimized objective functions of less than 1.0E−14 in 30 out of 35 cases and of less than or equal to 1.23E−12 in 33 out of 35 cases. If we are willing to treat 1.23E−12 as being approximately zero, which seems reasonable, then the test correctly indicates that the data are consistent with maximization of an additively separable utility function in 33 of the 35 cases we considered. In the other two cases, however, the solver was not able to minimize the objective function to approximately zero and, therefore, the test incorrectly rejects additive separability. We conclude that the additive separability test performs fairly well in our empirical exercise,

this alternative test rejected weak separability more than half of the time. This corroborates evidence in Barnett and Choi (1989) regarding the severe bias of the NONPAR test, which uses the same algorithm to compute Afriat indexes. 27 In the other two cases, the values are 8.87E−13 and 1.23E−12. 28 In the other case, the value is 1.87E−13.

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though not quite as well as the weak separability test, correctly identifying additively separable structure in all but two of the 35 cases we considered.

5. Conclusions Barnett and Choi (1989) carried out an extensive Monte Carlo simulation study based on the WS-branch utility tree model to examine the capability of four flexible functional form models (including the translog and generalized Leontief models) to make correct inferences about weak separability. Their main conclusion was that none of the parametric forms investigated were well suited for testing weak separability, even though in many cases these parametric models provided good elasticity estimates. They considered, but did not pursue, the use of nonparametric weak separability tests. The nonparametric test procedure that was available to them at the time failed to detect weak separability on data generated from a three good Cobb–Douglas utility function with no random disturbances and was, therefore, deemed highly unsatisfactory. In this chapter, we surveyed the current state of the art in the use of nonparametric separability tests. Significant progress has been made in addressing the problems with the use of these methods. In particular, the problem demonstrated by Barnett and Choi (1989) – namely, that nonparametric weak separability tests are highly biased towards rejecting separability – has been largely addressed. We generated data from the WS-branch utility tree model, under the assumption of additive separability, over a wide range of possible elasticities of substitution. We found that Fleissig and Whitney’s (2003) weak separability test and Jones and Stracca’s (2006) additive separability test were able to correctly detect separable structure (weak and additive respectively) in nearly all of the cases we examined. There seem to be at least two promising avenues for additional research in this area. First, Fleissig and Whitney (2003) consider the impact of random measurement errors on their weak separability test for data generated from Cobb–Douglas utility functions. Additional insights on how measurement errors affect nonparametric separability tests would be gained from a more comprehensive study, which considers a wide range of elasticities of substitution, along the lines of Barnett and Choi’s (1989) study. Second, although recent progress has been made on generalizing nonparametric revealed preference methods to explicitly account for random measurement error and other stochastic factors, more work needs to be done. In particular, much of that work has focused on determining the statistical significance of violations of the utility maximization hypothesis (i.e. of GARP violations), rather than explicitly focusing on weak separability tests. We suggest that this is a promising direction for additional work.

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Acknowledgements David Edgerton and Barry Jones wish to thank the Jan Wallander and Tom Hedelius foundation (Award no. J03/19) for support of this research. We also thank Adrian Fleissig for a helpful discussion regarding his recent paper on the statistical significance of GARP violations (Fleissig and Whitney, 2005).

References Barnett, W. (1987). The microeconomic theory of monetary aggregation. In Barnett, W. and Singleton, K. (Eds), New Approaches to Monetary Economics: Proceedings of the Second International Symposium in Economic Theory and Econometrics. Cambridge: Cambridge University Press, pp. 115–168. Barnett, W. and Binner, J. (2004). Functional Structure and Approximation in Econometrics. Amsterdam: Elsevier. Barnett, W. and Choi, S. (1989). A Monte Carlo study of tests of blockwise weak separability. Journal of Business and Economic Statistics 7, 363–377. Barnett, W. and Serletis, A. (2000). The Theory of Monetary Aggregation. Amsterdam: Elsevier. Belongia, M. and Chalfant, J. (1989). The changing empirical definition of money: Some estimates from a model of the demand for money substitutes. Journal of Political Economy 97, 387–397. Binner, J., Bissondeal, R., Elger, T., Jones, B. and Mullineux, A. (2006). The optimal level of monetary aggregation for the Euro area. Working paper. Blackorby, C., Primont, D. and Russell, R. (1978). Duality, Separability and Functional Structure: Theory and Economic Applications. New York: North-Holland. Blundell, R. and Robin, J.-M. (2000). Latent separability: Grouping goods without weak separability. Econometrica 68, 53–84. Chalfant, J. and Alston, J. (1988). Accounting for changes in taste. Journal of Political Economy 96, 391–410. Crawford, I. (2004). Necessary and sufficient conditions for latent separability. CeMMAP Working Paper CWP02/04. Deaton, A. and Muellbauer, J. (1980). Economics and Consumer Behavior. Cambridge: Cambridge University Press. de Peretti, P. (2005). Testing the significance of the departures from utility maximization. Macroeconomic Dynamics 9, 372–397. Diewert, W.E. (1976). Exact and superlative index numbers. Journal of Econometrics 4, 115–145. Drake, L., Fleissig, A. and Swofford, J. (2003). A semi-nonparametric approach to the demand for UK monetary assets. Economica 70, 99–120. Elger, T., Jones, B., Edgerton, D. and Binner, J. (2007). A note on the optimal level of monetary aggregation in the UK. Macroeconomic Dynamics, in press. Epstein, L. and Yatchew, A. (1985). Nonparametric hypothesis testing procedures and applications to demand analysis. Journal of Econometrics 30, 149–169.

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Fisher, D. and Fleissig, A. (1997). Monetary aggregation and the demond for assets. Journal of Money, Credit, and Banking 29, 458–475. Fleissig, A. and Whitney, G. (2003). A new PC-based test for Varian’s weak separability conditions. Journal of Business and Economic Statistics 21, 133–144. Fleissig, A. and Whitney, G. (2005). Testing for the significance of violations of Afriat’s inequalities. Journal of Business and Economic Statistics 23, 355–362. Fleissig, A., Hall, A. and Seater, J. (2000). GARP, separability, and the representative agent. Macroeconomic Dynamics 4, 324–342. Gross, J. (1995). Testing data for consistency with revealed preference. Review of Economics and Statistics 77, 701–710. Jones, B. and de Peretti, P. (2005). A comparison of two methods for testing the utility maximization hypothesis when quantity data is measured with error. Macroeconomic Dynamics 9, 612–629. Jones, B. and Stracca, L. (2006). Are money and consumption additively separable in the euro area? A non-parametric approach. European Central Bank Working Paper No. 704. Jones, B., Dutkowsky, D. and Elger, T. (2005a). Sweep programs and optimal monetary aggregation. Journal of Banking and Finance 29, 483–508. Jones, B., Elger, T., Edgerton, D. and Dutkowsky, D. (2005b). Toward a unified approach to testing for weak separability. Economics Bulletin 3(20), 1–7. Manser, M. and McDonald, R. (1988). An Analysis of Substitution Bias in Measuring Inflation, 1959–1985. Econometrica 56, 909–930. Patterson, K. (1991). A non-parametric analysis of personal sector decisions on consumption, liquid assets and leisure. Economic Journal 101, 1103–1116. Pollack, R. and Wales, T. (1992). Demand System Specification and Estimation. Oxford: Oxford University Press. Rickertsen, K. (1998). The demand for food and beverages in Norway. Agricultural Economics 18, 89–100. Spencer, P. (1997). Monetary integration and currency substitution in the EMS: The case for a European monetary aggregate. European Economic Review 41, 1403–1419. Swofford, J. and Whitney, G. (1987). Nonparametric tests of utility maximization and weak separability for consumption, leisure, and money. The Review of Economics and Statistics 69, 458–464. Swofford, J. and Whitney, G. (1994). A revealed preference test for weakly separable utility maximization with incomplete adjustment. Journal of Econometrics 60, 235–249. Varian, H. (1982). The nonparametric approach to demand analysis. Econometrica 50, 945–973. Varian, H. (1983). Non-parametric tests of consumer behaviour. Review of Economic Studies 50, 99–110. Varian, H. (1985). Non-parametric analysis of optimizing behavior with measurement error. Journal of Econometrics 30, 445–458. Varian, H. (1990). Goodness-of-fit in optimizing models. Journal of Econometrics 46, 125– 140. Zhou, J., Tits, A. and Lawrence, C. (1997). User’s guide for FFSQP version 3: A Fortran code for solving optimization programs, possibly minimax, with general inequality constraints and linear equality constraints, generating feasible iterates. Institute for Systems Research, University of Maryland, Technical Report SRC-TR-92-107r5, College Park, MD 20742.

Part II Flexible Functional Forms and Theoretical Regularity

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Chapter 4

Bayesian Estimation of Flexible Functional Forms, Curvature Conditions and the Demand for Assets Apostolos Serletisa,∗ and Asghar Shahmoradib a Department

of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4 b Faculty of Economics, University of Tehran, Iran

Abstract This chapter uses Bayesian inference to revisit the demand for money in the United States in the context of five popular locally flexible functional forms – the generalized Leontief, the basic translog, the almost ideal demand system, the minflex Laurent, and the normalized quadratic reciprocal indirect utility function. We pay explicit attention to the theoretical regularity conditions, impose these conditions using recently suggested procedures, and argue that a breakthrough from the current state of using locally flexible specifications that violate theoretical regularity to the use of such specifications that are more consistent with the theory will be through the use of Bayesian inference. We also provide a policy perspective, using parameter estimates that are consistent with full regularity, in that a very strong case can be made for abandoning the simple sum approach to monetary aggregation.

Keywords: generalized Leontief, translog, almost ideal demand system, minflex Laurent, normalized quadratic reciprocal indirect utility function, Bayesian estimation JEL: C3, C13, C51

1. Introduction This chapter focuses on the demand for money in the United States, building on a large body of recent literature, which Barnett (1997) calls the ‘high road’ literature, that takes a microeconomic- and aggregation-theoretic approach to the demand for money. This literature follows the innovative works by Chetty (1969), Donovan (1978) and Barnett (1980, 1983) and utilizes the flexible functional ∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18004-2

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forms approach to investigating the inter-related problems of monetary aggregation and estimation of monetary asset demand functions – see, for example, Ewis and Fisher (1984, 1985), Serletis and Robb (1986), Serletis (1987, 1988, 1991), Fisher and Fleissig (1994, 1997), Fleissig and Serletis (2002), Serletis and Rangel-Ruiz (2005) and Serletis and Shahmoradi (2005, 2007) among others. However, the usefulness of flexible functional forms depends on whether they satisfy the theoretical regularity conditions of positivity, monotonicity, and curvature, and in the older monetary demand systems literature there has been a tendency to ignore regularity – see, for example, Table 1 (and the related discussion) in Serletis and Shahmoradi (2007). In fact, as Barnett (2002, p. 199) put it in his Journal of Econometrics Fellow’s opinion article, without satisfaction of all three theoretical regularity conditions (of positivity, monotonicity and curvature) “. . . the second-order conditions for optimizing behavior fail, and duality theory fails. The resulting first-order conditions, demand functions, and supply functions become invalid.”. Motivated by these considerations, recently Serletis and Shahmoradi (2005, 2007) pay explicit attention to theoretical regularity. They argue that unless regularity is attained by luck, flexible functional forms should always be estimated subject to regularity, as suggested by Barnett (2002) and Barnett and Pasupathy (2003). In fact, Serletis and Shahmoradi (2005) use the globally flexible Fourier and AIM functional forms and impose global curvature, using methods suggested by Gallant and Golub (1984). Serletis and Shahmoradi (2007) follow Ryan and Wales (1998) and Moschini (1999) and, in the context of five locally flexible demand systems – the generalized Leontief (see Diewert, 1974), the translog (see Christensen et al., 1975), the almost ideal demand system (see Deaton and Muellbauer, 1980), the minflex Laurent (see Barnett, 1983), and the normalized quadratic reciprocal indirect utility function (see Diewert and Wales, 1988) – treat the curvature property as a maintained hypothesis. In this chapter we follow Terrell (1996) and take a Bayesian approach to incorporating the theoretical regularity restrictions into the locally flexible demand models used by Serletis and Shahmoradi (2007). In doing so, we use the Metropolis–Hastings algorithm to generate an initial sample from the posterior probability density function for the parameters for a prior that ignores theoretical regularity. For each parameter vector in the sample, we evaluate the theoretical regularity restrictions in the price domain over which inferences are drawn and use accept/reject sampling to generate a sample of the parameter vectors that is consistent with theoretical regularity. Then we use marginal posterior density functions consistent with theory to draw inferences about income elasticities, own- and cross-price elasticities, as well as the elasticities of substitution. We also evaluate the models in terms of their ability to describe US monetary demand in a manner that satisfies the restrictions imposed by microeconomic theory and gives rise to stable elasticity estimates.

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The chapter is organized as follows. Section 2 briefly sketches out the neoclassical problem facing the representative agent and Section 3 discusses the five parametric flexible functional forms that we use in this chapter. Sections 4 and 5 are devoted to econometric issues while in Section 6 we estimate the models and explore the economic significance of the results. The final section concludes the chapter.

2. The Monetary Problem and Data Following Serletis and Shahmoradi (2005, 2007), we assume that the representative money holder faces the following problem max f (x) x

subject to p x = y,

(1)

where x = (x1 , x2 , . . . , x8 ) is the vector of monetary asset quantities, described in Serletis and Shahmoradi (2005, Table 1); p = (p1 , p2 , . . . , p8 ) is the corresponding vector of monetary asset user costs – see Barnett (1978); and y is the expenditure on the services of monetary assets. We use the same quarterly data set (from 1970:1 to 2003:2, a total of 134 observations) for the United States, as in Serletis and Shahmoradi (2005). Because the functional forms which we use in this chapter are parameter intensive, we face the problem of having a large number of parameters in estimation. To reduce the number of parameters, we follow Serletis and Shahmoradi (2005) and separate the group of assets into three collections based on empirical pre-testing. Thus the monetary utility function in (1) can be written as f (x) = f fA (x1 , x2 , x3 , x4 ), fB (x5 , x6 ), fC (x7 , x8 ) , where the subaggregate functions fi (i = A, B, C) provide subaggregate measures of monetary services. Following Barnett (1980) we use Divisia quantity indices to allow for less than perfect substitutability among the relevant monetary components. In particular, subaggregate A is composed of currency, travelers checks and other checkable deposits including Super NOW accounts issued by commercial banks and thrifts. Subaggregate B is composed of savings deposits issued by commercial banks and thrifts and subaggregate C is composed of small time deposits issued by commercial banks and thrifts. Divisia user cost indexes for each of these subaggregates are calculated by applying Fisher’s (1922) weak factor reversal test.

3. Locally Flexible Functional Forms In this section we briefly discuss the five functional forms that we use in this chapter – the generalized Leontief, the basic translog, the almost ideal demand system,

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the minflex Laurent, and the normalized quadratic reciprocal indirect utility function. These functions are all locally flexible and are capable of approximating any unknown function up to the second order. 3.1. The Generalized Leontief The generalized Leontief (GL) functional form was introduced by Diewert (1973) in the context of cost and profit functions. Diewert (1974) introduced the GL reciprocal indirect utility function h(v) = a0 +

n

1 1/2 1/2 + βij vi vj , 2 n

1/2 ai v i

n

(2)

i=1 j =1

i=1

where v = [v1 , v2 , . . . , vn ] is a vector of income normalized user costs, with the ith element being vi = pi /y, where pi is the user cost of asset i and y is the total expenditure on the n assets. B = [βij ] is an n × n symmetric matrix of parameters and a0 and ai are other parameters, for a total of (n2 + 3n + 2)/2 parameters. Using Diewert’s (1974) modified version of Roy’s identity vi ∂h(v)/∂vi si = n , j =1 vj ∂h(v)/∂vj

(3)

where si = vi xi and xi is the demand for asset i, the GL demand system can be written as 1/2 1/2 1/2 ai vi + nj=1 βij vi vj si = n (4) , i = 1, . . . , n. n n 1/2 1/2 1/2 j =1 aj vj + k=1 m=1 βkm vk vm Because the share equations are homogeneous of degree zero in the parameters, we follow Barnett and Lee (1985) and impose the following normalization in estimation 2

n i=1

ai +

n n

βij = 1.

(5)

i=1 j =1

3.2. The Basic Translog The basic translog (BTL) flexible functional form was introduced by Christensen et al. (1975). The BTL reciprocal indirect utility function can be written as log h(v) = a0 +

n k=1

1 βj k log vk log vj , 2 n

ak log vk +

n

(6)

k=1 j =1

where B = [βij ] is an n × n symmetric matrix of parameters and a0 and ai are other parameters, for a total of (n2 + 3n + 2)/2 parameters.

Bayesian Estimation of Flexible Functional Forms

63

The share equations, derived using the logarithmic form of Roy’s identity, si = − are

∂ log h(v)/∂ log pi , ∂ log h(v)/∂ log m

i = 1, . . . , n,

ai + nk=1 βik log vk n n , k=1 ak + k=1 j =1 βj k log vk

si = n

i = 1, . . . , n.

(7)

3.3. The Almost Ideal Demand System The almost ideal demand system (AIDS) is written is share equation form (see Deaton and Muellbauer, 1980, for more details) as si = ai +

n

βik log pk + bi log y − log g(p) ,

i = 1, . . . , n,

(8)

k=1

where log g(p) is a translog price index defined by log g(p) = a0 +

n

1 βkj log pk log pj . 2 n

n

ak log pk +

k=1 j =1

k=1

In equation (8), si is the ith budget share, y is income, pk is the kth price, and (a, b, β) are parameters of the demand systemto be estimated. Symmetry n n = β for all i, j ), adding up ( (β ij j i n k=1 ak = 1, i=1 βij = 0 for all j , and n i=1 bi = 0), and homogeneity ( j =1 βij = 0 for all i) are imposed in estimation. With n assets the AIDS model’s share equations contain (n2 + 3n − 2)/2 free parameters. 3.4. The Minflex Laurent The minflex Laurent (ML) model, introduced by Barnett (1983) and Barnett and Lee (1985), is a special case of the Full Laurent model also introduced by Barnett (1983). Following Barnett (1983), the Full Laurent reciprocal indirect utility function is h(v) = a0 + 2

n

1/2

ai v i

+

−2

i=1

−1/2 bi v i

1/2 1/2

aij vi vj

i=1 j =1

i=1 n

n n

−

n n

−1/2 −1/2 vj ,

bij vi

(9)

i=1 j =1

where a0 , ai , aij , bi and bij are unknown parameters and vi denotes the income normalized price, as before.

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By assuming that bi = 0, bii = 0 ∀i, aij bij = 0 ∀i, j , and forcing the off diagonal elements of the symmetric matrices A ≡ [aij ] and B ≡ [bij ] to be nonnegative, (9) reduces to the ML reciprocal indirect utility function h(v) = a0 + 2

n

1/2 ai v i

+

i=1

+

n n

n

aii vi

i=1 1/2 1/2

−

aij2 vi vj

i=1 j =1 i=j

n n

−1/2 −1/2 vj .

2 bij vi

(10)

i=1 j =1 i=j

Note that the off diagonal elements of A and B are nonnegative as they are raised to the power of two. By applying Roy’s identity to (10), the share equations of the ML demand system are

n n 1/2 2 1/2 1/2 2 −1/2 −1/2 aij vi vj + bij vi vj si = ai vi + aii vi +

×

j =1 i=j n

1/2

+

ai v i

i=1

+

n

j =1 i=j

aii vi

i=1

n n

1/2 1/2 aij2 vi vj

+

i=1 j =1 i=j

n n

2 −1/2 −1/2 bij vi vj

−1 .

(11)

i=1 j =1 i=j

Since the share equations are homogeneous of degree zero in the parameters, we follow Barnett and Lee (1985) and impose the following normalization in the estimation of (11) n

aii + 2

i=1

n i=1

ai +

n

aij2 −

n

j =1 i=j

2 bij = 1.

(12)

j =1 i=j

Hence, there are 1+n+

n(n + 1) n(n − 1) + 2 2

parameters in (11), but the n(n − 1)/2 equality restrictions, aij bij = 0 ∀i, j , and the normalization (12) reduce the number of parameters in equation (11) to (n2 + 3n)/2.

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3.5. The NQ Reciprocal Indirect Utility Function Following Diewert and Wales (1988), the normalized quadratic (NQ) reciprocal indirect utility function is defined as n n n n 1 i=1 j =1 βij vi vj n bi v i + + θi log vi , h(v) = b0 + (13) 2 i=1 αi vi i=1

i=1

where b0 , b = [b1 , b2 , . . . , bn ], θ = [θ1 , θ2 , . . . , θn ], and the elements of the n × n symmetric B ≡ [βij ] matrix are the unknown parameters to be estimated. It is important to note that the quadratic term in (13) is normalized by dividing through by a linear function n

αi vi

i=1

and that the nonnegative vector of parameters α = [α1 , α2 , . . . , αn ] is assumed to be predetermined. As in Diewert and Wales (1988), we assume that α satisfies n

αj vj∗ = 1,

αj 0 ∀j.

(14)

j =1

Moreover, we pick a reference (or base-period) vector of income normalized prices, v∗ = 1, and assume that the B matrix satisfies the following n restrictions n

βij vj∗ = 0

∀i.

(15)

j =1

Using the modified version of Roy’s identity (3), the NQ demand system can be written as n n n 1 αi k=1 j =1 βkj vk vj j =1 βij vi n + θ + s i = v i bi − i 2 ( ni=1 αi vi ) ( i=1 αi vi )2 −1

n n n n 1 i=1 j =1 βij vi vj + bi v i + θi . × (16) 2 ( ni=1 αi vi ) i=1

i=1

Finally, as the share equations are homogeneous of degree zero in the parameters, we also follow Diewert and Wales (1988) and impose the normalization n

bj = 1.

(17)

j =1

Hence, there are n(n + 5)/2 parameters in (16), but the imposition of the (n − 1) restrictions in (15) and (17) reduces the number of parameters to be estimated to (n2 + 3n − 2)/2.

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4. Bayesian Estimation of Demand Systems In order to estimate share equation systems such as (4), (7), (8), (11) and (16), a stochastic version must be specified. Since these systems are in share form and only exogenous variables appear on the right-hand side, it seems reasonable to assume that the observed share in the ith equation deviates from the true share by an additive disturbance term ui . Furthermore, we assume u ∼ N (0, ⊗ IT ) where 0 is a null matrix and is the n × n symmetric positive definite error covariance matrix. With the addition of additive errors, the share equation system for each model can be written in matrix form as st = g(vt , θ ) + ut , ) ,

(18) (v, θ )) ,

where s = (s1 , . . . , sn g(v, θ ) = (g1 (v, θ ), . . . , gn θ is the parameter vector to be estimated, and gi (v, θ ) is given by the right-hand side of each of (4), (7), (8), (11) and (16). Following Judge et al. (1988) the likelihood function of the whole sample can be written as −1 1 −T /2

exp − (s − g) ⊗ IT (s − g) P (s|θ , ) ∝ || 2 1 −1 −T /2 ∝ || , exp − trc A 2 where A is an (n × n) matrix defined as follows A = (si − gi ) (sj − gj ) , i, j = 1, . . . , n. Assuming a priori independence of and θ, a constant prior probability density function for θ, and the conventional noninformative prior for , P () ∝ ||−n/2 , then the joint prior probability density function for all the unknown parameters can be written as P (θ, ) ∝ ||−n/2 . Using Bayes’ theorem, the joint posterior probability density function for all the parameters can be written as (the likelihood function of the sample, P (s|θ , ), times the prior probability density function for the parameters, P (θ , )) 1 −1 −(T +n)/2 . exp − trc A P (θ , |s) ∝ || (19) 2 As already noted, in a Bayesian investigation equation (19) is the source of all inferences about the unknown parameters. It can be used to obtain the marginal posterior probability density function for the parameters P (θ|s) = P (θ , |s) d ∝ |A|−T /2 (20)

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67

and calculate their posterior means and corresponding standard deviations. However, equation (19) is too complicated for analytical integration (to obtain the marginal posterior probability density function for each element of θ ). One solution to this problem is the use of simulation techniques, such as Gibbs sampling, introduced by Geman and Geman (1984), and the Metropolis–Hastings algorithm, due to early work by Metropolis et al. (1953) and Hastings (1970). Such simulation techniques provide a way of drawing observations from the joint posterior probability density function. These generated observations are then used to construct histograms and calculate sample means and variances to provide consistent estimates of the marginal posterior probability density functions and the posterior means and variances (that is, the Bayesian counterparts of sampling theory point estimates and variances) of the elements in θ – see, for example, Chib and Greenberg (1995, 1996) for a detailed discussion. In this chapter we use the Metropolis–Hastings algorithm, because Gibbs sampling is suitable for linear seemingly unrelated regression models. The steps involved in the Metropolis–Hastings algorithm are as follows – see Griffiths and Chotikapanich (1997, p. 333) for more details. 1. Select initial values for θ, say θ 0 . Perform the remaining steps with τ set equal to 0. 2. Compute a value for P (θ τ |s), based on Equation (20). 3. Generate z from N (0, κV), where V is an adjusted covariance matrix of the maximum likelihood estimates and κ is chosen in line with convention (so that the acceptance rate for θ ∗ is approximately 50%). 4. Compute θ ∗ = θ τ + z. 5. Compute a value for P (θ ∗ |s) and the ratio of the probability density functions r=

P (θ ∗ |s) . P (θ τ |s)

6. If r 1, set θ τ +1 = θ ∗ and return to step 2; otherwise proceed with step 7. 7. Generate a uniform random variable y from the interval (0, 1). If y r, set θ τ +1 = θ ∗ ; otherwise set θ τ +1 = θ τ and return to step 2. Clearly, the Metropolis–Hastings algorithm provides a means for drawing observations consistent with the marginal posterior probability density function for the parameters, P (θ |s). In particular, the vector z in step 3 represents a potential change from the last drawing of θ and the potential new value θ ∗ is given by the random walk process in step 4. In step 6 a new observation is accepted if it is more probable than the previous one; if it is less probable, it is accepted in step 7 with probability given by the ratio of the two probability density functions. Thus, as Griffiths and Chotikapanich (1997, p. 334) put it, “the procedure explores the posterior pdf yielding a relatively high proportion of observations in regions of

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high probability and a relatively low proportion of observations in regions of low probability”. In simulation procedures, like the Metropolis–Hastings algorithm, because observations are drawn artificially using computer software, we can make the estimated marginal posterior probability density functions as accurate as we like, by drawing as many observations as required. However, these generated observations are not independent and as a result the sample means and variances are not as efficient as they would be from uncorrelated observations. One way to produce independent observations is to run a large number of chains and select the last observation from each chain. Alternatively, we can run one long chain and select observations at a specified interval, say every tenth or twentieth observation. Here we take the latter approach and select every twentieth observation.

5. Empirical Evidence We start with unconstrained ML parameter estimates and their variance–covariance matrix as initials in the Metropolis–Hastings estimation, we follow steps 1–7 and run one chain with 20,200 draws and select every twentieth observation after deleting the first 200 samples to avoid sensitivity to the choice of θ 0 . As already noted, we choose the covariance weight κ = 0.23 in step 3 to get the 50% acceptance rate. Posterior probability density functions, as well as posterior means and variances (the Bayesian counterparts of sampling theory point estimates and variances) are then estimated from these observations for each of the parameters of the model – all Bayesian estimation in this chapter is performed in TSP/GiveWin 4.5. The posterior moments are presented in panel A of Tables 1–5, together with standard errors and 95% confidence intervals (calculated using the asymmetric approach discussed in Davidson and Mackinnon, 1993), as well as with information regarding positivity, monotonicity and curvature violations. As in the case with maximum likelihood estimation, although positivity and monotonicity are satisfied globally with each of the five locally flexible demand systems, curvature is violated at all data points with the AIDS and minflex models, at 93 observations with the GL, 72 observations with the BTL, and 98 observations with the NQ. As already noted, without satisfaction of all three theoretical regularity conditions (positivity, monotonicity and curvature), the resulting inferences are virtually worthless, since violations of regularity violate the maintained hypothesis and invalidate the duality theory that produces the estimated model. In what follows, we follow Terrell (1996) and incorporate the theoretical regularity restrictions of monotonicity and curvature into the prior distribution. We incorporate both monotonicity and curvature, because the imposition of curvature alone induces violations of monotonicity, as noted by Barnett and Pasupathy (2003).

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Table 1: GL Parameter Estimates Parameter

Number of violations

Mean

Standard error

95% Confidence interval

−1.688 −0.765 1.497 0.483 1.688 0.461 0.957 2.262

0.343 0.246 0.278 0.173 0.253 0.164 0.187 0.265

(−2.887, −1.213) (−1.343, −0.416) (1.092, 2.156) (0.237, 0.894) (1.335, 2.296) (0.215, 0.857) (0.654, 1.394) (1.749, 2.866)

−2.053 −1.414 1.621 1.027 1.776 0.739 1.419 2.057

0.134 0.115 0.107 0.084 0.099 0.097 0.102 0.146

(−2.292, −1.806) (−1.603, −1.197) (1.430, 1.808) (0.879, 1.167) (1.591, 1.955) (0.570, 0.939) (1.228, 1.601) (1.817, 2.365)

A. Unconstrained estimation a1 a2 b11 b12 b13 b22 b23 b33 Positivity Monotonicity Curvature

0 0 93

B. Constrained estimation a1 a2 b11 b12 b13 b22 b23 b33 Positivity Monotonicity Curvature

0 0 0

Note: Sample period, quarterly data 1970:1–2003:2 (T = 134).

In doing so, we define an indicator function h(θ ) which is equal to one if the specified theoretical regularity condition holds and zero otherwise, as follows 1, if the specified theoretical regularity condition holds, h(θ ) = 0, otherwise. Using this indicator function, we define the informative prior for our model parameters as P1 (θ ) = h(θ ) × P0 (θ ) ∝ h(θ ) × constant thereby assigning zero weight to estimated vectors which lead to violation of the specified theoretical regularity condition. After the Metropolis–Hastings algorithm generates a parameter vector, the monotonicity and curvature conditions are evaluated at all 134 observations. If a

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Table 2: BTL Parameter Estimates Parameter

Number of violations

Mean

Standard error

95% Confidence interval

0.413 0.291 0.718 0.054 0.401 0.208 0.177 0.467

0.004 0.003 0.030 0.037 0.050 0.031 0.041 0.050

(0.405, 0.421) (0.284, 0.298) (0.661, 0.779) (−0.014, 0.129) (0.323, 0.516) (0.150, 0.278) (0.107, 0.261) (0.388, 0.590)

0.414 0.295 0.279 −0.180 0.092 −0.007 0.008 0.215

0.003 0.003 0.062 0.084 0.067 0.063 0.061 0.039

(0.406, 0.421) (0.287, 0.302) (0.202, 0.469) (−0.257, 0.089) (0.008, 0.275) (−0.090, 0.182) (−0.070, 0.180) (0.154, 0.321)

A. Unconstrained estimation a1 a2 b11 b12 b13 b22 b23 b33 Positivity Monotonicity Curvature

0 0 72

B. Constrained estimation a1 a2 b11 b12 b13 b22 b23 b33 Positivity Monotonicity Curvature

0 0 0

Note: Sample period, quarterly data 1970:1–2003:2 (T = 134).

rejection occurs at any of the observations, the associated parameter vector is rejected and a new parameter vector is generated. We run the Metropolis–Hastings sampling until a full sample of 20,200 is obtained with no monotonicity and curvature violations. We delete the first 200 sample points and use the sample of 20,000 to calculate moments of the posterior density. We report posterior means and standard deviations, together with the violations of the theoretical regularity conditions when monotonicity and curvature are incorporated into the prior distribution, in panel B of Tables 1–5. 5.1. Income and Price Elasticities We now focus on the problem of interpreting the parameter estimates by computing the income elasticities, the price elasticities, the Allen elasticities of substitution, and the Morishima elasticities of substitution. We report the income

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Table 3: AIDS Parameter Estimates Parameter

Number of violations

Mean

Standard error

95% Confidence interval

2.911 0.228 1.823 −0.062 0.526 0.264 −0.276 1.054

0.462 0.090 0.178 0.031 0.045 0.020 0.052 0.107

(1.865, 3.626) (0.051, 0.399) (1.422, 2.095) (−0.127, 0.003) (0.435, 0.614) (0.228, 0.306) (−0.383, −0.174) (0.836, 1.243)

2.615 0.270 0.771 −0.026 0.364 0.203 −0.138 0.347

0.749 0.072 0.235 0.044 0.053 0.011 0.029 0.086

(1.386, 3.709) (0.068, 0.384) (0.356, 1.097) (−0.123, 0.059) (0.282, 0.478) (0.181, 0.231) (−0.215, −0.087) (0.184, 0.482)

A. Unconstrained estimation α0 α1 α2 β1 β2 b11 b12 b22 Positivity Monotonicity Curvature

0 0 134

B. Constrained estimation α0 α1 α2 β1 β2 b11 b12 b22 Positivity Monotonicity Curvature

0 0 0

Note: Sample period, quarterly data 1970:1–2003:2 (T = 134).

elasticities in panel A of Table 6 for the three subaggregates, A (M1), B (savings deposits) and C (time deposits), and the five models – the GL, BTL, AIDS, minflex and NQ – evaluated at the mean of the data. All elasticities reported in this chapter are based on the formulas used by Serletis and Shahmoradi (2005) and have been acquired using numerical differentiation. All expenditure elasticities, ηAm , ηBm and ηCm , are positive (with that for savings deposits, ηBm , being greater than 1) which is consistent with economic theory. However, there are differences among the models, but we have no a priori reason to reject any of these findings. In panel B of Table 6 we show the own- and cross-price elasticities for the five models and the three assets. In the first row we show the own-price elasticities in the first block for asset A (M1); the own-price elasticities for asset B (savings deposits) are presented in the second block of the second row and those for asset C (time deposits) in the third block of the third row. These are all negative

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Table 4: MINFLEX Parameter Estimates Parameter

Number of violations

Mean

Standard error

95% Confidence interval

−6.041 1.011 −7.666 8.601 2.244 5.889 −0.025 −0.001

2.326 0.920 3.735 3.201 0.502 2.780 0.212 0.009

(−10.134, −2.437) (−0.711, 2.951) (−14.451, −1.914) (3.561, 14.013) (1.351, 3.091) (1.510, 10.878) (−0.358, 0.317) (−0.107, 0.118)

−1.724 −0.145 −1.471 2.736 1.045 2.240 −0.283 0.003

0.486 0.263 0.660 0.685 0.178 0.766 0.051 0.005

(−2.623, −0.761) (−0.656, 0.366) (−2.537, −0.262) (1.258, 3.950) (0.690, 1.346) (0.927, 3.313) (−0.350, −0.146) (−0.102, 0.108)

A. Unconstrained estimation a1 a2 a3 a11 a13 a33 b12 b23 Positivity Monotonicity Curvature

0 0 134

B. Constrained estimation a1 a2 a3 a11 a13 a33 b12 b23 Positivity Monotonicity Curvature

0 0 0

Note: Sample period, quarterly data 1970:1–2003:2 (T = 134).

(as predicted by the theory), with the absolute values of these elasticities being less than 1, which indicates that the demands for all three assets are inelastic. For the cross-price elasticities (ηij ), economic theory does not predict any signs, but we note that most of the off-diagonal terms are negative, indicating that the assets taken as a whole are gross complements. 5.2. Allen and Morishima Elasticities of Substitution From the point of view of monetary policy, the measurement of the elasticities of substitution among the three monetary assets is of primary importance. As we already pointed out, the currently popular simple sum approach to monetary aggregation requires, in effect, that the elasticities of substitution be very high among the components of, especially, the aggregate M2. By ‘very high’ we mean infinite, of course, but since the policy literature has not addressed the question of

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Table 5: NQ Parameter Estimates Parameter

Number of violations

Mean

Standard error

95% Confidence interval

0.482 0.074 −0.029 0.085 −0.141 −0.186 0.034

0.038 0.034 0.011 0.009 0.017 0.025 0.025

(0.412, 0.560) (0.010, 0.146) (−0.053, −0.008) (0.067, 0.103) (−0.176, 0.108) (−0.234, −0.137) (−0.017, 0.086)

0.505 0.199 −0.037 0.048 −0.146 −0.093 −0.058

0.029 0.027 0.006 0.007 0.021 0.021 0.020

(0.453, 0.566) (0.145, 0.251) (−0.050, −0.026) (0.033, 0.065) (−0.188, −0.103) (−0.138, −0.053) (−0.102, −0.022)

A. Unconstrained estimation a1 a2 b1 b2 b11 b12 b22 Positivity Monotonicity Curvature

0 0 98

B. Constrained estimation a1 a2 b1 b2 b11 b12 b22 Positivity Monotonicity Curvature

0 0 0

Note: Sample period, quarterly data 1970:1–2003:2 (T = 134).

how high such an estimate should be to warrant a simple sum calculation ‘from a practical standpoint’, all we can do is report our results. The results are not encouraging for the simple sum method. An additional concern relates to the volatility of the elasticities. Specifically, if there is evidence of significant volatility in the elasticities of substitution the simple sum aggregates will surely be invalid and methods of aggregation that allow for variable elasticities of substitution will be preferable. There are currently two methods employed for calculating the partial elasticity of substitution between two variables, the Allen and the Morishima. The Allen elasticity of substitution is the traditional measure and has been employed to measure substitution behavior and structural instability in a variety of contexts. However, when there are more than two assets the Allen elasticity may be uninformative. For two assets the relationship is unambiguous – the assets must be substitutes. When there are more than two assets the relationship becomes complex and depends on things like the direction taken towards the point of ap-

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Table 6: Income and Price Elasticities Subaggregate i Model

A. Income elasticities

B. Price elasticities

ηi

ηiA

ηiB

ηiC

(A)

GL BTL AIDS Miniflex NQ

0.989 (0.099) 0.893 (0.099) 0.942 (0.098) 0.639 (0.068) 0.764 (0.039)

−0.566 (0.074) −0.566 (0.074) −0.531 (0.043) −0.389 (0.029) −0.469 (0.018)

−0.220 (0.032) −0.220 (0.032) −0.258 (0.036) −0.130 (0.020) −0.174 (0.025)

−0.106 (0.023) −0.106 (0.023) −0.152 (0.043) −0.119 (0.036) 0.101 (0.022)

(B)

GL BTL AIDS Miniflex NQ

1.264 (0.063) 2.036 (0.259) 2.440 (0.239) 2.006 (0.107) 1.560 (0.093)

−0.706 (0.022) −0.894 (0.152) −1.130 (0.125) −0.867 (0.054) −0.666 (0.045)

−0.728 (0.046) −0.855 (0.072) −0.843 (0.072) −0.768 (0.056) −0.710 (0.063)

−0.129 (0.036) −0.286 (0.084) −0.466 (0.092) −0.370 (0.037) −0.183 (0.043)

(C)

GL BTL AIDS Miniflex NQ

0.749 (0.084) 0.260 (0.132) 0.136 (0.230) 0.648 (0.099) 0.877 (0.072)

−0.152 (0.029) 0.121 (0.067) 0.252 (0.113) −0.209 (0.046) −0.226 (0.055)

0.012 (0.042) 0.206 (0.064) 0.254 (0.081) 0.001 (0.041) 0.005 (0.052)

−0.609 (0.057) −0.587 (0.068) −0.370 (0.100) −0.438 (0.084) −0.657 (0.057)

Note: Sample period, quarterly data 1970:1–2003:2 (T = 134). Numbers in parentheses are standard errors.

proximation. In that case the Morishima elasticity of substitution is the correct measure of substitution elasticity – see Blackorby and Russell (1989). Table 7 shows estimates of both the Allen and Morishima elasticities, evaluated at the means of the data. For panel A, we expect the three diagonal terms, representing the Allen own-elasticities of substitution for the three assets, to be negative. This expectation is clearly achieved. However, because the Allen elasticity of substitution produces ambiguous results off-diagonal, we will use the Morishima elasticity of substitution to investigate the substitutability/complementarity relation between assets. Based on the asymmetrical Morishima elasticities of substitution – the correct measures of substitution – as documented in panel B of Table 7, the assets are Morishima substitutes, with all Morishima elasticities of substitution being less than unity, irrespective of the model used. This clearly indicates difficulties for a simple-sum based monetary policy and helps explain why recent attempts to target and control the money supply (simple sum M2) in the United States have been abandoned in favor of interest rate procedures. Similar conclusions have been reached by Serletis and Shahmoradi (2005) based on the use of globally flexible functional forms and Serletis and Shahmoradi (2007) based on the use of locally flexible functional forms and sampling theoretic inference. Finally, in Figures 1–5 we show the marginal densities for the Morishima elasticities of substitution for the unconstrained and constrained Metropolis–Hastings

Table 7: Allen and Morishima Elasticities of Substitution Model

(A)

GL BTL AIDS Miniflex NQ

(B)

GL BTL AIDS Miniflex NQ

(C)

GL BTL AIDS Miniflex NQ

A. Allen elasticities

B. Morishima elasticities

a σiA

a σiB

a σiC

m σiA

−0.472 (0.032) −0.375 (0.108) −0.240 (0.046) −0.197 (0.044) 0.288 (0.044)

0.350 (0.048) 0.040 (0.105) −0.076 (0.101) 0.142 (0.047) 0.091 (0.070)

0.407 (0.058) 0.531 (0.093) 0.426 (0.108) 0.199 (0.090) 0.380 (0.076)

−1.348 (0.795) −1.282 (0.242) −0.887 (0.222) −0.926 (0.169) −1.101 (0.204)

0.795 (0.124) 1.062 (0.166) 0.870 (0.204) 0.645 (0.152) 0.901 (0.164)

0.473 (0.045) 0.342 (0.083) 0.206 (0.064) 0.279 (0.045) 0.318 (0.061)

−1.461 (0.154) −1.737 (0.194) −1.384 (0.247) −0.958 (0.241) −1.480 (0.188)

0.515 (0.056) 0.668 (0.087) 0.538 (0.109) 0.315 (0.089) 0.519 (0.075)

m σiB

m σiC

0.366 (0.029) 0.185 (0.085) 0.074 (0.046) 0.157 (0.029) 0.172 (0.042)

0.361 (0.034) 0.405 (0.079) 0.299 (0.063) 0.184 (0.060) 0.303 (0.048) 0.597 (0.069) 0.605 (0.098) 0.445 (0.099) 0.411 (0.077) 0.534 (0.089)

0.622 (0.077) 0.824 (0.108) 0.670 (0.139) 0.437 (0.105) 0.665 (0.103)

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Subaggregate i

Note: Sample period, quarterly data 1970:1–2003:2 (T = 134). Numbers in parentheses are standard errors.

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Figure 1: Morishima Elasticities of Substitution Based on the GL: Unconstrained Estimates (solid lines) and Constrained Estimates (dotted lines).

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Figure 2: Morishima Elasticities of Substitution Based on the BTL: Unconstrained Estimates (solid lines) and Constrained Estimates (dotted lines).

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Figure 3: Morishima Elasticities of Substitution Based on the AIDS: Unconstrained Estimates (solid lines) and Constrained Estimates (dotted lines).

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Figure 4: Morishima Elasticities of Substitution Based on the Minflex Laurent: Unconstrained Estimates (solid lines) and Constrained Estimates (dotted lines).

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Figure 5: Morishima Elasticities of Substitution Based on the NQ: Unconstrained Estimates (solid lines) and Constrained Estimates (dotted lines).

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81

estimations, demonstrating the need for a restrictive prior imposing monotonicity and curvature. In particular, for all models the imposition of the regularity conditions (of monotonicity and curvature) changes the means and standard errors for the elasticities. In fact, in some cases the relationship between assets switches from complementarity to substitutability – see for example the Morishima elasticity of substitution between assets B and A in Figure 4. It is also to be noted that the imposition of theoretical regularity reduces the variance of the elasticities.

6. Conclusion We have used Bayesian techniques to impose regularity conditions without the loss of flexibility associated with the global imposition of these properties. Our results concur with the evidence presented by Serletis and Shahmoradi (2005), who use the globally flexible Fourier and AIM functional forms and impose global curvature using methods suggested by Gallant and Golub (1984), and also with the evidence presented by Serletis and Shahmoradi (2007) using locally flexible functional forms and sampling theoretic inference. The evidence indicates that the elasticities of substitution among the monetary assets (in the popular M2 aggregate in the United States) are consistently and believably below unity. Even though model comparisons cannot be carried out realistically, we can offer some related advice on monetary policy, at least up to the point where the flexible models are fitted. As we noted, considerable research has indicated that the simple-sum approach to monetary aggregation, in the face of cyclically fluctuating incomes and interest rates (and hence user costs), cannot be the best that can be achieved. Our study corroborates the existence of these phenomena and hence concurs with the general preference for the use of chain-linked monetary aggregates based on, for example, the Divisia index. In addition, we should note that the collection of assets in the traditional M2 formulation is also not optimal, a result not generated in this chapter, but characteristic of our approach at any rate. This, too, would lead to imprecise estimates of the quantity of money for policy purposes.

Acknowledgement Serletis gratefully acknowledges financial support from the Social Sciences and Humanities Research Council of Canada (SSHRCC).

References Barnett, W.A. (1978). The user cost of money. Economics Letters 1, 145–149.

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Barnett, W.A. (1980). Economic monetary aggregates: An application of aggregation and index number theory. Journal of Econometrics 14, 11–48. Barnett, W.A. (1983). New indices of money supply and the flexible Laurent demand system. Journal of Business and Economic Statistics 1, 7–23. Barnett, W.A. (1997). Which road leads to a stable money demand? The Economic Journal 107, 1171–1185. Barnett, W.A. (2002). Tastes and technology: Curvature is not sufficient for regularity. Journal of Econometrics 108, 199–202. Barnett, W.A. and Lee, Y.W. (1985). The global properties of the minflex Laurent, generalized Leontief, and translog flexible functional forms. Econometrica 53, 1421–1437. Barnett, W.A. and Pasupathy, M. (2003). Regularity of the generalized quadratic production model: A counter example. Econometric Reviews 22, 135–154. Blackorby, C. and Russell, R.R. (1989). Will the real elasticity of substitution please stand up? American Economic Review 79, 882–888. Chetty, V.K. (1969). On measuring the nearness of near-moneys. American Economic Review 59, 270–281. Chib, S. and Greenberg, E. (1995). Understanding the Metropolis–Hastings algorithm. The American Statistician 49, 327–335. Chib, S. and Greenberg, E. (1996). Markov chain Monte Carlo simulation methods in econometrics. Econometric Theory 12, 409–431. Christensen, L.R., Jorgenson, D.W. and Lau, L.J. (1975). Transcendental logarithmic utility functions. American Economic Review 65, 367–383. Davidson, R. and Mackinnon, J.G. (1993). Estimation and Inference in Econometrics. Oxford, UK: Oxford University Press. Deaton, A. and Muellbauer, J.N. (1980). An almost ideal demand system. American Economic Review 70, 312–326. Diewert, W.E. (1973). Functional forms for profit and transformation functions. Journal of Economic Theory 6, 284–316. Diewert, W.E. (1974). Applications of duality theory. In Intriligator, M. and Kendrick, D. (Eds), Frontiers in Quantitative Economics. Contributions to Economic Analysis, Vol. 2. Amsterdam: North-Holland, pp. 106–171. Diewert, W.E. and Wales, T.J. (1988). Normalized quadratic systems of consumer demand functions. Journal of Business and Economic Statistics 6, 303–312. Donovan, D.J. (1978). Modeling the demand for liquid assets: An application to Canada. International Monetary Fund Staff Papers 25, 676–704. Ewis, N.A. and Fisher, D. (1984). The translog utility function and the demand for money in the United States. Journal of Money, Credit and Banking 16, 34–52. Ewis, N.A. and Fisher, D. (1985). Toward a consistent estimate of the substitutability between money and near monies: An application of the Fourier flexible form. Journal of Macroeconomics 7, 151–174. Fisher, D. and Fleissig, A.R. (1994). Money demand in a flexible dynamic Fourier expenditure system. Federal Reserve Bank of St. Louis Review 76, 117–128. Fisher, D. and Fleissig, A.R. (1997). Monetary aggregation and the demand for assets. Journal of Money, Credit and Banking 29, 458–475. Fisher, I. (1922). The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability. Boston: Houghton Mifflin.

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Fleissig, A.R. and Serletis, A. (2002). Semi-nonparametric estimates of substitution for Canadian monetary assets. Canadian Journal of Economics 35, 78–91. Gallant, A.R. and Golub, G.H. (1984). Imposing curvature restrictions on flexible functional forms. Journal of Econometrics 26, 295–321. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 609–628. Griffiths, W.E. and Chotikapanich, D. (1997). Bayesian methodology for imposing inequality constraints on a linear expenditure system with demographic factors. Australian Economic Papers 36, 321–341. Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrica 57, 97–109. Judge, G.G., Hill, R.C., Griffiths, W.E., Lütekpohl, H. and Lee, T.C. (1988). Introduction to the Theory and Practice of Econometrics. New York: Wiley. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics 21, 1087–1092. Moschini, G. (1999). Imposing local curvature in flexible demand systems. Journal of Business and Economic Statistics 17, 487–490. Ryan, D.L. and Wales, T.J. (1998). A simple method for imposing local curvature in some flexible consumer-demand systems. Journal of Business and Economic Statistics 16, 331–338. Serletis, A. (1987). The demand for Divisia M1, M2, and M3 in the United States. Journal of Macroeconomics 9, 567–591. Serletis, A. (1988). Translog flexible functional forms and substitutability of monetary assets. Journal of Business and Economic Statistics 6, 59–67. Serletis, A. (1991). The demand for Divisia money in the United States: A dynamic flexible demand system. Journal of Money, Credit and Banking 23, 35–52. Serletis, A. and Rangel-Ruiz, R. (2005). Microeconometrics and measurement matters: Some results from monetary economics for Canada. Journal of Macroeconomics 27, 307–330. Serletis, A. and Robb, A.L. (1986). Divisia aggregation and substitutability among monetary assets. Journal of Money, Credit and Banking 18, 430–446. Serletis, A. and Shahmoradi, A. (2005). Semi-nonparametric estimates of the demand for money in the United States. Macroeconomic Dynamics 9, 542–559. Serletis, A. and Shahmoradi, A. (2007). Flexible functional forms, curvature conditions, and the demand for assets. Macroeconomic Dynamics, in press. Terrell, D. (1996). Incorporating monotonicity and concavity conditions in flexible functional forms. Journal of Applied Econometrics 11, 179–194.

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Chapter 5

Productivity and Convergence Trends in the OECD: Evidence from a Normalized Quadratic Variable Profit Function Guohua Feng and Apostolos Serletis∗ Department of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4

Abstract This chapter empirically investigates aggregate productivity convergence among 15 European Union countries plus Canada and the United States, during the 1960–2002 period. Using a normalized quadratic variable profit function (within a multiple-output and multiple-input framework), we obtain total factor productivity estimates and then test for total factor productivity convergence, using both cross-section and panel data unit root tests. Our results support total factor productivity convergence for all countries. We also find that more countries converge to Germany than the United States, and that within the European Union more countries converge to Germany than France.

Keywords: flexible functional forms, normalized quadratic variable profit function, regularity conditions, productivity convergence, panel data unit root tests JEL: C22, C33

1. Introduction Typical neoclassical growth models in the tradition of Solow (1956) have the standard implication that, in terms of long-run macroeconomic behavior, poor economies catch up with richer economies. This catching up process has come to be known as the absolute convergence hypothesis. While most of the empirical studies in this literature concentrate on tests of convergence in per capita output (or income), convergence in total factor productivity (TFP) is equally or even more important. Romer (1990), Grossman and Helpman (1991) and Aghion and Howitt (1992) argue that rather than factor accumulation, technological progress and its diffusion are the major sources of economic growth. Dowrick and Nguyen

∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18005-4

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(1989) and Bernard and Jones (1996) further argue that it is very important to examine technological convergence by focusing on total factor productivity growth instead of convergence in per capita output. If technological convergence does not occur then countries and regions are not catching up, and per capita output growth in rich and poor countries will tend to lead to increased income dispersion. In this chapter we investigate total factor productivity convergence, using data for 17 OECD countries over the period from 1960 to 2002. The countries include 15 European Union countries – Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, the Netherlands, Portugal, Spain, Switzerland and the United Kingdom – plus two North American countries, Canada and the Unites States. The econometric methods that we use in testing the convergence (in total factor productivity) hypothesis fall into two classes. The first class consists of cross-section tests, where we estimate the ‘convergence equation’, as in Baumol (1986), Barro and Sala-i-Martin (1991), Islam (1995) and Sala-i-Martin (1996), among others. In this framework, a negative correlation between initial levels of total factor productivity and subsequent growth rates is taken as evidence of convergence. The second set of convergence tests are based on time series methods. Bernard and Durlauf (1995, 1996) suggest tests that examine the long-run behavior of differences in per capita output (or TFP) across countries. These tests define convergence to mean that these differences are always transitory in the sense that long-run forecasts of output differences between any pair of countries converge to zero as the forecast horizon grows to infinity. According to this approach, convergence requires that output differences between any pair of economies cannot contain unit roots or time trends and that output levels must be cointegrated with a cointegrating vector of (1, −1). The development of panel data unit root tests in recent years facilitates the research on convergence. Along this line, within a panel unit root test framework, productivity levels are said to be converging if differences in productivity levels across countries are stationary processes. What should be kept in mind is that the above two methods make different assumptions concerning the statistical properties of the data. While cross-section tests assume that the data under analysis are generated by economies far from a steady state, time series tests assume that the data possess well-defined population moments in either levels or first differences – see Bernard and Durlauf (1996). That is, the appropriateness of the two sets of tests depends on the property of the data under analysis. In any case, however, it is not an easy task to evaluate whether an economy is near or far from its steady state. In this chapter, we use both cross-section tests and two panel data unit root tests – namely, the Im et al. (2003) and Taylor and Sarno (1998) tests to evaluate whether (on average) our sample countries are converging or not. While they are commonly used in empirical studies, cross-section tests and most of the panel unit root tests, including the Im et al. (2003) and Taylor and Sarno (1998) tests, are not informative in the sense that they are not capable of

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determining which countries are converging and which are not. In the case of the panel unit root tests, the problem stems from the joint hypothesis that is tested. Under the joint null hypothesis, rejection of the nonstationary null (or nonconvergence null) does not provide information about how many panel members reject the null hypothesis and how many do not. In other words, both the Im et al. (2003) and Taylor and Sarno (1998) tests have an ‘all or nothing’ characteristic where all series are either stationary or nonstationary. Unfortunately, a similar limitation is also associated with the cross-section test. Because of this problem, in this chapter we also implement the Breuer et al. (2001) test as a complementary test to the cross-section test and the two panel unit root tests to identify how many panel members reject the null hypothesis and how many do not. While the methods of testing for total factor productivity convergence are important, an equally crucial and perhaps more complicated issue is how to estimate total factor productivity. The most commonly used functional forms for the production function are the Cobb–Douglas and the CES, with two inputs – capital and labor. However, these production functions imply serious limitations which may lead to biases in estimating total factor productivity. For example, two of the assumptions underlying the neoclassical growth model in an open economy context are that imports are final goods (openness) and that they are separable from the primary factors (separability). The assumption of openness conflicts with empirical evidence suggesting that the main bulk of imports consists of intermediate goods requiring further processing before delivery to final demand – see Burgess (1974). The assumption of separability implies that the marginal rate of substitution between capital and labor is independent of the quantity of intermediate inputs. It also implies that the elasticities of substitution between intermediate inputs and either capital or labor are equal. However, most empirical studies suggest that technology is not separable with respect to primary factors and intermediate inputs – see, for example, Burgess (1974). Hence, imports cannot be omitted from the neoclassical growth model without producing biased estimates. Moreover, by excluding imports and exports from the production function, we ignore the fact that international trade exerts an influence on the mechanism of aggregate productivity convergence through the transmission of technological knowledge and increased competition – see, for example, Dollar et al. (1988). Second, these production functions (although regular) suffer from limited flexibility – that is, they have too few parameters to make possible an independent representation of all economically relevant effects. As an example, the Cobb–Douglas production function, restricts the substitution elasticities between all factors and the elasticity of scale and size to unity. Thus, a more flexible functional form is needed in estimating total factor productivity. To address these problems, in this chapter we use a Normalized Quadratic (NQ) variable profit function in estimating total factor productivity. The NQ model, introduced by Diewert and Wales (1992), is flexible – that is, capable of approximating an

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arbitrary profit function to the second order – and also allows for the treatment of exports as one of the outputs and that of imports as one of the inputs. The rest of the chapter is organized as follows. Section 2 briefly discusses the NQ variable profit function and the methods for calculating total factor productivity while Section 3 describes the basic total factor productivity convergence and testing procedures. In Section 4 we discuss data issues, estimate the NQ model to obtain total factor productivity, perform the total factor productivity convergence tests and explore the economic significance of the results. The final section concludes the chapter with some suggestions for future research.

2. Total Factor Productivity Measurement As already noted, a flexible functional form that allows for exports and imports is needed in order to obtain correct estimates of total factor productivity. Although there is a large number of commonly used flexible functional forms that satisfy this criterion, such as, for example, the normalized quadratic (NQ), translog and generalized Leontief variable profit functions, here we choose to use the NQ, because as Diewert and Wales (1987) have shown it possesses the desirable property that correct curvature conditions can be imposed globally without destroying the flexibility of the functional form. The normalized quadratic variable profit function can be written for n+1 goods (including both multiple inputs and multiple outputs) as n n n n 1 i=1 j =1 βij pi pj k V (p, k, t) = a i pi + b i pi k + N 2 i=1 αi pi i=1

+

n

i=1

ci pi kt,

(1)

i=1

where V (p, k, t) denotes the variable profit (i.e., gross returns minus variable costs), p = (p1 , . . . , pn ) with pi being the price of the ith ‘output’ (with inputs treated as negative outputs), t is the time trend (over the entire sample period), k is the (n + 1)th good and denotes the minimum amount of fixed capital required to produce the vector of outputs and is assumed to be exogenously given in the short term, and a, b, β, c and α are parameters, usually with the α vector (α > 0) being predetermined. As can be seen, V (p, k, t) in (1) is linearly homogeneous in p. Further, we impose two restrictions on the B ≡ [βij ] matrix βij = βj i ∗

Bp = 0

for all i, j,

(2) ∗

for some p > 0.

(3)

The normalized quadratic variable profit function defined by (1)–(3) is technical progress flexible (TP flexible), as shown by Diewert and Wales (1992). In this

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chapter, we have two outputs (domestic output and exports), three variable inputs (imports, labor and reproducible capital) and one exogenously given input (being an aggregate of land, the stock of inventories and intellectual property). Differentiating (1) with respect to prices and using Hotelling’s lemma yields a system of net supply functions. By simultaneously estimating the system of net supply functions, we can obtain the estimates of all parameters appearing in (1). Then, differentiating the log of the profit function with respect to t, yields an index of total factor productivity for each t, n c i pi k ∇t V (p, k, t) At = (4) = i=1 . V (p, k, t) V (p, k, t)

3. Convergence Model and Test Procedures After obtaining total factor productivity estimates using the NQ variable profit function, we test the total factor productivity convergence hypothesis using procedures suggested by Bernard and Jones (1996). In particular, we assume that total factor productivity evolves according to i,t−1 + ln Ai,t−1 + εit , ln Ait = γi + λi ln A

(5)

i,t−1 is the catchwhere Ait is total factor productivity in country i (at time t), A up variable – the productivity differential with respect to the reference country i ∗ it = Ai ∗ t /Ait ). γi is the asymptotic (long-run) growth rate of Ait , λi cap(i.e., A tures the speed of catch-up for country i, and εit is a country-specific productivity shock. Equation (5) can be rewritten as1 it = (γi ∗ − γi ) + (1 − λi ) ln A i,t−1 + ln A εit ,

(6)

where εit = εi ∗ t − εit . Equation (6) is the basic model used in this chapter, implying that the total factor productivity gap between country i and the reference country i ∗ is a function of the lagged gap in total factor productivity. In the context of equation (6), if λi = 0 productivity levels will grow at different rates and there

1 For

the reference country, i ∗ , Equation (5) can be written as i ∗ ,t−1 + ln Ai ∗ ,t−1 + εi ∗ t . ln Ai ∗ t = γi ∗ + λi ∗ ln A

i ∗ ,t−1 = 0, the above equation can be written as Since ln A ln Ai ∗ t = γi ∗ + ln Ai ∗ ,t−1 + εi ∗ t . Subtracting (5) from the above equation yields Equation (6).

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it (the difference beis no tendency to catch-up asymptotically. In this case, ln A tween total factor productivity levels in the two countries) will contain a unit root. it will be stationary. λi > 0 provides an impetus for ‘catch-up’. If λi > 0, ln A As we discussed earlier, there are generally two approaches that can be used to test the convergence hypothesis, as represented by (6) – cross-section tests and panel unit root tests – to which we now turn. 3.1. The Cross-Section Test By repeated backward substitution, the difference equation in Equation (6) can be written as iT − ln A i0 i0 = − 1 − (1 − λi )T ln A ln A +

T (1 − λi )T −j (γi ∗ − γi + εi ),

(7)

j =0

i0 is the productivity differential with respect to the reference country where ln A at the beginning of the sample period. Dividing both sides of Equation (7) by T yields [1 − (1 − λi )T ] ln Ai0 T T 1 + (1 − λi )T −j (γi ∗ − γi + εi ), T

gi = −

(8)

j =0

where gi denotes the average growth rate relative to the reference country between time 0 and time T . Equation (8) is the familiar regression of long-run average growth rates on the initial level. Letting β ≡ −[1 − (1 − λi )T ]/T , convergence is said to occur if the estimate of β is negative and statistically different from 0. 3.2. Panel Unit Root Tests Instead of testing Equation (6) in an indirect way as we did in the cross-section test, we can test the null hypothesis of nonconvergence in the context of an augmented Dickey–Fuller (ADF) – see Dickey and Fuller (1981) – unit root test as follows it = αi + ρi ln A i,t−1 + ln A

k

i,t−j + cij ln A εit ,

(9)

j =1

where αi ≡ (γi ∗ − γi ), ρi = −λi , and the k extra regressors have been added to eliminate possible nuisance parameter dependencies of the test statistic caused by temporal dependencies in the disturbances. The optimal lag length, k, can

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be chosen using data-dependent methods that have desirable statistical properties. Testing the null hypothesis of a unit root in the productivity differentials, H0 : ρi = 0, is equivalent to testing the null hypothesis of nonconvergence in total factor productivity. A straightforward test procedure is to perform unit root tests for each country. In finite samples, however, unit root tests have low power against alternative hypotheses with highly persistent deviations from equilibrium – see, for example, Levin et al. (2002). To overcome this problem of low test power associated with univariate unit root test procedures, in this chapter we implement panel data unit root tests on the productivity differentials. The main advantage of the panel-based unit root test over individual series unit root tests is the increase in power against alternatives. This is because panel data unit root tests combine information from the time-series dimension with that obtained from the cross-sectional dimension. We apply two panel-based unit root tests to the pooled cross-section time series data of total factor productivity for the 17 OECD countries. The first test, proposed by Im et al. (2003), allows for heterogeneity in the value of ρi under the alternative hypothesis. The null hypothesis in the Im et al. (2003) test can be written as H0 : ρi = 0 for i against the alternative hypothesis HA : ρi < 0 for i = 1, 2, . . . , N1 and ρi = 0 for i = N1 + 1, N1 + 2, . . . , N . Thus under the it series in the panel are nonstationary processes; under null hypothesis, all ln A the alternative, a fraction of the series in the panel are assumed to be stationary. Im et al. (2003) propose to use the average of the tρi statistics from a singlecountry ADF tests to form the following t-statistic √ N (t N T − μT ) , t-bar = √ σT where tNT =

n 1 tρi , N i=1

and μT and σT are the mean and variance of each tρi statistic. In fact, t-bar is an adjusted group mean of the tρi statistics from single-country ADF tests. The values of μT and σT for different T and k (number of lags) are tabulated in Im et al. (2003, Table 2), based on stochastic simulations. They show that the distribution of t-bar weakly converges to a standard normal variate under the null hypothesis that ρi = 0 for all i and diverges under the alternative hypothesis that ρi < 0. A potential problem with the Im et al. (2003) procedure involves cross-sectional dependence. In particular, O’Connell (1998) demonstrated that, if the error terms in Equation (9) are contemporaneously correlated, size distortion will come in. Recognizing this potential problem, Sarno and Taylor (1998) and Taylor and Sarno (1998) suggest an approach to testing the unit root null hypothesis in panels, which allows both different ρ values under the alternative hypothesis and

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contemporaneous error correlations across the panel. They call this the multivariate augmented Dickey–Fuller (MADF) test. Following Taylor and Sarno (1998), the panel specification in our particular case (with country N being the reference country) can be written as 1t = α1 + ρ1 ln A 1,t−1 + ln A

k1

1,t−j + c1j ln A ε1t ,

j =1

.. . N−1,t−1 N−1,t = αN−1 + ρN−1 ln A ln A

kN−1

+

N−1,t−j + cN−1,j ln A εN−1,t .

j =1

In contrast to the Im et al. (2003) test, seemingly unrelated estimation is applied to the above system of equations, so that the covariance matrix of the εit ’s is used in estimating each ρi . Notice that this test also accommodates heterogeneous lags. The null hypothesis that ρi = 0 for i is tested against the alternative that at least one series is stationary, ρi = 0. A chi-squared form of Wald statistic is used in the MADF test. However the Wald statistic does not have a chi-squared (with N degrees of freedom) as a limiting distribution, and thus Monte Carlo simulation is need to calculate critical values. Taylor and Sarno (1998) demonstrate that this test has power properties that are significantly better than the single equation augmented Dickey–Fuller test. 3.3. Identification Issues As we mentioned earlier, a problem with most of the panel unit root tests, including both the Im et al. (2003) test and the Taylor and Sarno (1998) MADF test, is that while they have the power to reject a false nonconvergence null, they do not provide information about how many panel members reject the null hypothesis and how many do not. To address these issues, Breuer et al. (2001) advocate a panel data unit root test which involves estimating ADF regressions in a seemingly unrelated regression (SUR) framework and then testing for individual unit roots within the panel. The SURADF test is more powerful than independently estimated single equation ADF tests. Earlier SUR-based tests of O’Connell (1998) and Taylor and Sarno (1998) have an ‘all or nothing’ characteristic, as all series are either stationary or nonstationary. The Breuer et al. (2001) SURADF procedure allows ρi to differ across the series under the alternative hypothesis while exploiting information in error covariances to produce efficient estimators with potentially powerful test statistics. Thus, this procedure, which can be viewed as a multivariate version of the augmented Dickey–Fuller test, allows identification of how many and which members

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of the panel contain a unit root. The critical values, as in the MADF test case, are specific to the estimated covariance matrix for the system considered, the sample size, and the number of panel members, and thus must be derived through simulations.

4. Data and Empirical Evidence 4.1. The Data To obtain total factor productivity estimates, we use annual data for 17 OECD countries over the period from 1960 to 2002. The 17 OECD countries include 15 European Union (EU) countries – Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, the Netherlands, Portugal, Spain, Switzerland and the United Kingdom – plus two North American countries, Canada and the Unites States. The data consists of prices and quantities for two outputs, three variable inputs and one exogenously given input. Our treatment of imports as an input to the production sector is standard in this literature – see, for example, Kohli (1991) and Fox and Diewert (1999). The price and quantity aggregates for (domestic) output are derived from three components – private consumption, government consumption and investment. The price and quantity data for private and government consumption are from the OECD national accounts, whereas those for investment are constructed using the increase in stocks and gross fixed capital formation prices and quantities both taken from the OECD national accounts. Similarly, the price and quantity data for exports and imports are from the OECD national accounts. Following Fox (1997), price and quantity aggregates for labor (our fourth input) are derived from two components: wage earners and self employed plus unpaid family workers. The price aggregate, p4 , is calculated by dividing the compensation of employees by the number of wage earners, using OECD national accounts data for the former and Labor Force Statistics OECD data for the latter. The price of self-employed plus unpaid family workers is taken to be 0.4 times the price of employment, p4 . Hence, the compensation of self employed plus unpaid family workers is calculated as 0.4 × p4 × (civilian employment – wage earners), with the data for civilian employment (that is, total employment) taken from the Labor Force Statistics OECD. Regarding reproducible capital (our fifth variable, y5 ), its quantity is calculated as y5t+1 = (1 − δ)y5t + It , with the starting value of y5 being approximated by I1 /[1−(1−δ)/gI ], where gI is the average annual growth rate of investment over the sample period and δ the depreciation rate. Assuming that reproducible capital will depreciate to zero within T years and that the average annual investment growth rate is gI from (1960-T ) to 1960, then reproducible capital in 1960 is

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calculated as (1 − δ)T −1 I1 (1 − δ)T −2 I1 (1 − δ)T I1 + + + · · · + I1 T T −1 gI gI gIT −2 I1 ≈ . 1 − (1 − δ)/gI The price of reproducible capital, p5 , is calculated as (R + δ)pI , where pI is the price of investment, I , and R is the ex post rate of return. Finally, following Diewert (2001) we assume that the exogenously given input, y6 , is an aggregate of land, stock of inventories and intellectual property. We assume that initially land and fixed factors make up half of the residual input and inventories and intellectual capital make up the other half. Assuming that the latter grows at the GDP growth rate, gGDP , we set y6t+1 = y6t [0.5 + 0.5(1 + gGDP )], with the starting value of y6 (in 1960) approximated by p1 y1 + p2 y2 − p3 y3 − p4 y4 − p5 y5 . Finally, p6 is approximated by (p1 y1 + p2 y2 − p3 y3 − p4 y4 − p5 y5 )/y6 . 4.2. Total Factor Productivity Estimates In estimating total factor productivity for each of the 17 OECD countries (by the maximum likelihood method), we assumed that technology is constant returns to scale, by setting a = 0 in Equation (1), and also imposed correct curvature conditions globally. The total factor productivity estimates obtained using the normalized quadratic variable profit function, Equation (1), for each of the 17 OECD countries are shown in Table 1 and Figure 1. Compared to the total factor productivity estimates reported by Bernard and Jones (1996) using a Cobb–Douglas production, our estimates differ in two respects. First, Ireland has consistently been the best performer in our study, whereas the United States, which is the productivity leader in Bernard and Jones (1996), shows only a moderate performance throughout the sample period. Our results are not surprising, and have been confirmed by many previous studies using various different measurement approaches – see, for example, Coe and Helpman (1995). Second, while the total factor productivity estimates for all countries reported by Bernard and Jones (1996) show a general tendency to rise, our estimates show that the patterns of total factor productivity for different countries are quite diverse (see Figure 1). A partial explanation of these differences is that our NQ model is flexible and also allows for multiple outputs and multiple inputs as we discussed earlier, while the Cobb–Douglas production function used by Bernard and Jones (1996) does not possess these properties. Moreover, different sample periods may be another reason leading to different total factor productivity estimates. As can be seen in Figure 1, the total factor productivity series for the 17 OECD countries show substantial drop in variance throughout the sample period, pro-

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Table 1a: Estimates of TFP for 17 OECD Countries (1960–2002) Year

Austria

Belgium

Canada

Denmark

Finland

France

Germany

Greece

Ireland

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

0.0091 0.00998 0.00996 0.01013 0.01049 0.01069 0.01062 0.0103 0.01001 0.00962 0.00996 0.01014 0.00983 0.01031 0.00949 0.01023 0.01055 0.01008 0.00919 0.00974 0.00968 0.00904 0.0086 0.00958 0.00946 0.00942 0.00957 0.00982 0.0096 0.00936 0.0093 0.00928 0.00925 0.00929 0.00929 0.00956 0.00953 0.00956 0.00954 0.00959 0.00947 0.00959 0.00975

0.0175 0.01666 0.01628 0.01553 0.01555 0.01558 0.0158 0.01544 0.01501 0.01492 0.01473 0.0142 0.0141 0.01409 0.01401 0.01406 0.01383 0.014 0.01387 0.01416 0.01293 0.01277 0.01295 0.01328 0.0134 0.01353 0.01414 0.01409 0.01408 0.01397 0.01392 0.01413 0.01451 0.01506 0.01509 0.01485 0.01482 0.01481 0.01492 0.0148 0.01478 0.01505 0.01541

0.00224 0.00202 0.00186 0.00134 0.00147 0.00161 0.00167 0.00176 0.00168 0.00158 0.00163 0.0013 0.00141 0.00191 0.00303 0.00279 0.00283 0.0024 0.00202 0.00295 0.00391 0.00402 0.00404 0.00404 0.0038 0.0036 0.00329 0.00344 0.00343 0.00354 0.00357 0.00371 0.00378 0.00374 0.00367 0.00398 0.00407 0.00392 0.00362 0.00363 0.00393 0.00381 0.00364

0.02118 0.01978 0.01929 0.01879 0.01772 0.01689 0.01623 0.01513 0.01448 0.01421 0.01409 0.01328 0.01279 0.0124 0.0124 0.01262 0.01183 0.01172 0.0119 0.01135 0.01151 0.01157 0.01134 0.01131 0.01103 0.01077 0.0104 0.0102 0.00999 0.01015 0.01007 0.0099 0.01017 0.01016 0.00989 0.00986 0.00982 0.00986 0.00963 0.00932 0.00948 0.00942 0.00895

0.01513 0.01467 0.01395 0.01364 0.01438 0.01412 0.01358 0.0132 0.01368 0.0132 0.01261 0.01258 0.01187 0.01141 0.00629 0.0124 0.01275 0.01241 0.01227 0.01197 0.01105 0.01084 0.01068 0.01058 0.01075 0.01073 0.01059 0.01044 0.01018 0.00982 0.00987 0.01115 0.01242 0.01316 0.01312 0.01294 0.01259 0.01175 0.01122 0.01068 0.01009 0.01006 0.00992

0.00387 0.00430 0.00424 0.00463 0.00511 0.00508 0.00507 0.00517 0.00542 0.00542 0.00433 0.00487 0.00533 0.00549 0.00472 0.00392 0.00557 0.00449 0.00479 0.00544 0.00538 0.00482 0.00523 0.00459 0.00476 0.00532 0.00589 0.00629 0.00597 0.00577 0.00588 0.00587 0.00611 0.00631 0.00636 0.00644 0.00652 0.00666 0.00654 0.00649 0.00632 0.0064 0.00659

0.00334 0.00364 0.00382 0.00374 0.0039 0.00402 0.00413 0.0043 0.00439 0.00444 0.00502 0.00504 0.00501 0.00486 0.00474 0.00531 0.00514 0.00512 0.0052 0.0053 0.0051 0.00464 0.00474 0.0051 0.00507 0.00502 0.00527 0.00573 0.00574 0.00573 0.00581 0.00706 0.00734 0.0076 0.00775 0.00792 0.00799 0.00804 0.00815 0.00831 0.00839 0.00854 0.00875

0.0001 0.0002 0.00025 0.0003 0.00008 0.00048 0.00046 0.0006 0.00089 0.00116 0.00093 0.00103 0.00091 0.00122 0.00074 0.00095 0.00079 0.00157 0.00165 0.00159 0.00144 0.00208 0.0027 0.00302 0.00272 0.00303 0.00247 0.00311 0.0034 0.00368 0.0037 0.00365 0.00321 0.00317 0.00351 0.00395 0.00422 0.00466 0.00432 0.00426 0.00417 0.00471 0.00511

0.03906 0.03915 0.03911 0.03942 0.04064 0.04196 0.0424 0.04048 0.03961 0.03927 0.03272 0.03273 0.03284 0.0357 0.035 0.03346 0.03907 0.03699 0.03539 0.03362 0.03134 0.03103 0.03225 0.03466 0.03491 0.03457 0.03446 0.03364 0.03371 0.03376 0.02982 0.02931 0.02787 0.02886 0.02764 0.02581 0.02428 0.0225 0.02138 0.019 0.01812 0.01768 0.017

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Table 1b: Estimates of TFP for 17 OECD Countries (1960–2002) Year

Italy

Luxembourg

NL

Portugal

Spain

Switzerland

UK

US

STDEV

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

0.01316 0.0131 0.0125 0.01249 0.01341 0.01251 0.01236 0.01204 0.01166 0.01134 0.01152 0.01176 0.01206 0.01065 0.01046 0.01126 0.01059 0.01079 0.01051 0.01036 0.01012 0.0101 0.01053 0.01077 0.01073 0.01081 0.01117 0.01126 0.01139 0.0116 0.01185 0.01215 0.01231 0.01275 0.01279 0.0128 0.01291 0.01291 0.01284 0.01279 0.01256 0.01266 0.01308

0.03309 0.02941 0.02806 0.02461 0.02250 0.02292 0.02262 0.02257 0.02345 0.02273 0.02462 0.02134 0.02015 0.02098 0.02203 0.02089 0.02133 0.01781 0.01748 0.01781 0.01846 0.01981 0.02126 0.01969 0.01795 0.01787 0.01711 0.01632 0.01396 0.01330 0.01254 0.01212 0.01222 0.01230 0.01259 0.01277 0.01300 0.01247 0.01247 0.01201 0.01179 0.01169 0.01137

0.01281 0.01281 0.01216 0.01214 0.01126 0.01099 0.01049 0.01015 0.0098 0.00881 0.00868 0.00767 0.00733 0.00734 0.00725 0.00812 0.00747 0.00833 0.00784 0.007 0.00689 0.00625 0.00791 0.00813 0.0074 0.00794 0.00812 0.00723 0.00639 0.00661 0.0065 0.00646 0.00633 0.00626 0.00625 0.00627 0.00602 0.00595 0.00579 0.00533 0.00539 0.00546 0.00547

0.00591 0.00637 0.00661 0.00757 0.00626 0.00632 0.00595 0.00532 0.0083 0.00898 0.00814 0.00885 0.00874 0.00853 0.00996 0.01042 0.00961 0.00938 0.00957 0.00923 0.00853 0.00802 0.00816 0.00794 0.00829 0.00882 0.01033 0.01033 0.01005 0.01081 0.0111 0.01216 0.01291 0.01346 0.01415 0.01464 0.01446 0.01446 0.01388 0.01402 0.01403 0.01468 0.01552

0.00101 0.00132 0.0015 0.00179 0.00227 0.00375 0.00601 0.00807 0.00897 0.00923 0.00884 0.00922 0.00972 0.00948 0.00776 0.00869 0.0096 0.00947 0.01086 0.01151 0.01011 0.00933 0.00965 0.00885 0.00913 0.01026 0.01218 0.01209 0.01221 0.01216 0.01264 0.0134 0.01449 0.0154 0.01546 0.01526 0.01526 0.0149 0.01499 0.01442 0.01381 0.01387 0.01398

0.01428 0.01391 0.01296 0.01243 0.01180 0.01150 0.01108 0.01084 0.01087 0.01069 0.01064 0.01075 0.01038 0.01025 0.01055 0.01087 0.01085 0.01076 0.01047 0.01035 0.01017 0.01023 0.01009 0.01006 0.01006 0.01001 0.00992 0.00982 0.00991 0.00995 0.00975 0.00994 0.01026 0.01088 0.01097 0.01088 0.01084 0.01082 0.01064 0.01031 0.01041 0.01061 0.01035

0.01056 0.01083 0.01099 0.01067 0.01061 0.01062 0.01068 0.01081 0.01031 0.01042 0.0108 0.01104 0.01134 0.01005 0.00916 0.01051 0.01004 0.00996 0.0103 0.01048 0.01128 0.01179 0.01182 0.01172 0.01175 0.01159 0.01155 0.0114 0.011 0.01099 0.0117 0.01265 0.01337 0.01349 0.01322 0.01285 0.0127 0.01283 0.01303 0.01313 0.01303 0.01326 0.0134

0.00886 0.00884 0.00843 0.00801 0.00745 0.00712 0.00683 0.0069 0.00656 0.00645 0.00652 0.00641 0.00591 0.00554 0.00502 0.00519 0.00509 0.0047 0.00452 0.0043 0.00402 0.00428 0.00496 0.00532 0.00515 0.00516 0.00501 0.00482 0.00471 0.00466 0.00468 0.0049 0.00483 0.0048 0.0047 0.0046 0.00453 0.00446 0.0043 0.00412 0.00392 0.004 0.00401

0.01078 0.01020 0.00998 0.00964 0.00968 0.00998 0.00976 0.00927 0.00903 0.00888 0.00788 0.00751 0.00743 0.00798 0.00804 0.00756 0.00864 0.00800 0.00767 0.00728 0.00682 0.00689 0.00711 0.00743 0.00740 0.00726 0.00723 0.00696 0.00690 0.00688 0.00605 0.00594 0.00579 0.00607 0.00585 0.00545 0.00516 0.00481 0.00471 0.00438 0.00422 0.00419 0.00420

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Figure 1: Total Factor Productivity Estimates for 17 OECD Countries (1960–2002).

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viding evidence of convergence in total factor productivity. Roughly speaking, Figure 1 indicates that there are three sets of countries: countries converging from above average (Luxembourg, Denmark, Belgium, Finland, Sweden, the Netherlands and Ireland), countries converging from below average (Portugal, Spain, Greece and Canada), and countries that roughly remain at the average level throughout the sample period (Austria, France, Germany, Italy, the United Kingdom and the United States). This visual evidence of convergence of aggregate total factor productivity is further reinforced by Figure 2, which shows the cross-section standard deviation of total factor productivity. As can be seen from Figure 2, the dispersion has decreased steadily over the sample period from 1.08% in 1960 to 0.42% in 2002. However, formal tests are needed before we come to a conclusion concerning convergence in aggregate total factor productivity for our sample economies during the 1960–2002 period. We now turn to these tests. 4.3. Evidence from the Cross-Section Test We examine the convergence hypothesis, using the cross-section test, for two different panels – a full panel of all 17 OECD countries and an European panel of the 15 European Union countries. A related issue in testing (8) is how to choose benchmark countries. Asymptotically, this choice does not matter, but in small samples it is important. In this chapter we choose the largest two economies as the reference country in each panel. In particular, for the full panel of all 17 countries we choose the United States and Germany and for the panel of the 15 European Union countries we choose Germany and France. Ireland is not chosen due to its small economy size although it is the most productive country in our study. Table 2 reports the results from the cross-section regression of the long-run total factor productivity growth rate relative to the reference country on the total factor productivity differential with respect to the reference country in 1960, as follows i,1960 + ηi , gi = α + β ln A where gi is constructed as the trend coefficient from a regression of the productivity differential with respect to the reference country on a constant and a linear trend – see Bernard and Jones (1996). From the last column of Table 2, we see that all the R 2 values are greater than 88%, implying that a very large percentage of the variation in the total factor productivity growth rate relative to the benchmark country can be explained by differences in initial productivity differentials with respect to the benchmark country. Confirming the evidence from standard deviations (see Figure 2), we find a negative and significant coefficient on the initial total factor productivity differential for both panels – the 17 OECD economies and the 15 EU economies – regardless of the benchmark country. In fact, the point estimates of β from the 6 regressions are between −0.0165 and −0.0170, comparable to those reported by Bernard and Jones (1996), although they employ a different method in estimating total factor productivity and also a different

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Figure 2: Standard Deviation of Aggregate TFP for 17 OECD Countries (1960–2002).

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Table 2: Cross-section Convergence Tests Panel

Benchmark country

17 OECD countries

Germany United States Germany France

15 EU countries

β

t-statistic

p-value

λ

R2

−0.016662 −0.016500 −0.016541 −0.016947

−10.68 −12.77 −11.34 −12.80

0.001 0.001 0.001 0.001

0.012643 0.012547 0.012572 0.012811

0.8828 0.9153 0.9076 0.9261

dataset. Correspondingly, the point estimates of λ, the rates of catch up in total factor productivity, are around 1.25–1.30%. 4.4. Evidence from Panel Unit Root Tests Having confirmed the convergence hypothesis using the cross-section test, we now examine the convergence hypothesis for each of the two panels using the Im et al. (2003) test and the Taylor and Sarno (1998) MADF test. The benchmark countries are the same as in the cross-section test. Table 3a reports the results from the Im et al. (2003) test. As already noted, to compute the t-bar statistic we estimate Equation (9) individually for each of the countries in the panel (except for the reference country), and then construct the corresponding ADF t-statistics, tρi . These individual t-statistics are then averaged to obtain the t-bar statistic for the panel. The critical values for the t-bar statistic of the Im et al. (2003) test are from the standard normal distribution. Table 3b reports the results from the Taylor and Sarno (1998) MADF test. To calculate the critical values for this tests we first estimate the system of equations using SUR and restrict ρi to be equal to zero. We then generate 2000 simulated panel series of 100 + T observations using the restricted SUR parameter estimate ε ), where Σ ε is the restricted SUR estiαi and cij , we draw randomly from (0, Σ mate of the contemporaneous covariance matrix for the disturbances, and also set 1,t−1 and ln A i,t−j equal to zero. We drop the first 100 the initial values of ln A observations to yield simulated panel series of T observations, and then order the simulated Wald statistics and use the 20th, 100th and 200th values as 1%, 5% and 10% critical values, respectively. Confirming the evidence from the cross-section test, the results in Tables 3a and 3b show that the null hypothesis of nonconvergence (nonstationarity) is rejected for each of the two panels using both the Im et al. (2003) and Taylor and Sarno (1998) panel tests. In fact, the t-bar statistics of the Im et al. (2003) test are well below their respective critical values even at the 1% level and the Wald statistics of the Taylor and Sarno (1998) MADF test are well above their respective critical values at the 1% level.

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Table 3a: Panel Unit Root Tests (Im et al. (2003) Tests) Panel

17 OECD countries 15 EU countries

Benchmark country

Germany United States Germany France

Critical values

t-bar

−2.181 −2.207 −2.325 −2.105

p-value

10%

5%

1%

−1.78 −1.78 −1.81 −1.81

−1.85 −1.85 −1.89 −1.89

−1.98 −1.98 −2.05 −2.05

0.001 0.001 0.001 0.001

Table 3b: Panel Unit Root Tests (Taylor and Sarno (1998) Tests) Panel

17 OECD countries 15 EU countries

Benchmark country

Germany United States Germany France

Wald

212.3 224.1 212.3 200.4

Critical values 10%

5%

1%

169.2 168.6 167.1 141.0

181.8 181.0 180.1 153.8

209.8 207.9 206.2 184.4

4.5. Evidence from the Multivariate Test Having confirmed the convergence hypothesis using both cross-section and panel unit root tests, we now turn to identify the converging countries using the Breuer et al. (2001) multivariate test that accounts for cross-sectional dependence among the elements of the panel. Tables 4a–4d report the results for both the OECD panel and the European panel. As with the MADF test, we simulate critical values that are specific to lag structure and the estimated covariance matrix. As we noted earlier, the results from the SURADF tests show that the convergence result found by the cross-section test, the Im et al. (2003) test, and the Taylor and Sarno (1998) multivariate augmented Dickey–Fuller test are driven by the convergence behavior of some elements of the panel, not necessarily all of them. Tables 4a and 4b report the results from the SURADF tests for the OECD panel using Germany and the United States as the benchmark countries. There are two interesting findings regarding our SURADF tests for the full OECD panel. First, there are more countries converging to Germany than to the United States. From Table 4a, we can see that there are 8 countries converging to Germany at the 5% level, namely Belgium, Denmark, Italy, Portugal, Spain, Switzerland, the United Kingdom and the United States. Two more countries (Austria and the Netherlands) can be added to the convergence club if we use the 10% significance level. In contrast, there are only 4 countries converging to the United States at the 5% level, namely Denmark, Germany, Spain and the United Kingdom. And two more

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Table 4a: SURADF Tests (All 17 OECD Countries: Germany Is the Benchmark) Country

SURADF

1%

5%

10%

Austria Belgium Canada Denmark Finland France Greece Ireland Italy Luxembourg The Netherlands Portugal Spain Switzerland UK US

−4.177∗ −6.491∗ −1.623 −6.500∗ −2.050 −2.301 −1.441 −2.250 −5.690∗ −3.829 −3.431∗∗ −5.710∗ −7.214∗ −7.192∗ −4.668∗ −5.016∗

−5.216 −6.136 −4.695 −6.081 −4.289 −4.686 −4.145 −5.525 −5.597 −5.258 −4.527 −4.328 −4.900 −6.394 −5.228 −5.532

−4.475 −5.384 −4.015 −5.303 −3.601 −3.977 −3.450 −4.802 −4.910 −4.611 −3.771 −3.665 −4.189 −5.699 −4.484 −4.829

−4.066 −4.998 −3.665 −4.916 −3.252 −3.625 −3.119 −4.449 −4.524 −4.210 −3.404 −3.279 −3.790 −5.309 −4.105 −4.464

∗ indicates significance at the 5% level. ∗∗ indicates significance at the 10% level.

Table 4b: SURADF Tests (All 17 OECD Countries: US Is the Benchmark) Country

SURADF

1%

5%

10%

Austria Belgium Canada Denmark Finland France Germany Greece Ireland Italy Luxembourg The Netherlands Portugal Spain Switzerland UK

−3.049 −2.097 −0.618 −5.233∗ −3.216 −2.139 −3.991∗ −3.440∗∗ −1.019 −3.118 −2.496 −3.149∗∗ −1.483 −8.805∗ −1.480 −4.283∗

−4.612 −5.327 −4.818 −5.185 −4.281 −4.314 −4.376 −4.385 −4.451 −5.375 −4.709 −3.893 −4.432 −4.621 −5.470 −4.949

−3.960 −4.668 −4.176 −4.448 −3.654 −3.670 −3.708 −3.721 −3.800 −4.708 −4.003 −3.259 −3.763 −3.933 −4.755 −4.273

−3.584 −4.277 −3.783 −4.050 −3.275 −3.320 −3.324 −3.370 −3.442 −4.323 −3.631 −2.912 −3.419 −3.552 −4.353 −3.890

∗ indicates significance at the 5% level. ∗∗ indicates significance at the 10% level.

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Table 4c: SURADF Tests (15 EU Countries: Germany Is the Benchmark) Country

SURADF

1%

5%

10%

Austria Belgium Denmark Finland France Greece Ireland Italy Luxembourg The Netherlands Portugal Spain Switzerland UK

−4.449∗ −6.408∗ −6.591∗ −1.520 −2.713 −1.045 −2.663 −5.755∗ −4.163∗∗ −3.794∗∗ −5.488∗ −7.273∗ −7.128∗ −4.711∗

−5.012 −5.891 −5.793 −4.035 −4.496 −3.890 −5.322 −5.338 −5.003 −4.495 −4.118 −4.647 −6.130 −5.062

−4.306 −5.195 −5.137 −3.384 −3.818 −3.279 −4.708 −4.694 −4.337 −3.804 −3.474 −3.938 −5.407 −4.344

−3.952 −4.814 −4.752 −3.053 −3.450 −2.985 −4.347 −4.311 −3.951 −3.464 −3.139 −3.565 −5.075 −3.986

∗ indicates significance at the 5% level. ∗∗ indicates significance at the 10% level.

Table 4d: SURADF Tests (15 EU Countries: France Is the Benchmark) Country

SURADF

1%

5%

10%

Austria Belgium Denmark Finland Germany Greece Ireland Italy Luxembourg The Netherlands Portugal Spain Switzerland UK

−3.691 −5.802∗∗ −2.322 −5.723∗∗ −2.075 −2.001 −0.403 −6.313∗ −2.057 −3.166 −3.323 −9.010∗ −6.161∗ −5.778∗

−6.274 −6.668 −5.037 −6.600 −5.787 −4.224 −5.794 −6.380 −5.652 −6.080 −5.136 −5.526 −6.617 −6.123

−5.648 −6.115 −4.364 −5.939 −5.090 −3.621 −5.115 −5.768 −5.024 −5.436 −4.422 −4.771 −5.976 −5.480

−5.285 −5.745 −3.970 −5.582 −4.707 −3.299 −4.738 −5.423 −4.623 −5.051 −4.010 −4.363 −5.643 −5.115

∗ indicates significance at the 5% level. ∗∗ indicates significance at the 10% level.

countries (Greece and the Netherlands) are converging to the United States at the 10% level.

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Another interesting finding is that Canada does not converge to either of the two benchmark countries. These results show that convergence is more likely to occur within the EU countries than between EU member countries and the United States. To further investigate the convergence with the 15 European countries, we also perform the SURADF tests for the European panel using Germany and France as benchmark countries. The results are reported in Tables 4c and 4d. It can be clearly seen that there are 8 countries converging to Germany within the 15 EU countries at the 5% level and two more can be added if we use the 10% significance level. On the other hand, there are only 4 countries converging to France within the 15 EU countries at the 5% level and two more can be added if we use the 10% level. These results show that the convergence club within the 15 EU countries is formed around Germany rather than France.

5. Conclusion We used the normalized quadratic (NQ) variable profit function to estimate total factor productivity using annual data for 15 European Union countries plus Canada and the United States, during the 1960–2002 period. The NQ model is flexible (in the sense that it is capable of approximating an arbitrary profit function to the second order) and allows for the estimation of total factor productivity within a multiple-output and multiple-input framework. Moreover, we have treated imports as a separate input and exports as a separate output, thereby allowing for the influence of international trade on aggregate productivity convergence through the transmission of technological knowledge and increased competition. We also tested for productivity convergence, using methods suggested by Bernard and Jones (1996). In doing so, we applied both cross-section tests (β-convergence and σ -convergence) and two panel data unit root tests, proposed by Im et al. (2003) and Taylor and Sarno (1998). In general, our results support total factor productivity convergence for all countries. We also applied the multivariate tests proposed by Breuer et al. (2001) to identify the countries that converge and those that do not. We find that more countries converge to Germany than the United States, and that within the European Union more countries converge to Germany than France. We have investigated productivity growth, building on a large body of recent literature that takes a flexible functional forms modeling approach. In doing so, we have used the normalized quadratic variable profit function to estimate total factor productivity. Of course, alternative and perhaps more general and more robust specifications could be estimated. A particularly constructive approach would be to use flexible functional forms that possess global properties. One new globally flexible functional form has recently been introduced by Serletis and Shahmoradi (2006). This functional form is based on the Asymptotically Ideal Model (AIM), introduced by Barnett and Jonas (1983) and employed and explained in Barnett et

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al. (1991). The Serletis and Shahmoradi (2006) AIM(n)-TP cost function allows for technical progress and for the estimation of total factor productivity, technical progress biases and related price elasticities.

Acknowledgement Serletis gratefully acknowledges financial support from the Social Sciences and Humanities Research Council of Canada.

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Dollar, D., Wolff, E.N. and Baumol, W.J. (1988). The factor-price equalization model and industry labor productivity: An empirical test across countries. In Feenstra, R.C. (Ed.), Empirical Methods for International Trade. Cambridge, MA: MIT Press. Dowrick, S. and Nguyen, D.T. (1989). OECD comparative economic growth 1950–1985: Catch-up and convergence. American Economic Review 79, 1010–1030. Fox, K.J. (1997). Open economy data sets for 18 OECD countries, 1960–1992. University of New South Wales School of Economics, Discussion Paper 1997/26. Fox, K.J. and Diewert, W.E. (1999). Is the Asia Pacific region different? Technical progress bias and price elasticity estimates for 18 OECD Countries, 1960–1992. In Fu, T.-T., Huang, C.J. and Lovell, C.A.K. (Eds), Economic Efficiency and Productivity Growth in the Asia-Pacific Region. Cheltenham, UK; Northampton, MA: Edward Elgar Publishing, pp. 125–144. Grossman, G.M. and Helpman, E. (1991). Innovation and Growth in the Global Economy. Cambridge, MA: MIT Press. Im, K.S., Pesaran, M.H. and Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics 115, 53–74. Islam, N. (1995). Growth empirics: A panel data approach. Quarterly Journal of Economics 110, 1127–1170. Kohli, U. (1991). Technology, Duality and Foreign Trade. New York: Harvester Wheatsheaf. Levin, A., Lin, C.-F. and Chu, C.-S.J. (2002). Unit root tests in panel data: Asymptotic and finite-sample properties. Journal of Econometrics 108, 1–24. O’Connell, P.G.J. (1998). The overvaluation of purchasing power parity. Journal of International Economics 44, 1–19. Romer, P. (1990). Endogenous technical change. Journal of Political Economy 98, 71–102. Sala-i-Martin, X. (1996). The classical approach to convergence. Economic Journal 106, 1019–1036. Sarno, L. and Taylor, M. (1998). Real exchange rates under the recent float: Unequivocal evidence of mean reversion. Economics Letters 60, 131–137. Serletis, A. and Shahmoradi, A. (2006). The asymptotically ideal model and the estimation of technical progress. Mimeo, Department of Economics, University of Calgary. Solow, R.M. (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics 70, 65–94. Taylor, M. and Sarno, L. (1998). The behavior of real exchange rates during the postBretton Woods period. Journal of International Economics 46, 281–312.

Chapter 6

The Theoretical Regularity Properties of the Normalized Quadratic Consumer Demand Model William A. Barnett∗ and Ikuyasu Usui Department of Economics, University of Kansas, Lawrence, KS 66045, USA

Abstract We conduct a Monte Carlo study of the global regularity properties of the Normalized Quadratic model. We particularly investigate monotonicity violations, as well as the performance of methods of locally and globally imposing curvature. We find that monotonicity violations are especially likely to occur, when elasticities of substitution are greater than unity. We also find that imposing curvature locally produces difficulty in the estimation, smaller regular regions, and the poor elasticity estimates in many cases considered in the chapter. Imposition of curvature alone does not assure regularity, and imposing local curvature alone can have very adverse consequences.

Keywords: flexible functional form, Normalized Quadratic, regularity regions, monotonicity, curvature JEL: C14, C22, E37, E32

1. Introduction Uzawa (1962) proved that the constant elasticity of substitution (CES) model cannot attain arbitrary elasticities with more than two goods. As a result, the development of locally flexible functional forms evolved as a new approach to modeling specifications of tastes and technology. Flexible functional forms were defined by Diewert (1971) to be the class of functions that have enough free parameters to provide a local second-order approximation to any twice continuously differentiable function. If a flexible functional form has no more parametric freedom than needed to satisfy that definition, then the flexible functional form is called “parsimonious”. Barnett (1983a, 1983b) proved that a functional form satisfies

∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18006-6

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Diewert’s definition if and only if it can attain any arbitrary elasticities at any one predetermined point in data space. Most of the available flexible functional forms are based on quadratic forms derived from second-order series expansions. The translog model of Christensen et al. (1971) and the AIDS (almost ideal demand system) model of Deaton and Muellbauer (1980) use Taylor series expansions in logarithms, the generalized Leontief model of Diewert (1971) uses a Taylor series expansion in square roots, and the Laurent models of Barnett (1983a, 1983b) use the Laurent series expansion. As these flexible functional form models became available, applied researchers tended to overlook the maintained regularity conditions required by microeconomic theory. Regularity requires satisfaction of both curvature and monotonicity conditions. Simultaneous imposition of both of these conditions on a parsimonious flexible functional form destroys the model’s local flexibility property. For instance, Lau (1978) showed that imposition of global regularity reduces the translog model to Cobb–Douglas, which is not a flexible functional form and has no estimable elasticities. When regularity is not imposed, most of the estimated flexible functional forms in empirical applications exhibit frequent violations of regularity conditions at many data points.1 Since that fact became evident, information about violations of regularity conditions in empirical applications have become hard to find.2 An exception to the common neglect of regularity conditions was Diewert and Wales’ (1987) work on the Normalized Quadratic model. That model permits imposition of curvature globally, while remaining flexible. Since violations of curvature have more often been reported than violations of monotonicity, the imposition of curvature alone seems to merit consideration. In subsequent papers of Diewert and Wales (1992, 1993, 1995) and others, imposition of curvature globally, without imposition of monotonicity, has become a common practice with the Normalized Quadratic functional form. But once curvature is imposed without the imposition of monotonicity, the earlier observation may no longer apply. When global curvature is imposed, the loss of model-fit may induce spurious improvements in fit through violations of monotonicity. This problem could be especially common with quadratic models, which can have bliss points. It is possible that violations of monotonicity could be induced by imposition of curvature. With this model, it has become common not to check for monotonicity, after imposing global curvature. Diewert and Wales (1995) and Ryan and Wales (1998) have expected that monotonicity will be satisfied, as a result of the nonnegativity of the dependent variables. But nonnegativity of observed dependent variables

1 See,

e.g., Manser (1974) and Humphrey and Moroney (1975). noteworthy exception is Moroney and Trapani (1981), who confirmed the earlier findings of frequent violations of maintained regularity conditions.

2A

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does not assure nonnegativity of fitted dependent variables. In Kohli (1993) and Diewert and Fox (1999), the curvature condition is treated as the sole regularity condition. But without satisfaction of both curvature and monotonicity, the second-order condition for optimizing behavior fails, duality theory fails, and inferences resulting from derived estimating equations become invalid.3 Hence the common practice of equating regularity solely with curvature is not justified. Barnett (2002) and Barnett and Pasupathy (2003) confirmed the potential problem and found further troublesome consequences, when they checked regularity violations in their own previously published estimation of technology in Barnett et al. (1995). Initially, they imposed curvature globally, but monotonicity only at a central data point with the Normalized Quadratic production model. In addition to violations of monotonicity, they encountered induced curvature reversals of composite functions, along with nonunique isoquants and complex valued solutions. Even if curvature is imposed on both inner (category) production functions and weakly separable outer functions, the composite technology still can violate curvature, if monotonicity is violated. The evidence suggested the need for a thorough investigation of the global regularity property of the Normalized Quadratic model. We undertake this task in this chapter. A well-established approach to exploring regularity properties of a neoclassical function is to set the parameters of the model to produce various plausible elasticities, and then plot the regular regions within which the model satisfies monotonicity and curvature. We do so by setting the parameters at various levels to produce elasticities that span the plausible range, and then plot the regular region of the model when curvature is imposed but monotonicity not imposed. The intent is to explore the common practice with the Normalized Quadratic model. Such experiments have been conducted with the translog and the generalized Leontief by Caves and Christensen (1980) and with newer models by Barnett et al. (1985, 1987) and Barnett and Lee (1985).4 In our experiment, we obtain the parameter values of the Normalized Quadratic model by estimation of those parameters with data produced by another model at various settings of the elasticities. Jensen (1997) devised the experimental design, which closely follows that of Caves and Christensen, but Jensen applied the approach to estimate the coefficients of the Asymptotically Ideal Model (AIM) of Barnett and Jonas (1983).5 We adopt a similar experimental design, in which

3 The damage done to the inference has been pointed out by Basmann et al. (1983), Basmann et al. (1985), Basmann et al. (1990) and Basmann et al. (1994). 4 Other relevant papers include Wales (1977), Blackorby et al. (1977), Guilkey and Lovell (1980), White (1980), Guilkey et al. (1983), Barnett and Choi (1989). 5 The AIM is a seminonparametric model produced from a class of globally flexible series expansions. See also Barnett et al. (1991a, 1991b). Gallant’s Fourier flexible functional form (Gallant, 1981) is also globally flexible.

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(1) artificial data is generated, (2) the Normalized Quadratic model then is estimated with that data, and (3) finally its regular regions are displayed at the various elasticities used with the generating model. The functional form model that we investigate is the Normalized Quadratic reciprocal indirect utility function (Diewert and Wales, 1988b; Ryan and Wales, 1998). Globally correct curvature can be imposed on the model by imposing negative semi-definiteness of a particular coefficient matrix and nonnegativity of a particular coefficient vector, but at the added cost of losing flexibility. It has been argued that global curvature imposition forces the Slutsky matrix to be “too negative semi-definite”. In that sense, the method imposes too much concavity, and thereby damages the flexibility. Since concavity is required by economic theory, the model’s inability to impose full concavity without loss of flexibility is a serious defect of the model. If instead of the indirect utility function, the Normalized Quadratic is used to model the expenditure function, global concavity can be imposed without loss of flexibility. But the underlying preferences then are quasihomothetic and thereby produce linear Engel curves. Because of that serious restriction on tastes, we exclude that model from our experiment. Ryan and Wales (1998) suggested a procedure for imposing negative semidefiniteness on the Slutsky matrix, as is necessary and sufficient for the curvature requirement of economic theory. But to avoid the loss of flexibility, Ryan and Wales apply the condition only at a single point of approximation.6 With their data, they successfully found a data point such that imposition of curvature at that point results in full regularity (both curvature and monotonicity) at every sample point. They also applied the procedure to other earlier consumer demand systems.7 By imposing correct curvature at a point, the intent with this procedure is to attain, without imposition, the curvature and monotonicity conditions at all data points. We explore the regular regions of the models with these two methods of curvature imposition. The objective is to determine the extent to which imposition of global, local or no curvature results in regularity violations. Imposing curvature locally may induce violations of curvature at other points, in addition to violations of monotonicity. We find monotonicity violations to be common. With these models, the violations exist widely within the region of the data, even when neither global curvature nor local curvature is imposed. We believe that this problem is common with many nonglobally-regular flexible functional forms, and is not a problem specific to the Normalized Quadratic model. For example, one of the graphs for the AIM cost

6 Moschini

(1998) independently developed the identical procedure to impose local curvature on the semiflexible AIDS model. See Diewert and Wales (1988a) for the definition of the semiflexibility. 7 Moreover, Ryan and Wales (2000) showed effectiveness of the procedure when estimating the translog and generalized Leontief cost functions with the data utilized by Berndt and Khaled (1979).

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function in Jensen (1997), without regularity imposition, look similar to the one obtained below. Imposing curvature globally corrected the monotonicity violations globally in a case with complementarity among two of the goods. But that imposition produced some overestimation of cross elasticities of substitutions in absolute values. A pair of complementary goods became more complementary and a pair of substitute goods became stronger substitutes. Diewert and Wales (1993) similarly found that some method of imposing regularity can produce upper bounds on certain elasticities for the AIM and translog models.8 The chapter is organized as follows. Section 2 presents the model using the two methods of curvature imposition. Section 3 illustrates our experimental design, by which the artificial data is simulated, the model is estimated, and the regular region is displayed. Section 4 provides our results and discussion. We conclude in the final section.

2. The Model Central to the imposition of curvature is a quadratic term of the form v Bv, where v is a vector of the variables, and B is a symmetric matrix containing unknown parameters. With the Normalized Quadratic model, the quadratic term is normalized by a linear function of the form α v, so that the quadratic term can be written in the form v Bv/α v, where α is a nonnegative predetermined vector. According to Diewert and Wales (1987, Theorem 10), the Hessian matrix of the quadratic term is negative semi-definite, if the matrix B is negative semi-definite. Imposition of that matrix constraint ensures concavity of the normalized quadratic term. As a result, imposition of global curvature starts with imposition of negative semidefiniteness on the matrix B. We reparameterize the matrix B by replacing it by minus the product of a lower triangular matrix, K, multiplied by its transpose, so that B = −KK . Diewert and Wales (1987, 1988a, 1988b) used this technique, developed by Wiley et al. (1973) and generalized by Lau (1978). The Normalized Quadratic reciprocal indirect utility function of Diewert and Wales (1988b) and Ryan and Wales (1998) is defined as 1 v Bv + a log(v), h(v) = b v + (1) 2 α v

8 See also Terrell (1995). But it should be observed that more sophisticated methods of imposing regularity on AIM do not create that problem. In fact it is provable that imposition of global regularity on seminonparametric models, such as AIM, cannot reduce the span, if imposition is by the most general methods.

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where b is a vector containing unknown parameters, and v is a vector of prices, p, normalized by a scalar of total expenditure y, so that v ≡ p/y.9 A fixed reference point v0 is chosen, such that the matrix B satisfies Bv0 = 0

(2)

and the predetermined vector α satisfies α v0 = 1.

(3)

Using Diewert’s (1974) modification of Roy’s identity, the system of share equations is derived as s(v) =

V[v Bv/(α v)2 ]a + a Vb + V(Bv/(α v)) − (1/2) , v b + (1/2)[v Bv/(α v)] + ι a

(4)

where s = s(v) is a vector of budget shares, ι is a unit vector with 1 as each element and V is a diagonal matrix with normalized prices on the main diagonal and zeros on the off-diagonal. Homogeneity of degree zero in all parameters of the share equations (4) requires use of an identifying normalization. The normalization usually used is b v0 = 1.

(5)

The functional form (1) subject to restriction (2), (3) and (5) will be globally concave over the positive orthant, if the matrix B is negative semi-definite and all elements of the parameter vector a are nonnegative. Global concavity can be imposed during the estimation by setting B = −KK with K lower triangular, while setting ai = ci2 for each i, where c is a vector of the same dimension as a. We then estimate the elements of K and c instead of those of B and a. As mentioned above, this procedure for imposing global concavity damages flexibility. Imposition of curvature locally is at the point of approximation. Without loss of generality, we choose the v0 = 1 to be that point. For ease of estimation we impose the following additional restriction: a v0 = 0.

(6)

Using the other restrictions along with the restriction (6), the Slutsky matrix at the point v0 can be written as S = B− A + ab + ba + 2aa ,

(7)

where A = diag(a). Hence A is a diagonal matrix defined similarly to V. Imposing curvature locally is attained by setting S = −KK with K lower triangular,

9 Diewert

and Wales (1988b) include a level parameter b0 additively in Equation (1). But it is nonidentifiable and not estimatable, since it vanishes during the derivation of the estimating equations.

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and solving for B as B = −(KK ) + A − ab − ba − 2aa .

(8)

Ryan and Wales (1998) showed that the demand system described above is flexible. Moreover, the regular regions of this model and the unconstrained model with Equation (6) imposed will be exactly identical, when Caves and Christensen’s method is used. During estimation, the matrix B is replaced by the right-hand side of (8) to guarantee that the Slutsky matrix is negative semi-definite at the point of approximation. To see why imposing curvature globally damages the flexibility while imposing curvature locally does not, recall that the Slutsky matrix, S, is symmetric and satisfies Sp = 0 or equivalently Sv = 0. As a result, the rank of S is reduced by one, so that the number of the independent elements of S becomes equal to that of B. Therefore S in Equation (7) can be arbitrary determined by B, independently of a and b. But the Hessian matrix of the indirect utility function is usually full rank, unless linear homogeneity is imposed or attained empirically. In Diewert and Wales’ (1988b) approach to proof of local flexibility, the second partial derivatives at the point of approximation depend on both B and a. However, imposition of nonnegativity on a to attain global curvature reduces the number of independent parameters, and limits the span of B and a. As a result, imposing curvature globally on this model damages flexibility. Regarding local curvature imposition, the condition that the Slutsky matrix be negative semi-definite is both necessary and sufficient for correct curvature at the point of approximation.

3. Experimental Design Our Monte Carlo experiment is conducted with a model of three goods demand to permit different pairwise complementarities and substitutabilities. The design of the experiment is described below. 3.1. Data Generation The data set employed in the actual estimation process includes data for normalized prices and budget shares, defined as v ≡ p/y and s ≡ Vq where q is a vector of demand quantities and V is as defined previously. The data for demand quantities are produced from the demand functions induced by two globally regular utility functions: the CES functional form and the linearly-homogeneous Constant-Differences of Elasticities-of-Substitution (CDE) functional form.10

10 See

Hanoch (1975) and Jensen (1997) for details of the model.

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The CES indirect utility function with three goods is

3 −1/r ρ r U (p, y) = y pk , where r = . ρ−1

(9)

k=1

By applying Roy’s identity to (8), Marshallian demand functions are derived as qi (p, y) = ypir−1

3

pkr

(10)

k=1

for i = 1, 2 and 3. The CES utility function is globally regular if ρ 1. The values of ρ are chosen so that the elasticity of substitution σ = 1/(1 − ρ) covers a sufficiently wider range. Fleissig et al. (2000) also used this data generation model in comparing the performance of the Fourier flexible form, the AIM form, and a neural network in estimating technologies. The CDE indirect reciprocal utility function 1/u = g(p, y) is defined implicitly by an identity of the form: G(p, y, u) =

3

Gk (pk , y, u) ≡ 1,

(11)

k=1

with Gk = φk uθk (pk /y)θk . Parametric restrictions required for the implicit utility function (11) to be globally regular are φk > 0 and θk < 1 for all k, and either θk 0 for all k or 0 < θk < 1 for all k. In all cases, the φk ’s equal the corresponding budget shares at (p0 , y0 ) = (1, 1). Applying Roy’s identity, we derive the demand functions, θi φi uθi (pi /y)θi −1 qi (p, y) = 3 θk θk k=1 θk φk u (pk /y)

(12)

for i = 1, 2 and 3. The utility level u is set to unity without loss of generality, when generating simulated data. Our test bed consists of eight cases. Cases 1 to 4 use data simulated from demand functions of the Equation (10) CES form, and cases 5 to 8 use data from demand functions of the Equation (12) CDE form. Table 1 describes each case in terms of the elasticity of substitution and budget share settings at the reference point.11 It is convenient to construct the data such that the mean of the normalized prices is 1. We draw from a continuous uniform distribution over the interval [0.5, 1.5]

11 The values of those elasticities are computed as Allen–Uzawa elasticities of substitution. The Allen–Uzawa elasticity of substitution is the commonly used traditional measure. More complicated substitutability can be captured by the Morishima elasticity of substitution. Its superiority is maintained in Blackorby and Russell (1989).

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Table 1: True Underlying Preferences at a Point (p0 , y0 ) = (1, 1) in Terms of Budget Shares and Elasticities of Substitution Budget shares

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8

Elasticities of substitution

s1

s2

s3

σ12

σ13

σ23

0.333 0.333 0.333 0.333 0.300 0.300 0.395 0.409

0.333 0.333 0.333 0.333 0.300 0.300 0.395 0.409

0.333 0.333 0.333 0.333 0.400 0.400 0.211 0.182

0.200 0.700 2.000 4.000 1.500 0.500 0.061 −0.010

0.200 0.700 2.000 4.000 1.500 0.500 0.561 0.591

0.200 0.700 2.000 4.000 1.500 0.500 0.561 0.591

for price data and [0.8, 1.2] for total expenditure data. The sample size is 100, as would be a typical sample size with annual data.12 The stochastic data, adding noise to the model’s solved series, are constructed in the following manner. The noise vector ε is generated from a multivariate normal distribution of mean zero and covariance matrix μ cov(p), where μ is a constant and cov(p) is the covariance matrix of a generated price series. We arbitrarily set μ ∈ [0.0, 1.0] to adjust the influence of noise on the estimation. The price series incorporating noise is constructed as p∗ = p + ε, while making sure that the resulting prices are strictly positive with each setting of μ. We then use Equations (10) and (12), along with total expenditure y, to generate the data for quantities demanded, q∗ (p∗ , y). Using the noise-added data, we compute total expenditure y∗ = (q∗ ) (p∗ ), normalized prices v∗ = p∗ /y ∗ and budget shares s∗ = V∗ q∗ , ∗ ∗ ∗ where vˆ = diag(v ). We then have the data for the dependent variable, s and the noise-free independent variable, v = p/y. It is easier to ensure strictly positive noise-added data by this procedure, than by adding directly to the budget shares.13 3.2. Estimation Using our simulated data, we estimate the system of budget share Equation (4) with a vector of added disturbances e. We assume that the e’s are independently multivariate normally distributed with E(e) = 0 and E(ee ) = , where is constant across observations. Since the budget constraint causes to be singular, we

12 Unlike our design, Jensen’s (1997) design used price series of length 1000 with a factorial design at discrete points in the interval [0.5, 2.0]. Terrell (1995) used a grid of equally spaced data in evaluating the performance of the AIM production model. 13 Gallant and Golub (1984) used the same procedure for stochastic data generation with a production model.

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drop one equation, impose all restrictions by substitution, and compute the maximum likelihood estimates of the reparameterized model. Barten (1969) proved that consistent estimates can be obtained in this manner, with the estimates being invariant to the equation omitted. The unconstrained optimization is computed by MATLAB’s Quasi-Newton algorithm. A complete set of parameters is recovered using the associated restrictions. A priori there is no known optimal method for choosing the vector α, so we choose all elements of the vector to be equal. Some authors have experimented with alternative settings, such as setting α as weights to form a Laspeyres-like price index, but with no clear gain over our choice.14 Hence all elements of the vector α are set at 1/3, as a result of Equation (3) with v0 = 1. The number of Monte Carlo repetitions is 1000 for each case. We use boxplots to summarize the distribution of the estimated elasticities across the 1000 replicates. We follow the standard procedure for drawing boxplots. The box has lines at the lower quartile, median, and upper quartile values. The “whisker” is a line going through each end of the box above and below the box. The length of the whisker above and below the box equals 1.5× (the upper quartile value – the lower quartile value). Estimates above or below the whisker are considered outliers. Since we find these distributions of estimates to be asymmetric, standard errors alone cannot capture what is displayed in the boxplots. The average values of each parameter across replications are used to produce the regular regions of the models. To begin our iterations, we start as follows. We compute the gradient vector and the Hessian matrix of the data-generating function at the point (p0 , y0 ) = (1, 1). We set that vector and that Hessian of the Normalized Quadratic model to be the same as those of the data-generating model at that point. We then solve for corresponding parameter values of the Normalized Quadratic and use the solution as the starting values for the optimization procedure. Those starting parameter values produce a local second order approximation of the Normalized Quadratic to the generating function. Our starting values facilitate convergence to the global maximum of the likelihood function, since the global maximum is likely to be near the starting point. 3.3. Regular Region Following Jensen (1997), we plot two-dimensional sections of the regular region in the Cartesian plane. The x-axis represents the natural logarithm of (p2 /p1 ), and the y-axis the natural logarithm of (p3 /p1 ). Each axis ranges between log(0.2/5.0) ≈ −3.2189 and log(5.0/0.2) ≈ 3.2189. The sample range is defined as the convex hull of possible prices within the above intervals of relative values

14 See

Kohli (1993) and Diewert and Wales (1992).

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and is displayed in our figures as a rectangle in the center of the graph. Accordingly, each of the four sides of the rectangle range from log(0.5/1.5) ≈ −1.0986 to log(1.5/0.5) ≈ 1.0986, since our data is generated from the interval of [0.5, 1.5]. The entire section is divided into 150 × 150 grid points, at which the monotonicity and curvature conditions are evaluated. Each plot can be viewed as a display of a 2-dimensional hyperplane through the 3-dimensional space having dimensions log(p2 /p1 ), log(p3 /p1 ) and log(y). We section the regular region perpendicular to the log(y) axis at the reference point of y = 1.0. It is desirable to plot several hyperplanes at different settings of y, as done by Barnett et al. (1985, 1987) to investigate the full 3-dimensional properties of the model’s regular region. But since regularity is usually satisfied at the reference point and violations increase as data points move away from the reference point, the emergence of regularity violations on the single hyperplane with y fixed at the reference setting is sufficient to illustrate deficiencies of the Normalized Quadratic model. The monotonicity condition is evaluated using the gradient vector of the estimated Equation (1), ∇h(v). The model is required by theory to be strictly increasing in v. For each grid point at which the gradient is evaluated, the monotonicity condition is satisfied, if ∇h > 0. Our approach to evaluation of the curvature condition differs from that used in most studies. In those other studies, the curvature condition is judged to be satisfied, if the Allen elasticity of substitution matrix or the Slutsky substitution matrix is negative semi-definite.15 The problem is that satisfaction of the monotonicity condition is required for those matrix conditions to be necessary and sufficient for satisfaction of the curvature condition. Hence, we do not use substitution matrices to evaluate the curvature condition. We evaluate the quasiconcavity of the Equation (1) directly using the method proposed by Arrow and Enthoven (1961). Quasiconcavity is checked by confirming alternating signs of the principal minors of the bordered Hessian matrix, which contains the second partial derivatives bordered by first derivatives. This approach is general, regardless of whether there are any monotonicity violations. The appendix formalizes the procedure for checking quasiconcave. Each grid is filled with different gradations of black and white, designating the evaluation results for the regularity conditions. The completely black grid designates violations of both curvature and monotonicity; the very dark gray gird designates violation of only curvature; and the very light gray grid specifies violation of only monotonicity. The completely white regions are fully regular. There are 8 cases with 3 models each, resulting in 24 plots of regular regions.

15 For example, Serletis

and Shahmoradi (2005) computed the Cholesky values of the Slutsky matrix to evaluate its negative semi-definiteness. A matrix is negative semi-definite, if its Cholesky factors are nonpositive (Lau, 1978).

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4. Results and Discussion We confirm convergence to a global maximum of the likelihood function by comparing the estimated elasticity values with the true ones. If the discrepancy is large, we discard that run, and rerun the program. We do not seek to explain the cases of large discrepancies, other than to conclude that under such circumstances, an unresolved problem exists. Based on this criterion, we encounter substantial difficulty in the estimation of the model with local curvature imposed. When data are generated with elasticities of substitution greater than unity (cases 3, 4 and 5), the estimates converge to values far from the true ones. But when we try the somewhat lower elasticities of σ12 = σ13 = σ23 = 1.10 or 1.20, convergence to reasonable estimates is more successful, although some replications still often yield unreasonable estimates. The boxplots of case 4 in Figure 1 describe the distributions of 1000 estimates of the model’s elasticities of substitution with no curvature imposed (left), local curvature imposed (middle), and global curvature imposed (right).16 The estimates of the local-curvature-imposed elasticities not only have larger variations (described as longer whiskers) than those of the no curvature imposed and global-curvature-imposed elasticities, but also include a number of severe out-

Figure 1: Boxplots of the Distributions of Estimates for Case 4 with the Model with no Curvature Imposed Model (left), with Local Curvature Imposed (middle) and with Global Curvature Imposed (right).

data used for boxplots in this chapter are generated with μ = 0.20 setting of the noise adjustment constant.

16 All

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liers especially in the positive direction. Even when estimates of cross elasticities are reasonable, estimates of own elasticities (not included in this chapter) were often found to be far from the true values. In fact, the unconstrained and globalcurvature-imposed models both require very high values of a (around 400,000’s to 600,000’s for all elements) to attain maximization of the likelihood function. We conclude that constraint (6) is too restrictive. For the local-curvature-imposed model to approximate a symmetric function, as in cases 3 and 4, the optimal values of a should all be zeros, while satisfying the restriction (6). However, any statistical tests would reject the hypothesis that the restriction is valid. Moreover, as Diewert and Wales (1988b) observed, Equation (6) often renders global concavity to be impossible. The only way to achieve global curvature with (6) is to set a = 0, since globally concave requires nonnegativity of all elements of a. In addition, since B is a function of a as well as of b and K, any poor estimate of a will produce poor estimation of B, which are important parameters in attaining concavity. With all such problems and nonlinearity embedded in the likelihood function, the optimization procedure can search for wrong local maxima. We tried a few global optimization techniques, but without success.17 When the point estimates of elasticities are themselves poor, we view regularity violations to be a “higher-order problem” of lesser concern. For reference purposes, we plot in Figure 2 the regular region of the local-curvature imposed model (left) along with that of the global-curvature imposed model (right). However, the validity of conclusions drawn from that figure is questionable. In the Figure 2 display of the regular region of the model with global curvature imposed, the regularity violations occupy a large part of the area. Within the very light gray areas of the plots, the regularity violations are attributed entirely to monotonicity violations. Severe regularity violations resulted from using data produced with high elasticities of substitution, and therefore the most severe violations occurred in case 4. The plot of case 3 (omitted to conserve on space) displays similar shape of regularity violation regions to case 4, but with a somewhat wider regular region. In case 4, we obtained an almost identical figure with the unconstrained model. The phenomenon of monotonicity-induced regularity violations may be common with many nonglobally-regular flexible functional forms and should not be viewed as exclusive to the Normalized Quadratic model. In case 4 the model substantially underestimates true elasticities. The boxplot (right) in Figure 1 shows that the median of the 1000 sample estimates is near 3.0, while the true elasticity is 4.0. This finding is similar to those of Guilkey et

17 The genetic algorithm and the pattern search method are implemented. But the lack of convergence of the former and the very slow convergence of the latter produce substantial computational burden for any study that requires a large number of repeated simulations. Dorsey and Mayer (1995) provide empirical evaluation in econometric applications of the performance of genetic algorithms versus other global optimization techniques.

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Figure 2: Section through Regular Regions of the Model at y = 1.0 with Local Curvature Imposed and with Global Curvature Imposed for Case 4. al. (1983) and Barnett and Lee (1985), who found that the generalized Leontief model performs poorly when approximating the function with high elasticities of substitution. We find that the Normalized Quadratic model performs poorly both in estimating elasticities and in maintaining regularity conditions, when the data used was produced with high elasticities of substitution. For cases 1, 2 and 6, all models perform very well with no regularity violations. The Normalized Quadratic model performs well, when the data is characterized by low elasticities of substitution (below unity) and pairwise elasticities are relatively close to each other. The plot with case 5 is omitted to conserve on space, since the result is similar to case 4.18 With case 7, the plot on the left in Figure 3 displays a very dark gray cloud on the top of the plot, designating curvature violations for the unconstrained model. In this case, imposing curvature locally as well as globally eliminates all of the curvature violations within the region of the data, as shown by the entirely white region in the right plot. Figure 4 describes the distributions of estimates in case 7. For all three models, the median estimates are satisfactorily close to true elasticities. With global curvature imposed, elasticity estimates are severely downward biased when the true elasticities are high, and upward biased when the true elas-

18 With case 5, there are regularity violations inside the sample range, as in case 4, but to a milder extent, as in case 3, since both case 3 and case 5 use data with lower elasticities of substitutions than case 4. The plot of case 5 is slightly shifted from the center, as a result of the fact that the budget shares at the center point are slightly asymmetric. This plot is very similar to the case I of Jensen (1997). Our case 5 and his case I use the same data-generating setting.

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Figure 3: Section through Regular Regions of the Model at y = 1.0 with no Curvature Imposed and with Local and Global Curvature Imposed for Case 7.

Figure 4: Boxplots of the Distributions of Estimates for Case 7 with the Unconstrained Model (left), Local-Curvature-Imposed Model (middle) and Global-Curvature-Imposed Model (right).

ticities are low as outliers. It suggests that all pairwise elasticities become close to each other. As should be expected by outliers, the cause is not easy to determine, and we do not impute much importance to results with outliers. These problems do not arise in case 7, when we impose curvature locally, which succeeds in producing global regularity within the region of our data.

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Figure 5: Section through Regular Regions of the Model at y = 1.0 with no Curvature Imposed and with Local Curvature Imposed for Case 8. Figure 5 displays case 8 plots of the regular regions for the unconstrained (left) and the local-curvature constrained models (right). The top of the left plot has a wide thick cloud of curvature violations. That region intersects a small monotonicity-violation area on the right side. The resulting small intersection region designates the set within which both violations occur. Imposing local curvature does not shrink those regions, but rather expands them. On the right plot, the region of curvature violations now covers much of the 2-dimensional section, with the exception of the white convex regular region and a wide thick pillar of monotonicity violations on the left side. In the intersection of the two irregular regions, both violations occur. Notice that regularity is satisfied at the center point, at which correct curvature is imposed. This pattern of expansion and change of regularity-violation regions is hard to explain. Another disadvantage of the model is its failure to represent complementarity among goods. A middle boxplot in Figure 6 shows that the lower whisker for σ12 is strictly above zero. A typical estimate of a for the unconstrained model was a = (0.014, 0.014, 0.262).19 Although all elements are strictly positive values, they are not substantially different from a = 0, in contrast with case 4. Hence, we do not believe that the inability to characterize complementarity was caused by the restrictiveness of Equation (6). In case 8, the global-curvature-imposed model’s plot is identical to the right plot in Figure 3. However, imposing global regularity can decrease approximation accuracy (Diewert and Wales, 1987; Terrell, 1996). Comparing the Figure 6 boxplot of the unconstrained model’s elasticity estimates (left) with those of the

19 The

estimate was obtained using noise-free data with sample size of 500 instead of 100.

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Figure 6: Boxplots of the Distributions of Estimates for Case 8 with the Unconstrained Model (left), Local-Curvature-Imposed Model (middle) and Global-Curvature-Imposed Model (right). global-curvature imposed model (right), relative to the true elasticity values, we see that global-curvature-imposed model overestimates the elasticity of substitution, σ12 , for complementarity and the elasticities of substitution, σ13 and σ23 for substitutes. A possible cause is the imposition of negative semi-definiteness on the Hessian matrix. As in Figure 4 the outlier estimates cause the pairwise elasticities of substitution estimates to become closer to each other.

5. Conclusion We conduct a Monte Carlo study of the global regularity properties of the Normalized Quadratic model. We particularly investigate monotonicity violations, as well as the performance of methods of locally and globally imposing curvature. We find that monotonicity violations are especially likely to occur, when elasticities of substitution are greater than unity. We also find that imposing curvature locally produces difficulty in the estimation, smaller regular regions, and the poor elasticity estimates in many cases considered in the chapter. When imposing curvature globally, our results are better. Although violations of monotonicity remain common in some of our cases, those violations do not appear to be induced solely by the global curvature imposition, but rather by the nature of the Normalized Quadratic model itself. However, imposition of global curvature does induce a problem with complementary goods by biasing the estimates towards over complementarity and substitute goods towards over substitutability.

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With the Normalized Quadratic model, we find that both curvature and monotonicity must be checked with the estimated model, as has previously been shown to be the case with many other flexible functional forms. Imposition of curvature alone does not assure regularity, and imposing local curvature alone can have very adverse consequences.

Appendix: Theorem (Arrow and Enthoven, 1961) n . Define Let f be a twice differentiable function on the open convex set C ⊂ R+ the determinants Dk (x), k = 1, . . . , n, by ∂f ∂f 0 ··· ∂x1 ∂xk ∂f ∂2f ∂2f · · · ∂x1 ∂xk ∂x ∂x1 ∂x1 Dk (x) = . 1 (A.1) . .. . .. .. .. . . ∂2f ∂2f ∂f · · · ∂xk ∂xk ∂x1 ∂xk ∂xk

A sufficient condition for f (x) to be quasi-concave for x 0 is that the sign of Dk be the same sign of (−1)k for all x and all k = 1, . . . , n. A necessary condition for f to be quasi-concave is that (−1)k Dk 0, for k = 1, . . . , n, for all x.

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Part III Functional Structure and the Theory of Optimal Currency Areas

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Chapter 7

On Canada’s Exchange Rate Regime Apostolos Serletis∗ and Karl Pinno Department of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4

Abstract In this chapter we use a dynamic equilibrium model of the Canadian economy to investigate the degree of currency substitution between the Canadian dollar and the US dollar which potentially has implications for the theory of optimum currency areas and can be used to evaluate the desirability of a monetary union between Canada and the United States. In doing so, we include Canadian and US real money balances (adjusted for take-overs and acquisitions as discussed in Kottaras, 2003) in the representative agent’s utility function, to reflect the usefulness of both currencies in facilitating transactions, and estimate the degree of currency substitution between the Canadian dollar and the US dollar using Hansen’s (1982) Generalized Method of Moments (GMM) estimation procedure.

Keywords: currency unions, exchange rate regimes, currency substitution, generalized method of moments JEL: C22, F33

1. Introduction As the Bank of Canada’s former governor, Gordon Thiessen (2000–2001, p. 47), put it, “[o]ne of the issues that has often surfaced over the years is the exchange rate for the Canadian dollar. Indeed, over the past couple of years, it has been a topic of considerable public discussion. That discussion has revolved around such questions as: Should we continue floating, or should we peg our currency to the US dollar? In fact, should we even keep our own currency, or should we adopt the US currency?”.

The attention to the exchange rate regime stems from the long swings in the Canadian dollar per US dollar nominal exchange rate, over the recent flexible

∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18007-8

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exchange rate period – see, for example, Pinno and Serletis (2005) – and also from the recent creation of a single European currency (the euro) to replace the national currencies of member countries of the European monetary union. The debate in Canada has revolved around exchange rate alternatives and particularly around the issue of whether a floating currency is the right exchange rate regime for Canada or whether we should fix the exchange rate between the Canadian and US currencies, as we did from 1962 to 1970 – see, for example, Murray and Powell (2002) and Murray et al. (2003). A floating exchange rate gives Canada the flexibility to have different monetary conditions than the United States. In particular, a floating currency acts as a shock absorber, between the two economies, allowing us to respond differently to external economic shocks (such as, for example, fluctuations in world commodity prices) and domestic policy requirements. The costs of a floating currency come in two forms. First and most obviously, there are certain transactions costs which are large when the amount of cross-border and financial transactions is large, as is Canada’s case with the United States. A further cost is the fact that exchange rates fluctuate wildly in comparison with goods prices (in fact, almost as wildly as stock prices), although the effects of exchange rate volatility on macroeconomic quantities are difficult to be demonstrated. In this regard, as Fischer (2001, p. 21) recently put it, “. . . hard pegs are more attractive today, particularly when viewed from the asset markets, than had been thought some years ago. A small economy that depends heavily on a particular large economy for its trade and capital account transactions may wish to adopt that country’s currency. But it will need to give careful consideration to the nature of the shocks that affect it before the choice is made.”.

Noting the European developments, the trend towards currency unions and dollarization in Latin America and Eastern Europe, and Japan’s recent interest in exploring alternative monetary arrangements, in this chapter we investigate the issue of whether a floating currency is the right exchange rate regime for Canada or whether Canada should consider alternative monetary arrangements. We follow ˙Imrohoro˘glu (1994) and use a dynamic equilibrium (money-in-the-utilityfunction) model of the Canadian economy to estimate the degree of currency substitution between the Canadian dollar and the US dollar, using Hansen’s (1982) Generalized Method of Moments (GMM) estimation procedure. In doing so, we use recent monetary data adjusted for take-overs and acquisitions, as discussed in Kottaras (2003), and a slightly different econometric methodology than the one presented and used by ˙Imrohoro˘glu (1994). The rest of the chapter is organized as follows. In the next section we briefly discuss ˙Imrohoro˘glu (1994) dynamic equilibrium model of a small open (monetary) economy and present the Euler equations that describe optimal choices. In Section 3 we discuss the data and in Section 4 we discuss Hansen’s (1982) Generalized Method of Moments (GMM) procedure that we use to estimate the model.

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In Section 5 we present our empirical findings and discuss the policy implications of our results. The last section summarizes and concludes the chapter.

2. Theoretical Foundations We follow ˙Imrohoro˘glu (1994) and consider an economy made up of a large number of infinitely lived identical agents. At the beginning of each period, the representative domestic agent decides how much to consume, ct , how much to hold in the form of domestic balances, mt , and foreign balances, m∗t , and how much to save in the form of an internationally traded bond, bt∗ . We assume that money services are produced using a combination of domestic and foreign real balances in a Constant Elasticity of Substitution (CES) aggregator function, as follows xt = f (ht , h∗t ) −ρ ∗ −ρ −1/ρ mt mt = α + (1 − α) , pt pt∗

(1)

where 0 < α < 1, −1 < ρ < ∞, ρ = 0, and ht (= mt /pt ) and h∗t (= m∗t /pt∗ ) denote domestic and foreign real money balances, respectively. In the liquidity aggregator function (1), the elasticity of substitution is given by 1/(1 + ρ); α and (1 − α) denote the shares of domestic and foreign real balances (respectively) in the production of money services. Aggregator functions like (1) have been pioneered by Chetty (1969) and used by Husted and Rush (1984), Poterba and Rotemberg (1987) and ˙Imrohoro˘glu (1994), among others. We assume that the representative consumer’s preferences are given by mt m∗t (cσ x 1−σ )1−ψ − 1 u ct , (2) , ∗ = t t , pt p t 1−ψ where xt is the liquidity aggregate given by Equation (1). This utility function exhibits constant relative risk aversion in an aggregate of consumption and liquidity services. With these preferences and the liquidity aggregator function (1), the Euler equations for an interior solution are given by (see the second case presented in ˙Imrohoro˘glu, 1994, for details regarding the derivations) b ct+1 φ−1 ht+1 −ρ ∗ + (1 − α) α ∗ β(1 + rt ) ct ht+1 −ρ −b ∗ −ρb ht+1 ht × α ∗ + (1 − α) − 1 = ε1,t+1 , (3) ht h∗t βσ

ct+1 ct

−b b φ−1 −ρ ht ht+1 −ρ + (1 − α) + (1 − α) α ∗ α ∗ ht ht+1

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−1 −ρ pt ht + (1 − α) + α(1 − σ ) α h∗t pt+1 h∗t −ρ−1 ∗ −1 ht ht × − σ = ε2,t+1 , h∗t ct ×

h∗t+1

−ρb

−ρ b −ρ−1 ht ht α α ∗ + (1 − α) ht h∗t −ρ b ht − (1 − α) α ∗ + (1 − α) ht ∗ −ρ b ht+1 ct+1 φ−1 ht+1 −ρ − αβ + (1 − α) α ∗ ct ht h∗t −ρ−1 ht pt et+1 ct+1 a−1 × + (1 − α)β h∗t pt+1 et ct ∗ −ρ b −ρ h ht+1 pt t+1 × α + (1 − α) = ε3,t+1 , h∗t h∗t pt+1

(4)

(5)

where β ∈ (0, 1) is the subjective discount factor, rt∗ denotes the realized real interest rate on bt∗ , et is the nominal exchange rate (note that we do not impose purchasing power parity), φ = (1 − ψ)σ , b = −(1 − σ )(1 − ψ)/ρ, and εj,t+1 for j = 1, 2, 3 are the Euler equation errors. It should be noted that we also attempted to investigate the robustness of our empirical results (reported in Section 5 below) to alternative specifications of preferences and technology. In particular, we took direction from the approach employed by Poterba and Rotemberg (1987) and assumed that the representative agent faces portfolio adjustment costs that are proportional to the square of the percentage change in nominal asset holdings. Separately, we attempted to introduce a third asset (domestic personal savings deposits) and compare the degree of currency substitution to that of domestic asset substitution, as in Serletis and Rangel-Ruiz (2005). Both extensions, however, served to increase the nonlinearity of an already extremely nonlinear system and we were unable to successfully estimate either extension.

3. The Data We take Canada to be the small open (domestic) economy and the United States the world (foreign) economy. We use quarterly, seasonally adjusted data over the period from 1981:1 to 2003:1 (a total of 89 observations) on real per capita aggregate domestic consumption, domestic money balances and foreign money balances. Given that this is a representative agent model, to obtain real aggregate domestic consumption, we divide personal expenditures on nondurables and

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services (CANSIM II series V1992047 plus V1992119) by Canadian population fifteen years and older (CANSIM II series V2091030) and the Canadian consumer price index (retrieved from the IMF’s International Financial Statistics, series code 15664ZF). We use the M1 monetary aggregate, adjusted for take-overs and acquisitions as discussed in Kottaras (2003), to capture domestic nominal balances. This series is seasonally adjusted [using the SAMA command in TSP/GiveWin (version 4.5)] and converted to real per capita terms by dividing by population and the consumer price index, as we did for personal consumption.1 The amount of US currency held by Canadians is unobservable. Moreover, there is no breakdown of US dollar deposits held by Canadians into demand and time deposits – in fact, the distinction between demand and notice deposits is not so clear anymore, even for Canadian dollar deposits, and we think that most of the US dollar deposits held by Canadians are effectively demand deposits. For these reasons, in this chapter we use nonbank Canadian resident foreign currency deposits (US dollar deposits must account for almost all of this), again adjusted for take-overs and acquisitions as discussed in Kottaras (2003), as a proxy for currency and demand deposits denominated in US dollars and held by nonbank, nonofficial Canadians (whose mailing address is in Canada). Since this series (MB482, in Kottaras, 2003) is the value of the deposits in Canadian dollars, we seasonally adjust the series and convert it to real per capita terms by dividing by population and the consumer price index, as we did for Canadian M1. Finally, we use the three-month (constant maturity) Treasury bill rate in the United States as the nominal interest rate. This series is converted to a realized real interest rate series by dividing it by the gross inflation rate in the United States as measured by the rate of increase in the US consumer price index (retrieved from the IMF’s International Financial Statistics, series code 11164ZF).

4. GMM Estimation Let θ = (α, β, ρ, σ, ψ) be the vector of free parameters to be estimated. The theoretical relations that θ should satisfy are orthogonality conditions between a nonlinear function of θ , f (θ ), and a q-dimensional random vector of instrumental variables zt (referred to as the ‘information set’), expressed as follows E f (θ ) z = 0. The generalized method of moments estimator selects estimates of the parameter ˆ so that the sample correlations between f (θ) and z are as close to zero vector, θ,

1 The

continuity adjusted data is available as an R (http://cran.at.r-project.org) package at http://www.bankofcanada.ca/en

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as possible, as defined by the following (quadratic) criterion function

J (θ ) = g(θ ) Wg(θ ), where g(θ ) = f (θ) z and W is a symmetric positive definite weighting matrix. Any symmetric positive definite W will yield consistent parameter estimates, but as Hansen (1982) has shown, setting W equal to the inverse of the asymptotic covariance matrix of the sample moments, S, produces asymptotically efficient parameter estimates. We used the Bartlett kernel, as discussed by Newey and West (1987), to weight the autocovariances in computing the weighting matrix. All estimation is performed in TSP/GiveWin (version 4.5) using the GMM procedure – our programs are available upon request. As already noted, to obtain GMM estimates we need to express each moment condition as an orthogonality condition between an expression that includes the parameters and a set of instrumental variables. In doing so, we used the MASK command in TSP and matched the instruments (which are the variables lagged once) with the variables as they appear in the Euler equations, (3)–(5). This results in 20 orthogonality conditions which are used to minimize J (θ ) by choosing 5 parameters, α, β, ρ, σ and ψ. In this regard it should be noted that ˙Imrohoro˘glu (1994) applied every instrument from each of three instrument sets to each estimation equation. He then checked the robustness of his results by decreasing the number of instruments in subsequent sets from the base set. In contrast, we do not vary our instrument set in this manner, because as previously mentioned, by using the MASK command in TSP we are able to precisely match the instruments with the estimation equation within which they appear. When the number of instruments exceeds the number of parameters to estimate (as in our case), the estimation is overidentified and not all of the orthogonality conditions will be met. In our case we have 15 overidentifying restrictions and we can use the J -statistic to test the validity of the overidentifying restrictions. Under the null hypothesis that the overidentifying restrictions are satisfied, the J -statistic times the number of observations (in our case 86) is asymptotically distributed as χ 2 with degrees of freedom equal to the numbers of overidentifying restrictions (in this case 15).

5. Empirical Results The estimation results are reported in Table 1. As results in nonlinear estimation are often sensitive to the initial parameter values, we randomly generated 10,000 sets of initial parameter values, restricting each of α, β, ψ and σ in the interval [0, 1] and ρ in the interval [−1, 25]. We chose the starting θ that led to the lowest value of the objective function. Moreover, the starting θ was also subjected to random sensitivity analysis to ensure that there were no values in its neighborhood that would yield improvements to the objective function. The parameter estimates

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Table 1: GMM Estimates: Equations (3)–(5) Parameter α β σ ψ ρ J -statistic Cα (1) Cσ (1)

Estimate

p-value

0.99561 0.96360 0.99377 0.25633 3.01363

<0.000 <0.000 <0.000 0.970 <0.000

0.19204 16.51532 131.70850 491.75250

0.348 <0.000 <0.000

Note: Sample period, quarterly data 1981:1–2003:3.

reported in Table 1 are the ones that minimize the objective function and satisfy the theoretical relations, Equations (3)–(5), as closely as possible. The J statistic is small relative to the degrees of freedom (see the third last row of Table 1) and hence we cannot reject the overidentifying restrictions of the model at conventional significance levels. The estimate of α of 0.99561 (with a p-value of less than 0.000) is very close to 1 – recall that α is the share of domestic real balances in the production of (domestic) liquidity services. In fact, we tested the null hypothesis of α = 1 (against the alternative of α < 1), using the C-statistic proposed by Eichenbaum et al. (1988). This test is similar to a likelihood ratio test and the C-statistic is calculated as the difference between the restricted and unrestricted models’ J -statistics. The C-statistic is distributed as χ 2 with degrees of freedom equal to the number of restrictions (in this case 1). As shown in the second last row of Table 1, the C-test rejects the null hypothesis of α = 1, suggesting that foreign balances do have a statistically significant role in the production of domestic liquidity services. The estimate of the discount factor β is less than 1 although it is not statistically different from 1, consistent with previous research in this area. The estimate of σ is 0.99377 with a p-value of less than 0.000, suggesting that our included assets do yield utility – the null of σ = 1, corresponding to the case where the included assets yield no utility, is rejected by the Cσ (1) statistic reported in the last row of the table. Also, the estimate of the coefficient of relative risk aversion ψ is 0.25633 with a p-value of 0.970, suggesting that preferences are statistically different from logarithmic preferences (the case when ψ = 1). Consistent with the estimate of α, the estimate of ρ (of 3.01363 with a p-value of less than 0.000) implies an elasticity of currency substitution, 1/(1 + ρ), of 0.249, a bit lower than the highest estimate of 0.3037 reported by ˙Imrohoro˘glu (1994), using quarterly data over the period from January, 1974 to June, 1990. This lower estimate of the elasticity of currency substitution is consistent with

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Canada’s recent adoption of inflation targeting as its monetary policy regime. In particular, in February 1991 the Bank’s governor and the minister of finance jointly announced a series of declining inflation targets, with a band of plus and minus one percentage point around them. The targets were 3 percent by the end of 1992, falling to 2 percent by the end of 1995, to remain within a range of 1 to 3 percent thereafter. The 1 to 3 percent target range for inflation was renewed in December 1995, in early 1998, and again in May 2001, to apply until the end of 2006. Our finding of a low degree of currency substitution between the Canadian dollar and the US dollar is consistent with Canada’s recent low inflation record and implies that US dollar deposits in Canada do not provide a good substitute for the domestic (Canadian) currency. Moreover, low currency substitution has implications for the theory of optimum currency areas – see Mundell (1961), McKinnon (1963), Canzoneri and Rogers (1990) and Swofford (2000) – that has recently been used to evaluate the desirability of a monetary union between Canada and the United States. Although we made no attempt to establish or indicate what level of currency substitution will be indicative of a feasible optimum currency area, it seems that low substitutability between domestic and foreign assets indicates the absence of a viable single currency North American zone. Our results are similar (lower magnitude but same conclusions) to those obtained by ˙Imrohoro˘glu (1994) and, when considered jointly with his results, imply that currency substitution is actually decreasing over time. At first glance this appears counterintuitive; if for no other reason one might reasonably expect that Canadian consumers have been increasing their US-asset holdings as part of an international portfolio diversification strategy. However, as reported by Murray and Powell (2002), although there has been a trend towards increasing international diversification (particularly in US assets), US-dollar deposits held by Canadian residents in Canadian banks as a percentage of broad money (M3), were actually lower in the 1990s than they had been during the late 1970s and early 1980s. Our finding of a lower currency substitution than that reported by ˙Imrohoro˘glu (1994) is consistent with the conclusions of Murray and Powell (2002) that “[by] many measures Canada is less dollarized now than it was 20 years ago and bears little resemblance to economies that are typically regarded as truly dollarized.”.

6. Conclusion Although Canada’s experience with a flexible exchange rate over 46 of the past 54 years has been a good one, recently there has been a debate around exchange rate alternatives and particularly around the issue of whether a floating currency is the right exchange rate regime for Canada – see, for example, Schembri (2001). An alternative exchange rate arrangement is the adoption of a ‘currency board’, in which the domestic currency is backed 100% by a foreign (reserve) currency

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(such as the US dollar) and the exchange rate between the two currencies is fixed. A currency board is thus a variant of a fixed exchange-rate regime with an even stronger commitment mechanism, since domestic money can be issued only if it is fully backed by foreign reserves. In fact, a currency board arrangement is the modern day equivalent of a fully backed gold standard with foreign reserves taking the place of gold reserves. Currency boards have recently been adopted by countries such as Hong Kong (1983), Argentina (1991) and Lithuania (1994) with the US dollar, and Estonia (1992), Bulgaria (1997) and Bosnia (1998) with the euro. In addition, several countries in Eastern Europe and the former Soviet Union are considering adopting a currency board with the euro. Another possible exchange rate arrangement is ‘dollarization’ – one country’s use of another country’s money (which may not be the US dollar). Dollarization is another variant of a fixed exchange-rate regime, with an even better commitment device than a currency board. In particular, dollarization avoids the possibility of a speculative attack on the domestic currency and also eliminates the inflation-bias problem of discretionary policy (arising from attempts to stimulate the economy and incentives to monetize the public debt). However, dollarization is subject to the usual disadvantages of a fixed exchange-rate regime – it implies the loss of an independent monetary policy, the inability of the central bank to act as a lender of last resort, and the loss of seigniorage (the revenue that the government receives by issuing money). Recently, Ecuador adopted full dollarization and El Salvador announced its determination to do the same. Currency boards and dollarization, however, are strong measures that tend to be applied in extreme circumstances. They have been advocated as monetary policy strategies for emerging market countries, especially in parts of Latin America that have had a long history of monetary instability. In this chapter we have taken an optimum currency area approach to the problem of whether a floating currency is the right exchange rate regime for Canada or whether we should consider a currency union with the United States. Although our methodology is far removed from the usual criteria used to establish an optimum currency area (see, for example, Mundell, 1961; McKinnon, 1963; Canzoneri and Rogers, 1990), we have followed Swofford (2000) and Serletis and Rangel-Ruiz (2005) and assumed that a low degree of currency substitution is consistent with monetary independence and a high one with an optimum currency area. Our approach is also different from that recently taken by Murray and Powell (2002) and Murray et al. (2003) in investigating the same issue, but our results support Murray and Powell’s (2002, p. 11) conclusion that “there is no evidence that Canadians have lost faith in their currency and are beginning to adopt the US dollar. Moreover, as long as Canadian monetary policy continues to achieve its policy objective of low and stable inflation this is likely to remain the case.”.

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Acknowledgements We would like to thank Paul Gilbert of the Bank of Canada for his sharing and helpful comments. Serletis also gratefully acknowledges financial support from the Social Sciences and Humanities Research Council of Canada.

References Canzoneri, M.B. and Rogers, C.A. (1990). Is the European Community an optimal currency area? Optimal taxation versus the cost of multiple currencies. American Economic Review 80, 419–433. Chetty, V.K. (1969). On measuring the nearness of near-moneys. American Economic Review 59, 270–281. Eichenbaum, M.S., Hansen, L.P. and Singleton, K.J. (1988). A time series analysis of representative agent models of consumption and leisure choice under uncertainty. Quarterly Journal of Economics 103, 51–78. Fischer, S. (2001). Exchange rate regimes: Is the bipolar view correct? Finance & Development 38, 18–21. Hansen, L.P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054. Husted, S. and Rush, M. (1984). On measuring the nearness of near monies: Revisited. Journal of Monetary Economics 14, 171–182. ˙Imrohoro˘glu, S. (1994). GMM estimates of currency substitution between the Canadian dollar and the U.S. dollar. Journal of Money Credit and Banking 26, 792–807. Kottaras, J. (2003). The construction of continuity-adjusted monetary aggregate components. Working Paper 2003-22, Bank of Canada. McKinnon, R.I. (1963). Optimum currency areas. American Economic Review 53, 717– 725. Mundell, R.A. (1961). A theory of optimum currency areas. American Economic Review 51, 657–665. Murray, J. and Powell, J. (2002). Is Canada dollarized? Bank of Canada Review (Autumn), 3–11. Murray, J., Powell, J. and Lafleur, L.-R. (2003). Dollarization in Canada: An update. Bank of Canada Review (Summer), 29–34. Newey, W.K. and West, K.D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708. Pinno, K. and Serletis, A. (2005). Long swings in the Canadian dollar. Applied Financial Economics 15, 73–76. Poterba, J.M. and Rotemberg, J.J. (1987). Money in the utility function: An empirical implementation. In Barnett, W.A. and Singleton, K.J. (Eds), New Approaches to Monetary Economics. Cambridge: Cambridge University Press, pp. 219–240. Schembri, L. (2001). Conference summary: Revisiting the case for flexible exchange rates. Bank of Canada Review (Autumn), 31–37. Serletis, A. and Rangel-Ruiz, R. (2005). Microeconometrics and measurement matters: Some results from monetary economics for Canada. Journal of Macroeconomics 27, 307–330.

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Swofford, J.L. (2000). Microeconomic foundations of an optimal currency area. Review of Financial Economics 9, 121–128. Thiessen, G. (2000–2001). Why a floating exchange rate regime makes sense for Canada. Bank of Canada Review (Winter), 47–51.

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Chapter 8

Tests of Microeconomic Foundations of an Australia/New Zealand Common Currency Area James L. Swofforda,∗ , Lena Birkelöf b , Joshua R. Caldwella and John E. Filerc a Department

of Economics and Finance, University of South Alabama, Mobile, AL 36688, USA b Department of Economics, Umeå University, Umeå, Sweden c Department of Economics, Arizona State University, Tempe, AZ 85287, USA

Abstract The formation of a common currency area by some members of the European Union sparked interest in other possible common currency areas. One area could be an Australian/New Zealand common currency area. We present estimates of transactions costs savings associated with such a common currency and results from revealed preference tests of microeconomic foundations of an Australian/New Zealand common currency area. These revealed preference tests include the first tests of microeconomic foundations of a common currency area using quarterly data. Our results can be viewed as favorable toward the formation of an Australian/New Zealand common currency area.

Keywords: common currency areas, revealed preference JEL: F33, C14

1. Introduction In 2002, twelve European countries abandoned their individual currencies in favor of a single currency, the Euro. The countries adopting the Euro created the largest common currency area to date in modern history. The move to the Euro also raised the issue of what other areas or groups of countries might be successful or benefit from forming a common or optimal currency area. Two such countries that could possibly benefit from sharing a currency are New Zealand and Australia. ∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18008-X

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There are several reasons to suspect that Australia and New Zealand might form a common currency area. The two countries have formed a regional trading area with their Closer Economic Relations Agreement. Both countries already use a decimal based currency called the dollar.1 The two countries are geographically relatively close to each other and have a relatively large volume of trade between each other. The people of the two countries share a common language. The formation of a common or optimal currency area provides several benefits to the prospective countries. One benefit is the complete elimination of transaction costs, because with a single currency there are no exchange rate conversions. Having a single currency also removes the risk of economic exposure because there are no currency fluctuations. Another benefit to the formation of a common currency area is the characteristic of price transparency associated with having a single currency. With price transparency consumers will be able to comparison shop easily because all goods in both countries will be priced in the single currency. In addition, based on gravitational model results, Rose and van Wincoop (2001) argue that national money seems empirically to act as a significant barrier to international trade. This would mean that the gains from reduced transactions costs would be greater than those implied from merely looking at the current size of international trade among the potential members of a common currency area. While many of these benefits from forming a common currency area are difficult to measure, we can estimate the transactions costs savings that would result from adoption of a common currency. We are able to obtain cost estimates based upon actual cross-border foreign exchange transactions incurred by Australia and New Zealand during 2001. The Bank for International Settlements (2002) conducts a triennial survey of world central banks with respect to turnover in foreign exchange markets. Turnover is a measure of the volume of foreign exchange trading in a specified market, and is defined as the absolute gross value of all new deals entered into during the year and is measured in terms of the nominal amount of the contracts completed in the market. No distinction is made between sales and purchases (i.e., a purchase of NZ$ 7 million and a sale of NZ$ 12 million against Australian dollars amounts to gross turnover of NZ$ 19 million). Direct crosscountry transactions are counted as single transactions, and not double counted. Foreign exchange transactions are of three types, spot, forward and swap. A spot is a trade of cash for foreign exchange today (the actual settlement is two days from today). Spot trades are either bank-to-bank or customer-to-bank. Most spot trades are customer-to-bank, most bank-to-bank trades will be swaps and forward trades are mostly customer-to-bank. For both Australia and New Zealand in

1 Of

course a common currency is equivalent to a fixed exchange rate at a ratio of 1 to 1. Since the current exchange rate of Australian $ to New Zealand $ is about 0.86, the two countries’ price levels are such that adjustment to a common currency would be smaller than if the exchange rates indicated that the price levels were dramatically different.

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2001, Rosbourough (2001) found that spot trades were 25 percent of all foreign exchange transactions. The bid-ask spread quoted for interbank foreign exchange swaps during 2000 was stable at 10 basis points, and has increased, periodically, since that time depending upon market conditions. We use 10 basis points, or 0.1 percent, as a lower bound to price swap transactions costs. The typical credit card conversion rate is the interbank rate plus 2 percent, while the typical cash conversion rate is the interbank rate plus 4 percent. To price transactions costs for spot and forward exchange trades, we used the interbank rate plus 3 percent. In Table 1 we present total annual foreign exchange turnover between Australia and New Zealand for 2001 in US and local currencies. Foreign exchange trading between the two currencies averaged US$ 129 million per day, and was “steady” over the year.2 The potential cost savings from an Australian–New Zealand common currency area in US dollars was $355,524,000 in 2001. Also presented in Table 1 are the annual foreign exchange transactions costs for Australia–New Zealand trades in US dollars and local currencies.3 In local currencies, the present value of future transactions costs are AUS$ 10.6 billion to AUS$ 15.4 billion and NZ$ 13.0 billion to NZ$ 19.0 billion for Australia and New Zealand, respectively.4 Previous discussion on the existence of an optimal or common currency area concentrates heavily on the macroeconomic foundations that affect the formaTable 1: Australia and New Zealand Cross-Border Foreign Exchange Annual Turnover and Transactions Costs for 2001 Currency US Dollars Australian Dollars New Zealand Dollars

Annual total turnover

Annual transactions costs

$33,540,000,000 $64,852,944,000 $79,814,132,000

$355,524,000 $687,441,000 $846,030,000

Source: Turnover rate: Bank for International Settlement, Triennial Central Bank Survey, March 2002, p. 65. Transactions costs: L. Rosborough, Trends in foreign exchange trading, Reserve Bank of New Zealand Bulletin, Vol. 64, No. 4, October 2001, p. 30.

2 Bank

for International Settlements (2002), pp. 13, 65. “Steady” is the BIS description of the Australia/New Zealand foreign exchange market over the year, as contrasted with the BIS designators of “increasing” and “declining”. 3 Each amount is the total volume of Australia/New Zealand currency exchanged and total transactions costs resulting from all such cross-border trades for 2001 expressed in the three currencies. 4 The current forecast LIBOR rate over the coming year averages around 2 percent. The current forecast 30-year US bond rate averages around 5 percent. Transactions costs were discounted over a 30 year period using these two rates to bound the present value.

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tion of such areas. More specifically they consider mostly the political criteria that influence monetary policy. The direction of this thinking runs opposite to the idea that the determination of what actually constitutes money depends on the decisions of the people of a particular nation or nations. True to this line of thought, Swofford (2000) proposed microeconomic foundations for the existence of a common currency area and tested the Euro area for consistency. The basis for his foundation is that for a common currency area to exist, the people included in the area must use the same assets as money. Further Swofford (2005) tested North American data and found it consistent with a North American common currency area. The results in this chapter on Australia and New Zealand are the first test on microeconomic foundations of a common currency area using quarterly data. The basis for these microeconomic foundations is that for a common currency area to exist, the people included in the area must use the same asset or assets as money.

2. Existence of an Optimum Currency As Swofford (2000) discussed, a common currency area is any area in which the economic agents treat the same asset or group of assets as providing monetary services.5 For example, if the people of Australia use currency as money and the people of New Zealand use currency and checkable deposits as money, then the two countries do not form a common currency area. However, if the people of the two countries use only currency and checkable deposits as money, then the two countries can form a common currency area.6 Even if the people in Australia and New Zealand both use currency and checkable deposits as money and can form a common currency area, but the criteria of Mundell (1961) and McKinnon (1963) are not met, then the gains from forming a common currency area may not be large enough to offset the costs. In such case a political consensus to form a common currency area may not develop.7 The microeconomic content of this definition of a common currency area requires that the common currency be an asset or assets in economic agents’ optimizing function. If this common money is held by consumers for the liquidity services it provides, then it can be modeled in the consumer’s utility function: U = U (x, m),

(1)

5 The research by Patterson (1991) on the United Kingdom and the papers by Swofford and Whitney (1986, 1988) on the United States suggest that money is likely composed of more assets than just currency. 6 Machlup (1977) pointed out that a common currency area is not the same thing as affixed exchange rate regime unless the exchange rate were fixed at a ratio of one to one. 7 Of course as discussed above, Rose and van Wincoop (2001) have found national currencies act as significant barriers to trade and the gains from forming a common currency area are greater than that indicated from the current level of trade between countries.

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where U is a well-behaved utility function, x is a vector of nonmonetary goods and assets, and m is a vector of the asset or assets that provide liquidity services.8 If more than one asset is held as money, then the economic theory of aggregation over goods places additional restrictions on U (·). Similarly, in economies of more than one individual, the economic theory of aggregation over individuals places additional restrictions on U (·). These restrictions and their implications are discussed in turn. If more than one asset is held as money, then a common currency area must have a common economic monetary aggregate.9 For an economic monetary aggregate to exist, the assets combined into the aggregate must be at least weakly separable from all other items in the objective function. In monetary economics this criteria means that a monetary aggregate can be formed including only those financial assets that are at least weakly separable from all other goods in the agent’s preferences. Thus, if a common money of more than one asset is to exist it must be composed of assets at least weakly separable from all other goods. This restricts U (·) to be at least weakly separable in the monetary assets: U = U x, V (m) . (2) When a weakly separable sub-utility function such as V (m) exists, then the marginal rate of substitution between any two monetary goods in V (m) is independent of the level of consumption of any good in x.10 Less formally, the weak separability criteria for aggregation is a way to identify money as whatever people in the hypothesized optimum currency area treat as money. V (m) will contain currency and all other assets the agent treats in a similar manner. If the common currency of an optimum currency area is thought of as including both currency and other near monies, then the weak separability restrictions in Equation (2) must obtain. If this criterion is not met, then monetary policy in the area may be unstable due to the lack of a reliable monetary target. If more than one person is in a hypothesized optimum currency area, then (1) is restricted further by the conditions for aggregating over agents. The restrictions for aggregation over agents are more stringent than those for aggregation over goods. As Deaton and Muellbauer (1980) point out, aggregation over agents requires that the preferences of each agent be at least quasi-homothetic. Quasihomothetic preferences imply that each agent’s Engle curves are linear. Thus, quasi-homothetic representations of U (·) and V (·) are required for aggregation over agents. While quasi-homothetic Engle curves are linear, they need not pass

8 Feenstra

(1986) shows that the liquidity costs and the utility of money approaches to modeling money demand are functionally equivalent. 9 Barnett (1980) originated the concept of an economic monetary aggregate. 10 See Deaton and Muellbauer (1980) concerning aggregation over goods.

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through the origin as is the case for homothetic preferences. Still the quasihomothetic restrictions are very stringent and often times are finessed by assuming a representative agent.11 A common currency area still requires a political decision by the people within the hypothesized area. Thus the existence of a well-behaved utility function with an at least weakly separable sub-utility function containing money and other monetary assets can be viewed as a sufficient condition for the existence of a common money within an area. Approaches for testing for the existence of such a common currency area are discussed next.

3. Testing for the Existence of a Common Currency Area Revealed preference tests are used to test for the existence of an Australian/New Zealand common currency area. The revealed preference tests do not require the assumption of a particular functional form, and they can be used with limited data observations. However, revealed preference tests do not include random behavior. A detailed discussion of these nonparametic tests is presented in Varian (1982, 1983). Let p i = (p1i , pki ) be the ith observations for the prices of some k goods and assets and x i = (x1i , . . . , xki ) denotes the corresponding quantities of the k goods and assets. Afriat (1967) showed the following: Theorem. The following conditions are equivalent: 1. There exists a nonsatiated utility function that rationalizes the data. 2. The data satisfy the axiom of transitivity, that is p r x r p r x s , p s x s p s x t , . . . , p q x q p q x r implies p r x r = p r x s , p s x s = p s x t , . . . , p q x q = p q x r .12 3. There exists utility indexes U i and marginal utility indexes λi 0, i = 1, . . . , n, such that U i U j + λi p j (x i − x j ) for i, j = 1, . . . , n. 4. There exists a nonsatiated, continuous, monotonic, concave utility function that rationalizes the data. The significance of Afriat’s theorem is that if a data set can be shown to satisfy any of the first three conditions, then it can be rationalized by some well-behaved utility function as stated in condition 4. Since condition 3 is computationally burdensome, Varian (1982) developed an equivalent formulation of the transitivity

11 The

assumption of a representative agent is necessary unless micro or panel data exist. (1967) called this condition “cyclical consistency”.

12 Afriat

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axiom, condition 2. He called his condition the generalized axiom of revealed preference, henceforth GARP. Varian (1982) set forth the following definitions: Given an observation x i and bundle x, Definition 1. x i is directly revealed preferred to x written x i Dx, if p i x i p i x. Definition 2. x i is strictly directly revealed preferred to x written x i Sx, if p i x i p i x. Definition 3. x i is directly revealed preferred to x written x i Dx, if p i x i p i x j , p j x j p j x l , . . . , p m x m p m x l for some sequence of observations (x i , x j , . . . , x m ). In this case R is called the transitive closure of the relationship D. In terms of the notation and definitions used above GARP can be stated as: If x i Rx j then p j x j p j x i

for all i, j = 1, . . . , n.

Thus, a violation of GARP happens when for some x i Rx j , the condition x j Sx i is true or a violation of GARP happens if x i is shown to be revealed preferred to x j but x j is directly revealed preferred to x i . Further if the data are partitioned into two sets of goods and associated prices (p i , x i ), (r i , mi ), i = 1, . . . , n, then a utility function is weakly separable if a subutility function V (m) and a macro-utility function strictly increasing in V such that U (x, m) = U (x, V (m)) can be found. In this vein Varian (1983) showed, that: The following conditions are equivalent: 1. There exists a weakly separable nonsatiated continuous, concave, monotonic utility function that rationalizes the observed data. 2. There exist utility indices U i , V i and marginal utility indices λi > 0, μi > 0, i = 1, . . . , n, that satisfy the following Afriat inequalities: λj U i U j + λi p j x i − x j + j V i − V j , μ i i j j j j V V +μ r m −m for i, j = 1, . . . , n. 3. The data (r i − mi ) and (p i , 1/μi , x i , V i ) satisfy GARP for some choice of (ui , V i ) that satisfies the Afriat inequalities. To meet condition 3 the entire data set and any hypothesized weakly separable subgroup must satisfy GARP. Thus, consistency with GARP is a necessary condition for weak separability. The sufficient conditions checked for in this chapter are that the data satisfy GARP when the sub-utility function is calculated using the

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Afriat inequalities. That is, using the observed prices and quantities, the Afriat inequalities in 2 above are solved for utility levels, U i and V i , and marginal utilities, λi and μj , and used to construct an aggregate good for those in the hypothesized sub-utility function. This aggregate good is then included in the hypothesized overall utility function which is tested for consistency with GARP. If the data set including the aggregate good for the sub-utility function is consistent with GARP, then the original data are consistent with a well-behaved utility function weakly separable in the assets in the hypothesized sub-utility function. Hereafter, these sufficient conditions for weakly separable utility will be referred to as the Afriat sufficient conditions. The necessary test, consistency with GARP, and the Afriat sufficient test for a well-behaved utility function at least weakly separable in monetary assets collectively are a sufficient condition for the existence of a common monetary aggregate. If these conditions hold, then a common monetary aggregate exists in an area whether or not the people in the area politically decide to adopt a common currency. The data set used for these revealed preference tests is discussed in the next section.

4. Data The data used for this chapter are from the International Monetary Fund (2001) and data posted on the countries’ Central Bank websites. These data are annual and quarterly observations for the period 1991 to 2000. From these sources annual and quarterly data were gathered for Australia and New Zealand. Complete sets of annual data were taken from the IMF for Australia and New Zealand, while all quarterly data were gathered from the countries’ Central bank websites.13 The series of data taken for these two countries include population, private consumption (C), the consumer price index (P ), money (M), quasi-money (Q), the deposit rate (r), a benchmark interest rate (R), the lending rate except for quarterly Australia where it was the money market yield, and the dollar exchange rate. The money series consist of currency and demand deposits. The quasi-money series is made up of savings and time deposits. The consumer price index is used as the price of a unit of consumption in each country. To convert each consumption, money and quasi-money series into real terms the consumer price index and population series were used. To construct data series on areas broader than one country, the exchange rate is used to convert each series to United States dollars.

13 The

websites are http://www.rbnz.govt.nz/statistics/index.html for the New Zealand quarterly data and http://www.rba.gov.au/statistics/ for Australian quarterly data. Quarterly population figures for both countries were interpolated for annual series.

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The appropriate price for each category of financial assets is its user cost (Barnett, 1980). The user cost is a discounted interest rate differential, (R − r)/(1 + R). The differential is the opportunity cost of holding a particular asset rather than the bench mark asset and the bench mark asset is further used for discounting the differential back to the beginning of the period when the choice to hold assets is made. These data were converted into three ten-year annual time-series and three tenyear quarterly time series. These time-series are for each individual country both quarterly and annually as well as for aggregates of Australia and New Zealand quarterly and annually. The aggregates are a weighted average of the data for each country that comprises the respective area. The results from checking these data for consistency with the microeconomic criteria for the existence of a common currency area are presented in the following section.

5. Results The data described above were checked for consistency with the microeconomic criteria for the existence of common currency, using Varian’s (1985) three-step revealed preference test for weak separability that was described in Section 3. That is, the data on each individual country and the unified Australia and New Zealand area were checked for consistency with the microeconomic foundations of an optimum currency area. As presented in Table 2, for the New Zealand and Australian individual and aggregate annual data sets both the necessary and the Afriat sufficient conditions obtain. This suggests that people in Australia, New Zealand and Australia and New Zealand together consider currency, demand deposits and quasi-money in a similar way in their preferences. Thus, these annual results are consistent with a common currency for Australian and New Zealand. As presented in Table 3, the individual and aggregate quarterly data sets of both countries are consistent with the necessary conditions for the existence of Table 2: Revealed Preference Test Results on Annual Data Area

New Zealand Australia AU/NZ aggregate

Utility function

Sub-utility function

GARP

Necessary

Afriat sufficient

Y Y Y

Y Y Y

Y Y Y

Note: Y implies the condition is met and N means a condition is not met. The reader is reminded that the Afriat sufficient condition is not necessary and that other sufficient conditions might hold.

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Table 3: Revealed Preference Test Results on Quarterly Data Area

New Zealand Australia AU/NZ aggregate

Utility function

Sub-utility function

GARP

Necessary

Afriat sufficient

Y Y Y

Y Y Y

N N N

Note: Y implies the condition is met and N means a condition is not met. The reader is reminded that the Afriat sufficient condition is not necessary and that other sufficient conditions might hold.

a common currency area. The Afriat sufficient, but not necessary, condition for a common currency area does not hold with the quarterly data on Australia and New Zealand or Australia and New Zealand aggregate. That this condition is not necessary implies that these data may meet some other sufficient condition. In addition, that the annual data are consistent with an Australian and New Zealand common currency area and the quarterly data may not be could just be a result of incomplete adjustment within a quarter. Swofford and Whitney (1988) suggested incomplete or lagged adjustment as why quarterly data might not meet this test in situations where the tests obtain with annual data. Thus, overall, the data indicate Australia and New Zealand might form a common currency area. That the data are consistent with a common currency area does not overcome various macroeconomic and political issues that might arise. For example seigniorage and control of central banking institutions would need to be allocated between the two countries in such a common currency area.14

6. Summary and Conclusions A hypothesized Australian–New Zealand common currency area would result in substantial transactions costs savings. Such a common currency area would remove national currencies that might act as an implicit barrier to trade. We also have found that annual data for Australia, New Zealand and Australia and New Zealand together are consistent with the necessary and sufficient conditions for the microeconomic foundations of a common currency area. Further we found that quarterly data for Australia, New Zealand, and Australia and New Zealand together are consistent with the necessary conditions for a common currency area.

14 Clearly seigniorage could be allocated by population in the two countries, but such an agreement would need to be reached.

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Acknowledgements We would like to thank various referees for useful comments on this chapter. The usual caveat remains. Lena C. Birkelöf and Joshua R. Caldwell would like to thank the University of South Alabama Committee on Undergraduate Research for support for this project.

References Afriat, S.N. (1967). The construction of utility functions from expenditure data. International Economic Review 8, 66–77. Bank for International Settlements (2002). Triennial Central Bank Survey. Basel, Switzerland: Bank for International Settlement. Barnett, W.A. (1980). Economic monetary aggregate: An applications of index number and aggregation theory. Journal of Econometrics 14, 11–48. Deaton, A. and Muellbauer, J. (1980). Economics of Consumer Behavior. Cambridge: Cambridge University Press. Feenstra, R.C. (1986). Functional equivalence between liquidity costs and the utility of money. Journal of Monetary Economics 17(2), 271–291. International Monetary Fund (2001). International Financial Statistics Yearbook. Washington, DC: International Monetary Fund. Machlup, F. (1977). A History of Thought on Economic Integration. New York: Columbia University Press. McKinnon, R.I. (1963). Optimum currency areas. American Economic Review 51(4), 717– 725. Mundell, R.A. (1961). A theory of optimum currency areas. American Economic Review 51, 657–665. Patterson, K.D. (1991). A non-parametric analysis of personal sector decisions on consumption, liquid assets and leisure. The Economic Journal 101(408), 1103–1116. Rosbourough, L. (2001). Trends in foreign exchange trading. Reserve Bank of New Zealand Bulletin 64(4), 23. Rose, A.K. and van Wincoop, E. (2001). National money as a barrier to international trade: The real case for currency union. American Economic Review 91(2), 386–390. Swofford, J.L. (2000). Microeconomic foundations of an optimal currency area. Review of Financial Economics 9, 121–128. Swofford, J.L. (2005). Tests of microeconomic foundations of a North American common currency area. Canadian Journal of Economics 38, 420–429. Swofford, J.L. and Whitney, G.A. (1986). Flexible functional forms and the utility approach to the demand for money: A nonparametric analysis. Journal of Money, Credit, and Banking 18(3), 383–389. Swofford, J.L. and Whitney, G.A. (1988). Comparisons of nonparametic tests of weak separability for annual and quarterly data on consumption, leisure and money. Journal of Business and Economic Statistics 6, 241–246. Varian, H.R. (1982). The nonparametic approach to demand analysis. Econometrica 50, 945–973.

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Varian, H.R. (1983). Nonparametric tests of consumer behavior. Review of Economic Studies 50, 99–110. Varian, H.R. (1985). Nonparametric demand analysis. Software Program.

Part IV Functional Structure and Asymmetric Demand Responses

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Chapter 9

Food Demand in Sweden: A Nonparametric Approach Per Hjertstrand Department of Economics, Lund University, P.O. Box 7082, S-220 07 Lund, Sweden, e-mail: [email protected]

Abstract We study separability structures and the demand for food in the Swedish market using newly developed computationally intensive nonparametric tests for weak separability. For this purpose a data set covering the years 1963 to 2002 is used. In contrast to several previous studies that examine the separability structure of food commodities, we place no restrictions on neither the initial separability structure nor on functional form. A main finding is that traditionally employed groupings in food demand analysis cannot be used, as these constitute nonseparable groupings. We derive the most appropriate separability structure, consisting of separable groupings and use the implied structure to estimate elasticities in a demand system context.

Keywords: dynamic almost ideal demand system, nonparametric, weak separability JEL: C14, Q11

1. Introduction Studies of the market demand for food have important implications for consumers, farmers and other food processors. These decision makers use demand elasticities to understand the market and to plan their actions accordingly to the behavior of the market. Policy decisions based on accurate information are of critical importance in order to avoid price fluctuations associated with excess or deficit supply. Weak separability has vital implications for the formulation and estimation of demand systems. Indeed, separability assumptions provide powerful means of partitioning goods into groups, which greatly simplifies the estimation of large demand systems, see Deaton and Muellbauer (1980b), Pollack and Wales (1992), Edgerton et al. (1996) and Barnett and Serletis (2000). Consequently, the applied literature usually uses groupings of goods, implying a separability structure, in order to reduce the number of parameters to be estimated. Misspecifying the International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18009-1

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structuring scheme in demand systems may yield biased estimates and produce invalid inference on estimated parameters, see Deaton and Muellbauer (1980b) and Pollack and Wales (1992). The main purpose of this chapter is to find a theoretically appropriate partitioning of Swedish food commodities. More precisely, we examine structures for food in the Swedish market using newly developed nonparametric tests for weak separability that to our knowledge have not previously been applied to agricultural data. Moreover, the implied separability structure is used to estimate price and demand elasticities within the separable groups. For this purpose, we use a Dynamic Almost Ideal Demand System model. First, because of its extensive use in agricultural studies. Second, because of its well-known (local) approximation abilities. It is common practice in the demand literature to apply weak separability tests to some form of intuitive grouping. The exact groupings tested often also correspond to the groupings used in national accounts data. As pointed out by Edgerton et al. (1996), this raises some practical problems. The main being that there must be a reasonable a priori knowledge of the separability structure associated with the data. Instead of utilizing this approach, we place no initial intuitive restrictions on the grouping. More precisely, we derive a scheme of weakly separable groups of food commodities, that do not require any a priori knowledge of the structure. A majority of all international studies that address partitioning of food commodities use the parametric1 approach, for example, see Eales and Unnevehr (1988), Nayga and Capp (1994), Sellen and Goddard (1997), Eales and Wessells (1999) and Reed et al. (2005).2 In the parametric approach, one estimates the demand on the a priori specified groups and then tests if the weak separability assumption is satisfied. This approach is, however associated with a number of deficiencies. First, the parametric approach requires a given initial separability structure. Second, the parametric approach assumes a specific functional form for the utility function, which may cause misspecification problems. Consequently, rejection of weak separability might as well be a rejection of the functional form being used, see Barnett and Choi (1989). Third, in the parametric models usually Wald or likelihood-based tests are used. These test require estimations of the unrestricted model, which often is empirically impossible to obtain, due to the large number of parameters that need to be estimated. Consequently, there may be a problem with the degrees of freedom in the model, as more observations than estimated parameters are needed.

1 Parametric

tests assumes a specific functional form for the utility function. For a survey, see Blackorby et al. (1978). 2 Reed et al. (2005) however uses the generalized composite commodity theorem to find an optimal structure of food commodities.

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In contrast to parametric studies, we use newly developed nonparametric tests of weak separability. These nonparametric tests place no restrictions on neither the initial separability structure, the functional form of the utility function nor on the number of observations being used. Varian (1983) derives necessary and sufficient conditions for a data set to be consistent with neoclassical utility maximization, and also provides an algorithm (NONPAR) that tests weak separability using a sequential procedure. The algorithm uses a sequence of tests of the Generalized Axiom of Revealed Preference (GARP) in order to test necessary and sufficient conditions. Fleissig and Whitney (2003) provide another sequential test for weak separability that is shown to be more efficient than Varian’s initial test. A joint test of the necessary and sufficient conditions was proposed by Swofford and Whitney (1994), hence eliminating the bias that is associated with sequential tests, see Barnett and Choi (1989). There has only been a limited number of demand studies using Swedish data. Previous studies include Klevmarken (1979), Edgerton et al. (1996), Jörgensen (2001) and Assarsson (2004). Edgerton et al. (1996) studies the demand for food in all Nordic countries using yearly national accounts data for the period 1963– 1991. The authors found that own-price elasticities for the main part are negative, and that all goods are normal. Results from the parametric separability analysis indicate that the separability structure at the most aggregated level for Sweden may be misspecified. Moreover, the authors found evidence for changing the dynamic structure of the model. Hence, changing the model’s separability structure may improve the results, see Edgerton et al. (1996, p. 106). Jörgensen (2001) studies the demand for ecological foodstuffs using cross-sectional micro-data. Jörgensen (2001) found that there are positive income effects and negative price effects on the demand for ecological food commodities. Assarsson (2004) considers a two-stage budget system. The demand for food is measured as one good, hence obtaining an rather aggregated separability structure. Assarsson found evidence from the parametric analysis that food is separable from all other goods within the two-stage demand system. Moreover, the author found that food is a necessity, and that price elasticities for food is relatively low, while the demand for food is price inelastic. There has only been a few studies that use the nonparametric approach when testing for weak separability in a demand context. Rickertsen (1998) estimates the Norwegian demand for food and beverages using aggregated foodstuffs in a three-stage demand model. In an initial stage, Varians’s nonparametric test was used to check the separability structure of the model. Violations of the Generalized Axiom of Revealed Preferences (GARP) within the separability structure were attributed to measurement errors in the quantity data. This is in accordance to recent empirical evidence, see Elger and Jones (2004) and Fleissig and Whitney (2003). These authors stress that a low number of GARP violations in a sample indicates that the violations probably are due to measurement errors. For the data

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to satisfy GARP, Rickertsen (1998) adjusts the quantities, before conducting additional steps. He used what was considered to be the most intuitive plausible structure. Hence, modelling meat, fish, milk and eggs as one separable group, namely animalia, soft drinks, hot drinks and alcohol as another separable group, namely beverages and bread and cereals and fruit and vegetables as a separable group, namely vegetabilia, see Rickertsen (1998, p. 93). However, the author finds that the sufficient conditions for some of these separability specifications are violated. Moreover, Rickertsen uses the implied separability structure to estimate parameters and calculate conditional and unconditional elasticities in a static and Dynamic Almost Ideal Demand system. Various specification and misspecification tests are also performed in order to check the validity of the models. These results show that dynamic versions of the Almost Ideal Demand System perform better than their static counterparts. This chapter is organized as follows. The next section gives a short introduction to weak separability, as well as a short account of tests for weak separability. Section 3 provides a short discussion of the Dynamic Almost Ideal Demand System. Section 4 describes the data and method of obtaining the separability structure used in the demand analysis. Sections 5 and 6 present the results from the separability and demand analysis, while Section 7 concludes the chapter.

2. Weak Separability In this section, we begin by introducing notation and definitions. We provide necessary and sufficient conditions for weak separability and discuss how nonparametric tests may be used to study whether a data set can be rationalized by a weakly separable utility function. Finally, we describe the test procedure used in this study. 2.1. Notation and Definitions Assume that there are n goods on the market. Let pt = (p1,t , . . . , pn,t ) and xt = (x1,t , . . . , xn,t ) denote vectors of prices and quantities at time t; t = 1, . . . , T , respectively. Assume that the data set is partitioned into two subgroups with prices at time t: (rt ; qt ) = (r1,t , . . . , rm,t ; q1,t , . . . , qn−m,t ) and quantities (zt ; yt ) = (z1,t , . . . , zm,t ; y1,t , . . . , yn−m,t ). A utility function u is said to be weakly separable in the y goods if there exists a sub-utility function V (y), and a macro-function U (·), strictly increasing in V such that: u(x) = u(z, y) ≡ U (z, V (y)). Consequently, a group of goods is weakly separable from other goods, if the quantities demanded of all other goods are independent of the marginal rates of substitution between any pair of goods in the separable group.

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2.2. Necessary and Sufficient Conditions Varian (1982) showed that if a data set satisfies GARP, then there exist a utility function that rationalizes the data. Varian (1983) derives the necessary and sufficient nonparametric conditions for weak separability. Specifically, he shows that there exists a weakly separable, concave, monotonic, continuous nonsatiated utility function that rationalizes the data, for some choice of {Ut , Vt , λt , μt > 0, t = 1, . . . , T } that satisfies the Afriat inequalities: λs (Vt − Vs ), s, t = 1, . . . , T , Ut Us + λs rs (zt − zs ) + (1) μs Vt Vs + μs qs (yt − ys ), s, t = 1, . . . , T . (2) Conditions (1) and (2) are equivalent to that {((zt ; Vt ), (rt ; 1/μt )), t = 1, . . . , T } and {(yt , qt ), t = 1, . . . , T }, respectively, satisfies GARP, see Varian (1982). Moreover, conditions (1) and (2) imply together that the full data set {(xt , pt ), t = 1, . . . , T } are consistent with GARP. Hence, necessary conditions for weak separability are that {(yt , qt ), t = 1, . . . , T } and {(xt , pt ), t = 1, . . . , T } satisfies GARP, see Elger and Jones (2004). 2.3. Testing Weak Separability Equations (1) and (2) jointly imply that a sequential test for weak separability can be constructed. In the first step, test whether the full data set {(xt , pt ), t = 1, . . . , T } satisfies GARP. In the second step, test if {(yt , qt ), t = 1, . . . , T } satisfies GARP. If neither one of these necessary conditions are violated, proceed to t and μˆ t , that satisfies the Afriat inequalities in (2). Finally, construct numbers V test {(zt , Vt ), (rt , 1/μˆ t ), t = 1, . . . , T } for GARP consistency, see Varian (1982) and Elger and Jones (2004). If this final test does not reveal GARP violations, weak separability cannot be rejected. Varian implemented a sequential test procedure building on Varian (1982, 1983) in a software application called NONPAR that he made publicly available. This software has been used extensively in empirical research. The NONPAR is easy to implement but suffers from several deficiencies, one being that it is biased towards rejecting weak separability, see the simulation evidence in Barnett and Choi (1989). This bias arises as a consequence of the sequential implementation. More specifically, the possibly nonunique Afriat numbers satisfying (2) that are used in the test are constructed without reference to (1). The results in Barnett and Choi (1989) spurred further development of new procedures for testing weak separability. Two more recent tests are used in this chapter. We present these tests briefly below. Technical descriptions of the tests are given in Appendixes A and B. Fleissig and Whitney (2003) provide a new sequential test that utilizes superlative index numbers to obtain initial estimates of the Afriat numbers in (2). For

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various reasons discussed in Fleissig and Whitney’s paper, these estimates may not satisfy the inequalities. They therefore propose a linear programming technique to perturb the indexes to satisfy the inequalities. In the remainder of this chapter, we refer to the test as the LP-test. Using simulations, Fleissig and Whitney (2003) show that the LP-test performs better than Varian’s NONPAR test. However, being sequential, the test is still prone to be biased towards rejection. Empirical evidence in, for example, Jones et al. (2005b) suggests that the test yields similar results to more computationally challenging joint tests of the necessary and sufficient conditions for weak separability. Swofford and Whitney (1994) propose an unbiased joint test of the necessary and sufficient conditions for weak separability, which we refer to as the SW-test. This test allows for incomplete adjustment, by including an additional exogenous restriction on the expenditure for the separable y goods. The incomplete adjustment concerns the agents’ inability to fully adjust their optimal consumption of goods within the observed period. That is, the test accounts for such habit persistence as adjustment costs and the formation of expectations, see Swofford and Whitney (1994). Swofford and Whitney (1994) proposes to minimize the degree of incomplete adjustment as measured by a function of the shadow prices. If this can be minimized to exactly zero at all data points, then the necessary and sufficient conditions for weak separability in (1) and (2) are satisfied. The SW-test is computationally challenging as it employs nonlinear programming techniques with a large number of linear and nonlinear constraints even for modest sample sizes. 2.4. Accounting for Measurement Errors The nonparametric tests discussed so far are nonstochastic. This means that even small measurement errors in the data can cause rejections of weak separability. This has caused researchers to investigate how prone different tests are to measurement errors. Fleissig and Whitney (2003), for example, provide simulation evidence to illustrate that their LP-test is less sensitive to measurement errors than the NONPAR test. Other researchers have sought to augment the nonparametric framework to explicitly handle measurement errors, e.g. de Peretti (2005), Jones and de Peretti (2005) and Varian (1985). This chapter utilizes the framework developed by Varian (1985) which has even previously been used for testing weak separability, see, for example, Jones et al. (2005a, 2005b). The basic idea behind this approach, in the context of testing GARP, is to perturb the observed quantity data so that it satisfies GARP. Given assumptions on the distribution of the measurement errors, it is possible to calculate a bound statistic that can be used to determine whether or not the hypothesis that the data satisfies GARP should be rejected. The bound statistic is useful since it can be directly related to the researchers priors concerning the magnitude of measurement errors in the data. The

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exact implementation of Varian’s (1985) approach used in this chapter is detailed in Appendix C. 2.5. The JEED Test Procedure In this chapter, we follow Jones et al. (2005b) and test weak separability following, in spirit, what they refer to as the JEED test procedure, see also Jones et al. (2005a). Their procedure consists of three steps and tests if a subgroup of goods, (y) are weakly separable from the remaining, (z) goods. The first step is to test the necessary condition that the full data set, (x) = (y, z) satisfies GARP. We refer in the following to this as an overall GARP test. The next step is to test the necessary condition that the subgroup, (y) satisfy GARP, which is referred to as subgroup GARP. If any violations are detected at this stage, we employ Varian’s (1985) method to obtain perturbed quantities that satisfy GARP. Finally, we apply the LP-test to test if the subgroup, (y) is weakly separable from all other goods, (z) using either the observed or perturbed data. If the LP-test rejects, we employ the more powerful but also more computationally challenging exact SW-test.

3. Demand Estimation The purpose of this study is to identify a separable structure amongst the groupings used in national accounts data with an emphasis on groupings of foodstuffs, and to therefore estimate a demand system for this structure. The Almost Ideal Demand System, see Deaton and Muellbauer (1980a), henceforth referred to as AI, is widely used in studies concerning the demand for food, see Edgerton et al. (1996), Rickertsen (1998) and Sellen and Goddard (1997) and seems therefore a reasonable choice for the current study. An explanation for the extensive use of the AI in applied work is its ability to approximate (locally) systems from a broad class of data generating functions. 3.1. The Dynamic Almost Ideal Demand System The static AI model has been criticized on a number of different grounds. The most severe criticism has been that the static model lacks dynamic properties. A commonly used extension to the static model, that introduces dynamics by adding lagged expenditure shares into the model specification, is the Dynamic Almost Ideal Demand System (DAI), see Edgerton et al. (1996) and Rickertsen (1998). Another major criticism has been that the data generating process often is nonstationary, see Deaton and Muellbauer (1980a). To account for this problem, we follow Deaton and Muellbauer (1980a), and use first differences. The DAI has been proven to outperform the static AI in a number of studies, see in particular, Edgerton et al. (1996) and Rickertsen (1998).

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The dynamic almost ideal demand system in first difference form used in this chapter is: n

wi,t =ηi +

γi,j log pj,t +

j =1

n

θi,j wj,t−1

j =1

+ βi ( log χt − log Pt ),

i = 1, . . . , n, t = 1, . . . , T ,

(3)

where wi,t denotes the budget share of good i at time t, χt is the total expenditure at time t, pj,t is the price of the j th good at time t and Pt is a price index, defined in first difference form as: log Pt =

n

ak log pk,t +

k=1

θk,l (wl,t−1 log pk,t )

k=1 l=1

1 γk,l (log pk,t log pl,t ), 2 n

+

n n

n

t = 1, . . . , T .

k=1 l=1

η, a, β, θ and γ are parameters. The uncompensated own-price, eii , cross-price, eij and expenditure elasticities, Ei are given by, see Edgerton (1997):

eii,t

n n γii βi = − θi,k wk,t−1 + γik log pk,t − 1 ai + wi,t wi,t k=1

k=1

for i = 1, . . . , n,

n n γij βi − θi,k wk,t−1 + γik log pk,t eij,t = aj + wi,t wi,t k=1

k=1

for i, j = 1, . . . , n, i = j, Ei,t = 1 +

βi . wi,t

The compensated elasticities, e˜ij are calculated as: e˜ij = eij + wj Ei . For the purpose of estimating the parameters, we add an additive error term to (3) and impose the usual restrictions of symmetry, adding up and homogeneity, see, for example, Deaton and Muellbauer (1980a), Edgerton et al. (1996) and Serletis (2001). Moreover, we employ the iterated seemingly unrelated regression to estimate the DAI.

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Figure 1: Goods Used in This Study.

4. Data and Method 4.1. Data Our study is based on three national accounts data sets that comprise private consumption expenditures. The first data set consists of yearly observations, ranging from 1963 to 1991, while the second data set consists of quarterly data ranging from 1980Q1 to 1998Q4. The third data set consists of quarterly data from 1993Q1 to 2003Q2. The first and second data sets uses only old SNA and ESA3 definitions, with base years 1985 and 1998, respectively, while the third series uses the latest ESA 95 and COICOP4 definitions with base year 1995. Following Assarsson (2004), we construct a yearly series ranging from 1963 to 2002, with given base year 1985. Adjustments, similar to those in Assarsson (2004) were made in order to produce the best possible match, as the SNA and ESA definitions in some commodity groups have changed over time. To avoid well-known difficulties with durability we follow, among others, Rickertsen (1998) and Reed et al. (2005) and exclude durable as well as semi-durable goods. The data is divided into two stages, as shown in Figure 1. The second stage consists of ten Food-at-Home goods: Meat–Fish–Milk, Cheese and Eggs–Hot drinks–Soft drinks–Alcohol–Bread and Cereals–Fruit and Vegetables–Oils and Fats–Other Foodstuffs. The first stage consists, therefore of thirteen goods, all ten Food-at-Home goods, and the remaining goods, Other Nondurables–FoodAway-from-Home–Services. The aim of the preceding study is to find a structure, implying a partitioning of the goods into different groups at each stage.

3 ESA

is a European classification system for national accounts data. For a detailed description, see Statistical Office of the European Commision, Eurostat, http://www.europa. eu.int/comm/eurostat/ 4 COICOP and SNA are international accounting systems that classifies individual consumption expenditures according to purpose. For a more detailed description, see United Nations Statistics Division, http://unstats.un.org/unsd/default.htm

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The price: pi,t of good i at time t is constructed from an implicit price index by dividing the current (CCi,t ) by real expenditures (RCi,t ): pi,t =

CCi,t , RCi,t

i = 1, . . . , n, t = 1, . . . , T ,

while the unit of quantity is in real expenditure, measured per capita in fixed base year prices: RCi,t , i = 1, . . . , n, t = 1, . . . , T , POPt pi,t xi,t = n , i = 1, . . . , n, t = 1, . . . , T , i=1 pi,t xi,t

xi,t = wi,t

where wi,t is the expenditure share for good i at time t, and POPt is the home population at time t. 4.2. Method of Obtaining a Separable Structure We essentially follow the JEED procedure, described in Section 2.5. However, instead of placing an a priori restriction on the separability structure, we apply a data orientated search method that utilizes the JEED approach to find an appropriate structuring of the goods at each stage. Initially, we test all thirteen goods, stage 1, for overall GARP. Given that this condition is not violated, we proceed at both stages, by testing all possible subgroup combinations for GARP. More precisely, at each stage, all combinations of two goods, three goods, etc. are tested for consistency with GARP. At the second stage, this includes testing all subgroup combinations for ten goods, while at the first stage, it includes testing all subgroup combinations for the remaining three goods. Subgroups that fail to satisfy GARP are adjusted using Varian’s (1985) minimal perturbation procedure. Finally, we test, for both stages, all subgroup combinations for weak separability using either the original or perturbed data by applying the LP-test. If the number of LP violations for a particular grouping equal zero, then the grouping is said to be weakly separable from all other goods. If however, there are more than five LP violations for a subgroup, we reject weak separability for that particular grouping. For subgroups with one to five LP violations, we apply the SW-test given that the grouping is a candidate in a possible separability formation at the corresponding stage. As pointed out by Jones et al. (2005b), SW-test results tend to conform with LP-test results. We therefore, only apply the SW-test to groupings with a maximum of five LP violations. The SW-test either accepts weak separability with complete or incomplete adjustment or rejects weak separability.

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5. Results from Separability Analysis In this section, we present results from the separability analysis. We begin by presenting results from overall and subgroup GARP tests. We then proceed to test for weak separability by applying the LP-test and the SW-test. 5.1. Overall GARP A presumption for testing weak separability is that overall GARP is satisfied. Using the entire sample length, we obtain zero overall GARP violations. 5.2. Subgroup GARP and Measurement Adjustment In this section, results from subgroup GARP tests are reported. When testing for subgroup, we check, at each stage, the subset of goods that constitutes the group tested to be weakly separable for GARP. More precisely, we begin with the first stage and check all three possible subgroup combinations, to be consistent with GARP. Correspondingly, for the ten goods at stage 2, we check all possible subgroup combinations for GARP. For stage 1, we also compute the perturbed quantity adjustments for groupings that violated GARP, as well as Varian’s (1985) chi-square statistic. 5.2.1. Subgroup GARP Results for Stage 1 Results for stage 1 appear in Table 1. Table 1 shows that violations were only found for A1 and A5. Furthermore, the associated number of violations was also found to be small, 2 and 7, respectively. Empirical evidence has shown that a small number of GARP violations probably indicates measurement error, see Elger and Jones (2004) and Fleissig and Whitney (2003), implying that the GARP violations in the A1 and A5 groupings may possibly be attributed to measurement errors. Table 1: Subgroup GARP Group of goods tested for GARP

A1. {Food-at-home} A2. {Food-away-from-home} A3. {Services} A4. {Other nondurables; Food-away-from-home; Services} A5. {Other nondurables; Food-away-from-home} A6. {Other nondurables; Services} A7. {Food-away-from-home; Services}

Number of goods in grouping

GARP

Number of violations

Chi-square statistic

10 1 1 3

Fail Pass Pass Pass

2

0.224

2

Fail

7

1.294

2 2

Pass Pass

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The minimal error adjustments are relatively small, 0.224% for A1 and 1.294% for A5. According to Varian (1985), the hypothesis of no GARP violations should only be rejected if the prior belief of the true variance of the measurement errors is smaller than the chi-square statistic. Given the nature of the data, it seems unlikely that the variance of the measurement errors is lower than the reported adjustment statistics. That is, we find no support for rejecting the hypothesis of no GARP violations. 5.2.2. Subgroup GARP Results for the Stage 2 We apply the data orientated search method presented in Section 4.2. This method is a very computer intensive approach, since the number of combinations increases rapidly when including more goods in the separability analysis. The total i number of possible combinations is equal to n−3 i=0 2 (n − i), where n (3) denotes the number of goods in the sample. That is, for all 10 goods in stage 2, we obtain a total of 1012 different combinations. Table 2 reports percentage rejection rates for different groupings. We note that most of the combinations pass GARP. For example, 35.56% of the 45 possible two good combinations, 39.17% of the 120 possible three good combinations and 43.33% of the 210 possible four good combinations do not violate GARP. The results also indicate that the share of no GARP violations tend to increase as the number of goods in the different groupings increase. Hence, the data tends to favor broad groupings in terms of being consistent with GARP. We have that 53.33% and 80% of all eight and nine good combinations do not violate GARP, respectively. Furthermore, the number of violations tend to decrease as we increase the number of goods in the grouping. 20% and 15.83% of all two and three good combinations have 21 or more GARP violations, respectively, while only 1.67%, 2.22% and 0% of all seven, eight and nine good combinations have 21 or more GARP violations, respectively. Table 2: Subgroup GARP Number of goods in grouping 2 3 4 5 6 7 8 9 Total

Number of combinations

%0 violations

% 1–5 violations

% 6–10 violations

% 11–20 violations

% >21 violations

45 120 210 252 210 120 45 10

35.56 39.17 43.33 42.06 53.33 48.33 53.33 80

17.78 26.67 25.71 28.57 19.52 29.17 33.33 20

17.78 7.5 11.90 11.51 12.86 12.5 6.67 0

8.89 10.83 8.1 11.11 10.95 8.33 4.44 0

20 15.83 10.95 6.75 3.33 1.67 2.22 0

1012

45.65

25.59

11.46

9.58

7.71

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5.3. Fleissig and Whitney LP-Test Results In this section we report the total number of LP-test violations as well as the adjustment statistic based on the LP-procedure. These statistics measure the proportional distance between the superlative indexes and the Afriat indexes obtained from the LP-procedure. The statistic for the quantity index is denoted PRMSE(Q), and the corresponding statistic for the price index is denoted PRMSE(ϕ). Technical details are given in Appendix A. For stage 2, we report rejection frequencies at different violation intervals. However, due to the large number of possible combinations, we will only consider groupings that initially passed subgroup GARP. 5.3.1. LP-Test Results for Stage 1 First, we consider groupings A1–A7, defined in Table 1. LP-test results appear in Table 3. The groupings A1*, A2, A3 and A6 all pass weak separability, while the groupings A4, A5* and A7 fail to satisfy weak separability. A4 has only 5 violations, while A5* and A7 fail GARP with a large number of violations, 76 and 41, respectively. We also find A1 (Food-at-Home) to be weakly separable from all other goods. This result is in accordance with the separability structures traditionally employed, since these often model Food-at-Home as a separable grouping at the most aggregated stage in the separability tree. The PRMSE statistics for A1*–A7 are small, indicating that the estimated Afriat numbers tend to follow the superlative indices. We also note that the PRMSE statistic for the price index produce higher values than the corresponding statistic for the quantity index. That is, the estimated Afriat price index deviates more in relation to its superlative index than the estimated Afriat quantity index does in relation to its superlative index. Table 3: LP Test Results for Stage 1 Grouping A1∗ A2∗∗ A3∗∗ A4 A5∗ A6 A7

Weak separability

Number of violations

PRMSE(Q)

PRMSE(ϕ)

Pass Pass Pass Fail Fail Pass Fail

0 0 0 5 76 0 41

0.0044 0 0 0.0336 0.0128 0.0173 0.0474

0.1000 0 0 0.5660 0.2904 0.2802 0.5535

∗ denotes groupings that are quantity adjusted using Varian (1985) minimal perturbation procedure. ∗∗ denotes groupings with only one good. The LP-test implicitly never rejects weak separability for

groupings with only one good, as it lacks power to do so.

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Table 4: LP Results for Stage 2 Number of goods in grouping

Number of specifications

%0 violations

% 1–5 violations

% 6–10 violations

% 11–20 violations

% >21 violations

2 3 4 5 6 7 8 9

16 47 91 106 112 58 24 8

56.25 42.55 32.97 24.53 13.39 10.34 12.5 12.5

18.75 23.40 23.08 34.90 38.39 50 33.33 25

0 12.77 7.7 9.43 7.14 8.62 20.83 12.5

6.25 2.13 6.6 5.66 7.14 5.17 4.17 25

18.75 19.15 29.67 25.47 33.93 25.86 29.17 25

Total

462

23.81

33.33

9.1

6.1

27.71

5.3.2. LP-Test Results for Stage 2 Table 4 reports percentage LP rejection rates of weak separability for groupings at the second stage that initially passed subgroup GARP. The data tend to favor narrow specifications in terms of consistency of weak separability when applying the LP-test. We have that 56.25% and 42.55% of all possible two and three good combinations that passed subgroup GARP also passes weak separability, while only 12.5% of all eight and nine good combinations that passed subgroup GARP passes weak separability. The data also indicate that the share of 21 or more violations are robust to the number of goods in the grouping. We found that 462 of the total 1012 subgroup combinations in stage 2 passed subgroup GARP. However, since the number of parameters to be estimated increases as more goods are included, we restrict, the groupings to contain only two to five goods. That is, we only consider 157 different subgroups, 85 of these passed the LP-test, while 72 have one to five LP violations. Moreover, we can further restrict the number of subgroups to be considered, as not all of these 157 groupings can be used in possible separability structures. For example, if there are two separable groupings of four goods each, i.e. eight different goods in two separable groupings, then we have to find a separable grouping of the remaining two goods. We find that, after analyzing the 157 groupings that passed weak separability or have one to five LP violations, that 68 of the 85 groupings that passed weak separability using the LP-test can be excluded. The remaining 17 groupings with zero LP violations and their PRMSE statistics appear in Table 5. Correspondingly, we can exclude 54 of the 72 groupings that had one to five LP violations. The PRMSE(ϕ) statistics for some groupings tend to be relatively high. Groupings with fewer goods produce lower values of the PRMSE statistics than groupings with more goods. We find that groupings with two goods have average PRMSE(ϕ) and PRMSE(Q) values of 0.537675 and 0.019675, respectively, while groupings with three goods have average PRMSE(ϕ) and PRMSE(Q) values of 0.867588

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Table 5: LP Results for Groupings in Stage 2 Grouping {Meat; Hot drinks} {Meat; Alcohol} {Milk, Cheese and Eggs; Alcohol} {Bread and Cereals; Other Foodstuffs} {Meat; Hot drinks; Bread and Cereals} {Meat; Hot drinks; Other Foodstuffs} {Meat; Bread and Cereals; Other Foodstuffs} {Fish; Hot drinks; Fruit and Vegetables} {Fish; Hot drinks; Other Foodstuffs} {Fish; Bread and Cereals; Fruit and Vegetables} {Fish; Fruit and Vegetables; Other Foodstuffs} {Hot drinks; Soft drinks; Other foodstuffs} {Hot drinks; Bread and Cereals; Fruit and Vegetables} {Meat; Bread and Cereals; Fruit and Vegetables, Oils and Fats} {Fish; Hot drinks; Fruit and Vegetables; Oils and Fats} {Fish; Bread and Cereals; Fruit and Vegetables; Other Foodstuffs} {Fish; Fruit and Vegetables; Oils and Fats; Other Foodstuffs}

PRMSE(Q)

PRMSE(ϕ)

0.0109 0.0048 0.0158 0.0472 0.0029 0.0074 0.0220 0.1092 0.0878 0.0261 0.0462 0.0635 0.0390 0.0015 0.0650 0.0660 0.0567

0.2422 0.0526 0.2952 1.5607 0.0923 0.1637 0.7011 1.3641 1.0308 1.3516 0.5859 0.8188 1.7000 0.2067 1.2331 1.7919 0.9414

and 0.0449. The average PRMSE(ϕ) and PRMSE(Q) values for the four good groupings are 0.94644 and 0.04644, respectively. Moreover, most groupings with high PRMSE statistics include the goods Bread and Cereals and Other Foodstuffs. We cannot form any separability structure from the above given groupings in Table 5. Hence, in the next step, we apply the SW-test to the remaining 18 groupings that have one to five LP violations. 5.4. Swofford–Whitney Results In this section, results from the SW-tests5 are reported. All the results that appear in Tables 6 and 7 refer to weak separability with complete adjustment. We report if the algorithm found a feasible solution and also provide the number of violations. If a feasible solution was found, then weak separability is either accepted with complete or incomplete adjustment. In the case where weak separability is accepted with complete adjustment, the number of violations of weak separability is equal to zero. Consequently, if the algorithm found a feasible solution, but the number of violations is greater than zero, then there is weak separability with incomplete adjustment at a level that may be accepted or rejected, depending on ¯ which is a the amount of adjustment. In these cases, we report the statistic ψ,

5 The

data code for the Swofford and Whitney test was programmed in FORTRAN by Barry Jones and Thomas Elger. In order to solve the optimization problem Jones and Elger had to use solver FFSQP, see Zhou et al. (1997).

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measure of the average amount of incomplete adjustment. Following Jones et al. (2005b) the statistic is interpreted, “[a]s the increment of utility from spending an additional dollar on the separable y goods relative to the marginal utility of an additional dollar of total expenditure”, see also Swofford and Whitney (1994). A technical definition of the statistic is given in Appendix B. 5.4.1. SW-Test Results for Stage 1 SW results for groupings in stage 1, that failed to pass weak separability when applying the LP-test appear in Table 6. Feasible solutions were found for A4 and A5*, with 5 and 76 violations respectively. Weak separability with complete adjustment can therefore be rejected as the number of violations exceeds zero. The average amount of incomplete adjustment, measured by the statistic ψ¯ was found to be, 0.3% for A4 and 0.0125% for A5*. According to Jones et al. (2005b), an average incomplete adjustment of 0.3% is trivial and should therefore be regarded as very low. No feasible solution for A7 could be found, implying that weak separability for this particular grouping can be rejected. 5.4.2. SW-Test Results for Stage 2 SW results for groupings that failed the LP-test with one to five LP violations, and are potential candidates to form separability structures that add up to the total number of goods in stage 2, appear in Table 7. Feasible solutions were found for all groupings except for {Fish; Soft Drinks; Fruit and Vegetables}. Moreover, two groupings passed weak separability with complete adjustment, namely {Bread and Cereals; Fruit and Vegetables; Oils and Fats} and {Meat; Hot Drinks; Oils and Fats; Other Foodstuffs}. Most groupings that failed to pass weak separability with complete adjustment tend to produce low values of incomplete adjustment. There are only seven groupings with an average amount of incomplete adjustment larger than 0.1%.

Table 6: SW Results for Stage 1 Grouping

A4 A5∗ A7

Found a feasible solution

Number of violations

ψ¯ (%)

Yes Yes No

5 76 41

0.3000 0.0125 0.3741

∗ denotes which groupings that are quantity adjusted using Varian’s (1985) mini-

mal perturbation procedure.

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Table 7: SW-Test for the Remaining Groupings in Stage 2 Grouping

{Fish; Fruit and Vegetables} {Soft drinks; Fruit and Vegetables} {Meat; Fish; Hot drinks} {Meat; Hot drinks; Oils and Fats} {Fish; Soft drinks; Fruit and Vegetables} {Milk, Cheese and Eggs; Soft drinks; Alcohol} {Bread and Cereals; Fruit and Vegetables; Oils and Fats} {Bread and Cereals; Oils and Fats; Other Foodstuffs} {Meat; Fish; Hot drinks; Other Foodstuffs} {Meat; Fish; Oils and Fats; Other Foodstuffs} {Meat; Hot drinks; Bread and Cereals; Oils and Fats} {Meat; Hot drinks; Oils and Fats; Other Foodstuffs} {Meat; Bread and Cereals; Oils and Fats; Other Foodstuffs} {Fish; Bread and Cereals; Fruit and Vegetables; Oils and Fats} {Fish; Bread and Cereals; Fruit and Vegetables; Oils and Fats; Other Foodstuffs} {Fish; Milk, Cheese and Eggs; Bread and Cereals; Fruit and Vegetables; Oils and Fats} {Meat; Fish; Bread and Cereals; Fruit and Vegetables; Oils and Fats} {Meat; Fish; Hot drinks; Soft drinks; Fruit and Vegetables}

Found a feasible solution

Number of violations

ψ¯

Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

4 2 2 5 6 5 0 2 2 2 2 0 2 4 4

0.1536 0.0603 0.0230 0.2178 0.7489 0.0984 0.0780 4.6367 0.0253 0.0369 0.0705 0.0379 0.0743 0.2029 0.2734

Yes

5

0.2564

Yes Yes

2 2

0.0190 0.0266

5.5. The Separability Structure In this section, we analyze the results from the separability analysis and form separability structures for both stages. The implied separability structure at stage 2 is used in the demand analysis in the preceding Section 6. 5.5.1. Separability Structures for Stage 1 We find that three different structures with weakly separable groupings can be formed for stage 1. The first structure consists of all three goods in one grouping. The second and third structures consists of two weakly separable groupings each. The first grouping with two goods and the second grouping with one good, see Table 8. The second separability structure could be formed after applying the LP-test to the grouping. In contrast, the LP-test rejected weak separability for the first separability structure and also for the first grouping in the third structure. These groupings however passed weak separability with incomplete adjustment when applying the SW-test.

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Table 8: Separability Structure for Stage 1

1 2 3

Grouping 1

Grouping 2

{Other nondurables; Food-away-from-home; Services} {Other nondurables; Services} {Other nondurables; Food-away-from-home}

{Food-away-from-home} {Services}

Table 9: Separability Structure for Stage 2 Grouping 1 Meat Fish Bread and Cereals Fruit and Vegetables Oils and Fats

Grouping 2

Grouping 3

Hot Drinks Soft Drinks Other Foodstuffs

Milk, Cheese and Eggs Alcohol

5.5.2. Separability Structures for Stage 2 The SW and LP results reveal that seventeen different separability structures from groupings that pass weak separability can be formed for stage 2, Food-atHome. However, no separability structure could be formed after only applying the LP-test. That is, all possible separability structures in stage 2 consists of at least one grouping that is weakly separable from all other goods with incomplete adjustment. The choice of separability structure is based on two criteria. Firstly, how many groupings in the structure that passes weak separability when applying the LP-test. Secondly, we look at the average amount of incomplete adjustment for the groupings for the different structures. The separability structure with the lowest total average amount of incomplete adjustment across groupings appear in Table 9. The derived separability structure in Table 9 has a few similarities to the traditionally employed separability structure. In particular, Meat and Fish and Hot drinks and Soft drinks are modelled in the same groupings.

6. Results from Demand Analysis In this section, results from the demand analysis is reported. We apply the Dynamic Almost Ideal Demand System (DAI) on two different separability structures. First, we use the implied separability structure, presented in Table 9. Second, we apply the traditionally employed structure in order to illustrate the importance of specifying a correct separability structure. This structure is, for instance applied in Edgerton et al. (1996), Rickertsen (1998) and Reed et al. (2005).

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6.1. Demand Analysis from the Implied Separability Structure Classifying the goods into necessity and luxury goods, we observe that Meat and Other Foodstuffs are significant luxury goods, while Milk, Cheese and Eggs, Alcohol and Oils and Fats are significant necessities. Inconclusive results, that is, insignificant parameter estimates, but with positive estimates, implying luxury goods are given for Fish, Hot drinks, Soft drinks and Fruits and Vegetables, while the good Bread and Cereals is a necessity. All estimated elasticities are related to the expenditure and prices within each separable group. The uncompensated elasticity estimates, e are given in Table 10, while compensated elasticity, e˜ and expenditure elasticity E, estimates, are given in Table 11. The expenditure elasTable 10: Mean Uncompensated Elasticity Estimates e 1

2

3

4

5

1. Meat 2. Fish 3. Bread and Cereals 4. Fruit and Vegetables 5. Oils and Fats

−0.9216 0.0041 0.0869 −0.1292 −0.0810

0.0885 −0.2088 −0.2941 −0.1217 0.0130

−0.1268 −0.4306 −0.3466 −0.1099 −0.5960

−0.1484 −0.2953 −0.0903 −0.5210 −0.3049

0.0623 −0.2686 −0.0399 −0.0359 −0.6161

1. Hot drinks 2. Soft drinks 3. Other Foodstuffs

0.0243 −0.0260 −0.2407

−0.6360 −0.1090 −0.3041

−0.5997 −0.7994 −0.5293

1. Milk, Cheese and Eggs 2. Alcohol

−0.2043 −0.5952

−0.2319 −0.7954

Table 11: Mean Compensated Elasticity Estimates e˜

E

1

2

3

4

5

1. Meat 2. Fish 3. Bread and Cereals 4. Fruit and Vegetables 5. Oils and Fats

−0.5993 0.4083 0.3174 0.1799 0.4519

0.1865 −0.0966 −0.2301 −0.0358 0.1617

0.1228 −0.1444 −0.1833 0.1091 −0.2177

0.1269 0.0201 0.0896 −0.2794 0.1147

0.1331 −0.1874 0.0063 0.0261 −0.5106

1. Hot drinks 2. Soft drinks 3. Other Foodstuffs

0.1261 0.1202 −0.0725

0.0852 0.1117 −0.0506

−0.2114 −0.2320 0.1230

1. Milk, Cheese and Eggs 2. Alcohol

−0.0128 0.0119

0.0128 −0.0119

1.0727 1.2309 0.7017 0.9411 1.6208 0.6554 0.9584 1.1018 0.4493 1.4276

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ticities for all goods are positive. That is, we can exclude the possibility of any good being inferior. The own-price elasticities indicates that all goods, except Hot Drinks are elastic. However, the own-price elasticity estimates for Fish, Soft Drinks, and Milk, Cheese and Eggs are small and should be considered as insignificantly elastic. In the case of Hot Drinks, the own-price elasticity is positive, indicating that it is a Giffen good. However, the estimate is very close to zero, and should therefore be interpreted as a sign of misspecification, rather than an actual occurrence. We should also note that Edgerton et al. (1996) also found in their study that Hot Drinks is a Giffen good, not significantly different from zero. The cross-price elasticities are, as expected of varying signs. We have, for example that the cross-price elasticities between Meat and Fish are positive, implying that the goods are complements. These results are, however inconclusive, as the estimates seem highly insignificant, which applies for most cross-price elasticity estimates. The compensated elasticities are in most cases of the same sign as the compensated elasticities. However, the estimates tend to differ substantially in value. In the case of Hot Drinks, Soft Drinks and Other Foodstuffs, we get positive signs for the own-price elasticities. 6.2. Demand Estimates from the Traditionally Employed Separability Structure The traditionally employed separability structure usually models animalia products, beverages, vegetabilia products and miscellaneous products as separable groupings, as shown in Table 12. The structure, however, do not constitute separable groupings. The estimates may therefore be biased and the elasticity estimates misleading. We find, in the first grouping that Meat and Fish are luxury goods, significantly different from zero, while Milk, Cheese and Eggs is a significant necessity. The results also show that all goods in the second grouping, that is, Hot Drinks, Soft Drinks and Alcohol are luxury goods, however with insignificant parameter results. The goods in the third grouping, Bread and Cereals and Fruit and Vegetables are insignificantly necessity and luxury goods, respectively. Moreover, the goods in the forth grouping, Oils and Fats and Other Foodstuffs are significantly necessary and luxury goods, respectively. Conditional elasticity estimates appear in Table 12. The results differ substantially from Tables 10 and 11. We find, for example that Hot drinks is elastic and that Milk, Cheese and Eggs is a Giffen good. Moreover, in comparison to Table 10, we find higher elasticity estimates in absolute value for the goods Fish; Hot Drinks; Soft Drinks; Alcohol; Bread and Cereals; Fruit and Vegetables and lower estimates in absolute value for the goods Meat; Oils and Fats and Other Foodstuffs.

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Table 12: Mean conditional elasticity estimates e˜

e 1

2

3

1

2

E 3

1. Meat −0.4786 −0.0677 −0.6679 0.1238 0.0999 −0.2237 2. Fish 0.0522 −0.4842 −0.77696 0.6488 −0.3186 −0.3302 3. Milk, Cheese and Eggs −0.6933 −0.0927 0.2208 −0.4131 −0.0147 0.4278

1.2451 1.2334 0.5800

1. Hot drinks 2. Soft drinks 3. Alcohol

−1.4208 −0.2447 0.4563 −0.5677 −0.1544 −0.3416 0.2018 −0.1536 −0.9661

−1.2820 −0.0362 1.3182 −0.4442 0.0276 0.4168 0.3081 0.0038 −0.3119

1.2399 1.0904 0.9413

1. Bread and Cereals 2. Fruit and Vegetables

−0.4533 −0.2917 −0.4752 −0.7088

−0.0982 0.0982 0.0889 −0.0889

0.7645 1.2148

1. Oils and Fats 2. Other foodstuffs

−0.0332 −0.2567 −0.3051 −0.8910

0.0482 −0.0482 −0.0026 0.0026

0.3028 1.2276

7. Concluding Remarks The performance of demand system estimation depends on the specified separability structure. However, the separability structure is often misspecified in demand studies. Consequently, weak separability is therefore rejected, which frequently leads to inconsistent estimates of elasticities and other parameters. A majority of demand applications use the parametric approach when modelling the separability structure. This raises some practical considerations, the most common being that there must be a reasonable initial separability structure. In this chapter, we show that the most intuitive plausible structure, assuming that animalia products, beverages, vegetabilia products and other foodstuffs constitutes separable groupings, is a misleading assumption. Hence, we derive a scheme with separable groupings using nonparametric tests for weak separability that do not require any a priori knowledge of the structure. Our findings suggest grouping Meat and Fish, together with Bread and Cereals, Fruit and Vegetables and Oils and Fats. The results also supports that Hot Drinks and Soft Drinks together with Other Foodstuffs forms a separable group. Furthermore, the final grouping consists of Milk, Cheese and Eggs and Alcoholic beverages. The results also indicate that the data, in terms of being consistent with GARP tend to favor broad groupings. This is however not the case when testing for weak separability, as the data tend to reject broad groupings in favor of narrow groupings. The derived separability structure is used to calculate elasticities in a Dynamic nonlinear Almost Ideal Demand System. We find that the conditional elasticities are conformable with theory. Furthermore, we find, as expected, that all goods

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are normal. Results also indicate that all goods, except Hot drinks are elastic with respect to own price. We also estimate the demand for food using the intuitive plausible separability structure and compare these results to the results obtained with applying the derived structure. We find that elasticity results for some goods differ substantially, hence implying that the intuitive plausible separability structure is misspecified and therefore may yield biased results.

Acknowledgements I am very grateful to David Edgerton and Thomas Elger for valuable and helpful comments. Research funding from Jan Wallander and Tom Hedelius Foundation (J03/19) is gratefully acknowledged. Finally, I thank Barry Jones and Thomas Elger for allowing me to use their Fortran codes when conducting tests for weak separability.

Appendix A: The LP-test The LP-test is a linear programming algorithm that searches for a solution to the Afriat inequalities, Equation (2). The procedure minimizes the error correction of the superlative indices, that satisfies some linear restrictions, Equations (4)–(7). p p We define Qt and ϕt as the difference between the Afriat indices and the Sup p p p perlative indices, given that it is positive (Qt , ϕt > 0), and zero if Qt , ϕt < 0. p n n n Correspondingly, Qt and ϕt are defined as −Qt and −ϕt , respectively given that Qnt and ϕtn are positive, and zero otherwise. The LP algorithm minimizes the linear function: T t=1

p

Qt +

T t=1

Qnt +

T

p

ϕt +

t=1

T

ϕtn

t=1

subject to the linear restrictions: p

p

QVt + Qt − Qnt QVs + Qs − Qns + ϕs qs (yt − ys ), QVt p + ϕt − ϕtn , ϕt = incyt ϕ ϕt t ,

(4)

p QV QVt + Qt − Qnt t , p p Qt , Qnt , ϕt , ϕtn 0

(7)

(5) (6) (8)

for s, t = 1, . . . , T , where QVt is the quantity superlative index and an initial estimate of Vt and 1/ϕt is the group price index for the separable y goods. incyt is

Food Demand in Sweden: A Nonparametric Approach ϕ

179

QV

the expenditure on the separable y goods in period t and t , t are small positive p QV n numbers. The Afriat indices are defined as: QLP and t = QVt + Qt − Qt − t p ϕ QVt LP n ϕt = incyt + ϕt − ϕt − t , respectively. The usual procedure for sequential LP ), (q , 1 ), t = tests applies. Hence, weak separability is accepted if {(y , Q t

t

t

LP ϕ t

1, . . . , T } satisfies GARP. The proportional root mean squared statistic for the quantity index is defined as: T LP Qt − QVt 2 PRMSE(Q) = QVt t=1

while the corresponding statistic for the dual price index is defined as: T LP ϕt − ϕt 2 PRMSE(ϕ) = . ϕt t=1

Lower values of the statistics indicate that the Afriat numbers are proportionally close to the superlative indices.

Appendix B: The Swofford–Whitney Test Swofford and Whitney (1994) propose to minimize the quadratic function: F (θ1 , . . . , θT ) =

T

θt2 =

t=1

T

(μt φt − τt )2

(9)

s, t = 1, . . . , T ,

(10)

t=1

subject to: Vt Vs + μs qs (yt − ys ),

Ut Us + τs rs (zt − zs ) + φs (Vt − Vs ),

s, t = 1, . . . , T ,

(11)

Ut , Vt , μt , τs , φt > 0, where μ is the shadow price from the constrained optimization of the sub utility function V (y), while τ and θ are the shadow prices of the exogenous restrictions on the utility function U . Defining φ = (τ − θ )/μ, we find that the restrictions of the exogenous restrictions must equal: θt = μt φt −τt , t = 1, . . . , T . Swofford and Whitney (1994) and Elger and Jones (2004) propose to use the ratio: ψt = θt /τt as a measure of the amount of incomplete adjustment at each observation, and the ˆ estimate: ψ¯ = T1 Tt=1 | θτˆt | as a measure of the average incomplete adjustment. t Note that ψ¯ equals zero, when weak separability with complete adjustment is accepted. The utility function, U is weakly separable in the y goods if the objective function F (·) is minimized to zero. In this case, the expenditure is said to be optimally adjusted at all observations, see Elger and Jones (2004). The necessary and

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sufficient conditions, Equations (10) and (11) are violated if no feasible solution = 0. In the latter exist or if the objective function is not minimized to zero, F case, ψˆt = 0, indicating that the expenditure is not optimally adjusted. Jones et al. t ), (vt , 1/μˆ t ), t = 1, . . . , T } for GARP, (2005a) propose therefore to test {(zt , V t and μˆ t are estimated parameters from the minimization of F (·), Equawhere V tions (9)–(11). If GARP is violated, then the hypothesis of weak separability with complete adjustment is rejected. The optimization problem is very computationally burdensome, as there are T (T − 1) nonlinear and T + T (T − 1) linear inequality constraints and 4T sign restrictions.

Appendix C: Varian’s (1985) Minimal Perturbation Test Varian (1985) provides an algorithm allowing for measurement error in the quantity data. The following relation between the true data ζti and the observed data yti is assumed: ζti = yti 1 + εti , i = 1, . . . , n, t = 1, . . . , T , where εti ∼ iid(0, σ 2 ). Varian (1985) propose to minimize the function: n i T ξt − yti 2 min M = yti Vt ,μt ,ξti , t=1 i=1 i=1,...,n

subject to: Vt Vs + μs qs (ξt − ξs ),

s, t = 1, . . . , T ,

Vt , μt > 0. The measurement error in period t for good i is defined as: εti =

ξti − yti yti

,

i = 1, . . . , n, t = 1, . . . , T ,

where ξti is the perturbed quantity data that satisfies GARP. Varian also proposed a chi-square test for measurement errors. This test presumes that the observed data violates GARP because of measurement errors. In particular, the test assumes that the errors are Gaussian distributed with mean 0 and variance σ 2 . Varian (1985) suggested calculating the test statistic M/σ 2 and reject the hypothesis that the original data is measured with errors if M/σ 2 Cα , where Cα is the chi-square critical value with significance level α. The test is chisquare with T m degrees of freedom, where T denotes the sample length and m denotes the number of goods in the subgroup. However, as the variance is a la α and rejecting the hypothesis of tent variable, Varian proposed calculating M/C α. measurement errors if ones prior beliefs is that σ 2 is less than M/C

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181

References Assarsson, B. (2004). Consumer demand and labor supply in Sweden 1980–2003. Mimeo, Uppsala University. Barnett, W. and Choi, S. (1989). A Monte Carlo study of tests of blockwise weak separability. Journal of Business and Economic Statistics 7, 363–377. Barnett, W. and Serletis, A. (2000). The Theory of Monetary Aggregation. Amsterdam: North-Holland. Blackorby, C., Primont, D. and Russell, R. (1978). Duality, Separability, and Functional Structure: Theory and Economic Applications. Amsterdam: North-Holland. de Peretti, P. (2005). Testing the significance of the departures from utility maximization. Macroeconomic Dynamics 9, 372–397. Deaton, A. and Muellbauer, J. (1980a). An almost ideal demand system. The American Economic Review 70(3), 312–326. Deaton, A. and Muellbauer, J. (1980b). Economics and Consumer Behavior. Cambridge: Cambridge University Press. Eales, J. and Unnevehr, L. (1988). Demand for beef and chicken products: Separability and structural change. American Journal of Agricultural Economics 70, 521–532. Eales, J. and Wessells, C. (1999). Testing separability of Japanese demand for meat and fish within differential demand systems. Journal of Agricultural and Resource Economics 24(1), 114–126. Edgerton, D. (1997). Weak separability and the estimation of elasticities in multi-stage demand systems. American Journal of Agricultural Economics 79, 62–79. Edgerton, D., Assarsson, B., Hummelmose, A., Laurila, I., Rickertsen, K. and Vale, P. (1996). The Econometrics of Demand Systems: With Applications to Food Demand in the Nordic Countries. Boston/Dordrecht/London: Kluwer Academic Publishers. Elger, T. and Jones, B. (2004). A non-parametric test of weak separability with random measurement errors in the data. Mimeo, Lund University. Fleissig, A. and Whitney, G. (2003). A new PC-based test for Varian’s weak separability conditions. Journal of Business and Economic Statistics 21(1), 133–143. Jones, B. and de Peretti, P. (2005). A comparison of two methods for testing utility the maximization hypothesis when quantity data is measured with error. Macroeconomic Dynamics 9, 612–629. Jones, B., Elger, T., Edgerton, D. and Dutkowsky, D. (2005a). Toward a unified approach to testing for weak separability. Economics Bulletin 3(20), 1–7. Jones, B., Dutkowsky, D. and Elger, T. (2005b). Sweep programs and optimal monetary aggregation. The Journal of Banking and Finance 29, 483–508. Jörgensen, C. (2001). Prisbildning och efterfrågan på ekologiska livsmedel. Rapport 2001:1, Livsmedelsekonomiska Institutet, Lund, Sweden. Klevmarken, N.A. (1979). A comparative study of complete systems of demand functions. Journal of Econometrics 10, 165–191. Nayga, R. and Capp, O. (1994). Tests of weak separability in disaggregated meat products. American Journal of Agricultural Economics 76, 800–808. Pollack, R. and Wales, T. (1992). Demand System Specification and Estimation. NY: Oxford University Press. Reed, A., Levedahl, W. and Hallahan, C. (2005). The generalized commodity theorem and food demand estimation. American Journal of Agricultural Economics 87(1), 28–37.

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Rickertsen, K. (1998). The demand for food and beverages in Norway. Agricultural Economics 18, 89–100. Sellen, D. and Goddard, E. (1997). Weak separability in coffee demand system. European Review of Agricultural Economics 24(1), 133–144. Serletis, A. (2001). The Demand for Money. Boston/Dordrecht/London: Kluwer Academic Publishers. Swofford, J. and Whitney, G. (1994). A revealed preference test for weakly separable utility maximization with incomplete adjustment. Journal of Econometrics 60, 235–249. Varian, H. (1982). The nonparametric approach to demand analysis. Econometrica 50(4), 945–974. Varian, H. (1983). Non-parametric tests of consumer behavior. The Review of Economic Studies 50(1), 99–110. Varian, H. (1985). Non-parametric analysis of optimizing behavior with measurement error. Journal of Econometrics 30, 445–458. Zhou, J.L., Tits, A.L. and Lawrance, C.T.(1997). User’s guide for FFSQP version 3: A Fortran code for solving optimization programs, possible minimax with general inequality constraints and linear equality constraints, generating feasible iterates. Institute for Systems Research, University of Maryland Report SRC-TR-92-107r5, College Park, MD 20742, 1997.

Chapter 10

A Systems Approach to Modelling Asymmetric Demand Responses to Energy Price Changes David L. Ryan∗ and André Plourde Department of Economics, University of Alberta, Edmonton, AB, Canada T6G 2H4

Abstract Analyses of asymmetric demand responses to price changes have typically been undertaken within a single-equation framework. We generalize an approach involving decompositions of the price variables by extending its treatment to the case of multiple inter-related demands. The resulting systems approach is applied to the case of energy use in the residential sector of Ontario (Canada), alternately using real and relative prices as explanatory variables. The consequences on some standard properties of demand systems (homogeneity and symmetry) are also investigated. In addition, we outline and implement an approach to testing for the existence of asymmetric demand responses in the context of multiple energy sources. In the data set considered, our results suggest that demands for these energy sources were characterized by asymmetric responses to price changes, even after allowing for inter-fuel substitution.

Keywords: asymmetric responses, energy demand, residential sector, systems of equations JEL: Q41, C30

1. Introduction It has long been observed that quantity responses to price changes are not always symmetric, in the sense that equivalent price increases and decreases appear to elicit different sized quantity responses. Early attempts by researchers to model this phenomenon focused on supply responses to agricultural price variations (e.g., Tweeten and Quance, 1969). Initially, the approach that was used involved defining a dummy variable that equaled one if prices increased

∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18010-8

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between period (t − 1) and period t, and zero otherwise. Then in a singleequation specification of a supply function, the explanatory own-price variable was replaced with two terms, one in which price is multiplied by the dummy variable, and the other in which price is multiplied by one minus the dummy variable. In an influential contribution, Wolffram (1971) pointed out problems with this formulation, and suggested an alternative representation that involved separate cumulations of price increases and price decreases. This approach was adopted in subsequent applications – again in agriculture (e.g., Saylor, 1974; Houck, 1977), and was further refined by Traill et al. (1978), who argued that a key issue was whether the price change resulted in a new price that exceeded all previously observed levels. In other words, rather than reflecting differing responses to price increases and price decreases per se, these authors speculated that the observed asymmetry reflects different responses to price changes that lead to the establishment of a new maximum price, and all other price changes. In the early 1980s, as world oil prices fell, interest in modelling asymmetric demand responses to price changes spread to the energy field as the observed demand responses to these oil price decreases appeared to be substantially less than the responses that were observed earlier when these prices had increased. Early attempts to explain the observed sluggish response of oil demand were predominantly based on the models that had been developed for agricultural applications (e.g., Bye, 1986; Watkins and Waverman, 1987; Gately and Rappoport, 1988; Shealy, 1990; Brown and Phillips, 1991). The approach adopted by Dargay (1992) for the first time allowed for separate identification within a single-equation framework of different responses to price increases and price decreases as well as to the maximum price. This approach was further refined by Gately (1992), who demonstrated that these three effects could be captured through a respecification of the price variable. Specifically, current price was represented as the sum of the maximum price to date, cumulative price decreases, and cumulative price increases that do not establish a new maximum. Empirical implementation of this framework allows for straightforward testing of the existence of asymmetric responses to price changes, since evidence that the coefficients of the three price component series are not the same would indicate that there are different responses to variations in these three price components. This approach has been implemented in a series of papers (e.g., Gately, 1993a, 1993b; Hogan, 1993; Dargay and Gately, 1994, 1995; Haas and Schipper, 1998; Gately and Huntington, 2002), which have provided quantitative measures of the extent of the asymmetry that is present in demand responses in a number of different settings, including for various sectors of the economy and across different countries. However, in all cases the focus is on oil as a single fuel (or on total energy use), so that the empirical framework always utilizes a single-equation specification. Proceeding in this manner gives rise to a number of unresolved issues. In particular, the omission of any explicit allowance for inter-fuel substitution means that the effects of changes in the prices of alternative energy forms are

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not taken into consideration. Consequently, it is not possible to determine whether inter-fuel substitution accounts for any of the asymmetric effects detected in studies of oil demand and thus whether any such effects can be found once these substitution possibilities have been taken into account. On a similar note, the issue of whether this type of asymmetry can be identified for other energy sources has yet to be addressed. Finally, the single-equation approach does not take into consideration the inter-related demands for various alternative energy sources when allowing for asymmetric responses to energy price changes. Our primary concern in this paper is to outline an approach to modelling energy demand asymmetries that allows us to address some of these issues. We adopt a modelling strategy that explicitly allows for multiple energy sources, where spending on any one source – including oil – is seen as part of the overall pattern of energy expenditures. Possible asymmetries are captured via a generalization, introduced in Ryan and Plourde (2002), of the price decomposition popularized by Dargay and Gately. The consequences of these decompositions on some standard properties of demand systems (homogeneity and “standard” symmetry) are also discussed and investigated. We then undertake an empirical application of this approach for the case of three energy sources (electricity, natural gas and oil products), using data from the residential sector for the province of Ontario (Canada), and test for the existence of asymmetric demand responses to energy price changes in this dataset. The remainder of the chapter proceeds as follows. Section 2 provides an outline of the price decomposition proposed by Dargay and Gately, and considers its generalization to the case of multiple energy sources, alternately using real and relative prices. Some of the consequences of incorporating the Dargay–Gately price decomposition in multi-input types of frameworks are also highlighted. A description of the model used in the analysis is contained in Section 3, which also discusses the data and the econometric techniques used, and reports on estimation results. Both the procedures used to test for the existence of demand asymmetries and the results obtained are also described in this section. Section 4 concludes the chapter.

2. Modelling Energy Demand Asymmetries: Extending an Established Approach 2.1. From a Single-Equation Approach. . . Our starting point is the single-equation approach commonly used in this literature. Typically, oil demand has been modelled using a linear-in-logarithms functional form, where the per-capita consumption of oil (qoilpct ) depends on a measure of real per-capita income or economic activity (rgdppct ), the real price

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of oil (rpoilt ), and the previous period’s per-capita oil consumption (qoilpct−1 ): ln(qoilpct ) = β1 + β2 ln(rgdppct ) + β3 ln(rpoilt ) + β4 ln(qoilpct−1 ) + et .

(1)

In the approach popularized by Dargay (1992) and Gately (1992), to allow for asymmetric responses the logarithm of the real price term in (1) is replaced by a number of “components” that sum up to the original (logarithmic) price series. Three such component series are generated: the maximum historical values of the natural log of real prices (a nondecreasing series), cumulative sub-maximum recoveries in the natural log of real prices (a nondecreasing, nonnegative series), and cumulative decreases (or cuts) in the natural log of real price (a nonincreasing, nonpositive) series. This data transformation process yields the following breakdown into three component series for the natural logarithm of the real price of oil, ln(rpoilt ): ln(rpoilt ) = max ln(rpoilt ) + cut ln(rpoilt ) + rec ln(rpoilt ) , (2) where:

max ln(rpoilt ) = max ln(rpoil1 ), ln(rpoil2 ), . . . , ln(rpoilt ) , t min 0, max ln(rpoilm−1 ) − ln(rpoilm−1 ) cut ln(rpoilt ) = m=1

− max ln(rpoilm ) − ln(rpoilm ) ,

t max 0, max ln(rpoilm−1 ) − ln(rpoilm−1 ) rec ln(rpoilt ) = m=1

− max ln(rpoilm ) − ln(rpoilm ) .

Based on the decomposition in (2), ln(rpoilt ) would be replaced in the oil demand Equation (1) by the three components max(ln(rpoilt )), cut(ln(rpoilt )) and rec(ln(rpoilt )), and each of these terms would be permitted to have a different coefficient. Thus, the specification in (1) is replaced with: ln(qoilpct ) = β1∗ + β2∗ ln(rgdppct ) ∗ ∗ max ln(rpoilt ) + β3B cut ln(rpoilt ) + β3A ∗ rec ln(rpoilt ) + β4∗ ln(qoilpct−1 ) + et∗ . + β3C

(3)

A test for symmetry of price responses then involves testing whether the coefficients on the three components of ln(rpoilt ) are the same, that is, whether: ∗ ∗ ∗ = β3B = β3C . β3A

(4)

As is recognized in this type of work, the order of the steps in the data transformation process outlined above is important. Specifically, it is necessary to form

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the real price prior to identifying the maximum historical real price, since the ratio of two maximum historical series (here, a nominal oil price and a general price index) does not necessarily yield a nondecreasing series, nor must it yield the maximum historical values of the quotient of the two series concerned (i.e., the real price). In addition, to ensure that the series of cumulative price decreases (a nonpositive series) exists in logarithmic form, and for the three component series to add up to the original real price series, logarithmic transformation must follow division of nominal prices by a price index (such as the consumer price index or the GDP deflator), but precede the decomposition. Of course this data transformation process would be much simpler if nominal prices were used. However, during the time periods covered by most of these studies, nominal energy prices almost continually increased. Consequently, nominal prices for all energy sources tended to equal their maximum historical levels for a majority of the years considered, so that there would be relatively little information contained in series of cumulative recoveries and cumulative cuts derived from the nominal price data. It is presumably for this reason (and possibly the belief that consumers do not suffer from money illusion) that previous examinations of this issue have been based on a decomposition of real rather than nominal prices. As noted earlier, this single-equation approach does not allow an examination of the extent to which inter-fuel substitution may account for any (or all) of the observed asymmetries in oil demand responses, nor is it possible using this method to investigate whether and to what extent asymmetric responses to price changes also characterize the demands for other energy sources.1 To allow us to address these limitations we adopt a systems approach to modelling the inter-related demands for multiple energy sources. 2.2. . . . to a Systems Approach Although the application we consider later to illustrate this method pertains to energy use in the residential sector, the method that we develop here can be readily applied in a variety of contexts. To provide a concrete setting, we initially consider the generic form of a system of demand equations that could be derived in either a consumer setting (such as the residential sector) from, for example, the expenditure function due to Deaton and Muellbauer (1980) which yields the Almost Ideal Demand System (AIDS), or in a production setting (such as the industrial sector) from, for example, the translog cost function. In either of these widely used

1 Note

also that Walker and Wirl (1993), and more recently, Griffin and Shulman (2005) consider an alternative explanation of the observed asymmetric responses to energy price changes, one that is explicitly based on the role of technological change. For an earlier discussion of the possible role of technology (and other factors) in giving rise to these asymmetries, see Sweeney with Fenechel (1986).

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cases (and for a number of other specifications), the resulting system of equations, obtained via Shephard’s lemma as the derivative of the natural logarithm of the expenditure or cost function with respect to the natural logarithm of the ith good or input (here, energy) price, has the following form: sit = αi +

n

γik ln pkt + · · · ,

i = 1, . . . , n,

(5)

k=1

where sit is the expenditure or cost share of the ith energy source in period t, pkt is the price of the kth energy source in period t, αi and γik are parameters to be estimated, “. . . ” represents other terms in the share equations, and the subscripts i and k index the n different energy sources that are considered, so that i, k = 1,. . . , n. When estimating the resulting model, adding-up of the share equations ( i sit = 1 for all t) requires the following parameter restrictions: (6) αi = 1 i

and

γik = 0,

k = 1, . . . , n.

(7)

i

Although the prices in share equations such as (5) are typically expressed in nominal terms, if the demands satisfy the homogeneity condition (that is, they are homogeneous of degree zero in prices and total expenditure or total cost), so that a scaling of all prices and total expenditure or total cost (the denominator of the expenditure or cost share in (5)) does not affect the quantities that are demanded, then in (5): n

γik = 0.

(8)

k=1

pkt In this case the term nk=1 γik ln pkt in (5) can be rewritten as n−1 k=1 γik ln( pnt ). Consequently, any common price index that is used to convert nominal prices to real prices will cancel out when the price ratio terms (pkt /pnt ) are calculated. Hence, the use of real rather than nominal prices is justified provided the homogeneity condition holds. Since the homogeneity condition follows directly from the adding-up condition – that the sum of expenditures on (or costs of) each energy source equals total expenditure on (or cost of) energy – it is typically expected to hold in demand systems like (5) and would often be imposed. A second set of conditions that would be expected to hold in demand systems like (5) is what we refer to as the standard symmetry conditions, namely: γik = γki ,

i, k = 1, . . . , n.

(9)

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These conditions are required foridentification purposes, and follow from the n fact that the price term in (5), k=1 γik ln pkt , is obtained as the derivative with respect to the logarithm of the ith price of a cross-product term such as n 1 n ∗ ln p ln p ) that appears in the cost or expenditure function. ( γ j k j =1 k=1 j k 2 Technically this derivative equals nk=1 21 (γik∗ + γki∗ ) ln pkt , but since γik∗ and γki∗ always appear in the additive form (γik∗ + γki∗ )/2, neither is separately identified, so that this term is simply redefined as γik , and by definition, γik = γki , i, k = 1, . . . , n. In many circumstances these standard symmetry conditions are equivalent to the conditions required for Slutsky symmetry to hold, where the Slutsky symmetry conditions are the requirement that the derivative of the compensated demand for ith good with respect to the j th price is equal to the derivative of the j th compensated demand with respect to the ith price, in other words that the second derivatives of the cost or expenditure function are the same regardless of the order in which the derivatives are taken. However, we emphasize here that (9) are just identification conditions so that they would generally be imposed on (5). For subsequent analysis it is also important to note that due to the adding-up condition (7), the imposition of the standard symmetry conditions in (9) means that the homogeneity condition (8) will automatically be satisfied. 2.3. Real Price Decomposition A direct extension of the data transformation process proposed by Dargay and Gately to the case of n energy sources yields the following breakdown into the three standard component series for the natural logarithm of each real price: ln(pk )t = max(ln pk )t + cut(ln pk )t + rec(ln pk )t ,

(10)

where: max(ln pk )t = max ln(pk )1 , ln(pk )2 , . . . , ln(pk )t , cut(ln pk )t =

rec(ln pk )t =

t

min 0, max(ln pk )m−1 − ln(pk )m−1

m=1

− max(ln pk )m − ln(pk )m ,

t m=1

max 0, max(ln pk )m−1 − ln(pk )m−1 − max(ln pk )m − ln(pk )m

and the subscript k = 1, . . . , n indexes the different energy sources. For reasons similar to the single-fuel case, when more than one energy source is considered, decompositions are initially applied to logarithms of real prices and the same order of steps in the data transformation process will need to be respected for each real price series.

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To implement this decomposition in estimation, the term k γik ln pkt in (5) is replaced by: γik ln pkt = γik max(ln pk )t + γik cut(ln pk )t k

k

+

k

γik rec(ln pk )t .

(11)

k

To allow for the three real price components to have differing effects, we reparameterize the right-hand side of (11) as: (12) δik max(ln pk )t + θik cut(ln pk )t + φik rec(ln pk )t , k

k

k

which, once incorporated into (5), yields share equations that each contain 3n real price terms. When estimating the reparameterized model, adding-up of the share equations would require the following parameter restrictions in place of (7): δik = θik = φik = 0, i, k = 1, . . . , n. (13) i

i

i

Once the real prices of all energy sources are decomposed as in (10), the standard symmetry restrictions in (9) are applied to each price component to yield: δik = δki ,

θik = θki

and

φik = φki , i, k = 1, . . . , n; i = k.

(14)

In this context, testing for the existence of asymmetric demand responses to price changes amounts to testing for the equality of the coefficients on the three components of each real price in (12): δik = θik = φik ,

i, k = 1, . . . , n.

(15)

Under these conditions, no such asymmetry is detected in the data, and (12) reduces to (11). On the other hand, if some of these restrictions are rejected by the data, then the null hypothesis of no asymmetric demand responses to price changes will be rejected for some (or all) of the energy sources. A problem with this procedure is that an appropriate homogeneity restriction cannot be imposed or tested on a system of demand equations that incorporates different coefficients on the different components of logarithms of real prices. The reason why homogeneity cannot hold in general in such a setting derives from the fact that a scaling of all prices does not necessarily imply an equal scaling of each component of these prices. For example, if a real price pi is at its historical maximum, then ln(pi ) = max(ln pi ) and cut(ln pi ) = − rec(ln pi ). Now, a scaling of pi by some factor λ > 1 (and hence an increase in ln(pi ) of ln λ) will cause max(ln pi ) to be increased by ln λ, but will result in no change in either cut(ln pi ) or rec(ln pi ). However, if λ < 1 (so ln λ < 0), then both max(ln pi ) and rec(ln pi )

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will be unchanged, but cut(ln pi ) will be increased by ln λ. Alternatively, if a real price is below its historical maximum, max(ln pi ) will only be affected by scaling in those cases where the scaling factor λ is sufficiently large to cause the price to exceed its historical maximum. Thus, a scaling of real prices by some factor will have differing effects on the components of those prices depending on both the size of the scaling factor and the relationship between actual prices and their historical maximums at the time the scaling is applied. Consequently, unless for each real price the coefficients are the same – in which case there are no asymmetric demand responses to price changes – there are no parametric restrictions that can be imposed to ensure that the quantities demanded will not change in response to such scaling. An additional problem that arises when homogeneity cannot be imposed in this way concerns the standard symmetry conditions. As noted earlier, due to the adding-up condition in systems of equations, in the usual case where there are no asymmetric effects, imposition of the standard symmetry conditions typically requires additional parameter restrictions that are the same as those required for homogeneity, so that homogeneity will also be imposed. When asymmetric effects are allowed, imposing standard symmetry will still require additional parameter restrictions. These restrictions, which will apply to each price component, will be similar in character to the homogeneity condition in (8). Since homogeneity cannot hold or be imposed when there are different coefficients on the components of the logarithms of real prices, it is unlikely that the standard symmetry conditions will be satisfied in this case. 2.4. Relative Price Decomposition The argument developed above shows that homogeneity can neither be imposed nor tested on a system of demand equations if the Dargay–Gately decomposition is applied to real prices. However, since homogeneity is a direct implication of the budget constraint, and since failure to allow this property to hold may affect the results of tests for the existence of asymmetric price responses, it is desirable to identify an alternative breakdown of the price series that will allow for asymmetric price responses and permit homogeneity to be satisfied, so that the resulting demands can be consistent with a constrained optimization problem for a representative consumer (or firm). An approach that satisfies these requirements involves basing the decomposition in (10) on logarithms of relative prices. The use of relative price data also avoids problems with choosing between a decomposition based on real prices and one based on nominal prices, since any common price index used to deflate nominal series into real terms would cancel when relative prices are formed. In this case, the logarithm of each relative price is decomposed into three series as follows: pk = max(ln Pkr )t + cut(ln Pkr )t + rec(ln Pkr )t , ln (16) pr t

192

where:

D.L. Ryan and A. Plourde

pk ln Pkr = ln , k, r = 1, . . . , n; k = r, pr max(ln Pkr )t = max ln(Pkr )1 , ln(Pkr )2 , . . . , ln(Pkr )t , cut(ln Pkr )t =

rec(ln Pkr )t =

t

min 0, max(ln Pkr )m−1 − ln(Pkr )m−1

m=1

− max(ln Pkr )m − ln(Pkr )m ,

t

max 0, max(ln Pkr )m−1 − ln(Pkr )m−1

m=1

− max(ln Pkr )m − ln(Pkr )m .

Recall that with (10), homogeneity is not satisfied and cannot be imposed or tested unless no asymmetric demand responses to price changes are permitted. With (16), however, a scaling of all prices by a common factor will have no effect on relative prices and hence on the price component series. Homogeneity is automatically satisfied, and thus cannot be tested, using relative price decompositions. Since, as noted earlier, the presence or absence of homogeneity may affect the results of tests for asymmetric price responses, the empirical application that follows uses both decompositions alternately. To facilitate use of the relative price decomposition in (16), we begin by imposing the homogeneity condition (8) on the cost or expenditure function, and hence on the derived share equations, so that the price term in (5), k γik ln pkt , can be written as: pkt γik ln pkt = γik ln (17) , pnt k

k=n

and only relative prices appear in the share equations. For each of these (n − 1) relative price terms, (16) could now be substituted, yielding 3(n − 1) relative price components in each share equation. Unfortunately, during estimation, an additional complication arises: when one share equation is omitted due to the adding-up conditions, the results obtained will not be invariant to the particular equation that is left out. Essentially, as is shown below, this is caused by the omission of relevant information during estimation, and the effect of this omission differs for different equations. A respecification of (17) indicates the relevant information that is omitted, and provides a method for ensuring that the estimation results will be invariant to the omitted equation. Specifically, since the following relationship holds for the (undecomposed) relative prices: pj t pj t pkt = ln − ln , ln (18) pkt pnt pnt

Modelling Asymmetric Demand Responses

the right-hand side of (17) can be rewritten as: pj t pkt γik ln γj∗ik ln = , pnt pkt k=n

193

(19)

j =n k>j

∗i ∗i ∗i where γik = j =k γkj and γkj = −γj k . In general there would be no advantage to using the expression on the right-hand side of (19), which involves all n(n − 1)/2 relative prices, since – as shown by the left-hand side of (19) – only (n − 1) of these relative price terms are independent. Hence, there are n(n − 1)/2 − (n − 1) = (n − 1)(n − 1)/2 redundant relative price terms. As a result, in any share equation, the γj∗ik parameters would not be identified, and only the (n − 1) γik parameters could be estimated. However, the linear relationships (18) that hold among the (undecomposed) relative prices do not necessarily hold among the various components of these prices. For example, in the decomposition of relative prices in (16), where Pkr = (pk /pr ): max(ln Pkr )t = max(ln Pkn )t − max(ln Prn )t .

(20)

Thus, for max(ln Pkr )t , and similarly for cut(ln Pkr )t and rec(ln Pkr )t , all n(n − 1)/2 relative price component terms are independent. However, given (16) and (18), not all these 3n(n − 1)/2 components are jointly independent. Specifically, (16) must hold for each relative price, and (18) must hold among the undecomposed relative prices on the left-hand side of (16). Thus, there remain (n − 1)(n − 2)/2 redundant terms in the 3n(n − 1)/2 relative price components, so that only (n − 1)(n + 1) relative price components are independent. Therefore, to ensure that the estimation results do not depend on the particular equation that is omitted, it is necessary to include all of these (n − 1)(n + 1) relative price components in each share equation (rather than just the 3(n − 1) components that would result if the decomposition in (16) was applied only to the (n − 1) relative prices in (17)). In our empirical application, where n = 3, there are three possible relative price terms (i.e., ln(P12 )t , ln(P13 )t and ln(P23 )t ), so that eight of the nine relative price components are independent and are included in each equation. To incorporate (16), the term k γik ln pkt in (5) is replaced by: γik ln pkt = γjik max(ln Pj k )t + γjik cut(ln Pj k )t k

j =n k>j

+

γjin rec(ln Pj n )t .

j =n k>j

(21)

j =n

Here, the redundant terms are omitted by including only (n − 1) rec(ln Pj n )t components in each equation and the coefficients are redefined appropriately. To allow the different components to have differing effects, we reparameterize the right-

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hand side of (21) as: δji k max(ln Pj k )t + θji k cut(ln Pj k )t j =n k>j

j =n k>j

+

φji n rec(ln Pj n )t .

(22)

j =n

In estimating the model (5) that contains (22), adding-up of the share equations would require the following parameter restrictions in place of (7): δji k = θji k = φji n = 0, i

i

i

j = 1, . . . , (n − 1), k = 2, . . . , n, k > j.

(23)

2.5. Relative Price Decomposition: Standard Symmetry Conditions As discussed earlier, homogeneity is automatically satisfied with this formulation since all prices are expressed in relative terms. However, due to the inclusion of the additional relative price components, determination of the appropriate standard symmetry conditions (corresponding to (9)) is more complex. We begin with the relationship between terms involving nominal prices and terms involving relative prices. For the case where n = 3, the right-hand side of (19) can be written as:2 pj t ∗i γj k ln pkt j =n k>j p1 p1 p2 ∗i ∗i ∗i = γ12 ln ln ln + γ13 + γ23 p2 p3 p3 ∗i ∗i p1 p2 ∗i ∗i = γ12 ln ln + γ13 − γ12 + γ23 p3 p3 ∗i ∗i ∗i ∗i ∗i ∗i ln p1 + γ23 ln p2 + −γ13 ln p3 + γ13 − γ12 − γ23 = γ12 = γi1 ln p1 + γi2 ln p2 + γi3 ln p3 (24) (where j γij = 0) so that γij = k=j γj∗ik , where γj∗ik = −γkj∗i . Thus, with the third share equation omitted due to the adding-up conditions, the standard symmetry condition γ12 = γ21 becomes (in the absence of any relative price decomposition): ∗1 ∗2 ∗1 ∗2 γ23 − γ12 (25) = γ12 + γ13 .

2 Although

any n.

we develop the case where n = 3, the analysis can be readily generalized to

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195

With the price decomposition (16) applied to the (natural logarithms of) relative prices, and allowing for different coefficients on the different components, but prior to omitting redundant terms as in (22), we have: δj∗ik maxj k + θj∗ik cutj k + φj∗ik recj k j =n k>j

=

j =n k>j

j =n k>j

∗i ∗i ∗i δ12 max12 + δ13 max13 + δ23 max23 ∗i ∗i ∗i + θ12 cut12 + θ13 cut13 + θ23 cut23 ∗i ∗i ∗i + φ12 rec12 + φ13 rec13 + φ23 rec23 ,

(26)

where, to simplify the notation, max(ln Pj k ) = max(ln(pj /pk )) is written as maxj k , and similarly for cut(ln Pj k ) and rec(ln Pj k ). Using (16) and (18): (max12 + cut12 + rec12 ) = (max13 + cut13 + rec13 ) − (max23 + cut23 + rec23 )

(27)

so that: rec12 = (max13 + cut13 + rec13 ) − (max23 + cut23 + rec23 ) − (max12 + cut12 ). Thus, we can rewrite the right-hand side of (26) as: ∗i ∗i ∗i ∗i ∗i ∗i δ12 − φ12 max12 + δ13 max13 + δ23 max23 + φ12 − φ12 ∗i ∗i ∗i ∗i ∗i ∗i + θ12 − φ12 cut12 + θ13 + φ12 cut13 + θ23 − φ12 cut23 ∗i ∗i ∗i ∗i rec13 + φ23 rec23 , + φ13 + φ12 − φ12

(28)

(29)

which can be redefined as: i i i δ12 max12 + δ13 max13 + δ23 max23 i i i + θ12 cut12 + θ13 cut13 + θ23 cut23 i i + φ13 rec13 + φ23 rec23 ,

(30)

as in (22). Now, the standard symmetry conditions are obtained by applying (25) to the coefficients on each of the relative price components in (26). Thus, we require: ∗1 ∗2 ∗1 ∗2 δ23 − δ12 (31.1) = δ12 + δ13 , ∗1 ∗2 ∗1 ∗2 θ23 − θ12 = θ12 + θ13 , (31.2) ∗1 ∗1 ∗2 ∗2 φ23 − φ12 (31.3) = φ12 . + φ13 i = (φ ∗i − φ ∗i ) and φ i = (φ ∗i + φ ∗i ), (31.3) is Since, from (29) and (30), φ23 23 12 13 13 12 satisfied if: 1 2 = φ13 . φ23

(32.1)

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i = (θ ∗i − φ ∗i ), θ i = (θ ∗i − φ ∗i ) and θ i = (θ ∗i + φ ∗i ), so that Similarly, θ23 23 12 12 12 12 13 13 12 i i ∗i − θ ∗i ), while (θ i + θ i ) = (θ ∗i + θ ∗i ). Thus, (31.2) is satisfied (θ23 − θ12 ) = (θ23 12 12 13 12 13 if: 1 2 1 2 θ23 − θ12 (32.2) = θ12 + θ13 .

Similar relationships hold among the elements of δji k and δj∗ik , so that (31.1) is satisfied if: 1 2 1 2 δ23 − δ12 (32.3) = δ12 + δ13 . Thus, when n = 3, the standard symmetry conditions in the case where asymmetric demand responses are allowed in a model based on relative prices are provided by (32.1)–(32.3). 2.6. Conditions for the Absence of Asymmetric Demand Responses to Relative Price Changes If there are no such asymmetries, the coefficients on maxj k , cutj k and recj k in (26) are the same for each relative price. From (26), this requires the following: ∗i ∗i ∗i δ12 = θ12 = φ12 ,

(33.1)

∗i ∗i ∗i δ13 = θ13 = φ13 ,

(33.2)

∗i δ23

(33.3)

=

∗i θ23

=

∗i φ23 .

From (29) and (30), it can be seen that (33.1) is satisfied if: i i δ12 = θ12 = 0,

(34.1)

while (33.2) and (33.3) will be satisfied if: i i i δ13 = θ13 = φ13 ,

(34.2)

i i i = θ23 = φ23 . δ23

(34.3)

Therefore, when n = 3, (34.1)–(34.3) jointly provide the restrictions under which asymmetric demand responses to relative price changes are absent.

3. Empirical Analysis The purpose of our empirical analysis is threefold. First, using the models outlined in the previous sections, which take account of multiple energy sources and the possibility of inter-fuel substitution, we test for evidence of asymmetric demand responses to energy price changes. Second, we investigate the sensitivity of our

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197

results concerning asymmetric price responses to the choice of real rather than relative prices in the energy price decomposition, to the imposition of the standard symmetry conditions, and to variations in the stochastic specification and estimation method. Third, we examine the nature of any asymmetric effects that are detected, in order to determine the magnitudes and directions of these effects, and assess their potential implications for energy demand analysis, and to evaluate the importance of accounting for inter-fuel substitution. 3.1. A Model of Residential Energy Demand Since our empirical application pertains to the residential sector, we use the expenditure function due to Deaton and Muellbauer (1980), which yields the Almost Ideal Demand System (AIDS): ln Et (ut , pt ) = α0 + αj ln pj t j

+

1 2

j

k

γij ln pj t ln pkt + ut β0

β

pj tj ,

(35)

j

where ut is household utility in period t, pt = (p1t , . . . , pnt ) is the vector of prices prevailing in period t, and αj , βj , γj k are parameters. Here, expenditures on energy are treated as forming a separable sub-group, so that the prices of energy sources are the only ones relevant to the analysis, while the appropriate expenditure measure is total (per-household) energy expenditure. Based on the assumption of expenditure-minimizing behavior on the part of households, a system of share equations describing residential demands for the various energy sources is derived from (35) using Shephard’s lemma and equating Et (ut , pt ) with observed per-household energy expenditures. To simplify the empirical analysis, the nonlinear price index that appears in these equations is replaced with the Stone price index, ln Pt∗ = k skt ln pkt , where skt is the expenditure share for the kth energy source in period t. This yields the popular Linear Approximation to the Almost Ideal Demand System (or LAIDS), which has been estimated frequently in empirical demand applications (Buse, 1994). As is often the case in studies of residential energy demand, the extent of weatherinduced need for space heating and cooling (e.g., Dunstan and Schmidt, 1988) is incorporated in the model through the use of heating degree-days and cooling degree-days as additional explanatory variables. Analysis of residential energy demand must also take into account the fact that changes in decisions to use particular energy sources cannot be enacted instantaneously. The most common method to allow for this feature is via a partial adjustment mechanism. Here, expenditure shares in the current period are assumed to adjust only partially to their desired level from the previous (last period)

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level: st − st−1 = (st∗ − st−1 ),

(36)

where st = (s1t , . . . , snt ) is a vector of the expenditure shares of the n different energy sources in period t, st∗ is a vector of desired shares derived from expenditure-minimizing behavior, and is an (n × n) matrix of adjustment coefficients. In the simplest form of this specification is a constant, diagonal matrix, so that the adjustment for each energy source depends only on its own desired and previous levels, and – since both the actual and desired shares sum to unity – the speed of adjustment is restricted to be the same across all energy sources. To avoid this limitation, the general form in (36) is used, although only (n−1) lagged shares will appear in each equation, since the lagged shares also sum to unity.3 The basic model used in estimation, prior to the incorporation of the price components as in (10) or (16), is thus the system of expenditure share equations corresponding to the LAIDS model, supplemented with two weather-related variables and a partial adjustment specification. The share equations corresponding to this formulation have the following form:

Et γik ln pkt + βi ln ∗ sit = αi + Pt k λij sj,t−1 , + ci ln hddt + di ln cddt +

(37)

j

where sit is the expenditure share of the ith energy source in period t, pkt is the price of the kth energy source in period t, Et is the observed per-household expenditure on residential energy in period t, Pt∗ is the Stone price index, defined as: ln Pt∗ = k skt ln pkt , hddt is heating degree-days in period t (degree-days below 18 ◦ C), cddt is cooling-degree-days in period t (degree-days above 18 ◦ C) and αi , λik , ci , di , λij are parameters to be estimated, where i, k = 1, . . . , n; j = 1, . . . , (n − 1).

3A

number of criticisms have been leveled at the use of partial adjustment formulations in energy demand analysis. Berndt et al. (1981) indicate that partial adjustment mechanisms are not based on optimizing behavior, and that the resulting estimated long-run elasticities do not necessarily exceed their corresponding short-run values. Hogan (1989) also identifies a potential misspecification that results when the partial adjustment process is expressed in terms of expenditure shares rather than quantities. In the alternative specification that Hogan (1989) suggests, the coefficients of the lagged shares are themselves functions of prices. Unfortunately, like the model with endogenous adjustment that Berndt et al. (1981) propose, this specification would prove problematic with the price decompositions in (10) or (16) that are incorporated in our empirical analysis.

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199

When estimating the resulting model, adding-up of the share equations requires the following parameter restrictions: αi = 1, (38) i

i

βi =

i

ci =

di = 0,

(39)

i

λij = 0,

i = 1, . . . , n, j = 1, . . . , (n − 1),

(40)

γik = 0,

k = 1, . . . , n.

(41)

i

and

i

This basic model is now modified by incorporating either the decomposition based on logarithms of real prices, as in (10), or the decomposition based on logarithms of relative prices, as in (16). In both cases, the price components are used in the term k γik ln pkt in (37). Although prices also appear in the Stone index, ln Pt∗ , in (37), since the simplification obtained by using the LAIDS model is achieved by taking this index as given, and since this index involves no estimable coefficients, we do not consider asymmetric responses involving the prices that make up this term. Note that when the real price decomposition is used, (41) is replaced by (13), while (23) replaces (41) when the relative price decomposition is used. 3.2. Data The model is estimated using data on energy use by the residential sector in Ontario, Canada’s most populous province, for the period 1962 to 1994. This is an interesting period over which to study asymmetric demand responses since it incorporates sub-periods with sharp and sustained increases and decreases in world oil prices and in North American natural gas prices. Figure 1 presents real, aftertax prices to final users for the three energy sources (or fuels) considered, namely electricity, natural gas and oil products.4 Information on the share of residential energy expenditures attributable to each energy source, the dependent variables in the estimation work that follows, is presented in Figure 2. Figures 1 and 2 jointly reveal that residential energy usage in Ontario exhibits the pattern that has led researchers elsewhere to suspect the existence of asymmetries in oil demand. Specifically, while increases in the real price of oil products were accompanied by reductions in consumption until the mid-1980s, the ensuing drop in real prices

4 The

consumer price index (CPI) for Ontario serves as deflator. See the data appendix for a more detailed discussion of the series used in estimation.

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Figure 1: Real Energy Prices (Residential Sector, Ontario – 1962–1994).

Figure 2: Energy Expenditure Shares (Residential Sector, Ontario – 1962–1994). did not lead to a resurgence in oil consumption levels. Indeed, since 1985, the expenditure share of oil products has fallen faster than the real, after-tax price facing final users of oil products.

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201

3.3. Stochastic Specification and Estimation Procedures To complete the specification of the system of share Equations (37), incorporating either the real price decomposition as in (11) and (12) or the relative price decomposition as in (21) and (22), along with the associated parameter restrictions, additive error terms eit are appended to each share equation. These error terms are allowed to follow a first-order autoregressive process, eit = ρei,t−1 + εit , where the vector of error terms, εt = (ε1t , ε2t , ε3t ) has a multivariate normal distribution with mean zero and a covariance matrix that is constant across time. As shown by Berndt and Savin (1975), due to the adding-up conditions it is necessary that the autocorrelation coefficient be the same in each equation. We estimate the system of share equations for the various formulations of the model both with and without the autocorrelation parameter included in order to assess the robustness or sensitivity of the results concerning asymmetric demand responses to this aspect of the specification. Since the covariance matrix is singular, when autocorrelation is not included we drop an equation and maximize the likelihood corresponding to each specification of the system of share equations. When autocorrelation is included, due to the appearance of lagged shares among the explanatory variables, iterative three-stage least squares (I3SLS) estimation is used. Here, all the variables act as their own instruments except for the lagged shares. Additional instruments included are the proportion of personal income that comprises wages, salaries and other supplementary income; and the proportion of provincial GDP that the government spends on goods and services. As with maximum likelihood (ML) estimation, I3SLS takes account of the covariance structure between the error terms in different equations. With the specifications and parameter restrictions that are used here, both estimation methods will yield parameter estimates that are invariant to the particular equation that is omitted from the system that is estimated. In the estimation work that follows, we omit the electricity equation. 3.4. Homogeneity and Standard Symmetry As noted earlier, in view of the adding-up conditions that apply to this system of share equations, with no price decomposition imposition of the standard symmetry conditions implies that the homogeneity conditions will also be satisfied.5 Thus, these symmetry conditions cannot be tested without also (jointly) testing

5 Note

that in the case of the real price specification, the standard symmetry conditions are provided by (9) when there is no price decomposition, and by (14) when each real price is broken down into three components. Similarly, for the relative-price specification with three energy sources, (25) provides the standard symmetry restrictions in the absence of price decomposition, and (32.1)–(32.3) do so when prices are decomposed according to the approach used in this chapter.

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Table 1: Tests of Standard Symmetry and Homogeneity Conditions Specification Relative prices Standard symmetry Auto (3SLS) No auto (ML) Real prices Standard symmetry Auto (3SLS) No auto (ML) Homogeneity Auto (3SLS) No auto (ML)

No price decomposition

Price decomposition

W(1) = 1.14 LR(1) = 3.84∗ W(1) = 4.07∗

W(3) = 1.63 LR(3) = 2.13 W(3) = 2.20

W(3) = 24.38∗∗ LR(3) = 15.49∗∗ W(3) = 50.67∗∗

W(9) = 101.27∗∗ LR(9) = 19.00∗ W(9) = 33.92∗∗

W(2) = 21.51∗∗ LR(2) = 11.64∗∗ W(2) = 37.32∗∗

Notes: 1. LR refers to the Likelihood Ratio test statistic, while W refers to the Wald test statistic. 2. Numbers in parentheses are the degrees of freedom for that test. ∗ indicate significance at the 1% level. ∗∗ indicate significance at the 5% level.

homogeneity. However, for the model involving real prices, homogeneity cannot hold when the price decomposition is incorporated in the specification. Rather, in this case, some homogeneity-like parameter restrictions are required when the standard symmetry conditions are imposed. In view of this requirement, it is perhaps not surprising that, as reported in Table 1, the standard symmetry restrictions are clearly rejected when the real price decomposition is incorporated in the model. This result holds both when the errors are allowed to follow an autoregressive process and I3SLS estimation is used, and when the errors are assumed to be well behaved and estimation is by ML. Interestingly, this rejection of the standard symmetry conditions also holds for both estimation procedures when real prices are used with no price decomposition. When using relative prices, the standard symmetry conditions are satisfied when the relative price decomposition is incorporated in the specification, and this result also holds at the 1% significance level when no price decomposition is used. Since the use of relative prices with no decomposition is equivalent to imposing homogeneity on the corresponding model that uses real prices, a Likelihood Ratio (LR) test of the homogeneity conditions in the real price model can be based on a comparison of these two models. Alternatively, these homogeneity conditions can be evaluated based on a Wald (W) test of parameter restrictions in the real price model with no price decomposition where the standard symmetry conditions

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203

are not imposed. The results for these tests, reported in the bottom section of Table 1, indicate that for both estimation procedures the homogeneity restrictions are rejected in the real price model. In view of these findings, in the remainder of our analysis the standard symmetry conditions are maintained when using the relative price version of the model, but are omitted when real prices are used. Similarly, homogeneity is not imposed when using the model based on real prices, but it is automatically imposed (and therefore cannot be tested) in the model that uses relative prices. A comparison of the results concerning asymmetry of demand responses in the real price and relative price models will thus provide some evidence as to the sensitivity of these findings to the relatively common practice of imposing these homogeneity and standard symmetry restrictions in demand system estimation. 3.5. Tests of Asymmetric Demand Responses Table 2 reports tests of the null hypothesis that there is no asymmetric response in Ontario’s residential sector to changes in the price of any energy source (basically, that the appropriate model is (37)), while the alternative hypothesis is one of asymmetric responses to all prices. As noted earlier, this amounts to tests of the restrictions in (15) for the real-price specification, and of those in (34.1) to (34.3) when relative prices are considered. The results are unambiguous: the null hypothesis is rejected at the 1% level for both types of specification. Further, this result holds both in models incorporating a first-order autoregressive process in the error terms when I3SLS estimation is used, and in models where this autoregressive process is omitted and estimation is by ML. All in all, this suggests that specifications allowing for asymmetric responses to the prices of all energy sources considered are preferable to those allowing for no such effects. The results in Table 2 address the issue of asymmetric responses to price changes only from a system-wide perspective, testing models in which no asymmetric responses are allowed against models that allow asymmetric responses to the prices of all energy sources (or fuels) that are considered. The tests underlying the results in Table 3 are designed to complement this view with a fuel-by-fuel assessment. Since the autocorrelation parameter is significant whether the real price Table 2: Tests of Asymmetric Price Responses Specification

Relative prices (test of (34.1)–(34.3))

Real prices (test of (15))

Auto (3SLS) No auto (ML)

W(10) = 130.54∗∗ LR(10) = 59.43∗∗ W(10) = 112.38∗∗

W(12) = 103.04∗∗ LR(12) = 27.14∗∗ W(12) = 39.79∗∗

∗∗ indicate significance at the 5% level.

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Table 3: Tests of Asymmetric Price Responses for Different Fuels and Fuel Prices in Models Allowing for Autocorrelation Equation

Relative prices: Test of restrictions in (34.1)–(34.3)

Real prices: Test of restrictions in (15)

Oil price/ Gas price/ Oil price/ All rela- Oil price Gas price Elec price All prices Elec price Elec price Gas price tive prices (df = 2) (df = 2) (df = 2) (df = 6) (df = 2) (df = 2) (df = 2) (df = 6) Oil products 2.07 Natural gas 33.40∗∗ Electricity 13.74∗∗

4.10 63.06∗∗ 17.49∗∗

0.67 14.18∗∗ 4.95

27.46∗∗ 0.06 111.67∗∗ 26.26∗∗ 32.49∗∗ 6.12∗

2.71 17.15∗∗ 9.51∗∗

4.76 29.21∗∗ 15.45∗∗

59.50∗∗ 45.14∗∗ 40.12∗∗

Notes: 1. Numbers in the table are Wald statistics for a test of no asymmetric price response of the fuel in each row with respect to the fuel price in each column. 2. df indicates the degrees of freedom for each test. 3. Results for electricity are obtained indirectly from the estimates for the other two equations. ∗ indicate significance at the 1% level. ∗∗ indicate significance at the 5% level.

or relative price decomposition is used (see Table 4a), for conciseness the results in Table 3 refer to the specifications that allow for autocorrelation and where estimation is by I3SLS. Each cell in the table gives the Wald statistic for a test of no asymmetric price responses for the energy source in each row with respects to the price in each column. The general pattern of results is quite similar for both real and relative price specifications. These suggest that demand for both natural gas and electricity by Ontario’s residential sector was characterized by price asymmetries with respect to changes in the real prices of each energy source (and in each relative price) individually and as a whole.6 Results for the oil product expenditure share equation are less straightforward. Here, in both real and relative price specifications, it is not possible to reject the hypothesis of no asymmetric response of oil-product consumption to changes in any of the prices individually. However, in both specifications the hypothesis of no asymmetry is rejected in favor of the alternative that there are asymmetric responses of oil-product consumption to changes in all (real or relative) prices (last column of Table 3). This would tend to suggest that there may be important inter-relationships among the variables, and that a fuel-byfuel/equation-by-equation testing strategy does not take into consideration some of the information used in system-wide estimation and testing.

6 The

only exception is with the relative price specification of the electricity equation, in which no asymmetry is detected for the price of oil products relative to that of natural gas.

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205

3.6. Parameter Estimates In view of our finding of asymmetric demand responses to price changes, the next issue we consider concerns the magnitudes and directions of these asymmetric effects, and their potential implications for energy demand analysis. We first consider the parameter estimates for two specifications (using real prices and using relative prices) both with and without the price decomposition included, as presented in Table 4a and 4b. Here the subscripts 1, 2 and 3 refer to oil products, natural gas and electricity, respectively. Columns [1] and [2] of Table 4 report the results of estimations using real prices, while columns [3] and [4] report those of estimations using relative prices. There is no decomposition of the price variables in the estimations underlying the results presented in columns [1] and [3], but columns [2] and [4] report the results of estimations incorporating the price decompositions in (11) and (12), and in (21) and (22), respectively. Note that in all reported cases, the estimated coefficients in the electricity equation are obtained indirectly using the parameter restrictions. When the relative-price specification

Table 4a: Parameter Estimates Parameter

α1 α2 β1 β2 c1 c2 d1 d2 λ11 λ12 λ21 λ22 ρ

Using real prices

Using relative prices

No decomposition [1]

Decomposition [2]

No decomposition [3]

Decomposition [4]

−0.120 −0.989∗∗ 0.110 −0.114∗∗ 0.064 0.106∗∗ −0.022∗∗ 0.004 0.807∗∗ −0.041 −0.066 0.061 0.618∗∗

0.171 −2.163∗∗ 0.139 −0.131∗∗ 0.035 0.323∗∗ 0.003 0.001 0.479∗∗ −0.177 −0.049 0.722∗∗ −0.767∗∗

0.486 −1.501∗∗ 0.177∗ −0.234∗∗ 0.002 0.178∗∗ −0.021∗∗ 0.004 0.739∗∗ −0.051 −0.169∗∗ 0.085 0.395

0.622 −2.118∗∗ 0.108∗ −0.259∗∗ −0.006 0.213∗∗ −0.022∗∗ 0.002 0.209 −0.218 0.082 0.058 0.504∗∗

Notes: 1. Subscripts 1, 2 and 3 refer to oil products, natural gas and electricity, respectively. 2. All estimated price coefficients in the electricity equation, as well as those on p1 /p3 in the gas equation when relative prices are used, are obtained indirectly. 3. Reported results refer to the case where the standard symmetry conditions are imposed when using relative prices but not when using real prices. ∗ indicate significance at the 1% level. ∗∗ indicate significance at the 5% level.

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Table 4b: Price Coefficients Using real prices No decomp. [1] γ

Oil products p1 0.033 p2 −0.075 p3 0.002

Using relative prices

Decomposition [2] δ (max)

[2] θ (rec)

No decomp. [3] γ

[3] φ (cut)

0.118∗ 0.139∗ 0.081 −0.164 −0.014 −0.088 0.044 0.101 −0.121

p1 /p3 −0.026 p2 /p3 0.006 p1 /p2

Natural gas 0.136∗∗ −0.089∗∗ 0.602∗∗ p1 /p3 p1 0.011 p2 0.148∗∗ −0.025 0.150∗∗ 0.173∗∗ p2 /p3 p3 −0.036 −0.332∗∗ 0.041 −0.306∗∗ p1 /p2 Electricity p1 −0.044 p2 −0.073 p3 −0.035

Decomposition [4] δ (max) −0.018 −0.067 0.010

[4] θ (rec)

[4] φ (cut)

0.066 0.042 0.048

−0.138 −0.050

0.006 0.017 0.209∗∗ −0.050 0.135∗∗ −0.062 −0.034 0.282∗∗ −0.094∗∗ −0.215∗∗

−0.254∗∗ −0.049 −0.683∗∗ p1 /p3 0.020 0.189 −0.136∗ −0.085 p2 /p3 −0.141∗∗ 0.288∗∗ −0.142∗ 0.426∗∗ p1 /p2

0.001 0.128 0.084

−0.275∗∗ 0.188 −0.008 −0.232∗∗ 0.167∗

Notes: 1. Subscripts 1, 2 and 3 refer to oil products, natural gas and electricity, respectively. 2. All estimated price coefficients in the electricity equation, as well as those on p1 /p3 in the gas equation when relative prices are used, are obtained indirectly. 3. Reported results refer to the case where the standard symmetry conditions are imposed when using relative prices but not when using real prices. ∗ indicate significance at the 1% level. ∗∗ indicate significance at the 5% level.

is used, the same is true for the estimated coefficients on the (log) price of oil products relative to that of electricity (p1 /p3 ) in the natural gas equation. While the parameter estimates are of little direct interest, a number of observations seem in order. First, in all cases except the relative price specification with no price decomposition (column [3]), there is evidence of autocorrelation at standard levels of statistical significance. Second, especially with the real price specification, the additional flexibility offered by the price decompositions appears to be welcome by the data. In particular, a much greater proportion of the coefficients in the real price model are statistically significant at a 5% level or higher when the three components of each real price are allowed to have different coefficients. Third, in contrast to the other two equations, the price coefficients in the oil equation tend to be mainly insignificant. This may account for our earlier nonrejection of the hypothesis that there is no asymmetric response of oil product consumption to changes in any of the prices individually. Finally, a comparison of columns [1] and [3] indicates the effects of imposing homogeneity and standard symmetry

Modelling Asymmetric Demand Responses

207

restrictions. Although, as shown in Table 1, these restrictions are rejected, they appear to have only minor effects on the statistically significant parameter estimates. 3.7. Price Elasticities Since the parameter estimates themselves provide relatively little information about either the direction of the asymmetries detected in the data or their relative importance, we turn to a consideration of various price elasticities that are based on these parameter estimates. From this analysis we wish to determine, in particular, whether demand is more responsive to price increases or price decreases, and by how much demand responses differ when prices increase rather than decrease. In the absence of any price decomposition, the (short-run) price elasticities for the various energy sources (using either the real or relative price specifications) can be calculated from the estimated parameters using the relationship:7 ηik =

γik βi sk − − ωik , si si

(42)

where ωik = 1 if i = k, and ωik = 0 otherwise. Since these elasticities depend on the shares (and are evaluated using estimated values of these shares), they differ for each observation. Here we evaluate the elasticities for the years 1964, 1974, 1984 and 1994. These years correspond to different phases in the evolution of energy prices. As Figure 1 shows, in 1964 the real prices of all energy sources that we consider were either constant or falling. Ten years later marks the trough in real electricity prices, and the beginning of a decade of increases in all real energy prices. In 1984, real prices of natural gas and oil products are either at or close to their highest point during our sample (1962 to 1994), while the real price of electricity had leveled off after many years of increases. Finally, by 1994, prices of both natural gas and oil products had fallen from their peak and had experienced a long period during which they were relatively constant. Real electricity prices, on the other hand, which increased in the early 1990s appeared to have stabilized by the end of our sample period. 3.7.1. Responses to Changes in Real Prices When the model using real prices includes the decomposition in (11) and (12), the elasticities will differ according to whether the price changes causes a change in max, rec or cut. We consider each of these alternatives separately, and in each case

7 See

Buse (1994) for an evaluation of the various possible elasticity expressions that can be used with the LAIDS model. This expression is the most widely used and, according to Buse’s results, is marginally the best.

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D.L. Ryan and A. Plourde

evaluate the elasticity as if the price change only causes a change in that component of price. Thus, for example, a change in the real price of energy source k that leads to the establishment of a new maximum is viewed as causing max(ln pk ) to change by an amount equal to the logarithm of this change in real price. Similarly, a change in the real price of energy source k that causes rec(ln pk ) to change (since that price is currently below its historical maximum) is viewed as changing rec(ln pk ) by the logarithm of this change in real price.8 Hence, the elasticities corresponding to an increase in either max or rec (due to an increase in the real price) or a decrease in cut (due to a fall in the real price) can be obtained by alternately substituting δik , φik or −θik , respectively, in place of γik in (42). Elasticities for the model estimated using real prices, both with and without the real price decomposition incorporated, are presented in Table 5. The ownprice elasticity of the demand for oil products tends to be smaller (in absolute value) when price decomposition is allowed, whether max, rec or cut is used as the basis for comparison. Furthermore, contrary to the evidence provided by much of the existing literature on the responsiveness of oil demand to own-price changes (e.g., Dargay and Gately, 1994), our results do not support the existence of a stronger demand response to price recoveries and to changes in the maximum price, than to price cuts – point estimates of elasticities are quite similar for all three components. However, the demand for oil products in Ontario’s residential sector becomes less responsive to own-price changes, whether or not the real price is decomposed, as time progresses, consistent with results for nontransport energy demand for various countries reported in Ryan and Plourde (2002). This suggests that any change in oil price has a smaller proportional effect on the demand for oil products, the closer it occurs to the end of our sample period. On the other hand, our results provide some support for the existence of asymmetric responses of oil demand to changes in natural gas prices. In particular, it appears that the demand for oil products is more responsive to increases (than to decreases) in the price of natural gas, and that changes in the maximum historical price exert a slightly larger influence. And this pattern is maintained throughout our sample period. Finally, electricity prices do not appear to have much of an effect on oil demand in our sample. This is not too surprising since oil products and natural gas compete for space- and water-heating markets, with little electricity penetration for these uses in Ontario. Electricity, on the other hand, is used for lighting and to power many consumer durables (such as televisions and microwave ovens), where all of the available energy-using equipment is specific to this energy source. Changes in real oil prices do appear to give rise to asymmetric effects, but in Ontario’s residential sector these seem largely felt by the demand for natural gas

8 It

is possible that a sufficiently large increase could cause the real price to exceed its previous maximum, but this possibility can be ignored when marginal price changes are considered.

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209

Table 5: Estimated Elasticities – Real Price Model Year

Effect of oil price on demand for: Oil −1.024∗∗

Gas

Elec

Effect of natural gas price on demand for: Oil

0.213∗ −0.130

Gas

Elec

Oil

Gas

Elec

−0.314∗∗ −0.210

−0.096

0.016

−0.907∗∗

1964 −0.268 ↑price ↑ max −0.860∗∗ 0.855∗∗ −0.708∗∗ −0.462∗ −0.982∗∗ 0.539∗ ↑ rec −0.947∗ 2.937∗∗ −1.921∗∗ −0.282∗ −0.095 −0.236 ↓price ↓ cut 0.810∗∗ 0.153 0.130 0.106 0.200 0.380∗ 1974 −1.018∗∗ ↑price ↑ max −0.814∗∗ ↑ rec −0.916 ↓price ↓ cut 0.756∗∗ 1984 −0.897∗∗ ↑price ↑ max −0.434 ↑ rec −0.654 ↓price ↓ cut 0.307 1994 −0.599 ↑price ↑ max 0.691 ↑ rec 0.119 ↓price ↓ cut −1.021

0.245∗ −0.108

−0.272

Effect of electricity price on demand for:

−0.193

−0.174

−0.013 −0.402

−1.274∗∗ −0.179 −1.157∗∗ 0.212

−0.122

−0.392∗∗ 1.394∗∗

−0.124

0.809∗∗ −0.611∗∗ −0.539∗ −0.980∗∗ 0.464∗ −0.036 2.863∗∗ −1.655∗∗ −0.330∗ −0.106 −0.203 −0.491 0.185

0.113

0.124

0.081

−0.091

−0.733

0.209

0.327∗ −0.121

−0.466∗∗ −0.152

−0.336

0.440∗∗ −0.533∗∗ −1.282∗ −0.939∗∗ 0.405∗ −0.132 1.737∗∗ −1.439∗∗ −0.827∗ −0.387∗∗ −0.174 −1.118 0.188

0.101

0.380

0.062

−0.069

−1.664

0.452∗∗ 0.282∗ −0.210 −0.385∗∗ −0.116

−1.057

0.057

−0.923∗∗

−1.224∗∗ −0.292 −1.109∗∗ 0.045 −0.419∗∗ 1.338∗∗ 0.057

−0.934∗∗

−0.750∗∗ −0.385 −0.678∗∗ −0.093 −0.288∗∗ 1.292∗∗ 0.124

−0.951∗∗

0.489∗∗ −0.396∗∗ −3.190∗ −0.955∗∗ 0.299∗ −0.700 2.065∗∗ −1.067∗∗ −2.009∗ −0.283∗∗ −0.130 −3.260

−0.838∗∗ −0.543∗∗ −0.750∗∗ −0.326

0.274∗

−0.423∗∗ 1.214∗∗

0.076

0.850

0.362∗∗ 0.209∗ −0.186

Notes: 1. Elasticity estimates are based on the estimated parameters reported in columns [1] and [2] of Table 4. 2. For each year, the first line reports elasticities based on the model with no price decomposition. Subsequent rows for each year refer to elasticities calculated when the price change causes a corresponding change in the component “max”, “rec” or “cut”, respectively. ∗ indicate significance at the 1% level. ∗∗ indicate significance at the 5% level.

and electricity. In particular, throughout our sample period natural gas demand seems to be much more responsive to oil price increases, and especially to real oil price recoveries (that do not set a new historical maximum). A similar pattern is observed (in absolute value) for electricity demand, even though the signs of the estimated coefficients suggest that oil products and electricity are complements in consumption for this sector. Note as well that the cross-price elasticities asso-

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D.L. Ryan and A. Plourde

ciated with the individual real oil price components are more much likely to be statistically significant than are those associated with the undecomposed price. Our estimates also suggest that changes in natural gas prices give rise to asymmetric responses in the demand for that energy source. Here, increases in the maximum real price are associated with larger proportional responses in natural gas demand than are price recoveries or price cuts. And this basic pattern holds for the four years examined, even if real price recoveries and cuts seem to play a more important role as time progresses. Also, note that changes in the undecomposed real price of natural gas appear to give rise to similar proportional changes in the demand for that energy source than do real price cuts. In addition, electricity demand by Ontario’s residential sector seems to be more sensitive to increases in the maximum historical real price of natural gas and to reductions of this price than to price recoveries. This pattern of effects can only be captured if the real price of natural gas is decomposed. As far as electricity prices are concerned, our results suggest that price cuts exert much more influence on electricity demand than does either of the two other components. Asymmetric effects are also detected as far as natural gas demand is concerned. Here, increases in real electricity prices have a proportionately larger effect on natural gas demand than do real price cuts, but our results suggest that the differences in responsiveness to increases in the maximum historical electricity price have rather similar effects on natural gas demand than do recoveries in that price. Overall, our results lend support to the existence of a rich pattern of asymmetric demand responses to real energy price changes in the residential sector of the province of Ontario. In particular, the demand for natural gas seems to respond in an asymmetric fashion to changes in the real prices of all energy sources considered. Indeed, natural gas demand seems particularly responsive to oil price recoveries, increases in the maximum historical price of natural gas, and all increases in real electricity prices. Own-price increases exert much less of an effect on electricity demand than do own-price decreases. On the other hand, demand for oil products is much more responsive to increases in real natural gas prices than to decreases in that price. As the results in Table 5 remind us, none of these effects can be detected when the model used in estimation does not allow for the decomposition of real energy prices. 3.7.2. Responses to Changes in Relative Prices When the decomposition in (21) and (22) is incorporated in the model that is estimated using relative prices, the price elasticities will again differ according to whether the price change causes a change in max, rec or cut. However, since each of these terms refers to a relative price, and since (for example) max(ln p1 − ln p2 ) = max(ln p1 ) − max(ln p2 ), there are many possibilities that need to be considered when elasticities are calculated. For example, an increase in ln(p1 )

Modelling Asymmetric Demand Responses

211

Table 6: Relative Price Components that Change when Individual Prices Change Price change

Effect of price change on components of relative prices ln(p1 /p3 ) max13

p1 p1 p1 p1 p1

increase increase increase increase decrease

p2 p2 p2 p2

increase increase decrease decrease

p3 p3 p3 p3 p3

increase decrease decrease decrease decrease

+ +

rec13

ln(p2 /p3 ) cut13

max23

ln(p1 /p2 ) cut23

max12 +

+ +

+ − +

+ +

rec23

− + +

+ +

+

− −

+

rec12

cut12

+ +

− − −

+

− + +

Notes: A “+” indicates that the price change in that row causes the price component in that column to increase, while a “−” indicates that the price component in that column decreases.

that causes max(ln P13 ) to increase, will either cause max(ln P12 ) or rec(ln P12 ) to increase by a similar amount. The same effect occurs for an increase in ln(p1 ) that causes rec(ln P13 ) to increase. Conversely, a decrease in ln(p1 ) will cause both cut(ln P13 ) and cut(ln P12 ) to decrease. Similar types of results hold for changes in ln(p2 ) and ln(p3 ). As Table 6 shows, there are 14 different combinations of changes in the price components that occur depending on which one of the three energy prices changes, and on whether that particular price increases or decreases. Table 7 contains the various price elasticities for the models that are estimated using relative prices for the years 1964, 1974, 1984 and 1994. Examining this table we see that increases in oil prices tend to exert a stronger influence on the demand for oil products by Ontario’s residential sector than do own-price cuts. By the end of our sample period, however, none of the own-price components seem to have a statistically significant effect on the demand for oil products. As time progresses, oil price increases seem to play a more important role in the evolution of natural gas demand, while the importance of oil price cuts seems to wane. As far as electricity demand is concerned, price cuts are the most influential movement in oil prices, at least until the end of our sample period. Given the price decomposition effected, only changes in the maximum historical price of natural gas seem to affect the demand for oil products by Ontario’s residential sector. However, all of the own-price components are important in explaining the evolution of natural gas demand, especially in the second half of our

212

Table 7: Estimated Elasticities – Relative Price Model Effect of oil price on demand for:

↓ poil ↓ c13 , ↓ c12

1974 ↑ poil ↑ m13 , ↑ m12 ↑ m13 , ↑ r12 ↑ r13 , ↑ m12 ↑ r13 , ↑ r12 ↓ poil ↓ c13 , ↓ c12

Gas

−1.241∗∗

0.430∗∗

−0.008

−1.128∗∗ −1.154∗∗ −1.436∗∗ −1.462∗∗

0.100 0.483∗∗ −0.176 0.207

0.070 −0.159 0.582 0.354

0.815∗∗

−0.388∗∗

−1.252∗∗

0.386∗∗

0.001

−1.130∗∗ −1.158∗∗ −1.466∗∗ −1.494∗∗

0.075 0.516∗∗ −0.242 0.199

0.071 −0.124 0.507 0.312

0.788∗∗

−0.407∗∗

Effect of gas price on demand for:

Elec

0.456∗∗

0.377∗∗

Oil

Gas

1964 ↑ pgas ↑ m23 , ↓ c12 ↑ r23 , ↓ c12

−0.088

−0.187

−0.425∗∗

−0.361∗∗ −0.320

−0.112 1.296∗∗

−0.207 −1.192∗∗

↓ pgas ↓ c23 , ↑ m12 ↓ c23 , ↑ r12

−0.014 −0.039

1974 ↑ pgas ↑ m23 , ↓ c12 ↑ r23 , ↓ c12

↓ pgas ↓ c23 , ↑ m12 ↓ c23 , ↑ r12

0.498∗∗ 0.881∗∗

0.350∗ 0.121

−0.098

−0.169

−0.362∗∗

−0.384∗∗ −0.339

−0.018 1.601∗∗

−0.165 −1.003∗∗

−0.024 −0.052

0.461∗∗ 0.902∗∗

Effect of elec price on demand for:

Elec

0.286∗ 0.092

Oil

Gas

Elec

1964 ↑ pelec ↓ c13 , ↓ c23

−0.109

−0.240∗

−0.725∗∗

−0.378∗∗

−0.327∗∗

−0.379∗∗

↓ pelec ↑ m13 , ↑ m23 ↑ m13 , ↑ r23 ↑ r13 , ↑ m23 ↑ r13 , ↑ r23

−0.115 −0.074 −0.423 −0.382

−0.571∗ 0.838∗∗ −0.847∗∗ 0.562∗∗

1.504∗∗ 0.519∗∗ 2.016∗∗ 1.032∗∗

1974 ↑ pelec ↓ c13 , ↓ c23

−0.159

−0.183

−0.774∗∗

−0.432∗∗

−0.298∗

−0.494∗∗

↓ pelec ↑ m13 , ↑ m23 ↑ m13 , ↑ r23 ↑ r13 , ↑ m23 ↑ r13 , ↑ r23

−0.106 −0.061 −0.442 −0.397

−0.734∗ 0.885∗∗ −1.052∗∗ 0.568∗∗

1.451∗∗ 0.614∗∗ 1.887∗∗ 1.050∗∗

D.L. Ryan and A. Plourde

1964 ↑ poil ↑ m13 , ↑ m12 ↑ m13 , ↑ r12 ↑ r13 , ↑ m12 ↑ r13 , ↑ r12

Oil

Table 7: (Continued.)

Oil 1984 ↑ poil ↑ m13 , ↑ m12 ↑ m13 , ↑ r12 ↑ r13 , ↑ m12 ↑ r13 , ↑ r12 ↓ poil ↓ c13 , ↓ c12

Gas

−1.346∗∗

0.119

−1.154∗∗ −1.213∗∗ −1.863∗ −1.922∗∗

−0.094 0.174 −0.286∗ −0.019

0.433

−0.108

Effect of gas price on demand for:

Elec 0.023 0.122 −0.051 0.512 0.338 0.278∗∗

Oil

Gas

Elec −0.328∗∗

1984 ↑ pgas ↑ m23 , ↓ c12 ↑ r23 , ↓ c12

−0.367

−0.384∗∗

−0.899∗∗ −0.803

−0.302∗∗ 0.681∗∗

−0.191 −0.940∗∗

↓ pgas ↓ c23 , ↑ m12 ↓ c23 , ↑ r12

0.036 −0.023

0.571∗∗ 0.839∗∗

0.299∗∗ 0.126

Effect of elec price on demand for: Oil

Gas

Elec

1984 ↑ pelec ↓ c13 , ↓ c23

−0.433∗

−0.073

−0.812∗∗

−0.944∗∗

−0.143

−0.563∗∗

↓ pelec ↑ m13 , ↑ m23 ↑ m13 , ↑ r23 ↑ r13 , ↑ m23 ↑ r13 , ↑ r23

−0.192 −0.097 −0.901 −0.805

−0.483∗∗ 0.500∗∗ −0.676∗∗ 0.307∗

1.420∗∗ 0.670∗∗ 1.810∗∗ 1.060∗∗

Modelling Asymmetric Demand Responses

Effect of oil price on demand for:

213

214

Table 7: (Continued.) Effect of oil price on demand for: Gas

Elec

−1.556∗∗

0.076

0.025 0.120∗ −0.011 0.415∗ 0.284

1994 ↑ poil ↑ m13 , ↑ m12 ↑ m13 , ↑ r12 ↑ r13 , ↑ m12 ↑ r13 , ↑ r12

−1.257 −1.448 −3.550 −3.741

−0.202∗∗ 0.098 −0.419∗∗ −0.118

↓ poil ↓ c13 , ↓ c12

−1.074

−0.024

0.182∗∗

Effect of gas price on demand for: Oil

Gas

Effect of elec price on demand for:

Elec

1994 ↑ pgas ↑ m23 , ↓ c12 ↑ r23 , ↓ c12

−0.654

−0.297∗

−0.245∗∗

−2.829∗ −2.519

−0.249∗ 0.855∗∗

−0.135 −0.702∗∗

↓ pgas ↓ c23 , ↑ m12 ↓ c23 , ↑ r12

0.037 −0.154

0.550∗∗ 0.851∗∗

0.217∗∗ 0.086

Oil −1.358∗

Gas

Elec

0.033

−0.869∗∗ −0.707∗∗

1994 ↑ pelec ↓ c13 , ↓ c23

−3.373∗∗

−0.032

↓ pelec ↑ m13 , ↑ m23 ↑ m13 , ↑ r23 ↑ r13 , ↑ m23 ↑ r13 , ↑ r23

−0.303 0.007 −2.596 −2.286

−0.672∗∗ 0.432∗∗ −0.888∗∗ 0.216

1.354∗∗ 0.788∗∗ 1.649∗∗ 1.082∗∗

Notes: 1. Elasticity estimates are based on the estimated parameters reported in columns [3] and [4] of Table 4. 2. For each year, the first line reports elasticities based on the model with no price decomposition. Subsequent rows for each year refer to elasticities calculated for a particular type of price change – an increase (↑) or decrease (↓) in the price of oil (poil), natural gas (pgas) or electricity (pelec) – when that price change causes the specified changes in the relative price components. The prefixes “m”, “r” and “c” refer to “max”, “rec” and “cut”, respectively, so that, for example, ↑ m12 refers to an increase in max12 , defined as max(ln(p1 /p2 )) or max(ln(P12 )). The subscripts 1, 2 and 3 refer to oil products, natural gas and electricity, respectively. 3. In rows associated with an increase in the price of a particular energy source, values in cells show the proportional change in demand due to a proportional increase in price. In rows associated with a decrease in the price of a particular energy source, values in cells show the proportional change in demand due to a proportional decrease in price. ∗ indicate significance at the 1% level. ∗∗ indicate significance at the 5% level.

D.L. Ryan and A. Plourde

Oil

Modelling Asymmetric Demand Responses

215

sample period. Here, our estimates suggest that demand is proportionately more responsive to price cuts than to increases in the maximum historical price. Surprisingly, natural gas price recoveries would appear to spur demand for that energy source. As far as oil demand is concerned, it would appear that natural gas prices exert a statistically significant influence only when increases in the latter give rise to an increase in the maximum historical price of natural gas relative to that of oil products. Natural gas price recoveries seem to play an important role in the evolution of electricity demand, as do price cuts when these again lead to an increase in the maximum historical price of natural gas relative to that of oil products. Increases seem to be the only movements in electricity prices that affect the demand for oil products – and seem to do so increasingly strongly as time progresses. The effect of electricity price increases on natural gas demand seems to have waned over time. However, the same cannot be said about the consequences of electricity price cuts. And here the pattern of effects depends on whether the drop in the price of electricity establishes a new historical maximum for the price of natural gas relative to electricity. If a new historical maximum is established, then the electricity price reduction leads to increased natural gas demand. But if the electricity price reduction leads instead to a recovery in the price of natural gas relative to electricity, then our results suggest that natural gas demand would rise. Finally, electricity demand in Ontario’s residential sector responds in a statistically significant manner to changes in all components of relative electricity prices, and appears to do so most strongly to reductions in electricity prices that bring about a new maximum historical price of natural gas relative to electricity. As was the case with the results based on estimations of real price models, when our empirical investigation of the demand for various energy sources by the residential sector of the province of Ontario allows for different components of relative prices to have different effects, a rich pattern of asymmetric responses to relative price changes is revealed. Needless to say, estimation of relative price models that do not incorporate this decomposition cannot capture this pattern of effects. 3.7.3. Comparisons of Response Patterns A comparison of Tables 5 and 7 reveals patterns of elasticities that are similar under real price and relative price estimation, when no decomposition is implemented. Once price decompositions are implemented, however, the resulting estimated elasticities show no systematic pattern of the kind obtained in early contributions, where demand is typically shown to be most responsive to changes in the maximum price, and more responsive to (sub-maximum) price recoveries than to price cuts.9 Our results provide some support for the notion that increases in

9 Note

that comparisons across real and relative price models are not straightforward since there is not a one-to-one correspondence in the effects of price changes. For example, in

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D.L. Ryan and A. Plourde

maximum historical prices result in proportionately larger demand responses. But for neither real nor relative prices do we systematically find estimated elasticities for max to exceed those for rec, with even smaller (in absolute value) responses resulting from changes in cut. With the real price model, for example, estimated electricity demand responses are proportionately larger when a decrease in electricity price does not result in the establishment of a new historical maximum value for the price of oil products relative to that for electricity. Even if, as noted earlier, no systematic ranking of elasticities emerges from our results, there tend to be clear differences in the (absolute) values of the estimated elasticities associated with the various price components. And this holds for both real and relative price models. A good illustration of this observation emerges from a comparison of the estimated responses to changes in the components of the price of natural gas (the middle set of columns in Tables 5 and 7). To the extent that demand asymmetries are detected in the data (Tables 2 and 3), our estimated elasticities suggest that differences in responses to changes in the various price components are not only systematic, but that such differences are relatively large. Finally, the results reported in Tables 5 and 7 show that, in the case of Ontario’s residential sector, demand asymmetries are not limited to the market for oil products, and also remind us of the importance of accounting for inter-fuel substitution. In the real price model, for example, oil price movements clearly give rise to asymmetric responses in the demand for natural gas, while in the relative price model the same can be argued about the effects of natural gas price changes on electricity demand, among others. In our empirical applications, models that explicitly take inter-fuel substitution into account thus allowed us to detect rich patterns of asymmetric responses for all of the energy sources considered to changes in their own prices as well as to changes in the prices of other energy sources. Given the context in which decisions about energy use are made, we would expect these types of results to be observed in a broad range of circumstances – in other jurisdictions and in other sectors.

4. Summary and Conclusion Previous analysis concerning asymmetric responses to price changes, both in the agriculture and energy fields, has been undertaken within a single-equation framework. Typically, this work has involved the modification of the own-price term to facilitate the modelling of asymmetric responses observed in the data. In this

real price models, an increase in the maximum historical price of natural gas can only be caused by a (sufficiently large) increase in the price of that energy source. An increase in (log) max(pgas/poil) however, can follow from an increase in the price of natural gas or from a decrease in the price of oil products.

Modelling Asymmetric Demand Responses

217

chapter we have extended this approach to a framework with multiple inter-related demands. Specifically, we embedded the price decomposition popularized by Dargay and Gately in a system of demand equations, and developed strategies to test for asymmetric responses to price changes. In doing so, we have addressed a number of issues that arise concerning the homogeneity and standard symmetry conditions associated with such systems. This led us to consider specifications that alternately included real and relative prices, and to investigate the consequences in each of these cases when price decompositions are incorporated. We show that in the case of real price models, homogeneity can be neither imposed nor tested when real prices are decomposed along the lines suggested by Dargay and Gately. When a decomposition of relative prices is carried out, however, we show that homogeneity is automatically satisfied, and thus cannot be tested. To explore the potential usefulness of our proposed approach, we undertake an empirical examination of the demand for three energy sources (electricity, natural gas and oil products) by Ontario’s residential sector, over the period 1962 to 1994. Our results indicate that asymmetric demand responses to energy price changes are detected for the system as a whole, and for most energy sources individually (interestingly, in our empirical application support for the existence of asymmetric responses is least compelling for oil demand). This pattern of results applies for both real price and relative price decompositions. Our strategy to broaden the scope of the analysis to include multiple energy sources thus appears to be fruitful. Asymmetric demand responses to price changes have been detected for energy sources other than oil products, and demands for individual energy sources have been shown to respond asymmetrically to movements in the prices of other fuels. We then try to gauge the importance of the detected asymmetric responses in explaining energy demand patterns by using information obtained in estimation to calculate own- and cross-price elasticities (and standard errors). In contrast to much of the existing literature in this area, our results are not consistent with the existence of a systematic ranking of the responses to changes in maximum historical prices, price recoveries and price cuts. In some cases, estimated demand responses are proportionately largest (in absolute value) for price recoveries (as with the effect of real oil price changes on the demand for electricity), while in other cases (own-price elasticities of electricity demand, for example) real price cuts exert the largest proportional effect (in absolute value). While there is no clear ranking in terms of the responsiveness associated with individual price components, whenever asymmetric responses are detected, there are generally noticeable differences in the (absolute) values of the estimated elasticities. By adopting a modelling strategy that explicitly considers the inter-related demands for three energy sources, we were able to uncover in Ontario’s residential sector a pattern of asymmetric demand responses to real and relative energy price changes that goes beyond that detected by single-equation approaches that typically focus on oil. At this stage, there is a need to assess whether our approach to modelling and testing for the existence of energy demand asymmetries will

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prove equally useful for other sectors and jurisdictions, and to explore the policy implications of these results – tasks to which we will turn in subsequent work.

Appendix: Data Annual data on expenditures by energy source are measured on an after-tax basis and were constructed using information on annual quantities (in petajoules) taken from the database maintained by the National Energy Board of Canada. The quantity series for “oil products” was formed by adding those for kerosene and light fuel oil (LFO). For electricity and natural gas, measures of annual expenditures were constructed by combining the quantity information with implicit pre-tax price series derived from Electric Power Statistics, Annual Statistics (Statistics Canada, catalogue #57-202) and Gas Utilities (#57-205), respectively. Note that, while the province of Ontario never taxed residential energy use during our sample period, the federal goods and services tax (GST) has applied to all such energy sales since the beginning of 1991. We incorporate this information in all price and expenditure series so that all energy sources are treated on the same after-tax basis. In the case of oil products, monthly after-tax prices of LFO were assembled for a number of cities from Consumer Prices and Price Indexes (Statistics Canada, catalogue #62-010-XPB) and predecessors. These were aggregated into annual series by using monthly quantities from Refined Petroleum Products (#45-004) and annual populations for these cities from Census of Canada information (with interpolations for years between censuses). The resulting annual after-tax price series were again combined with quantity information from the National Energy Board to form the expenditure series for oil products used in this study. Since complete series of kerosene prices could not be obtained, we assumed that kerosene and LFO have the same after-tax prices. This assumption is also embodied in the energy demand models maintained by Natural Resources Canada and the National Energy Board. The price variables used in estimation are after-tax marginal prices, which in the case of oil products is the same series as that described above. In the case of electricity and natural gas, however, these were calculated using information on city-specific monthly pre-tax prices (contained in numerous utility publications and price schedules, and decisions of the Ontario Energy Board). Annual after-tax prices for both electricity and natural gas were then obtained applying information on monthly quantities from Gas Utilities (Statistics Canada, catalogue #57-205), annual city populations from the Census of Canada (again, with interpolations for years between censuses), and applying the GST to the monthly pre-tax price series described in the previous sentence. All price series used in estimation were expressed in dollars per unit of energy content (e.g., gigajoule), using conversion factors taken from Statistics Canada’s Quarterly Report on Energy

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Supply/Demand in Canada (#57-203), which, along with Environment Canada, is also the source of the data series on heating and cooling degree-days. An application of the Dargay–Gately approach to the decomposition of the real prices of LFO, natural gas, and electricity yields the original and the component series shown in Figures 3–5. The component series for each of the energy sources are similar to those reported in studies such as Dargay and Gately (1995, p. 63), where the only price variable decomposed is a measure of the real price of oil. The results of applying the Dargay–Gately decomposition to the relative prices of residential energy in Ontario are reported in Figures 6–8. To illustrate how these prices move in relation to one another, these figures present the original and the component series for each of these relative prices (LFO and natural gas relative to electricity, and LFO relative to natural gas, respectively). Again, all of the component series exhibit the standard pattern characteristic of the Dargay– Gately decomposition, given that the price of electricity has exceeded that of LFO and natural gas throughout our sample period. Finally, as noted in Section 3.3, one of the estimation procedures adopted requires the use of instrumental variables. Population data from Ontario were taken from the electronic version of CANSIM, Statistics Canada’s main database. The series for the length of the natural gas pipeline distribution system is from Gas Utilities (#57-205). Housing completions were obtained from Canada Mortgage and Housing Corporation. Series used to construct all of the remaining instrumental variables (i.e., the proportion of personal income that comprises wages,

Figure 3: Dargay–Gately Decomposition – Real Price of LFO.

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Figure 4: Dargay–Gately Decomposition – Real Price of Natural Gas.

Figure 5: Dargay–Gately Decomposition – Real Price of Electricity.

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Figure 6: Dargay–Gately Decomposition – Price of LFO/Electricity.

Figure 7: Dargay–Gately Decomposition – Price of Natural Gas/Electricity.

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Figure 8: Dargay–Gately Decomposition – Price of LFO/Natural Gas. salaries, and other supplementary income; and the proportion of provincial GDP that governments spend on goods and services) were taken from Provincial Economic Accounts (Statistics Canada, catalogue #13-213).

References Berndt, E.R. and Savin, N.E. (1975). Estimation and hypothesis testing in singular equation systems with autoregressive disturbances. Econometrica 43(5–6), 937–957. Berndt, E.R., Morrison, C.J. and Watkins, G.C. (1981). Dynamic models of energy demand: An assessment and comparison. In Berndt, E.R. and Field, B.C. (Eds), Modeling and Measuring Natural Resource Substitution. Cambridge, MA: MIT Press, pp. 259– 287. Brown, S.P.A. and Phillips, K.R. (1991). U.S. oil demand and conservation. Contemporary Policy Issues 9(1), 67–72. Buse, A. (1994). Evaluating the linearized almost ideal demand system. American Journal of Agricultural Economics 76(4), 781–793. Bye, T. (1986). Non-symmetric responses in energy demand. In Proceedings of the Eighth Annual International Conference of the International Association of Energy Economists. Washington: IAEE, pp. 354–358. Dargay, J.M. (1992). The irreversible effects of high oil prices: Empirical evidence for the demand for motor fuels in France, Germany and the U.K. In Hawdon, D. (Ed.), Energy Demand: Evidence and Expectations. London: Surrey University Press, pp. 165–182.

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Dargay, J.M. and Gately, D. (1994). Oil demand in the industrialized countries. The Energy Journal 15(Special Issue), 39–67. Dargay, J.M. and Gately, D. (1995). The imperfect price reversibility of non-transport oil demand in the OECD. Energy Economics 17(1), 59–71. Deaton, A. and Muellbauer, J. (1980). An almost ideal demand system. American Economic Review 70(3), 312–326. Dunstan, R.H. and Schmidt, R.H. (1988). Structural changes in residential energy demand. Energy Economics 10(3), 206–212. Gately, D. (1992). Imperfect price-reversibility of U.S. gasoline demand: Asymmetric responses to price increases and declines. The Energy Journal 13(4), 179–207. Gately, D. (1993a). The imperfect price-reversibility of world oil demand. The Energy Journal 14(4), 163–182. Gately, D. (1993b). Oil demand in the US and Japan: Why the demand reductions caused by the price increases of the 1970’s won’t be reversed by the price declines of the 1980’s. Japan and the World Economy 5(4), 295–320. Gately, D. and Huntington, H. (2002). The asymmetric effects of changes in price and income on energy and oil demand. The Energy Journal 23(1), 19–55. Gately, D. and Rappoport, P. (1988). Adjustment of U.S. oil demand to the price increases of the 1970s. The Energy Journal 9(2), 93–107. Griffin, J.M. and Shulman, C.T. (2005). Price asymmetry in energy demand models: A proxy for energy-saving technical change. The Energy Journal 26(2), 1–21. Haas, R. and Schipper, L. (1998). Residential energy demand in OECD countries and the role of irreversible energy efficiency improvements. Energy Economics 20(4), 421–442. Hogan, W.W. (1989). A dynamic putty–semi-putty model of aggregate energy demand. Energy Economics 11(1), 53–69. Hogan, W.W. (1993). OECD oil demand dynamics: Trends and asymmetries. The Energy Journal 14(1), 125–157. Houck, J.P. (1977). An approach to specifying and estimating nonreversible functions. American Journal of Agricultural Economics 59(3), 570–572. Ryan, D.L. and Plourde, A. (2002). Smaller and smaller? The price responsiveness of nontransport oil demand. Quarterly Review of Economics and Finance 42(2), 285–317. Saylor, R.G. (1974). Alternative measures of supply elasticities: The case of São Paulo coffee. American Journal of Agricultural Economics 56(1), 98–106. Shealy, M.T. (1990). Oil demand asymmetry in the OECD. In Energy Supply/Demand Balances: Options and Costs. Proceedings of the Twelfth Annual North American Conference of the International Association for Energy Economics. Washington: IAEE, pp. 154–165. Sweeney, J.L. and Fenechel, D.A. (1986). Price asymmetries in the demand for energy. In Proceedings of the Eighth Annual International Conference of the International Association for Energy Economists. Washington: IAEE, pp. 218–222. Traill, B., Colman, D. and Young, T. (1978). Estimating irreversible supply functions. American Journal of Agricultural Economics 60(3), 528–531. Tweeten, L.G. and Quance, C.L. (1969). Positivistic measures of aggregate supply elasticities: Some new approaches. American Journal of Agricultural Economics 51(2), 342–352. Walker, I.O. and Wirl, F. (1993). Irreversible price-induced efficiency improvements: Theory and empirical application to road transportation. The Energy Journal 14(4), 183– 205.

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Watkins, G.C. and Waverman, L. (1987). Oil demand elasticities: The saviour as well as the scourge of OPEC? In The Changing World Energy Economy. Papers and Proceedings of the Eighth Annual North American Conference of the International Association of Energy Economists. Cambridge, MA: IAEE, pp. 223–227. Wolffram, R. (1971). Positivistic measures of aggregate supply elasticities: Some new approaches – some critical notes. American Journal of Agricultural Economics 53(2), 356–359.

Part V Seasonality, Liquidity Constraints and Hopf Bifurcations

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Chapter 11

Rational Seasonality Travis D. Nesmith Board of Governors of the Federal Reserve System, 20th & C Sts., NW, Mail Stop 188, Washington, DC 20551, USA, e-mail: [email protected]

Abstract Seasonal adjustment usually relies on statistical models of seasonality that treat seasonal fluctuations as noise corrupting the ‘true’ data. But seasonality in economic series often stems from economic behavior such as Christmas-time spending. Such economic seasonality invalidates the separability assumptions that justify the construction of aggregate economic indexes. To solve this problem, Diewert (1980, 1983, 1998, 1999) incorporates seasonal behavior into aggregation theory. Using duality theory, I extend these results to a larger class of decision problems. I also relax Diewert’s assumption of homotheticity. I provide support for Diewert’s preferred seasonally-adjusted economic index using weak separability assumptions that are shown to be sufficient.

Keywords: seasonality, separability, aggregation, index numbers, consumer decision theory JEL: C43, D11, E31

1. Introduction Economic indexes are often treated as given; the complicated aggregation theory underlying the construction of the index is ignored in empirical research. But aggregation and statistical index number theory has returned the favor and largely ignored the consensus that seasonal fluctuations, due to by phenomena such as seasonal patterns in the growing cycle, Christmas shopping, etc., are endemic to economic time series including economic indexes. Relatively little work has attempted to incorporate seasonal fluctuations into the theory, even though seasonality can invalidate the separability assumptions that justify the construction of aggregate economic indexes. Seasonality has usually been addressed econometrically. Standard econometric approaches view seasonality as an undesirable characteristic of the data. Consequently, the bulk of the research on seasonality has treated seasonal fluctuations as noise that is corrupting the underlying signal. Econometric research International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Published by Elsevier B.V. ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18011-X

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has focused on how to smooth or remove seasonal fluctuations. Econometric seasonal-adjustment techniques – ranging from the inclusion of seasonal dummies in regression analysis to the complicated procedures, such as the X-12 procedure, implemented by statistical agencies to produce seasonally-adjusted data – rely on statistical models of seasonality. No matter how statistically sophisticated, these models share a fundamental weakness in that they have little or no connection to economic theory. Diewert (1996b, p. 39) describes such models as “more or less arbitrary”. Except for a few exceptions (Ghysels, 1988; Miron and Zeldes, 1988; Miron, 1996; Osborn, 1988), research on seasonal-adjustment explicitly or implicitly assumes that seasonality is not the result of economic behavior. Grether and Nerlove (1970) acknowledge that seasonal phenomena in economic data is generated by customs and institutions, and should be expected to be much more complex that meteorological phenomena. Nevertheless, the main approaches to econometric seasonal adjustment are based on unobserved component models historically developed to model astronomical phenomena. In a series of papers Diewert (1980, 1983, 1996b, 1998, 1999) argues that much of the seasonality in economic time series is produced by the behavior of economic agents, and that such behavior should consequently be modeled with economics rather than econometrics. Diewert focuses on the fact that many economic time series are constructed as statistical index numbers. The construction of statistical index numbers is justified, in the economic approach to index number theory, by their connection to specific economic models. Diewert stresses that these models do not account for behavior that varies across seasons, and, consequently, the economic indexes are not valid in the presence of seasonality. He examines two different ways that seasonal behavior of economic agents can be rationalized in a neoclassical framework, and concludes that only one of these possibilities is consistent with the economic approach to constructing index numbers. Seasonal behavior can be rational if the agent is optimizing a time varying objective function. However, a time varying objective function generally cannot be tracked by an economic index.1 Alternatively, the agent’s objective function is not separable at the observed seasonal frequency. The implications of this lack of separability on the functional structure of the agent’s decision is more amenable to analysis that general time variability. Diewert (1980) concludes that research into seasonal behavior should focus on decision problems that are not time separable at seasonal frequencies; in his subsequent papers he adapts a standard utility maximization problem to account for seasonality. His seasonal decision problem can be used to construct economic indexes from data that contains seasonality; Diewert (1998, p. 457) describes his research as filling a gap:

1 Time-varying

patterns that are only a function of the season as in Osborn (1988) are a special case of seasonal inseparability which is the second approach Diewert examines.

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“The problem of index number construction when there are seasonal commodities has a long history. However, what has been missing is an exposition of the assumptions on the consumer’s utility function that are required to justify a particular formula. We systematically list separability assumptions on intertemporal preferences that can be used to justify various seasonal index number formulas from the viewpoint of the economic approach to index number theory.”.

Diewert’s approach deseasonalizes statistical index number by their construction. Diewert’s models and the resulting indexes have an obvious advantage over econometric models; their connection to economic theory obviates the development of econometric criteria for evaluating different adjustment methods. Economic theory directly justifies the index number approach to seasonal adjustment. In this chapter, I further extend the aggregation theory approach to seasonal adjustment. While this chapter extends Diewert’s line of research, the focus is slightly different. I focus more on defining seasonal aggregates, rather on the resulting index formulae, as defining aggregates is logically prior to defining the indexes that track them. In addition, although I am weakening the conditions necessary to rationalize seasonal aggregates the resulting indexes are the same as in Diewert (1999) so focusing on the index formulae would be redundant. Using duality theory, I extend Diewert’s results to a larger class of decision problems. I also relax Diewert’s assumption of homotheticity. The most novel result is a justification of Diewert’s moving year index, which is his preferred seasonal index, using only separability assumptions. The derivation, which follows from a theorem of Gorman (1968), is not only sufficient, it is also shown to be necessary. The remainder of this chapter is organized as follows. Section 2 briefly discusses econometric seasonal adjustment methods. Section 3 reviews the index number approach to seasonal adjustment developed by Diewert. Section 4 presents different types of separability for the expenditure and distance functions. Section 5 provides conditions that support the construction of seasonal indexes. In particular, Diewert’s moving year index is derived from a separability assumption. An argument for why these particular separability assumptions are reasonable is also advanced. Although no empirical analysis of the index number approach to seasonality is provided, Section 6 comments on some empirical implications of the theory. The last section concludes.

2. Econometric Adjustment This section briefly discusses econometric adjustment techniques; see Nerlove et al. (1979), Bell and Hillmer (1984), Hylleberg (1992) and Miron (1996) for more extensive reviews. The discussion focuses on how difficult it is to establish criteria for determining how to econometrically adjust series for seasonality. The lack of criteria makes the choice of which method to use subjective. The situation is similar to the difficulties in choosing a statistical index number formula solely on the basis of their axiomatic properties.

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The majority of seasonal adjustment techniques are based on decomposing a series, or multiple series, into unobserved components. Grether and Nerlove (1970, p. 686), in discussing the “desiderata” of seasonal adjustment, note that the unobserved components methods originated in astronomy, and state, “It is of course, quite debatable whether the idea of unobserved components, appropriate as it may be in the analysis of astronomical observations, is usefully applied to economic data or even to meteorological data. Nonetheless, we believe that this idea lies behind both present methods of seasonal adjustment and the desire for seasonally adjusted time series.”

Grether and Nerlove (1970) and Nerlove et al. (1979) show that ‘optimal’ econometric seasonal adjustment depends on both the model of seasonality and the model in which the data are to be used. Nerlove et al. (1979, p. 171) conclude, “. . . it is clear that (a) no single method of adjustment will be best for all potential users of the data and (b) it is essential to provide economic time series data in unadjusted form”. They additionally conclude that despite the increase in sophistication of the econometric techniques, “. . . in terms of modeling explicitly what is going on, there seems to have been remarkably little progress”. This conclusion remains valid. The lack of an empirical standard leads Bell and Hillmer (1984) to conclude that seasonal adjustment methods should be judged on whether the model of seasonality implicit in the method is consistent with the observed seasonality in the data. This would suggest that different adjustment methods should be applied to different data series, so there is no unique ‘optimal’ method. Seasonal adjustment has also been characterized as a signal extraction problem in the frequency domain. Grether and Nerlove (1970) argue against evaluating adjustment methods using empirical criteria based on the spectral properties of the adjustment, although they do not discount its usefulness for characterizing the effects of different methodologies. The lack of a consensus on how to seasonally adjust has led some authors to focus on the effects on the statistical properties of the data when the data is seasonally adjusted using the wrong statistical model (Wallis, 1974, discusses this issue). Several authors, notably Lovell (1963, 1966) and Jorgenson (1964) try to derive a set of axioms that a seasonal adjustment method should satisfy. Jorgenson’s approach is to specify that the adjustment method should satisfy the properties of the unique minimum variance, linear, estimator. While this seems reasonable, it still provides an indeterminate solution, because other statistical models, for example a minimum distance estimator or weighted least squares, are just as sensible. In addition, Lovell (1966) showed that Jorgenson’s method does not satisfy Lovell’s orthogonality axiom, so the adjusted series is correlated with the seasonal adjustment component. The approach in Lovell (1963) is perhaps the most intriguing relative to Diewert’s approach, because it is reminiscent of the axiomatic approach to index number theory developed in Fisher’s (1922) seminal work. Just like the axiomatic

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approach to index numbers,2 this axiomatic approach to seasonal adjustment is flawed by the fact that sensible sets of axioms are inconsistent with each other. Lovell is up-front about the difficulty. Lovell (1963, p. 994) shows in Theorem 2.1 that the only operators that preserve sums, in the sense that xta + yta = (xt + yt )a , and preserve products, in the sense that xta yta = (xt yt )a are trivial in that either xta = xt or xta = 0. These two axioms are intuitive because the first one implies that accounting identities are unchanged by the adjustment, and the second one implies that the relationship between prices, quantities, and expenditure are not altered by the adjustment. Consequently, this result shows that two of the most intuitive axioms for seasonal adjustment are inconsistent; Lovell (1963, p. 994) characterizes this result as ‘disturbing’ and concludes that “it suggests that two quite simple criteria rule our the possibility of a generally acceptable ‘ideal’ technique for adjusting economic time series”. The solution to the inconsistency of the axiomatic approach to index numbers is the economic approach. The economic approach allows evaluation of index formulae by appealing to theory. Indexes that have a stronger connection to economic theory under weaker assumptions are judged to be superior. The usefulness of such criteria can be seen in how superlative indexes (Diewert, 1976) are now accepted as the definitive approach to constructing index numbers, not only by theorists, but also by statistical agencies. Diewert’s approach to constructing seasonal index numbers by defining seasonal economic aggregates can similarly answer how indexes should be seasonally adjusted by appealing to economic theory.

3. Review of Diewert’s Approach Diewert (1996b, 1998, 1999) treats the problem of seasonality as part of the economic approach to constructing bilateral index numbers and justifies different seasonal index number formulae on the basis of different separability assumptions.3 The theoretical basis of this work allows it to be used as a standard for seasonal adjustment. This section reviews Diewert’s approach. It focuses on three of his definitions of seasonal indexes: Annual, Year-over-year and Moving Year. The notation largely follows Diewert’s, but a different separability definition will be used. For exposition, a number of simplifying assumptions are made. First, the consumption space will be assumed to be of constant dimension in each season.

2 See

Swamy (1965). model can be easily adapted to represent a representative firm that produces a single output from multiple inputs. Multiple output firms introduce further complications (Fare and Primont, 1995).

3 The

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Diewert (1998) divides seasonal commodities into type-1 and type-2. Type-1 commodities are goods that are not available in every season. These are type of goods are particularly problematic for index number theory. The assumption that the dimension of the commodity space does not change in a season means that type-1 goods are not allowed to be randomly missing. Note that if a good is not consumed, it does not necessarily mean it was unavailable. It could be that the price of the good was above its reservation price. This case is observationally equivalent to the first, however, and in aggregate data it seems reasonable to assume that if a good is not consumed it is unavailable, so the focus is on type-2 goods. Diewert (1998) also assumes that type-2 seasonal commodities can be further divided into type-2a and type-2b commodities. Type-2a commodities are commodities whose seasonal fluctuations corresponding to rational optimizing behavior over a set of seasons where prices fluctuate but preferences for the commodity remain unchanged. Type-2b commodities are those where this does not apply. Type-2a commodities can be aggregated under normal aggregation assumptions In the following, I do not differentiate between the type-2 sub-commodities; it is implicitly assumed that any group of commodities contains at least one type-2b commodity. The effect of inflation is also ignored. Consequently, current period prices are used rather than spot prices. Thus, the cost indexes are futures price indexes rather than spot price indexes (Pollak, 1975). In a low inflation environment using the current period prices is not a major concern and it removes a level of complexity from the exposition. The simplifying assumptions can be relaxed without much difficulty following Diewert (1998, 1999). Some notation is needed to define the seasonal decision problem: Notation 1. Let m = 1, . . . , M denote the season, where M is the number of seasons, typically 4 or 12. Each season m has Nm commodities for each tm ] be the vector of positive year t ∈ {0, 1, . . . , T }. Let p tm = [p1tm , . . . , pN m tm tm tm = [q1 , . . . , qNm ] be the vector of commodities consumed in prices and q season m of year t. Annual vectors of prices and consumption are defined by pt = [p t1 , . . . , p tM ] and q t = [q t1 , . . . , q tM ], respectively. Let Ω denote the complete consumption space which is equal to RT (N1 +···+NM ) . Let x · y denote the standard inner product for vectors. The (representative) agent is assumed to have a transitive, reflexive, complete and continuous preference ordering on Ω. Preferences are also assumed to be nondecreasing and convex. Under these assumptions, preferences can be represented by a real-valued utility function U : Ω → R that satisfies: Condition 2. Continuity, positive monotonicity and quasi-concavity.

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The following decision problem then represents a basic utility maximization problem adapted to the seasonal notation: Problem 3 (Utility maximization). The (representative) agent solves the following intertemporal utility maximization problem where the utility function U (·) satisfies: T 0 1 T t t max (1) σt p · x W , U x ,x ,...,x x 0 ,x 1 ,...,x T

t=0

tm · x tm , σ is a strictly where x i has the same dimension as q i , p t · x t = M t m=1 p positive discount factor, and W is the discounted present value of intertemporal wealth at t = 0. Assume the vector [q 0 , . . . , q T ] solves the intertemporal utility maximization problem. Then W = Tt=0 σt p t · q t . Remark 4. The assumptions on preferences imply that the superior set, defined as S(u) ≡ {q | q ∈ Ω ∧ U (q) u}, is closed and convex. These properties of the superior set are important for duality, as they imply that preferences can be equivalently represented by a expenditure function. The dual representation is valid, because a closed convex set can be equivalently represented by the intersection of the closed half-spaces that contain it (Luenberger, 1969, Theorem 5). Diewert makes a series of structural assumptions on this general utility maximization problem to define annual, year-over-year, and moving year seasonal aggregates and economic indexes. In order to define annual economic indexes, Diewert (1998, 1999) assumes that the utility function in (1) takes the form U x0, x1, . . . , xT = F f x0 , f x1 , . . . , f xT , (2) where f (·) is positively linearly homogeneous (PLH) and satisfies Condition 2. The annual aggregator function f (·) treats each good in a different season as a different good. From Theorem 5.8 in Blackorby et al. (1978, pp. 206–207), the annual aggregator functions satisfy additive price aggregation, and define annual economic quantity aggregates because of their homogeneity. The dual unit expenditure function is the annual economic price aggregate. Annual Konüs (1939) true cost-of-living indexes and Malmquist (1953) economic quantity indexes can be defined: Definition 5 (Annual economic indexes). Annual economic price and quantity indexes are defined by K A (t, s) =

e(p t ) e(p s )

and M A (t, s) =

f (q t ) f (q s )

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for 0 s < t T , where e(·) is the unit expenditure function.4 Remark 6. The assumptions necessary to define the annual economic index are the weakest that address seasonality. These annual indexes can be tracked using standard index number theory. The resulting index number is deseasonalized by construction. The problem is that the index only provides a single measure per year, which is not frequent enough for many applications. The deseasonalization of the annual index is a by-product of the time aggregation that takes place. The annual indexes also represents Diewert’s (1980) preferred method for time aggregating economic data, as these assumptions place the fewest restrictions on intertemporal preferences. This method was implemented in constructing annual indexes from monthly data in Anderson et al. (1997a); the annual indexes calculated from seasonally adjusted and nonseasonally adjusted data are indistinguishable.5 Year-over-year indexes, which were suggested by Mudgett (1955) and Stone (1956), give a measure for each season, but require further assumptions. Diewert (1999, p. 50) assumes that the annual aggregator function, for each t ∈ {0, . . . , T }, takes the form f x t1 , x t2 , . . . , x tM = h f 1 x t1 , f 2 x t2 , . . . , f M x tM , (3) where f m (·) for m = 1, . . . , M is a seasonal aggregator function, with dimension Nm , of the annual aggregator function f (·). Under this assumption f (·) is an annual aggregator function over seasonal aggregator functions, f m (·). Note that since f (·) is PLH, so are the seasonal aggregators. The f m (·) are clearly valid seasonal aggregates, and are used to define year over year seasonal indexes: Definition 7 (Year-over-year seasonal economic indexes). For every season, denoted by m ∈ {1, . . . , M}, year-over-year seasonal economic price and quantity indexes are defined by K m (t, s) =

em (p tm ) em (p sm )

and M m (t, s) =

f m (q tm ) f m (q sm )

for 0 s < t T , where em (·) is the dual unit expenditure function for the season. Remark 8. The seasonal indexes are still comparing one season to a season in a previous year.

4 Homogeneity

of the annual aggregator functions implies the existence of annual unit expenditure functions. 5 The data are available from the MSI database on FRED at http://www.stls.frb.org

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The separable assumptions imply that the solution achieved by solving the general problem in (1) will also be the solution to the following multistage decision problem: in the first stage, the consumer chooses the optimal amount of wealth to allocate to each year to maximize the overall utility function U (·); in the second stage, for each year, the consumer chooses the optimal amount of the allocated wealth from the first stage to allocate to expenditure in each season to maximize h(·); and in the third stage, the consumer chooses the optimal quantities of the different seasonal goods subject to the allocated wealth to maximize f m (·). The multistage decision justifies defining annualized year-over-year indexes, by first constructing year-over-year indexes, suitably normalized in the base period, and then constructing an annual index from the seasonal indexes.6 Clearly, the annual index calculated in stages generally requires stronger assumptions than the actual annual indexes. Superlative indexes constructed in such a two-stage algorithm will not generally equal a superlative annual index, because superlative indexes only approximately satisfy consistency in aggregation (Diewert, 1978). Diewert’s (1999) last type of index is the moving year index.7 Diewert makes the additional assumption that U (·) satisfies U x 01 , . . . , x 0M ; . . . ; x T 1 , . . . , x T M T M m tm −1 , βm ψ f x =ψ (4) t=0 m=1

where βm are positive parameters that allow cardinal comparison of the transformed seasonal utilities ψ[f m (x tm )] and ψ[·] is a monotonic function of one positive variable defined by α if α = 0, z ψ(z) ≡ fα (z) ≡ (5) ln z if α = 0. This assumption implies that the intertemporal utility function U (·) is a constant elasticity of substitution (CES) aggregator of the seasonal aggregator functions f m (·). It also implies that the annual aggregator functions are CES in the seasonal aggregator functions: M 1 t1 2 t2 m tm M tM −1 h f x ,f x ,...,f x (6) =ψ βm ψ f x m=1

6 The

dual price index can be calculated by factor reversal. The effect of discounting is ignored in this discussion. In practice the effect of intertemporal discounting could be minimized by chaining the indices. 7 In Section 3 of his paper, Diewert (1999) discusses short-term season–season indexes, which are defined over subsets of nonseasonal commodities. Since seasonal behavior is excluded from these indexes, they are not covered here.

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for t = 0, . . . , T . A well-known result in index number theory, due to Sato (1976), is that the Sato–Vartia quantity index is exact for the CES functional form.8 Under the CES assumptions, the change in the annual aggregates can be tracked through the same two-stage method discussed previously. The difference is that at the second stage the aggregator functions are assumed to have the restricted CES form. As Diewert (1996a) notes, the strong assumption that U (·) is CES might be puzzling. Its usefulness is that aggregation to be extended to noncalendar years; under the CES assumption there exists an annual aggregator function for any sequential run of the M seasons. Thus, for each season, an annual index can be calculated from the previous M − 1 seasons (e.g. in July, an index could be calculated over the monthly data from the previous August through July). These moving year annual indices are already seasonally adjusted by construction. The notation will be simplified by the following lag function: Definition 9 (Lag function). The function for time t is defined by t if x 0, Lt (x) = t − 1 if x < 0. With this function the moving year indexes can be written as follows: Definition 10 (Moving year annual seasonal economic indexes). For each season m in year t, moving year annual seasonal economic price and quantity indexes are defined by −1 i Lt (m−i)i )]} ψ{ M i=1 βi ψ [e (p K(m, t, s) = (7) −1 i Ls (m−i)i )]} ψ{ M i=1 βi ψ [e (p and

i Lt (m−i)i )]} ψ −1 { M i=1 βi ψ[f (q M(m, t, s) = i Ls (m−i)i )]} ψ −1 { M i=1 βi ψ[f (q

(8)

for 0 s < t T , where em (·) is the dual unit expenditure function for the season. Remark 11. The moving year indexes provide an annual measurement for each season. Similar indexes could also be constructed for shorter or longer sequential runs. As a further sophistication, Diewert (1999) suggests centering the noncalendar years. The lag function is no longer sufficient; the following centering function will be used: 8 The

Sato–Vartia index was first defined by Vartia (1976a, 1976b) as the Vartia II index.

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Definition 12 (Centering sons M is defined by t +1 t (x) = t CM t −1

237

function). The function for time t and number of seaif x > M, if 0 x M, if x < 0.

The following definition centers the moving year indexes assuming there are an even number of seasons as is the norm. Effectively, the index is calculated by taking the M − 1 terms centered around the season m and adding half of the value of two extra terms: m + M/2 seasons ahead and m − M/2 seasons prior. Notationally, this accomplished by averaging two sequences of M terms where the second sequence is lagged one season relative to the first. Definition 13 (Centered moving year annual seasonal economic indexes). For each season m in year t, centered moving year annual seasonal economic price and quantity indexes are defined by M t 1 C K (m, t, s) = ψ βi ψ −1 ei p CM (m+M/2−i)i 2 i=1 M t 1 + βi ψ −1 ei p CM (m−1+M/2−i)i 2 i=1

M s 1 × ψ βi ψ −1 ei p CM (m+M/2−i)i 2 i=1 −1 M C s (m−1+M/2−i)i 1 −1 i e p M βi ψ + (9) 2 i=1

and

M (m, t, s) = ψ C

−1

t 1 βi ψ f i q CM (m+M/2−i)i 2 M

i=1

M t 1 + βi ψ f i q CM (m−1+M/2−i)i 2 i=1

M s 1 × ψ −1 βi ψ f i q CM (m+M/2−i)i 2 i=1

−1 M i C s (m−1+M/2−i)i 1 + βi ψ f q M 2 i=1 (10)

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for 0 s < t T , where em (·) is the dual unit expenditure function for the season. Remark 14. If there are an odd number of seasons, the notation for a centered index is much simpler. Remark 15. In practice, Diewert (1999) suggests calculating the annual indices as superlative indices also, as they can provide a second-order approximation to any aggregator function including the CES specification. Also, superlative indexes are usually chained, so that the reference period advances and is always one lag of the current period.9 The seasonal indexes reviewed in this section are connected to economic theory by their derivation from the utility maximization problem. The various indexes were derived by assuming more and more about the structure of the utility function. The section followed Diewert’s in that separability was only implicitly mentioned as the rationale for the functional structures. This seeming oversight is justified by the assumption that the nested utility functions are homothetic. Implicitly, Diewert is assuming that preferences are homothetically strictly separable at the annual and seasonal frequencies. Homotheticity, which Swamy (1965) called a ‘Santa Claus assumption’, allows the most elegant treatment of aggregation and statistical index number theory. But, homotheticity is a strong assumption and not necessary.

4. Duality and Separability Diewert implicitly connected the seasonal structures and indexes defined in Section 3 to an agent’s preferences through assuming homothetic strict separability. The seasonal indexes that Diewert derived can be supported under weaker conditions than he used; weakening the Diewert’s conditions provides broader theoretical support for the seasonal indexes and helps inoculate the theoretical approach to seasonality from criticism that claims the assumptions are unrealistic. As telegraphed at the end of the previous section, the first step to weakening Diewert’s conditions is to weaken the homotheticity assumption. Relaxing homotheticity leads naturally to focusing on the expenditure and distance function representation of preferences. The benefit of beginning with the expenditure and distance function is twofold. First, these two dual representations are always homogeneous in prices and quantities respectively. This property led Konüs (1939)

9 See

Anderson et al. (1997b) for a discussion of chaining.

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239

to define the true cost of living index through the expenditure function. Similarly, Malmquist (1953) originally used the distance function to define economic quantity indexes. The weakest conditions that support the various seasonal structures are naturally specified on the functions that are used in the definition of the indexes. Clearly, this argues for using the expenditure function; the similar argument for the distance function is obscured by the Diewert’s assumption of homotheticity. Second, the duality between the expenditure and distance functions is stronger than between other representation of preferences. Imposing functional structure on the expenditure function implies that the distance function will have the same property and vice versa. This is not generally true for other representations of preferences. In particular, assuming the utility function has a separable structure does not generally imply that the expenditure function will have the same structure, unless homotheticity is also imposed. These statements are clarified in the first subsection, which presents the expenditure and distance function and discusses their duality. The second step to weakening Diewert’s conditions is to make the separability assumptions explicit. This will make clear what Diewert is implicitly assuming when deriving the different functional structures that account for seasonal decision-making. The second subsection presents a variety of definitions of separability. Using these definitions, the weakest conditions that rationalize the annual and seasonal indexes can be established. These definitions also set up the subsequent section which discusses the moving year indexes. 4.1. The Dual Expenditure and Distance Functions The strong connection between the expenditure and distance function stems from the fact that they are both conic representation of preferences. The expenditure function, which is the negative of the support function, and the distance function are equivalent mathematical representations of a convex set. For a utility level, both functions are positive linearly homogeneous convex functions.10 Gorman (1970, p. 105) refers to the pair as ‘perfect’ duals as they always share the same properties. Section 2.5.3 in Blackorby et al. (1978) provides some further intuition for the strong connection between the expenditure and distance function by showing that the functions switch roles with regard to the indirect utility function; the distance function can be viewed as an indirect expenditure function and the expenditure function can be viewed as an indirect distance function. The two functions can be defined in terms of the utility problem in (1) as follows: Problem 16 (Expenditure minimization problem). Let R(U ) denote the range of U (·) with the infimum excluded and Ω+ the positive orthant of Ω. The expendi10 In the theory of convex functions, such functions are called gauge functions (Eggleston, 1958).

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ture function, e : Ω+ × R(U ) → R+ , that is dual to the utility function in (1) is defined as e p0 , . . . , pT , u T M tm 0 t = min (11) σt ptm · x U x , . . . , x u . x 0 ,...,x t

t=0 m=1

Problem 17 (Distance minimization problem). The distance function, d : Ω+ × R(U ) → R is defined by d q 0 , . . . , q T , u = min λ ∈ R+ | U q 0 /λ, . . . , q T /λ u . (12) λ

Although the distance function has been used in economics since at least Debreu (1954), it is less familiar. For a given u, the distance function measures the amount that q ∈ Ω must be scaled up or down such that q is in the boundary of the superior set: i.e. q/λ ∈ ∂S(u). For more discussion, see Deaton and Muellbauer (1980). Given the prior assumptions made on preferences in defining the utility function, the expenditure function will have the following properties: continuity in (p, u); nondecreasing, and concave in p; and increasing in u, where p ∈ Ω+ and u ∈ R(U ). The expenditure function has an additional property that is extremely useful in aggregation and statistical index number theory, positive linear homogeneity (PLH) in p, which means that ∀θ > 0, ∀(p, u) ∈ Ω+ × R(U ),

e(θp, u) = θ e(Π, u).

The PLH of the expenditure function holds without any such similar property holding for U (·). The properties of the expenditure function are referred to as: Condition 18. Joint continuity in (p, u), strict positive monotonicity in u, and positive monotonicity, positive linear homogeneity and concavity in p. As per our discussion, the distance function has the same properties except that it is strictly negatively monotonic in u and q takes the role of p: Condition 19. Joint continuity in (q, u), strict negative monotonicity in u, and positive monotonicity, positive linear homogeneity and concavity in q. The duality of the expenditure and distance functions can be made clearer using the fact that away from points of global satiation, U (q) u if and only if d(q, u) 1. Consequently, the expenditure function can be defined as e(p, u) = min p · q | q ∈ Ω ∧ d(q, u) 1 . q

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Similarly, the distance function can be defined as d(q, u) = min p · q | p ∈ Ω ∧ e(p, u) 1 . p

The two functions have identical functional form except that the roles of prices and quantities are reversed. The relationship to the utility function is clarified by noting that if preferences are homothetic then e(p, u) = e(p, 1)u

and d(q, u) = d(q, 1)u.

The homotheticity of U (·) implies that the unit distance function d(q, 1) is the PLH cardinalization of U (·) and is itself a utility function. As noted, the key properties of the expenditure and distance functions is their PLH and their functional equivalence. This first implies that separability assumptions can be applied to support the construction of seasonal indexes without assuming homotheticity. Without homotheticity, separability assumptions do not necessarily commute from one representation of preferences to another. The second property enables this to be avoided. 4.2. Separability The basic definition of separability used here is originally due to Bliss (1975). This definition is more general than the familiar definition developed independently by Sono (1961) and Leontief (1947a, 1947b) as it does not require differentiability. In addition, strict separability (Stigum, 1967) is used; strict separability is equivalent to Gorman’s (1968) definition of separability. Finally, complete (strict) separability is defined. Homothetic versions of the various forms of separability are also discussed. The definitions will be presented for the expenditure function. Equivalent definitions exist for the distance function with quantities replacing prices. Similar definitions also exist for the utility function, but let me reiterate that imposing separability on the utility function does not generally imply anything about the expenditure function and vice versa. The definition of separability depends on the existence of a collection of subsets being nested. Let B = {B1 , B2 , . . .} be a collection of subsets of some set. The collection is nested if ∀Bi , Bj ∈ B either Bi ⊆ Bj or Bi ⊇ Bj . Some further notation is required: Notation 20. Let I = {1, 2, . . . , T (N1 + · · · + NM )} denote the set of integers that identify variables over which preferences are defined. Define an n-partition of the set I to be a division of I into n subsets such that: n I [n] = I (1) , I (2) , . . . , I (r) , . . . , I (n) = I (j ) , j =1

(13)

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where ∀j , k, I (j ) ∩ I (k) = ∅; and ∀j , I (j ) = ∅. Corresponding to I [n] , Ω can be expressed as the Cartesian product of n subspaces: Ω=

Ω (j ) , × j =1 n

where for every j , the cardinality of Ω (j ) is given by I (j ) .11 The goods vector can then be written as q = (q (1) , q (2) , . . . , q (n) ) and the price vector p = (p (1) , p (2) , . . . , p (n) ) where the n categories denote general sectors, which are years or seasons in this chapter; if qi is in the kth sector then qi is a component of (k) q (k) ∈ Ω (k) and pi is a component of p (k) ∈ Ω+ . For simplicity, only the case where a sector is separable from its complement in Ω is presented. The generality lost by making this assumption is not a problem for the seasonal structures. The assumption means that the partition used in definitions of separability assumes n = 2 rather than the fully general case where n = 3. The general case can be found in Blackorby et al. (1978). The following function will be used: (r) Definition 21. Define γ r : Ω+ × R(U ) → ℘ (Ω+ ), where ℘ denotes the power set, to be a mapping whose image is γ r p (j ) , p (r) , u¯ (r) = pˆ (r) ∈ Ω+ e p (j ) , pˆ (r) , u¯ e p (j ) , p (r) , u¯ . (14)

This function defines a collection of subsets (j ) (r) ¯ = γ p (j ) , p (r) , u¯ p (j ) ∈ Ω+ ∧ p (r) ∈ Ω+ Γ r (u)

(15)

for a fixed scalar u¯ ∈ R(U ). Properties of the sets defined in (15) are used to define separability, strict separability, and complete (strict) separability: Definition 22 (Separability). The set of variables indexed by I (r) is separable in e(·) from its complement in I [n] if Γ r (u) ¯ is nested for every u¯ ∈ R(U ); Definition 23 (Strict separability). The set of variables indexed by I (r) is strictly separable in e(·) from its complement in I [n] if γ r p (j ) , p (r) , u = γ r p˜ (j ) , p (r) , u (j )

(r)

for all (p (j ) , p (r) , u) ∈ Ω+ × Ω+ × R(U ); 11 The

goods are trivially assumed to be conveniently ordered so that Ω is equal to the Cartesian product.

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and Definition 24 (Complete (strict) separability). The expenditure function is completely (strictly) separable in the partition I [m] ⊆ I [n] if every proper subset of I [m] is separable from its complement in I [m] . Remark 25. Separability is implied by either strict separability or complete separability, but the converse is not generally true. Similarly, strict separability is implied by complete strict separability but not the converse. Remark 26. Multiple separable sectors are not precluded; defining multiple separable sectors simply requires repeated application of the appropriate definition. Remark 27. The definition of complete (strict) separability is sensible only if there are at least three separable sectors in I [m] . Consequently, the definition of (strict) separability is implicitly being applied at least three times to define at least three (strictly) separable sectors in I [m] prior to considering their proper subsets. Defining homothetic (strict) separability for the expenditure function is more complicated as the function is already PLH in p. Note that both separability and (r) strict separability define a preference ordering on Ω+ for every u¯ ∈ R(u); pˆ (r) is (r) ¯ e(p (j ) , pˆ (r) , u) ¯ preferred to p (r) on Ω+ conditionally on u¯ if e(p (j ) , p (r) , u) (j ) (j ) ∈ Ω+ . If a consumer is indifferent between pˆ (r) and p (r) for for every p this conditional preference ordering, they are always indifferent between λpˆ r and λp r for every λ > 0. To see this, suppose it is not true. Then there exists pˆ (r) , p (r) and λ > 0 such that the consumer conditionally strictly prefers either λpˆ (r) or λp (r) although they are indifferent between pˆ (r) and p (r) . Without loss of generality, suppose λpˆ (r) is strictly preferred to λp (r) . This implies ¯ > e(p˜ (j ) , λpˆ (r) , u). ¯ Multhat that there exists a p˜ (j ) , such that e(p˜ (j ) , λp (r) , u) tiply both sides by 1/λ. Homogeneity of the expenditure function implies that ¯ > e( λ1 p˜ (j ) , pˆ (r) , u). ¯ This violates the assumption that the cone( λ1 p˜ (j ) , p (r) , u) (r) sumer is indifferent between pˆ and p (r) . Consequently, indifference between pˆ (r) and p (r) implies that the consumer is indifferent between λpˆ (r) and λp (r) for (r) (r) every λ > 0 for all (pˆ (r) , p (r) ) ∈ Ω+ × Ω+ and the conditional preference (r) ordering on Ω+ is always homothetic for a particular u¯ ∈ R(U ). There is however a sensible definition of homothetic (strict) separability for the expenditure function. Generally, the conditional preference ordering on the rth sector depends on u. If it does not depend on u, the sector is defined to be homothetically (strictly) separable. The definition will need the following:

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Notation 28. Define 0 I = I (0) ∪ I = {0, 1, 2, . . . , T (N1 + · · · + NM )}. Let n

[n] = 0I

I (j )

j =0

represent an extended partition. The following definition uses this extended partition: Definition 29 (Homothetic (strict) separability). The rth sector is homothetically (strictly) separable if it is (strictly) separable from its complement in 0 I [n] . This condition implies that the conditional preference ordering is not dependent on u. The rationale for calling this condition homothetic (strict) separability is that it implies, and is implied by, homothetic (strict) separability of the utility function. As noted previously, homothetic separability is an exception to the statement that, in general, (strict) separability of one of the representations of preferences has no implications for separability of the other dual representations. It should be apparent that Diewert is implicitly assuming homothetic separability, so that he can use the dual unit expenditure function.

5. Seasonal Decision-Making With the separability apparatus developed in the previous section, the seasonal indexes developed in Section 3 can be revisited. First, note that a current year annual aggregate can be defined if the expenditure function is separable. Theorem 30. Let e(·) satisfy Condition 18. Then e(·) is separable in I [m] ⊆ I [n] if and only if there exist m + 1: (r)

er : Ω+ × R(U ) → R+ , and

eˆ :

× m

r=1

r = 1, . . . , m,

r (j ) R e × Ω+ × R(U ) → R+

each satisfying the following regularity conditions in prices only, (i) continuity, (ii) positive monotonicity, (iii) positive linear homogeneity, and (iv) concavity,12

12 e(·) ˆ

satisfies these properties if it has these properties in (e1 (p (1) , u), . . . , em (p (m) , u)).

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such that e(p, u) = eˆ e1 p (1) , u , . . . , er p (r) , u , . . . , em p (m) , u , p (c) , u . (16) ˆ is continuous and Furthermore if e(·) is strictly separable in I [m] ⊆ I [n] , e(·) there exists an appropriate normalization of the expenditure function such that er p (r) , u = e p¯ 1 , . . . , p¯ (r−1) , p (r) , p¯ (r+1) , . . . , p¯ (m) , p¯ (c) , u , (j )

where p¯ (j ) ∈ Ω+ , j = 1, . . . , r − 1, r + 1, . . . , m, c are arbitrary reference vectors. Moreover, the following apply: 1. 2. 3. 4.

e(·, ˆ u) is increasing; each er (·) satisfy Condition 18; e(·, ˆ u) and each er (·, u) inherit (partial) differentiability in prices;13 and each er (p (r) , ·) inherits (strict) convexity and positive linear homogeneity in u.

Proof. Follows from Theorems 3.4 and Corollaries 3.5.2 and 4.1.4 in Blackorby et al. (1978, pp. 70, 80, 112). By the strong duality of the expenditure and distance functions, the equivalent theorem holds for the distance function. The following corollary is immediate: Corollary 31. The expenditure and distance functions are separable at annual frequencies if and only if e p 0 , . . . , p T , u = eˆ e0 p 0 , u , e1 p 1 , u , . . . , eT p T , u , u and d q 0 , . . . , q T , u = dˆ d 0 q 0 , u , d 1 q 1 , u , . . . , d T q T , u , u . An annual separability assumption on either the expenditure or distance function is sufficient to define annual economic indexes, albeit ones that depend on u. This is significantly weaker than assuming homothetic strict separability. Unfortunately, separability is not quite enough, because the annual economic indexes cannot be guaranteed to satisfy weak factor reversal under only separability even though strong factor reversal holds. Consequently, separability of the expenditure function is sufficient to define economic aggregates, but strict separability is necessary (and sufficient) to define annual economic indexes. Not surprisingly, since

13 If

the parent function is directionally differentiable in p, then the sectoral functions are partially differentiable in p(r) .

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the difference between separability and strict separability disappears under homotheticity, if homothetic separability is assumed weak factor reversal holds – in fact homotheticity makes weak and strong factor reversal equivalent. Strict separability is still a much weaker assumption than homothetic separability. Similar assumptions rationalize year-over-year aggregates. Define a partition I [M,t] = {I (1,t) , I (2,t) , . . . , I (M,t) } where for every t, I (m,t) indexes p tm for m ∈ {1, . . . , M}. Furthermore, define the partition {I [M,0] , C} where C indexes the complement of I [M,0] . Corollary 32. The expenditure and distance functions are separable at seasonal frequencies if and only if the expenditure function is separable in I [M,0] from its complement so that e p 0 , . . . , p T , u = eˆ e01 p 01 , u , . . . , e0M p 0M , u , . . . , eT 1 p T 1 , u , . . . , eT M p T M , u , u and the distance function is separable in I [M,0] from its complement so that d q 0 , . . . , q T , u = dˆ d 01 q 01 , u , . . . , d 0M q 0M , u , . . . , d T 1 q T 1, u , . . . , d T M q T M , u , u . The discussion about annual aggregates and indexes is appropriate here as well. Consequently, year over year seasonal indexes can be rationalized by only the imposition of strict separability. Note that there is another generalization here. The seasonal indexes are not nested inside an annual index, so the assumptions on preferences are relaxed somewhat. Of course, the same result could be applied to Definition 7. In this case, the seasonal pattern of the decision problem does not imply that it is not separable at frequencies higher than a year. It simply implies that preferences are not stationary.14 This type of time variation can be handled by the index approach, at least to some extent. At least some of the time-varying utility functions that have been used to model seasonal behavior fit into this framework: for example, Osborn (1988). The results so far demonstrate that homotheticity is not necessary to rationalize constructing seasonal index numbers. As discussed previously, separability in the utility and expenditure functions are not generally related to each other. Consequently, the developments in this section extend the class of preferences that can be used to justify seasonal aggregates and indexes. The most interesting extension addresses the moving year seasonal indexes, however.

14 It

might be sensible to refer to these kind of preferences as cyclostationary. See Gardner and Franks (1975) for a definition of cyclostationarity for random variables.

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247

In order to derive the moving year index, Diewert assumed that the utility function had a constant elasticity form. Diewert’s modus operandi was to “systematically list separability assumptions on intertemporal preferences” to rationalize the index numbers. The CES assumption does not seem to follow from any separability condition, so the assumption appears out of place. This apparent problem can be rectified. Either one of the following conditions is sufficient: Condition 33. e(·) is differentiable such that ∀i, ∂e(p, u)/∂pi > 0 for all p ∈ Ω+ , and that each sectoral function er (·) can be chosen to be differentiable; or ¯ implies Condition 34. For all prices, pˆ (r) ∈ γ r (p (j ) , p (r) , u) e∗ p (j ) , pˆ (r) , u¯ < e∗ p (j ) , p (r) , u¯ for all p (j ) ∈ Ω (j ) and for each u¯ ∈ R(U ) where e∗ (·) denotes the extension of the expenditure function to the boundary by continuity from above.15 The following theorem gives a representation for complete strict separability: Theorem 35 (Complete strict separability representation). Let the expenditure function, e(·) be completely strictly separable in I [n] .16 If e(·) satisfies Condition 18 and either Condition 33 or Condition 34 then there exists a function Γ : R(U ) → R+ and n functions, (r)

er : Ω+ × R(U ) → R+ all satisfying regularity conditions 1–4 from Theorem 30, such that either

n 1/ρ(u) ρ(u) r (r) e(p, u) = Γ (u) e p ,u , 0 = ρ(u) 1, r=1

or

e(p, u) = Γ (u)

where

m

r=1 ρ

n

r

e p

(r)

,u

ρ r (u)

,

ρ r (u) > 0 ∀r,

r=1 r (u)

= 1.

To prove this theorem, the following lemma is needed:

15 This condition rules out thick indifference curves for the conditional preordering on Ω (r) . 16 Notice that the complement of the union of the separable sectors is of zero dimension.

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Lemma 36. Assume that e(·) is continuous and that the commodities indexed by I (r) are separable from their complement. Then letting e∗ (·), eˆ∗ (·) and er∗ (·) denote the extensions of e(·), e(·) ˆ and er (·) in (16), respectively, to Ω × R(U ), (j ) r∗ (r) Ω × R(e ) × R(U ), Ω × R(U ) by continuity from above, ∀p ∈ Ω, e∗ (p, u) = eˆ∗ p (j ) , er∗ p (r) , u , u .

(17)

Moreover, er∗ (·) satisfies conditions 1–4 from Theorem 30. Proof. Suppose that (17) is false under the assumptions for p ∈ ∂(Ω). For a given arbitrary u, let {ps } be a sequence in {p ∈ Ω+ | e(p, u) e(p , u)} converging to p . Then (j ) lim e(ps , u) = lim eˆ ps , er ps(r) , u , u s→∞ (j ) = eˆ lim ps , er lim ps(r) , u , u

s→∞

s→∞

= eˆ

∗

s→∞

(j )

ps , er∗ ps(r) , u , u

= e∗ (p , u) which contradicts the continuity of e∗ (·) from above. Since u was arbitrary, this establishes (17). The properties of er∗ (·) follow from the properties of er (·) by a similar argument. Complete Strict Separability Representation. Under Condition 33 the result follows from Theorem 4.9 of Blackorby et al. (1978, pp. 143–147). To prove the theorem under Condition 34, note that complete strict separability of e(·) in I [n] implies, by Corollary 4.8.4 in Blackorby et al. (1978, p. 142) that e(·) can be written as:

n ∗ r∗ (r) e p ,u ,u , e(p, u) = e (18) r=1

where e∗ (·) is increasing and each er ∗ (·) is homothetic. By Theorem 30, e(p, u) = eˆ e1 p (1) , u , . . . , en p (n) , u , u . By repeated application of Lemma 36, this representation can be extended to the boundary of Ω+ . The condition implies that the representation extended to the boundary can be taken to be strictly separable rather than just separable. Consequently, ∀r the sectoral utility function can be chosen as er p (r) , u = e p (r) , 0c , u ,

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where 0c is the zero element of the complement of Ω (r) .17 From the properties of the expenditure function, this equation implies that er (·) is PLH in p (r) . Substituting from (18) into this equation produces

n (r) r ∗ r∗ s r∗ (r) e p ,u = e e 0 , u + e p , u , u , r = 1, . . . , m. s=1 s=r

Let

Since

n r∗ s s=1 s=r e (0 , u)

(19)

= ar (u) for r = 1, . . . , n. Then (19) can be written as

er p (r) , u = e∗ er∗ p (r) , u + ar (u), u .

(20)

is PLH, this implies that, ∀λ > 0 and ∀r = 1, . . . , n, e∗ er∗ λp (r) , u + ar (u), u = λe∗ er∗ p (r) , u + ar (u), u .

(21)

er (·, u)

Homotheticity of each er∗ (·, u) implies that φ r e˙r p (r) , u = er∗ p (r) , u , r = 1, . . . , n,

(22)

where each φ r (·) is increasing and each e˙r (·) is PLH. Substituting this equation into (21) yields, for each r, e∗ φ r λe˙r p (r) , u + ar (u) , u = λe∗ φ r e˙r p (r) , u + ar (u), u . (23) Letting λ−1 = e˙r (p (r) , u), for each r, e∗ φ r 1 + ar (u) , u =

1 e˙r (p (r) , u)

e∗ φ r e˙r p (r) , u + ar (u), u . (24)

Rearranging terms, this implies, for each r, e∗ φ r e˙r p (r) , u + ar (u), u = e˙r p (r) , u e∗ φ r 1 + ar (u) , u . (25) Call the right-hand side of (25) e(Π ˜ r , u). Inverting (25) for fixed u, yields −1 φ r e˙r p (r) , u = e∗ e˜r p (r) , u , u − ar (u).

(26)

Using (22), substitute (26) into (18) to get

n −1 ∗ ∗ r (r) e(p, u) = e e˜ p , u , u + A(u), u , e

(27)

r=1

17 This is a slight abuse of notation as e(·) and er (·) are now referring to the extension to the boundary; this abuse will be continued throughout this proof, as it simplifies notation.

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where A(u) = − nr=1 ar (u). The fact that e(·) and each e(·) ˜ are PLH in user costs implies that

n −1 ∗ ∗ r (r) λe e˜ p , u , u + A(u), u e r=1

= e∗

n

e

∗−1

r (r) e˜ λp , u , u + A(u), u

r=1

for each u. Since u was arbitrary, this argument holds for every u ∈ R(U ), which implies that e(·) ˙ is a quasi-linear PLH function of the arguments r 1 e˙ Π , u , . . . , e˙r Π 1 , u . (28) This completes the proof by a theorem of Eichhorn (1974, p. 24).

The immediate corollary provides a basis for the moving year seasonal indexes using complete strict separability: Corollary 37. Define the partition I [M,T ] = I [1,0] , . . . , I [M,0] , . . . , I [m,t] , . . . , I [1,T ] , . . . , I [M,T ] . Assume, in addition to satisfying Condition 18, the expenditure function satisfies either Condition 33 or Condition 34. Then it has a CES functional form in the seasonal aggregates if it is completely strictly separable in I [M,T ] . These sufficient conditions may not be necessary. Nevertheless, they would seem to be the weakest separability conditions sufficient to rationalize the moving year indexes that can be expected to hold. This result gives some insight into the discussion in Section 3. It is not surprising that an index can be defined for noncalendar years if it is completely separable in seasons. Remember that complete separability means that any subset of the partition is also completely strictly separable. Thus, complete strict separability allows us to define aggregates over arbitrary partitions of the seasons. Noncalendar years are just one of the possibilities. For example, econometric seasonal adjustment is often done using filters that contain more than just 12 leads or lags. The CES or alternatively the complete strict separability assumption may seem overly strong, but there is a sensible argument for this condition. The calendar year is not necessarily intrinsically special. For example, the fiscal year may be more important economically. In the discussion at the start of Section 2, the consumer was normally assumed to re-optimize or re-plan at the beginning of the period. I adapted this to re-optimizing at the beginning of each year in order to finesse how strong the separability conditions needed to be. However, there is nothing

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intrinsic to seasonality to suggest that the consumer can not still be allowed to re-plan every period rather than sticking with his plan for an entire year. The presumption that seasonality in the data implies that decision problem is not separable at periods shorter than a year still seems reasonable. So the model is that the consumer solves a problem in say the first month of the year, where his or her preferences are separable over the year but not for any shorter timeperiod. Then in the next period, the consumer resolves a problem, where his or her preferences are separable over the year but not for any shorter time-period. This is called ‘rolling plan optimization’. But, these are not the same years. The first year runs from January through December and the second run from February through January. If this is viewed as being embedded in a larger, possibly infinite horizon problem, then this implies the existence of overlapping separable sectors. If each sector is strictly separable and January in either the first or second years is strictly essential, then Gorman’s (1968) overlapping theorem implies that January commodities in year 0, February through December commodities, and January year commodities in year 1 are all strictly separable. In fact, the theorem states that they are completely strictly separable. If this thought experiment is iterated, it implies that each month’s commodities are completely strictly separable. To the extent that it seems sensible that consumers plan over an annual horizon and re-plan throughout the year the CES assumption seems plausible.

6. Empirical Implications This chapter contains no empirical analysis, but there are some interesting implications of the index number approach to seasonality. Diewert (1999) brings up some practical reasons to favor the index number method. First, the method is perhaps less arcane than current econometric practices, and could be applied more easily. Second, the data indexes could be produced in a timely manner. Third, the data would be subject to fewer historical revisions, perhaps only those associated with switching to the centered version from a preliminary noncentered index after six months. These are cogent arguments for using index number method. However, most statistical agencies will require substantial empirical analysis before they would consider switching methods, so a few suggestions for future research seem warranted. A natural way to analyze seasonal adjustment is in the frequency domain. Examining the index formula advanced here should make it apparent that these formula remove all power at frequencies higher than annual. Consequently, it might be interesting to view the index numbers as acting like a low pass filter. This is in contrast to some seasonal adjustment methods, which are more like a notch filter: see Nerlove (1964). One well-known problem in finite filtering theory is that the optimal low pass filter is not realizable. An interesting question is whether seasonal indexes approximate the ideal filter by effectively pooling data.

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This is almost a stochastic index number viewpoint. Given the perspective that the index number formula effectively clips all higher order power, the indexes should be relatively smooth. Consequently, the moving year indexes, which average these seasonal indexes, should be expected to be exceptionally smooth. The moving year indexes should be expected to be isolating largely the long-run trend. In addition, the econometric adjustment literature often takes linearity as a desirable property for seasonal adjustment, despite the fact that the X-12 procedure, its predecessors and related methods are not, generally, linear. A fair amount of work has been undertaken trying to demonstrate that these procedures are approximately linear. The index number method suggests that the linearity criterion is misguided. Clearly, the index number adjustment is nonlinear. In fact, looking at Lovell’s axioms, it preserves products by definition, so it cannot preserve sums in general. An interesting question is whether or not a linear method can approximate the index number methods. If not, an open question would be whether there are nonlinear econometric methods that can approximate the index number approach. Finally, the fact that the index number approach satisfies the product preserving axiom suggests that economic indexes, if not adjusted using the index number methods, should be adjusted by techniques that are also product preserving rather than sum preserving. Furthermore, many economic time series are not indexes, so the index number approach is not applicable. Consequently, the development of econometric techniques that approximate the output of index number methods as closely as possible would be useful to maintain consistency.

7. Conclusion This chapter has further developed the rationale behind the index number approach. The class of preferences that can rationalize the seasonal indexes advocated by Diewert (1998, 1999) were extended. In particular, sufficient conditions for the moving year index based on a separability assumption were developed. Additionally, a heuristic argument was proposed based on Gorman’s (1968) overlapping theorem that supports this separability assumption if an agent reoptimizes over a new time horizon each period. Judging among econometric seasonal adjustment methods is confounded by the lack of obvious criteria; different optimality criteria lead to alternative techniques. The index number approach to seasonality solves this indeterminacy problem. By referring to the economic theory as a arbiter, it provides a criterion for judging among different techniques. It does not obviate the need or probably the desire to have econometric techniques for seasonal adjustment; series that are not economic indexes can not be adjusted this way. However, because it provides a standard, it could also be used to judge among econometric methods in situations where it

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itself is not directly applicable. In order for the index number approach to serve as an arbiter, empirical characterizations of its properties are required.

Acknowledgements I wish to thank William Barnett, Erwin Diewert, Barry Jones and Heinz Schättler for helpful discussions and comments. The views presented are solely my own and do not necessarily represent those of the Federal Reserve Board or its staff.

References Anderson, R.G., Jones, B.E. and Nesmith, T.D. (1997a). Building new monetary services indexes: Concepts, data. Federal Reserve Bank of St. Louis Review 79(1), 53–82. Anderson, R.G., Jones, B.E. and Nesmith, T.D. (1997b). Monetary aggregation theory and statistical index numbers. Federal Reserve Bank of St. Louis Review 79(1), 31–51. Bell, W.R. and Hillmer, S.C. (1984). Issues involved with the seasonal adjustment of economic time series. Journal of Business and Economic Statistics 2(4), 291–320. Blackorby, C., Primont, D. and Russell, R.R. (1978). Duality, Separability, and Functional Structure: Theory and Economic Applications. New York: North-Holland. Bliss, C.J. (1975). Capital Theory and the Distribution of Income, Vol. 4. Amsterdam, New York: North-Holland. Deaton, A. and Muellbauer, J. (1980). Economics and Consumer Behavior. Cambridge, New York: Cambridge University Press. Debreu, G. (1954). Representation of a preference ordering by a numerical function. In Thrall, R., Coombs, C. and Davis, R. (Eds), Decision Processes. New York: Wiley, pp. 159–195. Diewert, W.E. (1976). Exact and superlative index numbers. Journal of Econometrics 4(2), 115–145. Diewert, W.E. (1978). Superlative index numbers and consistency in aggregation. Econometrica 46(4), 883–900. Diewert, W.E. (1980). Aggregation problems in the measurement of capital. In Usher, D. (Ed.), The Measurement of Capital. Chicago: University of Chicago Press, pp. 433–528. Diewert, W.E. (1983). The treatment of seasonality in a cost-of-living index. In Diewert, W.E. and Montmarquett, C. (Eds), Price Level Measurement. Ottawa: Statistics Canada, pp. 1019–1045. Diewert, W.E. (1996a). Axiomatic and economic approaches to international comparisons. National Bureau of Economic Research, Inc., NBER Working Papers 5559. Diewert, W.E. (1996b). Seasonal commodities, high inflation and index number theory. Department of Economics Discussion Paper No. 96-06. University of British Columbia, UBC Departmental Archives. Diewert, W.E. (1998). High inflation, seasonal commodities, and annual index numbers. Macroeconomic Dynamics 2(4), 456–471. Diewert, W.E. (1999). Index number approaches to seasonal adjustment. Macroeconomic Dynamics 3(1), 48–68.

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Eggleston, H.G. (1958). Convexity, Vol. 47. Cambridge: Cambridge University Press. Eichhorn, W. (1974). Characterization of the CES production functions by quasilinearity. In Eichhorn, W., Henn, R., Opitz, O., Shepard, R.W., Beckman, M. and Kunzi, H.P. (Eds), Production Theory: Proceedings of an International Seminar Held at the University of Karlsruhe, May–July 1973, Vol. 99. Berlin, New York: Springer-Verlag. Fare, R. and Primont, D. (1995). Multi-Output Production and Duality: Theory and Applications. Boston: Kluwer Academic. Fisher, I. (1922). The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability, Vol. 1. Boston, New York: Houghton Mifflin Company. Gardner, W.A. and Franks, L.E. (1975). Characterization of cyclostationary random signal processes. IEEE Transactions on Information Theory 21(1), 4–14. Ghysels, E. (1988). A study toward a dynamic theory of seasonality for economic time series. Journal of the American Statistical Association 83(401), 168–172. Gorman, W.M. (1968). The structure of utility functions. Review of Economic Studies 35(4), 367–390. Gorman, W.M. (1970). Quasi-separability. London School of Economics. Grether, D.M. and Nerlove, M. (1970). Some properties of ‘optimal’ seasonal adjustment. Econometrica 38(5), 682–703. Hylleberg, S. (Ed.) (1992). Modelling Seasonality. Oxford: Oxford University Press. Jorgenson, D.W. (1964). Minimum variance, linear, unbiased seasonal adjustment of economic time series. Journal of the American Statistical Association 59(307), 681–724. Konüs, A.A. (1939). The problem of the true index of the cost of living. Econometrica 7(1), 10–29. Leontief, W. (1947a). Introduction to a theory of the internal structure of functional relationships. Econometrica 15(4), 361–373. Leontief, W. (1947b). A note on the interrelation of subsets of independent variables of a continuous function with continuous first derivatives. Bulletin of the American Mathematical Society 53, 343–350. Lovell, M.C. (1963). Seasonal adjustment of economic time series and multiple regression. Journal of the American Statistical Association 58(304), 993–1010. Lovell, M.C. (1966). Alternative axiomatizations of seasonal adjustment. Journal of the American Statistical Association 61(315), 800–802. Luenberger, D.G. (1969). Optimization by Vector Space Methods. New York: Wiley. Malmquist, S. (1953). Index numbers and indifference surfaces. Tradajos de Estatistica 43, 209–242. Miron, J.A. (1996). The Economics of Seasonal Cycles. Cambridge, MA: MIT Press. Miron, J.A. and Zeldes, S.P. (1988). Seasonality, cost shocks, and the production smoothing models of inventories. Econometrica 56(4), 877–908. Mudgett, B.D. (1955). The measurement of seasonal movements in price and quantity indexes. Journal of the American Statistical Association 50(269), 93–98. Nerlove, M. (1964). Spectral analysis of seasonal adjustment procedures. Econometrica 32(3), 241–286. Nerlove, M., Grether, D.M. and Carvalho, J.L. (1979). Analysis of Economic Time Series: A Synthesis. New York: Academic Press. Osborn, D.R. (1988). Seasonality and habit persistence in a life cycle model of consumption. Journal of Applied Statistics 3(4), 255–266.

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Pollak, R.A. (1975). Subindexes in the cost of living index. International Economic Review 16(1), 135–150. Sato, K. (1976). The ideal log-change index number. Review of Economics and Statistics 58(2), 223–228. Sono, M. (1961). The effect of price changes on the demand and supply of separable goods. International Economic Review 2(3), 239–271. Stigum, B.P. (1967). On certain problems of aggregation. International Economic Review 8(3), 349–367. Stone, R. (1956). Quantity and price indexes in national accounts. Paris: Organisation for European Economic Co-operation. Swamy, S. (1965). Consistency of Fisher’s tests. Econometrica 33(3), 619–623. Vartia, Y.O. (1976a). Ideal log-change index numbers. Scandinavian Journal of Statistics 3, 121–126. Vartia, Y.O. (1976b). Relative changes and economic indices. Technical report. Helsinki: Research Institute of the Finnish Economy. Wallis, K.F. (1974). Seasonal adjustment and relations between variables. Journal of the American Statistical Association 69(345), 18–31.

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Chapter 12

Hopf Bifurcation within New Keynesian Functional Structure William A. Barnett∗ and Evgeniya Aleksandrovna Duzhak Department of Economics, University of Kansas, Lawrence, KS 66045

Abstract Grandmont (1985) found that the parameter space of the most classical dynamic models are stratified into an infinite number of subsets supporting an infinite number of different kinds of dynamics. But Grandmont provided his result with a model in which all equilibria are Pareto optimal. Hence he was not able to reach conclusions about policy relevance. In this chapter we explore bifurcation within the class of New Keynesian models. We provide background theory relevant to locating bifurcation boundaries in log-linearized New Keynesian models with Taylor policy rules or inflation-targeting policy rules. Further relevant theory is available in Barnett and Duzhak (2006). Empirical implementation will be the subject of a future paper.

Keywords: bifurcation, new Keynesian, Hopf bifurcation, robustness, macroeconometrics, policy JEL: C14, C22, E37, E32

1. Introduction 1.1. The History Grandmont (1985) found that the parameter space of even the simplest, classical models are stratified into bifurcation regions. This result changed prior views that different kinds of economic dynamics can only be produced by different kinds of structures. But he provided that result with a model in which all policies are Ricardian equivalent, no frictions exist, employment is always full, competition is perfect, and all solutions are Pareto optimal. Hence he was not able to reach conclusions about the policy relevance of his dramatic discovery. Years of controversy ∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18012-1

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followed, as evidenced by papers appearing in Barnett et al. (2004) and Barnett et al. (2005). The econometric implications of Grandmont’s findings are particularly important, if bifurcation boundaries cross the confidence regions surrounding parameter estimates in policy-relevant models. Stratification of a confidence region into bifurcated subsets seriously damages robustness of dynamical inferences. The dramatic transformation of views precipitated by Grandmont’s paper was criticized for lack of policy relevance. As a result, Barnett and He (1999, 2001, 2002) investigated a continuous-time traditional Keynesian structural model (the Bergstrom–Wymer model), and found results supporting Grandmont’s conclusions. Barnett and He found transcritical, codimension-two, and Hopf bifurcation boundaries within the parameter space of the Bergstrom–Wymer continuoustime dynamic macroeconometric model of the UK economy. That highly regarded Keynesian model was produced from a system of second-order differential equations. The model contains frictions through adjustment lags, displays reasonable dynamics fitting the UK economy’s data, and is clearly policy relevant. See Bergstrom and Wymer (1976), Bergstrom (1996), Bergstrom et al. (1992, 1994) and Bergstrom and Nowman (2007). Barnett and He found that bifurcation boundaries cross confidence regions of parameter estimates in that model, such that both stability and instability are possible within the confidence regions. The Lucas critique has motivated development of Euler equations models. Hence, Barnett and He (2006) chose to continue the investigation of policy relevant bifurcation by searching the parameter space of the best known of the policy relevant Euler equations macroeconometric models: the Leeper and Sims (1994) model. The results further confirm Grandmont’s views, but with the finding of an unexpected form of bifurcation: singularity bifurcation. Although known in engineering, singularity bifurcation has not previously been encountered in economics. Barnett and He (2004) have made clear the mathematical nature of singularity bifurcation and why it is likely to be common in the class of modern Euler equation models rendered important by the Lucas critique. Recently, interest in policy in some circles has moved away from Euler equations models to New Keynesian models, which have become common in monetary policy formulations. As a result, in this chapter we explore bifurcation within the class of New Keynesian models. We study forward-looking and current-looking models and hybrid models having both forward and current looking features. We provide theory relevant to detecting Hopf bifurcation, with the setting of the policy parameters influencing the existence and location of the bifurcation boundary. No other form of bifurcation is possible in the three equation log-linearized New Keynesian models that we consider. Further relevant theory is available in Barnett and Duzhak (2006). In a future paper, we shall report on our results solving numerically for the location and properties of the bifurcation boundaries and their dependency upon policy-rule parameter settings.

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Beginning with Grandmont’s findings with a classical model, we continue to follow the path from the Bergstrom–Wymer policy-relevant Keynesian model, then to the Euler equation macroeconometric models, and now to New Keynesian models. At this stage of our research, we believe that Grandmont’s conclusions appear to hold for all categories of dynamic macroeconomic models, from the oldest to the newest. So far, our finding suggest that Barnett and He’s initial findings with the policyrelevant Bergstrom–Wymer model appear to be generic. 1.2. Bifurcation Background During the past 30 years, the literature in macroeconomics has moved from comparative statics to dynamics, with many such dynamic models exhibiting nonlinear dynamics. The core of dynamics is bifurcation theory, which becomes especially rich in its possibilities, when the dynamics are nonlinear. The parameter space is stratified into subsets, each of which supports a different kind of dynamic solution. Since we do not know the parameters with certainty, knowledge of the location of the bifurcation boundaries is of fundamental importance. Without knowledge of the location of such boundaries, there is no way to know whether the confidence region about the parameters’ point estimates might be crossed by such a boundary, thereby stratifying the confidence region itself and damaging inferences about dynamics. There are different types of bifurcations, such as flip, fold, transcritical and Hopf. Hopf bifurcation is the most commonly seen type among economic models, since the existence of a Hopf bifurcation boundary is accompanied by regular oscillations in an economic model, where the oscillations may damp to a stable steady state or may never damp, depending upon which side of the bifurcation boundary the point estimate of the parameters might lie.1 The first theoretical work on Hopf bifurcation is in Poincaré (1892). The first specific study and formulation of a theorem on Hopf bifurcation appeared in Andronov (1929), who, with his coauthors, developed important tools for analyzing nonlinear dynamical systems. A general theorem on the existence of Hopf bifurcation was proved by Hopf (1942). While the work of Poincaré and Andronov was concerned with two-dimensional vector fields, the theorem of Hopf is valid in n dimensions. When parameters cross a bifurcation boundary such that the solutions change from stable to limit cycles, it is common in mathematics to refer to the resulting bifurcation as Poincaré–Andronov–Hopf bifurcation. Hopf bifurcations have been encountered in many economic models, such as Benhabib and Nishimura (1979). Historically, optimal growth theory received the most attention as the subject of bifurcation analysis. Hopf bifurcations were also

1 See,

e.g., Benhabib and Nishimura (1979), Kuznetsov (1998) and Seydel (1994).

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found in overlapping generations models.2 These studies show that the existence of a Hopf bifurcation boundary results in the existence of closed curves around the stationary state, with the solution paths being stable or unstable, depending upon which side of the bifurcation boundary contains the parameter values. New Keynesian models have become increasingly popular in policy analysis. The usual New-Keynesian log-linearized model consists of a forward-looking IScurve, describing consumption smoothing behavior, and a New Keynesian Phillips curve, derived from price optimization by monopolistically competitive firms in the presence of nominal rigidities. This chapter pursues our first steps towards a bifurcation analysis of New Keynesian functional structure. We study the system using eigenvalues of the linearized system of difference equations and find the possibility of existence of a Hopf bifurcation. We also investigate different monetary policy rules relative to bifurcation.

2. Model Our analysis is centered on the New Keynesian functional structure described in this section. The main assumption of New Keynesian economic theory is that there are nominal price rigidities preventing prices from adjusting immediately and thereby creating disequilibrium unemployment. Price stickiness is often introduced in the manner proposed by Calvo (1983). The model below, used as the theoretical background for our log linearized bifurcation analysis, is based closely upon Walsh (2003), Section 5.4.1, pp. 232–239, which in turn is based upon the monopolistic competition model of Dixit and Stiglitz (1977).3 It is assumed that there is a continuum of firms of measure 1, and firm j ∈ [0, 1] produces good cj at price pj . Since all goods are differentiated in the monopolistically competitive manner, each firm has pricing power over the good it sells. The composite good that enters the consumers’ utility func1 (θ−1)/θ tions is Ct = ( 0 cj t dj )θ/(θ−1) , and its dual price aggregator function is 1

1/(1−θ) , where θ > 1 is the price elasticity of demand for Pt = [ 0 pj1−θ t dj ] each individual good, assumed to be the same for each good j .4 As θ → ∞, the individual goods become closer and closer substitutes, and as a consequence, individual firms have less market power. Price rigidity faced by the firm is modeled as follows: a random fraction, 0 < ξ < 1, of firms does not adjust price in each period. The remaining firms

2 Aiyagari

(1989), Benhabib and Day (1982), Benhabib and Rustichini (1991), Gale (1973). 3 Other relevant references include Shapiro (2006) and Woodford (2003). 4 The duality proof can be found in Walsh (2003, p. 233).

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adjust prices to their optimal levels pj∗t , j ∈ [0, 1]. Accordingly, it follows from the formula for the price aggregator function that the aggregate price in period t satisfies the equation: 1−θ Pt1−θ = (1 − ξ )(Pt∗ )1−θ + ξ Pt−1 ,

(2.1)

where ξ is the probability that a price will remain unchanged in any given period 1 and Pt∗ = [ 0 pj∗1−θ dj ]1/(1−θ) is the optimal aggregate price at time t. t Therefore, the aggregate price level in period t is determined by the fraction, 1−ξ , of firms that adjust and charge a new optimal price pj∗t and by the remaining fraction of firms that charge the previous period’s price. 2.1. Households Consumers derive utility from the composite consumption good, Ct , real money balances, and leisure. We define the following variables for period t: Mt Nt Bt Wt it Πt

money balances, labor quantity, real balances of one-period bonds, wage rate, interest rate, total profits earned by firms.

Consumers supply their labor in a competitive labor market and receive labor income, Wt Nt . Consumers own the firms producing consumption goods and receive all profits, Πt . The representative consumer can allocate wealth to money and bonds and choose the aggregate consumption stream by solving the following problem: max Et

∞ i=0

βi

1+η Nt+i γ Mt+i 1−b + −χ 1−σ 1 − b Pt+i 1+η 1−σ Ct+i

subject to Mt Bt Ct + + = Pt Pt

(2.2)

Mt−1 Wt Bt−1 Nt + + Πt + (1 + it−1 ) Pt Pt Pt (2.3)

with scaling parameters γ and χ along with parameters: β time-discount factor, σ degree of relative risk aversion, b−1 interest elasticity of money demand, η−1 wage elasticity of labor supply.

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In practice, the decision of a “representative consumer” is for per capita values of all quantities. The households’ first-order conditions are given by Pt −σ −σ Ct = β(1 + it )Et (2.4) C , Pt+1 t+1 (Mt /Pt )−σ it γ (2.5) = , −σ 1 + it Ct η

χ

Nt Wt −σ = P . Ct t

(2.6)

Equations (2.4)–(2.6) are Euler equations for consumption, money and labor supply, respectively. Following solution of (2.2) subject to (2.3), the representative consumer, in a second stage decision, allocates chosen aggregate consumption, Ct , over the continuum of goods, cj t , j ∈ [0, 1], to minimize the cost of consuming Ct .5 Let πt be the inflation rate at time t. Following Walsh (2003, p. 244), we log-linearize the households’ first-order condition (2.4) around the steady state inflation rate, π = 0. With aggregate output by firms equaling aggregate consumption, Ct , in the steady state, we get 1 yˆt = Et yˆt+1 − (2.7) (it − Et πt+1 ), σ where yˆt is the percentage deviation of output from its steady state. Writing (2.7) in terms of output gap, we get 1 xt = Et xt+1 − (it − Et πt+1 ) + ut , σ

(2.8)

f

where xt = (yˆt − yˆt ) is the gap between actual output percentage devif ation, yˆt , and the flexible-price output percentage deviation, yˆt , and where f f ut ≡ Et yˆt+1 − yˆt . Equation (2.8) can be viewed as describing the demand side of the economy, in the sense of an expectational, forward-looking IS curve. 2.2. Firms Firms hire labor and produce and sell consumption goods in a monopolistically competitive market. The production functions for goods cj t , j ∈ [0, 1], have the following form: cj t = Zt Nj t ,

5 The

first stage decision allocating over individual goods, conditionally upon composite goods demand, can be found in Walsh (2003, p. 233).

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where Nj t is time spent on production of good cj t during period t, and Zt is labor’s average product, assumed to be random with mean E(Zt ) = 1. Labor’s average product is drawn once for all industries, so has no subscript j . Firms make their production and price-setting decisions by solving the following two problems: Cost Minimization Problem For each period t, firm j selects labor employment, Nj t , to minimize labor cost, (Wt /Pt )Nj t , subject to the production functions’ constraints on technology. The resulting Lagrangian, with Lagrange multipliers ϕj , is Wt (2.9) Nj t + ϕj (Cj t − Zt Nj t ), j = 1, . . . , J, Pt which is minimized to solve for Nj t . The first-order condition to solve (2.9) is: Wt /Pt (2.10) . Zt As is usual for Lagrange multipliers, (2.10) can be interpreted as a shadow price. In this case, ϕj is the shadow price, or equivalently the real marginal cost, of producing Cj t . ϕt =

Pricing Decision Each firm j maximizes the expected present value of its profits by choosing price pj t . Recall that θ is the price elasticity of demand and is the parameter in the consumer quantity and price aggregator functions. Since that elasticity of demand is the same for all goods, the following relationship exists between consumption of each good and aggregate consumption: cj t pjθ t = Ct Ptθ . Using that result, the profit maximization problem for firm j can be written as:6 1−θ −θ ∞ p p j t j t max Et w i Δi,t+1 − ϕt+i (2.11) Ct+i , Pt+i Pt+i i=0

where Δi,t+1 = β i (Ct+1 \Ct )−σ is the discount factor; and the consumer price indexes, Pt+i , are taken as given by the firm for all i = 0, . . . , ∞. This yields the following first-order condition, which shows how adjusting firms set their prices, conditional on the current price level: ∞ i i 1−σ Et i=0 ξ β Ct+i ϕt+i (Pt+i /Pt )θ pt∗ θ (2.12) = . i i 1−σ θ−1 Pt θ − 1 Et ∞ i=0 ξ β Ct+i (Pt+i /Pt )

6 See,

e.g., Walsh (2003, p. 235).

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As in Walsh (2003, p. 237), we log-linearize (2.1) and (2.12) around the zeroinflation steady state equilibrium to get the following expression for aggregate inflation: πt = βEt πt+1 + κ˜ ϕˆ t ,

(2.13) f

) where κ˜ = (1−ξ )(1−βξ and where ϕˆt = (wˆ t − pˆ t ) − (yˆt − nˆ t ) = γ (yˆt − yˆt ) is ξ real marginal cost, expressed in terms of percentage deviations around the steady state. In particular, wˆ t is the percentage deviation of the wage rate around its steady state, pˆ t is the percentage deviation of Pt around its steady state, and nˆ t is the percentage deviation of Nt around its steady state. We can rewrite the relation for inflation, in terms of the output gap. Then (2.13) becomes

πt = βEt πt+1 + κxt ,

(2.14)

f yˆt

where xt = yˆt − is the gap between actual output-percentage-deviation from steady state and the flexible-price output percentage deviation from steady state, ) . with κ = γ κ˜ = γ (1−ξ )(1−βξ ξ We now have two equations. The first equation, (2.8), provides the demand side of the economy. It is a forward-looking IS curve that relates the output gap to the real interest rate. Equation (2.14) is the New-Keynesian Phillips curve, which represents the supply side, by describing how inflation is driven by the output gap and expected inflation. The resulting system of two equations has three unknown variables: inflation, output gap, and nominal interest rate. We need one more equation to close the model. The remaining necessary equation will be a monetary policy rule, in which the central bank uses a nominal interest rate as the policy instrument. Numerous types of monetary policy rules have been discussed in the economics literature. Two main policy classes are targeting rules and instrument rules. A simple instrument rule relates the interest rate to a few observable variables. The most famous such rule is Taylor’s rule. Taylor demonstrated that a simple reaction function, with a short-term interest-rate policy instrument responding to inflation and output gap, follows closely the observed path of the Federal Funds rate. His original work was followed by a large literature, in which researchers have tried to modify Taylor’s rule to get a better fit to the data.7 We initially center our analysis on the following specification of the current-looking Taylor rule: it = a1 πt + a2 xt ,

(2.15)

7 See, e.g., Clarida et al. (1999), Gali and Gertler (1999), McCallum (1999) and Taylor (1999).

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where a1 is the coefficient of the central bank’s reaction to inflation and a2 is the coefficient of the central bank’s reaction to the output gap. We also consider the forward-looking and the hybrid Taylor rule. Among targeting rules, the recent literature proposes many ways to define an inflation target.8 We consider inflation targeting policies of the form: it = a1 πt ,

(2.16)

which is a current-looking inflation targeting rule. Forward-looking inflation targeting will also be considered. When we use the current-looking Taylor rule, we are left with these three equations. 1 xt = Et xt+1 − (it − Et πt+1 ), σ πt = βEt πt+1 + κxt , it = a1 πt + a2 xt . This 3-equation system constitutes a New Keynesian model. 2.3. Stability Analysis Continuing with the current-looking Taylor rule, we reduce the system of three equations to a system of two log-linearized equations by substituting Taylor’s rule into the consumption Euler equation. The resulting system of expected difference equations has a unique and stable solution, if the number of eigenvalues outside the unit circle equals the number of forward looking variables (see Blanchard and Kahn, 1980). That system of two equations has the following form: 1 + aσ2 − aσ1 Et xt+1 xt 1 σ1 = , 0 β Et πt+1 −k 1 πt which can be written as AEt xt+1 = Bxt , where xt = [ πxtt ], A = [ 10

a2

σ ] and B = [ 1 + −k β 1 σ

Premultiply the system by the inverse matrix 1 1 − βσ A−1 = , 1 0 β we get

− aσ1 ]. 1 −1 A ,

Et xt+1 = Cxt

8 See

Bernanke et al. (1999), Svensson (1999) and Gavin (2003).

266

or

W.A. Barnett and E.A. Duzhak

Et xt+1 Et πt+1

=

a2 β+k σβ − βk

a1 β−1 σβ 1 β

1+

xt , πt

where C = A−1 B. We have two forward-looking variables, xt+1 and πt+1 . Therefore uniqueness and stability of the solution require both eigenvalues to be outside the unit circle. The eigenvalues of C are the roots of the characteristic polynomial p(λ) = det(C − λI) a2 β + k 1 σβ + a2 β + ka1 β = λ2 − λ 1 + . + + σβ β σβ 2 Defining D as a2 β + k σβ + a2 β + ka1 β 1 2 D = 1+ −4 , + σβ β σβ 2 we can write the eigenvalues as a2 β + κ λ1 = 0.5 1 + + σβ a2 β + κ λ2 = 0.5 1 + + σβ

√ 1 +4 D , β √ 1 −4 D . β

(2.17) (2.18)

(2.19)

(2.20)

It can be shown that both eigenvalues will be outside the unit circle, if and only if (a1 − 1)γ + (1 − β)a2 > 0.

(2.21)

Equivalently, (2.21) holds if a1 > 1. Interest rate rules that meet this criterion are called active. This relationship is also known as Taylor’s principle, which prescribes that the interest rate should be set higher than the increase in inflation. Monetary policy satisfying the Taylor’s principle is thought to eliminate equilibrium multiplicities. Assuming uniqueness of solutions, the dynamical properties of the system can be explored through bifurcation analysis.

3. Bifurcation Analysis The New Keynesian model has both a continuous time and a discrete form. To define our notation for the discrete form, we consider a continuously differentiable map x → f(x, α),

(3.1)

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267

where x ∈ n is a vector of state variables, α ∈ m is a parameter vector, and f : n × m → n is continuously differentiable. We will study the dynamic solution behavior of x as α varies. System (3.1) undergoes a bifurcation, if its parameters pass through a critical (bifurcation) point, defined as follows. Definition 3.1. Appearance of a topologically nonequivalent phase portrait under variation of parameters is called a bifurcation. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. There are two possible bifurcation analyses: local and global. We look at small neighborhoods of a fixed point, x∗ = f(x∗ , α), to conduct local bifurcation analysis. Definition 3.2. A local bifurcation is a bifurcation that can be analyzed purely in terms of a change in the linearization around a single invariant set or attractor. The bifurcations of a map (3.1) can be characterized by the eigenvalues of the Jacobian of the first derivatives of the map, computed at the bifurcation point. Let J = f(x, α)x be the Jacobian matrix. The eigenvalues, λ1 , λ2 , . . . , λn , of the Jacobian are also referred to as multipliers. Bifurcation will occur, if there are eigenvalues of J on the unit circle that violate the following hyperbolicity condition. Definition 3.3. The equilibrium is called hyperbolic, when the Jacobian J has no eigenvalues on the unit circle. Nonhyperbolic equilibria are not structurally stable and hence generically lead to bifurcations as a parameter is varied. There are three possible ways to violate the hyperbolicity condition. They give rise to three codimension-1 types of bifurcations. Definition 3.4. Bifurcation associated with the appearance of λi = 1 is called a fold (or tangent) bifurcation. Definition 3.5. Bifurcation associated with the appearance of λi = −1 is called a flip (period-doubling) bifurcation. Definition 3.6. Bifurcation corresponding to the presence of a pair of complex conjugate eigenvalues, λ1 = e+iθ0 and λ2 = e−iθ0 , for 0 < θ0 < π, is called a Hopf bifurcation. In the 2-dimensional case, we shall need the following theorem, based upon the version of the Hopf bifurcation theorem in Gandolfo (1996, Ch. 25, p. 492).

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Theorem 3.1 (Existence of Hopf bifurcation in 2 dimensions). Consider a map x → f(x, α), where x has 2 dimensions; and for each α in the relevant region, suppose that there is a continuously differentiable family of equilibrium points x∗ = x∗ (α) at which the eigenvalues of the Jacobian are complex conjugates, λ1 = θ (x, α) + iω(x, α) and λ2 = θ (x, α) − iω(x, α). Suppose that for one of those equilibria, (x∗ , α ∗ ), there is a critical value αc for one of the parameters, αi∗ , in α ∗ such that: (a) The modulus of the eigenvalues becomes unity at α = α ∗ , but the eigenvalues √are not roots of unity. Formally, λ1 , λ2 = 1 and mod(λ1 ) = mod(λ2 ) = + θ 2 + ω2 = 1. Also suppose that ∂|λj (x∗ ,α ∗ )| |αi∗ =αc = 0 for j = 1, 2. (b) ∂α ∗ i

Then there is an invariant closed curve Hopf-bifurcating from α ∗ .9 Condition (b) implies that the eigenvalue crosses imaginary axes with nonzero speed. This theorem only works for a 2 × 2 Jacobian. We use it for the analysis of the reduced 2 × 2 model, AEt xt+1 = Bxt . The more general case requires the rest of the eigenvalues to have a real part less than zero. The 3 × 3 case requires different tools of investigation. In the three equation case with current-looking or forward looking policy rules, it can be shown that the only form or bifurcation that is possible with the linearized model is Hopf bifurcation. The broader range of possible types of bifurcation possible with the nonlinear model will be the subject of future research. Also we do not consider backwards looking models in this chapter, since there currently is more interest in the current looking and future looking models. But in future research, we shall consider the backwards looking models, since they raise the possibility of codimension-2 bifurcation, even with the log-linearized model. 3.1. Current-Looking Taylor Rule The matrix C was the Jacobian of the New Keynesian model presented above. We now change the notation so that the Jacobian is: β+k a1 β−1 1 + a2σβ σβ . J= 1 − βk β We apply the Hopf bifurcation existence Theorem 3.1 to the Jacobian of the loglinearized New Keynesian model, AEt xt+1 = Bxt . The characteristic equation of the Jacobian is: λ2 − bλ + c = 0, 9 Note that we use the notations mod(λ ) and |λ | interchangeably to designate modulus j j of a complex variable.

Hopf Bifurcation within New Keynesian Functional Structure

where

269

1 a2 β + k + , b = 1+ σβ β σβ + a2 β + ka1 β c= . σβ 2

In order to get a pair of complex conjugate eigenvalues, the discriminant D must be strictly negative:

a2 β + k 1 + D = b − 4c = 1 + σβ β 2

2 −4

σβ + a2 β + ka1 β < 0. σβ 2

Given the sign of the parameters, the discriminant could be either positive or negative. We assume that the discriminant is negative, so that the roots of the characteristic polynomial are complex conjugate:10 λ1 = θ + iω

and λ2 = θ − iω,

√ where θ = 12 b is the real part, iω is the imaginary part, and ω = 12 D = √ 1 2 2 b − 4c. We need to choose a bifurcation parameter to vary while holding other parameters constant. The model is parameterized by: ⎛ ⎞ β ⎜σ ⎟ ⎜ ⎟ α = ⎜ k ⎟. ⎝ ⎠ a1 a2 Candidates for a bifurcation parameter are coefficients for the monetary policy rule, a1 and a2 . In Barnett and Duzhak (2006), we provide and prove a theorem needed to determine whether or not Hopf bifurcation can appear in this case and to locate its boundary, if it exists. 3.2. Forward-Looking Taylor Rule A forward-looking Taylor rule sets the interest rate according to expected future inflation rate and output gap, in accordance with the following equation: it = a1 Et πt+1 + a2 Et xt+1 .

(3.2)

10 This assumption can be satisfied during the numerical procedure for locating the bifurcation regions.

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The resulting Jacobian has the form σ (1+a1 ) (1+a1 ) + κσ − σ(1−a (1−a2 )β 2 )β J = σ −a2 . 1 − βκ β The characteristic equation is λ2 − bλ + c = 0,

(3.3)

κσ (1+a1 ) (1−a2 )β

+ + and c = det(J). where b = In order to get a pair of complex conjugate eigenvalues, the discriminant D must be strictly negative: σβ + σ − a2 4σ κσ (1 + a1 ) 2 D= − + < 0. β(σ − a2 ) (1 − a2 )β β(σ − a2 ) σ σ −a2

1 β

Given the sign of the parameters, the discriminant could be either positive or negative. We assume that the discriminant is negative, so that the roots of the characteristic polynomial are complex conjugate: λ1 = θ + iω

and λ2 = θ − iω,

√ is the real part, iω is the imaginary part, and ω = 12 D. where θ = We need to choose a bifurcation parameter to vary while holding other parameters constant. The model is parameterized by: ⎛ ⎞ β ⎜σ ⎟ ⎜ ⎟ α = ⎜ k ⎟. ⎝ ⎠ a1 a2 Candidates for a bifurcation parameter are coefficients, a1 and a2 , for the monetary policy rule. In Barnett and Duzhak (2006), we provide and prove a theorem needed to determine whether or not Hopf bifurcation can appear in this case and to locate its boundary, if it exists. 1 2b

3.3. Hybrid Taylor Rule Consider the Taylor rule of the following form: it = a1 Et πt+1 + a2 xt ,

(3.4)

where the interest rate is set according to forward-looking inflation and currentlooking output gap. A rule of that form was proposed in Clarida et al. (2000). This form of the rule is intended to capture the central bank’s existing policy. Substituting Equation (2.4) into the consumption Euler Equation (2.8), we acquire the Jacobian, 1) 1 − 1−a 1 + aσ2 + κ(1−a βσ σβ , J= 1 − βκ β

Hopf Bifurcation within New Keynesian Functional Structure

271

with the associated characteristic polynomial λ2 − bλ + c = 0,

(3.5)

βa2 −κ(a1 −1) σβ

a2 σβ .

and c = det(J) = + where b = + 1 + In order to get a pair of complex conjugate eigenvalues, the discriminant D must be strictly negative: σ (1 + β) + βa2 − κ(a1 − 1) 2 4(σ + a2 ) − D= < 0. σβ σβ 1 β

1 β

Given the sign of the parameters, the discriminant could be either positive or negative. We assume that the discriminant is negative, so that the roots of the characteristic polynomial are complex conjugate: λ1 = θ + iω

and λ2 = θ − iω,

√ is the real part, iω is the imaginary part, and ω = 12 D. where θ = We need to choose a bifurcation parameter to vary while holding other parameters constant. The model is parameterized by: ⎛ ⎞ β ⎜σ ⎟ ⎜ ⎟ α = ⎜ k ⎟. ⎝ ⎠ a1 a2 Candidates for a bifurcation parameter are coefficients for the monetary policy rule, a1 and a2 . In Barnett and Duzhak (2006), we provide and prove a theorem needed to determine whether or not Hopf bifurcation can appear in this case and to locate its boundary, if it exists. 1 2b

3.4. Current-Looking Inflation Targeting Using the inflation targeting equation it = a1 πt ,

(3.6)

instead of the Taylor rule, as the third equation for New Keynesian model, produces the following Jacobian: σβ+k a1 1 σβ σ − σβ J= 1 − βk β with characteristic equation λ2 − bλ + c = 0, where σ +κ , b =1+ βσ

(3.7) 1 1 βσ + κ κ c= + a1 − . β σβ σβ β

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In order to get a pair of complex conjugate eigenvalues, the discriminant D must be strictly negative: σβ + σ + κ 2 4(σβ + κa1 β) − < 0. D= σβ σβ 2 Given the sign of the parameters, the discriminant could be either positive or negative. We assume that the discriminant is negative, so that the roots of the characteristic polynomial are complex conjugate: λ1 = θ + iω

and λ2 = θ − iω,

√ where θ = is the real part, iω is the imaginary part, and ω = 12 D. We need to choose a bifurcation parameter to vary while holding other parameters constant. The model is parameterized by: ⎛ ⎞ β ⎜σ ⎟ α = ⎝ ⎠. k a1 1 2b

A candidate for a bifurcation parameter is the coefficient, a1 , of the monetary policy rule. In Barnett and Duzhak (2006), we provide and prove a theorem needed to determine whether or not Hopf bifurcation can appear in this case and to locate its boundary, if it exists. 3.5. Forward-Looking Inflation Target Rule Using the following forward-looking inflation targeting rule, it = a1 Et πt+1

(3.8)

instead of the current-looking rule, as the third equation for New Keynesian model, (2.8), (2.14), (3.8), produces the following Jacobian: κ 1 (a1 − 1) σβ (a1 − 1) 1 − βσ J= 1 − βκ β with characteristic equation λ2 − bλ + c = 0, where 1+β κ − (a1 − 1), β σβ βσ − κ(a1 − 1) κ(a1 − 1) − . c= σβ 2 σβ 2 b=

Hopf Bifurcation within New Keynesian Functional Structure

273

In order to get a pair of complex conjugate eigenvalues, the discriminant D must be strictly negative: σ (β + 1) − κ(a1 − 1) 2 4 D= − < 0. σβ β Given the sign of the parameters, the discriminant could be either positive or negative. We assume that the discriminant is negative, so that the roots of the characteristic polynomial are complex conjugate: λ1 = θ + iω

and λ2 = θ − iω,

√ is the real part, iω is the imaginary part, and ω = 12 D. where θ = We need to choose a bifurcation parameter to vary, while holding other parameters constant. The model is parameterized by: ⎛ ⎞ β ⎜σ ⎟ α = ⎝ ⎠. k a1 1 2b

In Barnett and Duzhak (2006), we provide and prove a theorem needed to determine whether or not Hopf bifurcation can appear in this case and to locate its boundary, if it exists.

4. Conclusion In dynamical analysis, it is essential to employ bifurcation analysis to detect whether any bifurcation boundaries exist close to the parameter estimates of the model in use. If such a boundary crosses into the confidence region around the parameter estimates, robustness of dynamic inferences is seriously compromised. Our ongoing bifurcation analysis of New Keynesian functional forms is detecting the possibility of Hopf bifurcation. This chapter provides the relevant initial background leading up to the theorems provided by Barnett and Duzhak (2006). Empirical results will appear in a subsequent paper based upon the theoretical foundations in these two papers.

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Barnett, W.A. and Duzhak, E.A. (2006). Non-robust dynamic inferences from macroeconometric models: Bifurcation stratification of confidence regions. University of Kansas, Working paper. Barnett, W.A. and He, Y. (1999). Stability analysis of continuous-time macroeconometric systems. Studies in Nonlinear Dynamics and Econometrics 3, 169–188. Barnett, W.A. and He, Y. (2001). Nonlinearity, chaos, and bifurcation: A competition and an experiment. In Negishi, T., Ramachandran, R. and Mino, K. (Eds), Economic Theory, Dynamics and Markets: Essays in Honor of Ryuzo Sato. Amsterdam: Kluwer Academic, pp. 167–187. Barnett, W.A. and He, Y. (2002). Stabilization policy as bifurcation selection: Would stabilization policy work if the economy really were unstable? Macroeconomic Dynamics 6, 713–747. Barnett, W.A. and He, Y. (2004). Bifurcations in macroeconomic models. In Dowrick, S., Pitchford, R. and Turnovsky, S. (Eds), Economic Growth and Macroeconomic Dynamics: Recent Developments in Economic Theory. Cambridge, UK: Cambridge University Press, pp. 95–112. Barnett, W.A. and He, Y. (2006). Robustness of inferences to singularity bifurcation. In Proceedings of the Joint Statistical Meetings of the 2005 American Statistical Society 100. American Statistical Association, February. Barnett, W.A., Deissenberg, C. and Feichtinger, G. (2004). Economic Complexity: NonLinear Dynamics, Multi-Agents Economies, and Learning. Amsterdam: North-Holland. Barnett, W.A., Geweke, J. and Shell, K. (2005). Economic Complexity: Chaos, Sunspots, Bubbles, and Nonlinearity. Cambridge, UK: Cambridge University Press. Benhabib, J. and Day, R.H. (1982). A characterization of erratic dynamics in the overlapping generations model. Journal of Economic Dynamics and Control 4, 37–55. Benhabib, J. and Nishimura, K. (1979). The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth. Journal of Economic Theory 21, 421–444. Benhabib, J. and Rustichini, A. (1991). Vintage capital, investment and growth. Journal of Economic Theory 55, 323–339. Bergstrom, A.R. (1996). Survey of continuous time econometrics. In Barnett, W.A., Gandolfo, G. and Hillinger, C. (Eds), Dynamic Disequilibrium Modeling. Cambridge, UK: Cambridge University Press, pp. 3–26. Bergstrom, A.R. and Nowman, K.B. (2007). A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends. Cambridge, UK: Cambridge University Press. Bergstrom, A.R. and Wymer, C.R. (1976). A model of disequilibrium neoclassic growth and its application to the United Kingdom. In Bergstrom, A.R. (Ed.), Statistical Inference in Continuous Time Economic Models. Amsterdam: North-Holland, pp. 267–327. Bergstrom, A.R., Nowman, K.B. and Wymer, C.R. (1992). Gaussian estimation of a second order continuous time macroeconometric model of the United Kingdom. Economic Modelling 9, 313–352. Bergstrom, A.R., Nowmann, K.B. and Wandasiewicz, S. (1994). Monetary and fiscal policy in a second-order continuous time macroeconometric model of the United Kingdom. Journal of Economic Dynamics and Control 18, 731–761. Bernanke, B.S., Laubach, T., Mishkin, F.S. and Posen, A.S. (1999). Inflation Targeting: Lessons from the International Experience. Princeton, NJ: Princeton University Press.

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Blanchard, O.J. and Kahn, C.M. (1980). The solution of linear difference models under rational expectations. Econometrica 48, 1305–1312. Calvo, G. (1983). Staggered prices in a utility-maximizing framework. Journal of Monetary Economics 12, 383–398. Clarida, R., Galí, J. and Gertler, M. (1999). The science of monetary policy: A New Keynesian perspective. Journal of Economic Literature 37, 1661–1707. Clarida, R., Galí, J. and Gertler, M. (2000). Monetary policy rules and macroeconomic stability: Evidence and some theory. Quarterly Journal of Economics 115(1), 147–180. Dixit, A. and Stiglitz, J.E. (1977). Monopolistic competition and optimum product diversity. American Economic Review 67, 297–308. Gale, D. (1973). Pure exchange equilibrium of dynamic economic models. Journal of Economic Theory 6, 12–36. Gali, J. and Gertler, M. (1999). Inflation dynamics: A structural econometric analysis. Journal of Monetary Economics 44, 195–222. Gandolfo, G. (1996). Economic Dynamics (3rd edition). New York and Heidelberg: Springer-Verlag. Gavin, W.T. (2003). Inflation targeting: Why it works and how to make it work better? Federal Reserve Bank of Saint Louis Working Paper 2003-027B. Grandmont, J.M. (1985). On endogenous competitive business cycles. Econometrica 53, 995–1045. Hopf, E. (1942). Abzweigung einer periodischen Lösung von einer stationaren Lösung eines Differetialsystems. Sachsische Akademie der Wissenschaften MathematischePhysikalische, Leipzig 94, 1–22. Kuznetsov, Y.A. (1998). Elements of Applied Bifurcation Theory. New York: SpringerVerlag. Leeper, E. and Sims, C. (1994). Toward a modern macro model usable for policy analysis. NBER Macroeconomics Annual, 81–117. McCallum, B.T. (1999). Issues in the design of monetary policy rules. In Taylor, J.B. and Woodford, M. (Eds), Handbook of Macroeconomics. Amsterdam: North-Holland. Poincaré, H. (1892). Les methodes nouvelles de la mechanique celeste. Paris: GauthierVillars. Seydel, R. (1994). Practical Bifurcation and Stability Analysis. New York: SpringerVerlag. Shapiro, A.H. (2006). Estimating the New Keynesian Phillips curve: A vertical production chain approach. Federal Reserve Bank of Boston Working Paper No. 06-11. Svensson, L.E.O. (1999). Inflation targeting as a monetary policy rule. Journal of Monetary Economics 43, 607–654. Taylor, J.B. (1999). A historical analysis of monetary policy rules. In Taylor, J.B. (Ed.), Monetary Policy Rules. Chicago: University of Chicago Press for NBER, pp. 319–340. Walsh, C.E. (2003). Monetary Theory and Policy (2nd edition). Cambridge, MA: MIT Press. Woodford, M. (2003). Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, NJ: Princeton University Press.

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Chapter 13

Insurance and Asset Prices in Constrained Markets Pamela Labadie Department of Economics, George Washington University, 300 Academic Building, Mount Vernon Campus, Washington, DC 20052, USA, e-mail: [email protected]

Abstract The effects of liquidity and borrowing constraints on intermediation and consumption insurance are examined in a stochastic version of Kehoe and Levine (2001). Agents have limited trading opportunities because of a pattern of trade based on the Townsend (1980) turnpike model. Trading occurs through a clearing house offering consumption insurance and intermediary services. Three types of constrained economies are examined: pure insurance with no intermediation, borrowing constrained, and liquidity constrained. The resulting allocations are compared with the first best allocations. I show the impact of the constraints on asset prices and consumption allocations generally depend on whether the aggregate shock is i.i.d. or Markov.

Keywords: liquidity, borrowing constraint, turnpike model JEL: E44, G13

Liquidity constrained or borrowing constrained agents have a future income stream that cannot be easily converted into consumption today. In a deterministic setting, the constraint prevents agents from smoothing consumption over time. In a stochastic setting, the liquidity or borrowing constraint also affects an agent’s ability to insure against income shocks. I examine the implications for risk sharing of several types of constraints in a framework equivalent to that of Kehoe and Levine (2001). The model has the feature that agents have limited trading opportunities because any pair of agents is located at the same site only once, if at all. Hence, in the absence of a clearing house or some tradable asset and assuming a single nonstorable endowment good, no exchange takes place. Exchange takes the form of contingent claims contracts, which are traded through a clearing house. There are three steps to implementing a contingent claims contract: first agents need to communicate with agents at other sites to originate the contract. Second, the counter parties need to ensure delivery on the International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18013-3

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contract in the event they are located at the different sites on the delivery date. Third, if an agent can enter into multiple contracts, there must be record keeping and some netting scheme – an accounting procedure by which the contracts are summed – to ensure the agent’s lifetime budget constraint is satisfied. The clearing house facilitates contingent claims trading at each step. Borrowing constraints and liquidity constraints emerge when either communication across sites or delivery on contracts is restricted or if the netting scheme is limited. I examine three versions of the model under different assumptions about communication across sites, restrictions on delivery, and restrictions on netting schemes: pure insurance with no borrowing or lending, a borrowing-constrained model, and a liquidityconstrained model using short sales constraints. As a convenient benchmark and starting point, I solve the unconstrained central planning problem and construct the competitive equilibrium. To achieve this allocation requires full communication, unrestricted delivery, and broad netting across all sites and agents. Specifically, agents located at different sites at time t must be able to communicate with agents at other sites. There must be unrestricted delivery on the contracts, meaning the counter parties in the contract can be located at different sites at the delivery date. Finally, the broad netting implies that there is no limit on an agent’s net indebtedness to a subset of agents, as long as his lifetime budget constraint is satisfied. At the other extreme, when there is no communication across sites, so that agents can enter into contracts only with agents at the same site, there is no trade and the only solution is autarchy. If there is full communication but delivery on a contract can take place only if the counter parties are located at the same site on the delivery date, then even though an agent can enter into a countable infinity of contingent claim contracts, there may be only partial insurance against aggregate risk. This model is referred to as the pure insurance economy, in that agents are allowed to issue contingent claims, but the outstanding value of the contingent claim portfolio vis-à-vis another agent is restricted to equal zero in value. This is an example of bilateral netting. In the pure insurance economy, if the aggregate shock is i.i.d., then consumption allocations are identical to the first best allocations. However, when the aggregate shock is first-order Markov, the restriction on the portfolio value at the end of the period impacts consumption allocations and these allocations are no longer first best. I then allow a clearing house to facilitate delivery on contracts, but assume that there are technological constraints on record keeping. I show these record-keeping constraints are equivalent to either borrowing constraints or liquidity constraints. A borrowing-constrained model places a lower bound on the value of the end-ofperiod portfolio held by an agent. The pure-insurance economy is a special case of the borrowing-constrained economy. The borrowing constraint has no impact on consumption allocations when the aggregate shock is i.i.d. but impacts consumption allocations when the aggregate shock is Markov. The liquidity-constrained

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279

economy essentially imposes state-contingent short sales constraints on agents, which are important regardless of the time series properties of the aggregate shock. Finally, the model is linked with the Alvarez and Jermann (2000) model of endogenous solvency constraints.

1. Description of the Model An agent is indexed by his type, his location, the date, and the history of the system. The model is based on Townsend’s (1980) turnpike model. There are two types of agents: type E (east-moving) and type W (west-moving). There is a countable infinity of each type of agent. At time t, the type E or W agent is at a location i ∈ I , where I is the set of integers. In period t + 1, an E-type will move to site i + 1 while the W-type agent will move to site i − 1. If, at time t a type E agent is located at site i and a type W agent is located at site j , then the following set of potential interactions is possible. If j < i, then the two agents never meet in the future. If i = j , then the agents are present at the same site at the same point of time, but never meet again. If j > i, then the two agents may potentially be at the same site at the same time. If j − i is an even (and positive) number, then the agents are at the same site at time t + j −i 2 . If j − i is odd, then the two agents are never at the same site at the same point in time. All agents at all locations act as price takers. At each site and in each time period, each type of agent receives a stochastic and exogenous endowment. The exogenous endowments follow a stationary, firstorder Markov chain. Let st ∈ S = {1 , . . . , n }. A type E agent at site i has a nonstorable endowment yei : S → Y = [y, y], ¯ where y 0. Type W agent at site i has nonstorable endowment ywi : S → Y . Denote y¯ i (s) = yei (s) + ywi (s) as total endowment in state s at site i. The endowment is nonstorable and cannot be moved across sites during the time period it is received. Moreover, let yei = ye and ywi = yw for all i, so that type E agents are identical across sites, as are type W. In much of the discussion, the location index will be dropped when there is no ambiguity. Define πi,j = prob(st+1 = j |st = i ) for i, j = 1, . . . , n. Define Π as the n × n matrix of transition probabilities with (i, j )-element π(sj |si ), where summation across a row equals one. Finally, let πˆ (s) denote the unconditional probability of being in state s, equal to the sum of a column of the matrix Π . denote the vector of unconditional probabilities. Let s t = (s1 , . . . , st ) be Let Π the history of realizations up to time t and let πt (s t ) denote the probability of s t , where s t ∈ S t = 'S × ·() · · × S*. Hence, at time t an agent is characterized by his t

type, location i where i ∈ I , and the common history s t .

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A type E agent has preferences over consumption bundles described by ∞

β t πt s t U (ct ),

(1)

t=0 s t

where 0 < β < 1. Let cti (s t ) denote the consumption of a type E agent at time t, location i when the history is s t . The type W agent has preferences over consumption bundles described by ∞

β t πt s t W (ηt ).

(2)

t=0 s t

Let ηti (s t ) denote the consumption of a type W agent in location i at time t when history is s t . The functions U , W are assumed to be strictly increasing, strictly concave, and twice continuously differentiable. Let U1 , W1 denote the first derivatives and assume the Inada conditions hold: limc→0 U1 (c) = ∞ and limc→∞ U1 (c) = 0 for U = U, W . 1.1. First-Best Solution It is convenient to start with the central planner’s problem and then construct the competitive equilibrium under the assumptions that there is full communication across sites, no restrictions on the delivery on contracts over different sites, and broad netting. Full communication means that an agent can enter into contracts with agents at different sites. Unrestricted delivery on contracts means that delivery can be guaranteed even if the two counter parties are not present at the same site on the delivery date. Broad netting means that net indebtedness of an agent to any subset of agents is unrestricted as long as his lifetime budget constraint is satisfied. The central planner allocates resources, but is subject to the technology constraint that the consumption good is location-specific and cannot be transported across sites within a period. Let φei denote the time t Pareto weight attached to a type E agent, located at site i when t = 0 and let φwi denote the time-t Pareto weight attached to a type W, located at site i at t = 0. All type E agents are viewed as identical, as are type W, so that φe = φei and φw = φwi . Only stationary allocations are examined. The central planner solves max (3) β t πt s t φe U cti+t s t + φw W ηti+t s t , I

t

st

subject to the resource constraint y¯ h (st ) = cth s t + ηth s t ,

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281

where h ∈ I . Notice that y(s) ¯ = y¯ h (s) for all h ∈ I . Let λht (s t ) denote the Lagrange multiplier for the resource constraint. The first-order conditions are β t πt s t φe U1 cth s t = λht s t , (4) t h t t h t β πt s φw W1 ηt s = λt s . (5) Eliminate the multiplier and η using the resource constraint to obtain U1 (cth (st )) W1 (y(s ¯ t ) − cth (st ))

=

φe . φw

To find a solution, first let K > 0 be given and solve for c in U1 (c) = K. W1 (y(s) ¯ − c) The left-hand side is strictly decreasing in c because U , W are strictly concave and twice continuously differentiable. The inverse function theorem can be applied to define a function g such that c = g(s, K), where gK < 0, so c is decreasing in K. For the central planning problem, the stationary solution is φe " . c (s) = g s, φw The associated competitive equilibrium is now examined. Competitive Equilibrium In the competitive equilibrium with full communication, unrestricted delivery, and broad netting, agents located at different sites can enter into contracts and guarantee delivery on those contracts even if the counter parties are at different sites on the delivery date. If the clearing house allows contracts without delivery restrictions, so that counter parties can be located at different sites at the delivery date, then, as will be shown below, the clearing house is acting as an intermediary and facilitates borrowing and lending. The clearing house then serves two purposes: to facilitate consumption insurance and to facilitate borrowing and lending. While goods cannot be transported across sites at a point in time, under full communication and unrestricted delivery on contracts, contingent claims can be bought and sold in a centralized market at time 0. Let qti (s t ) denote the time 0 price of a unit of consumption at site i at time t contingent on history s t . The type E agent who is located at site i at t = 0 has a budget constraint 0=

∞ t=0 s t

qti+t s t ye (st ) − cti+t s t .

(6)

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The type E agent maximizes (1) subject to (6) by choosing consumption sequences. Let μe denote the Lagrange multiplier. The first-order condition is β t πt s t U1 cti+t s t = μe qti+t s t . (7) The budget constraint for a type W agent who starts at location i at t = 0 is 0=

∞

qti−t s t yw (st ) − ηti−t s t .

(8)

t=0 s t

The type W agent maximizes (2) subject to (8) by choosing consumption sequences. Let μw denote the Lagrange multiplier. The first-order condition is β t πt s t W1 ηti−t s t = μw qti−t s t . (9) At site i, the market-clearing condition is y(s ¯ t ) − cti s t − ηti s t = 0.

(10)

The first-order conditions for the two agents located at site i at time t can be solved for the price to obtain qti (s t ) U1 (cti (s t )) W1 (ηti (s t )) = . = β t πt (s t ) μw μe

(11)

Consider stationary solutions of the form ci : S → Y for i ∈ I . With market clearing at each site, (11) can be rewritten as U1 (ci (st )) μe . = W1 (y(s ¯ t ) − ci (st )) μw

(12)

Define μ ≡ μμwe . Observe that the left-hand side is strictly decreasing in c. The stationary solution is c(st ) = g(st , μ). To determine the value of μ, substitute for the equilibrium price into the type E’s budget constraint 0=

∞

β t πt s t U1 g(st , μ) ye (st ) − g(st , μ) .

(13)

t=0 s t

The right-hand side is strictly increasing in μ, hence there exists a unique solution μ" . Let c" (st ) = g(st , μ" ). As is well known, this solution has the property that the marginal rate of substitution (MRS) between the two agents at time t is equal across states for all states st ∈ S. Moreover, the MRS is equal to a constant μ" , regardless of history s t−1 .

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283

There are several implications for prices. First, observe that the price is proportional to W1 y(s ¯ t ) − c" (st ) μ" = U1 c" (st ) , so that the time t price will depend only on the state st , and not the history s t . Next, observe that the ratio of prices,

qt+1 (s t+1 ) β t+1 πt+1 (s t+1 )

qt (s t ) β t πt (s t )

−1

=

βπ(st+1 |st )U1 (c" (st+1 )) , U1 (c" (st ))

indicates that marginal utility over time is linked only through the Markov probabilities since the optimal consumption c" is a function only of the current state st . The associated dynamic programming problem illustrates the implications of full communication and unrestricted delivery for liquidity and borrowing constraints. Let q(s ˆ t+1 , st ) =

qt+1 (s t+1 ) qt (s t )

and let ze (st ) denote a contingent claim held by a type E agent. For the type E agent located at site i at time t, the Bellman equation is + i e e π(st+1 |st )Ve st+1 , z (st+1 ) Ve st , z = max U ct + β (14) st+1

subject to ye (st ) + ze = cti +

qˆt+1 (st+1 , st )ze (st+1 ).

(15)

st+1

If contingent claims holdings are restricted such that

qˆt+1 (st+1 , st )ze (st+1 ) = 0,

st+1

then agents are allowed to enter into only pure insurance contracts. There is restricted delivery and the clearing house facilitates communication across sites but does not act as an intermediary. If

qˆt+1 (st+1 )ze (st+1 ) = 0,

st+1

then the clearing house is acting as an intermediary, facilitating delivery contracts in which counter parties are at different sites on delivery dates.

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2. Pure Insurance Economy In this section, no borrowing or saving is allowed; however households can purchase consumption insurance. This allows the agents to pool consumption risk, but the ability to do so through intertemporal reallocation of consumption is prevented, generally limiting risk sharing. Agents can freely communicate with agents at other sites, but delivery on a contract is guaranteed only if the counter parties are located at the same site on the delivery date. If trading of contingent claims is to occur, then an agent located at site i at time must be able to communicate with sites other than the adjacent sites i − 1, i + 1. To see this, suppose that type E agents at site i can enter into contingent contracts with type W agents at site i + 1. Under the restriction on the delivery site, the two agents will not enter into a contract because the type W will be located at site i at t + 1 while the type E will be located at site i + 1. Hence the type E at site i at time t will enter into a contract with a type W at site i + 2 at time t. Similarly, the type W at site i will enter into a contract with a type E at site i − 2 at time t. The type E agent at site i can also enter into contracts with other type W agents, besides W(i +2). Since agent E(i) will cross paths with W(i + 4), W(i + 6) and into the infinite future, E(i) may enter into contracts with the restriction that the end-of-period portfolio with delivery on a particular date and specific site have zero value. Hence an agent is able to enter into a countable infinity of contracts into the future with delivery at different sites, but each period is constrained in terms of borrowing (or lending) against future income in that the value of the portfolio of contingent claims for a specific delivery site held at the end of any period must have zero value. This is equivalent to requiring bilateral netting of claims to sum to zero: the value of claims of type E at site i at time t + n are just offset by the value of claims of type W at site i at time t + n. i+1 t+1 (s ) denote the time 0 price of a unit of consumption delivered to Let qt+1 site i + 1 at time t + 1 in state s t+1 . Agent E at site i in state s t is restricted such that i+1 i+1 t+1 st+1 |s t ye (st+1 ) − ct+1 s , qt+1 0= (16) st+1

where (16) must hold for each s t ∈ S t so that the net transfer of resources across all possible states must sum to zero, conditional on s t . The type E agent at site i at time 0 maximizes his objective function (1) subject to (16) and his lifetime constraint qti+t s t ye (st ) − cti+t s t . 0 (17) t

st

t Let μe denote the Lagrange multiplier for (17) and let λi+t t (s ) be the multiplier for (16). The first-order conditions are t−1 i+t t s qt s . β t πt s t U cti+t s t = μe + λi+t−1 (18) t−1

Insurance and Asset Prices in Constrained Markets

The type W agent maximizes (2) subject to 0 qti−t s t yw (st ) − ηti−t s t , t

0=

285

(19)

st

i−1−t i−1−t t+1 st+1 , s t yw (st+1 ) − ηt+1 s . qt+1

(20)

st+1

Let μw and ψti−t (s t ) denote the Lagrange multipliers for (19) and (20). The firstorder conditions are i−t+1 t−1 i−t t s qt s . β t πt s t W ηti−t s t = μw + ψt−1 (21) The market clearing condition at each site is (10). Solve for the price at site j in time t and history s t , qt (s t ) W (ηt (s t )) U (ct (s t )) = . = j −1 j +1 β t πt (s t ) μe + λt−1 (s t−1 ) μw + ψt−1 (s t−1 ) j

j

j

(22)

Rewrite (22) as j −1

μe + λt−1 (s t−1 ) j +1

μw + ψt−1 (s t−1 )

U (ct (s t )) j

=

j

W (ηt (s t ))

.

(23)

As before, only stationary solutions are considered. Notice from (23) that the ratio of multipliers is a function of st−1 . Define ˆ t−1 ) ≡ λ(s

j −1

μe + λt−1 (s t−1 ) j +1

μw + ψt−1 (s t−1 )

(24)

,

where the location index is dropped for convenience. To find the solution, let λˆ ˆ recall that the function g be given. Using the market clearing condition and λ, satisfies ˆ U (g(st , λ)) , λˆ = (25) ˆ W (y(s ¯ t ) − g(st , λ)) ˆ Let λˆ be the solution to where g is strictly decreasing in λ. 0=

βπ(st |st−1 )U1 (g(st , λˆ )) st

λˆ

ye (st ) − g(st , λˆ ) .

(26)

Observe that β and λˆ cancel from the right side of (26). There exists a unique ˆ There are two solution λˆ because the right-hand side is strictly increasing in λ. cases: • If the borrowing constraint is nonbinding for both types, so λˆ = μμwe , then consumption is a function of st and not st−1 , regardless of whether or not the aggregate shock is Markov.

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• If the borrowing constraint is binding for either agent in some state, observe that λ or ψ may depend on the aggregate shock last period st−1 . This dependence of the multiplier λˆ on st−1 occurs only if the aggregate shock is a first-order Markov process. If s were independently and identically distributed (i.i.d.), then λˆ would not depend on st−1 , even if the intermediation constraint is binding. Specifically, if π(st |st−1 ) = π(st ), then the solution to (26) is state-invariant. Hence the lack of intermediation, which limits the ability of agents to borrow or save, has no effect on the consumption allocations when there is consumption insurance and the aggregate shock is i.i.d. As a result, if the clearing house uses only bilateral netting, mitigating borrowing and lending opportunities, the agents are still able to achieve full-risk sharing because the function c" (st ) satisfies the budget constraint (26). When the exogenous aggregate shock is Markov, then the constraint on intermediation does affect consumption allocations. To see this, observe in (26) that λˆ depends on st−1 and this dependence occurs through the Markov probability π(st |st−1 ). Hence, the benefits from intermediation arise from the ability to smooth consumption intertemporally when aggregate risk is correlated over time. Under full communication, contracts in which both counter parties are at the same site at the delivery date can be arranged privately. Moreover, each type of agent at any site can enter into a countable infinity of such contracts. Regardless, the restriction on the delivery limits risk sharing when risk is Markov.

3. Borrowing and Liquidity Constrained Households In the pure insurance economy, in which delivery on a contract is guaranteed only if the counter parties are present at the same site on the delivery date, contingent claims trading provides only partial insurance against aggregate risk that is Markov. Inefficient risk sharing can be eliminated only if the clearing house facilitates delivery when agents are located at different sites on the delivery date. Unrestricted delivery allows borrowing and lending among agents. To achieve full communication and unrestricted delivery equilibrium described earlier, the intermediary must keep records on all trades at all sites, contingent on the history of the state. The borrowing constraints and liquidity constraints emerge in this model by assuming the clearing house has limits on its record keeping ability. Limits on the ability to keep records is discussed by Kocherlakota and Wallace (1998), for example. This is the limiting case of a model in which the clearing house intermediates loans between agents that are located a fixed number of sites apart at time t. Since the sites are identical, the site index is dropped below for convenience.

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287

3.1. Borrowing-Constrained Households Typically models with borrowing constraints, such as in Hugget (1993), Aiyagari (1994), or Ljungqvist and Sargent (2004), impose a restriction on the amount that can be borrowed in addition to ruling out the existence of contingent claims markets. To examine the impact of the borrowing constraint when there are contingent claims, the borrowing constraint is modeled as a lower bound on the value of the portfolio at the end of the period. The constraint is best understood in a sequential model using dynamic programming. Let Ve (st , ze ) denote the value function of a type E agent at location i in state st who holds contingent claims ze at the beginning of the period. The type E agent solves e e π(st+1 |st )Ve st+1 , z (st+1 ) , Ve st , z = max U (ct ) + β (27) st+1

subject to ye (st ) + ze = ct +

q(s ˆ t+1 , st )ze (st+1 ),

(28)

st+1

and D

q(s ˆ t+1 , st )ze (st+1 ),

(29)

st+1

where D 0, places a restriction net indebtedness since the constraint implies ct ye (st ) + ze (st ) − D. Hence, the constraint restricts both borrowing and the ability to roll over debt. Let μe (st ) denote the Lagrange multiplier for the budget constraint (28) and let λe (st ) denote the multiplier for the borrowing constraint (29) for a type E agent. The first-order conditions and envelope condition are U1 (ct ) = μe (st ),

ˆ t+1 , st ) − λe (st ) = βπ(st+1 |st )V2 st+1 , z (st+1 ) , μe (st )q(s V2 st , ze (st ) = μe (st ). e

(30) (31) (32)

These conditions simplify as ˆ t+1 , st ) − λe (st ) = βπ(st+1 |st )U1 (ct+1 ). U1 (ct )q(s

(33)

Market-clearing requires that ze (st+1 ) + zw (st+1 ) = 0. Observe that the perfect insurance economy is a special case of this economy in which D = 0. Hence, the first implication is that, if the aggregate shock is i.i.d.,

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then the constraint D < 0 is nonbinding, and more generally the borrowing constraint has no impact on consumption allocations for all of the reasons described in the previous section. For this reason, aggregate risk is assumed to be Markov in the rest of the discussion. A second case is that the constraint binds for one of the agents. Notice that it can only bind for one type of agent in each time period. If the constraint is binding for a type E agent, then consumption is c(st ) = ye (st ) + ze (st ) − D so that λe (st ) = max 0, U1 ye (st ) + ze (st ) − D qˆ i (st+1 , st ) − βπ(st+1 |st )U1 (ct+1 ) . If λe (st ) > 0, then observe that the equilibrium consumption of a type W agent is η(st ) = yw (st ) + zw + D. This borrowing constraint model will be interpreted as a netting scheme or record keeping technology used by the clearing house. Borrowing Constraints as Netting Schemes For intermediation to occur, the clearing house must facilitate delivery on contracts between counter parties located at different sites on the delivery date. To prevent Ponzi schemes from occurring, the clearing house will need to maintain records all of the contingent claims contracts entered into by agents W(i) or E(i), i ∈ I . As discussed earlier, there are a countable infinity of such contracts. In this section, the clearing house maintains records but is technologically constrained from maintaining records and deliveries on all contracts at all sites over time. In particular, an agent at site i at time t will be allowed to borrow from agents that are no more than n sites away from site i. There is an upper bound on how much the agent can borrow, equal to the expected discounted present value of his income for the next n periods. Agents are allowed limited opportunities to roll this debt over and, over his lifetime, the expected discounted present value of expenditures cannot exceed the expected discounted present value of income. Suppose that the clearing house nets deliveries at site i + 1 at time t + 1 with deliveries at site i − 1. If E(i) borrows from W(i + 2) at time t, resources are shifted from W(i) to E(i) for the loan to take place. Agent W(i) will be repaid at site i − 1 from payments made by E(i − 2) and W(i + 2) will be repaid at site i + 2 by E(i) for resources that W(i + 2) shifted to E(i + 2). Although multiple agents are involved in the transaction, the initial loan takes place at two sites (i, i + 2) and the repayment takes place at two sites (i − 1, i + 1). Rolling the debt over will expand the number of agents and sites involved in the transaction. Define An,t s t = qt s t ye (st ) + πt+1 (st+1 )An−1,t+1 s t+1 , st+1

≡ 0, which equals the discounted present value of endowment for where A0,t n periods in the future, including the current period, measured in time 0 prices. (s t )

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289

Similarly for the type W agent define B1,t s t ≡ qt s t yw (st ) and

Bn,t s t = qt s t yw (st ) + πt+1 s t+1 Bn−1,t+1 s t+1 . st+1

For general netting schemes, the problem solved by a type E is to maximize (1) subject to 0 (34) qti+t s t yei+t (st ) − cti+t s t , t

0

Ai+t n,t

st

t s − qti+t s t cti+t s t .

(35)

The first constraint is the standard lifetime budget constraint, while the second constraint states that the agent’s borrowing is capped by the expected discount present value of income for the next n periods. This places a limit on the amount of debt that an agent can acquire. It also has the following interpretation: that claims against an agent at time t in state s t , site i, when netted against agents at sites i, . . . , i +n are bounded above by the agent’s ability to repay in the next n periods. Suppose that (35) binds in period t. Then next period, the agent faces (35) updated one time period and can borrow, as long as (34) is satisfied over his lifetime. This formulation allows a more general and flexible borrowing constraint than is typically used. Let μe denote the multiplier for the lifetime constraint for a type E agent and let λne,t (s t ) denote the multiplier for the borrowing constraint (35). The first-order condition is β t πt s t U cti+t s t = μe + λne,t s t qti+t s t . (36) If n = 1, then the clearing house will not intermediate any loans between the agents at site i at time t and the equilibrium allocation is autarchy. If n = 2, then the clearing house will intermediate loans but only for agents that are no more than 2 sites apart at time t + 1. Clearly, as n grows large, the constraint is less likely to bind because the lifetime budget constraint must be satisfied. As n → ∞, the constraint is equivalent to the no-Ponzi scheme condition. 3.2. Liquidity Constrained Households Kehoe and Levine (2001) define liquidity constraints as a restriction on short-sales of an asset. In their paper there are two types of assets: a tradable asset which is a claim to physical capital and a nontradable asset. A short-sale constraint in this model takes the form ye (st ) − c(st ) D(st ).

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The implications of this constraint are discussed next and then related to the Alvarez and Jermann (2000) model of endogenous debt constraints. The east and west traveling agents maximize their objective functions subject to their lifetime budget constraints and the short sale constraint of the form yi (st ) − ci (st ) D(st ),

(37)

where i = e, w and D(st ) < 0. Notice that the constraint is generally statedependent, and is stronger than the borrowing constraint model above since it not only limits the end of period portfolio qt+1 s t+1 yi (st+1 ) − ci (st+1 ) qt+1 s t+1 D(st+1 ) st+1

st+1

but also it limits indebtedness in each state, a point made by Kehoe and Levine (2001). Observe also that the short sale constraint can be binding even if the aggregate shock is i.i.d., unlike the pure insurance models or the borrowing constraint model. Let φi (st ) denote the Lagrange multiplier for the short sale constraint and let μi denote the multiplier for the lifetime budget constraint. The first-order condition for a type E agent is πt s t β t U1 ct s t = μe qt s t + φe (st ). (38) If φe > 0 then c(st ) = ye (st ) − D(st ). From market-clearing, it follows that η(st ) = yw (st ) + D(st ). If the east-type agent has φe (st ) > 0, then φw (st ) = 0 and conversely. The short-sale constraint can be binding for only one type of agent in equilibrium. If φe > 0 then the equilibrium price will satisfy qt (s t ) U1 (ye (st ) − D(st )) − φe (st ) W1 (yw (st ) + D(st )) = = . t t πt (s )β μe μw

(39)

3.3. Debt-Constrained Economies Alvarez and Jermann (2000) derive a constrained efficient equilibrium based on endogenous solvency constraints. Their work builds on earlier work by Kocherlakota (1996) and Kehoe and Levine (1993, 2001), who construct equilibria in endowment economies where there are participation constraints. Agents can always opt to revert to the autarchy solution and so any efficient allocation with market participation must take this into account. Alvarez and Jermann show that the participation constraints can be interpreted as endogenous solvency constraints. Agents can choose to default and revert to the autarchy solution. They derive endogenous borrowing constraints such that the agent, while having the option of default, will in equilibrium never choose default. Define Vea (st ) as Vea (st ) = U ye (st ) + β (40) π(st+1 |st )Vea (st+1 ) st+1

Insurance and Asset Prices in Constrained Markets

so that V a is the value of the endowment under autarchy. Alvarez and Jermann examine the following economy e e Ve st , z = max U (ct ) + β π(st+1 |s)Ve st+1 , z (st+1 )

291

(41)

st+1

subject to ye (st ) + ze = ct +

q(s ˆ t+1 , s)ze (st+1 )

st+1

and ze (st+1 ) D(st+1 ), which is dynamic programming version of the short sale constrained model in the previous section. Let μe (st ) denote the multiplier on the budget constraint and let λe (st+1 ) denote the multiplier on the borrowing constraint. The first-order conditions and the envelope condition are U1 (ct ) = μe (st ),

(42)

ˆ t+1 , st ) = λe (st+1 ) + βπ(st+1 |st )V2,e st+1 , ze (st+1 ) , μe (st )q(s

(43)

V2,e = μe (st+1 ).

(44)

They show that the solvency constraint is not too tight if Ve st , D(st ) = Vea (st ).

(45)

As long as the short-sale constraint is not too tight, the agent will never choose to default on debt. The endogenous solvency constraint corresponds to a netting scheme.

4. Conclusion In the absence of an outside asset, or some mechanism by which contingent claims can be traded with agents at other sites, the only equilibrium in the stochastic turnpike model is the autarchy solution. The introduction of a clearing house facilitates the use of consumption insurance and intermediation. To focus on the implications for insurance, I examine the case where the clearing house facilitates trade in state contingent claims, but the clearing house takes no risk by restricting the value of a household’s contingent claim portfolio to zero. I then extend this to the case of limited borrowing. Essentially, the clearing house introduces liquidity into the system in a limited way. If there is full communication so that agents at different sites can enter into contracts, then the location of delivery must be specified because goods cannot be transferred across sites within a period. If delivery

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P. Labadie

is restricted such that the counter parties must be located at the same site when delivery is made, then there is partial risk sharing, even though each agent enters into a countable infinity of contracts. When there is a clearing house facilitating delivery on contracts, then the clearing house will necessarily require some netting scheme over transactions to prevent Ponzi schemes. I examine various netting schemes, and show that these schemes (which are related to record keeping) are equivalent to borrowing constraints or short sales constraints. Finally I examine the Alvarez–Jermann model of endogenous solvency constraints and the associated netting scheme in which there is no default in equilibrium.

References Aiyagari, S.R. (1994). Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 17(1), 659–684. Alvarez, F. and Jermann, U. (2000). Efficiency, equilibrium, and asset pricing with risk of default. Econometrica 68(4), 775–797. Hugget, M. (1993). The risk free rate in heterogeneous-agent, incomplete-insurance economies. Journal of Economic Dynamics and Control 16(1), 79–92. Kehoe, T.J. and Levine, D.K. (1993). Debt-constrained asset markets. Review of Economic Studies 60(4), 865–888. Kehoe, T.J. and Levine, D.K. (2001). Liquidity constrained markets versus debt constrained markets. Econometrica 69(3), 575–598. Kocherlakota, N. (1996). Implications of efficient risk sharing without commitment. Review of Economic Studies 63(4), 595–609. Kocherlakota, N.R. and Wallace, N. (1998). Incomplete record-keeping and optimal payment arrangements. Journal of Economic Theory 81(2), 272–289. Ljungqvist, L. and Sargent, T.J. (2004). Recursive Macroeconomic Theory (2nd edition). Cambridge, MA: MIT Press. Townsend, R.M. (1980). Models of money with spatially separated agents. In Kareken, J.J. and Wallace, N. (Eds), Models of Monetary Economies. Minneapolis: Federal Reserve Bank of Minneapolis, pp. 265–303.

Part VI Fiscal Policy and Real Business Cycles in Open Economies

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Chapter 14

Labor Adjustment Costs, Shocks and the Real Business Cycle Zuzana Janko Department of Economics, University of Calgary, 2500 University Drive, N.W., Calgary, AB, Canada T2N 1N4, e-mail: [email protected]

Abstract In this chapter we analyze a real business cycle model with labor adjustment costs. The model is calibrated under the specification of variable and constant capital utilization rate, given both technology and government spending shocks. Labor adjustment costs introduce a propagation mechanism that improves the propagation mechanism of both shocks, leading to impulse response functions that are hump-shaped and persistent. Furthermore, the statistical properties of the model are well matched by a model with labor adjustment costs. Overall, we find labor adjustment costs to be an important propagation mechanism for shocks.

Keywords: real business cycle, labor adjustment costs JEL: E13, E32, J20

1. Introduction In this chapter we build on existing literature to assess the importance of labor adjustment costs in the propagation of supply and demand shocks. Cogley and Nason (1995) find that a prototypical RBC model with labor adjustment costs generates qualitatively similar impulse response functions of output to the data when subjected to transitory shocks.1 Burnside and Eichenbaum (1996) find that labor hording acts as an important propagation mechanism for shocks, however qualitatively the impulse response function of output do not match the humpshape response of output as observed in the data. In contrast, we find that labor adjustment costs are an important transmission mechanism for shocks. The impulse response functions we obtain are qualitatively able to match the data and

1 In their model they are unable to match the impulse responses quantitatively. Specifically,

they find the magnitudes to be much too small. International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18014-5

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the magnitudes are strong as well. Furthermore, the model matches the statistical properties of the data well. Empirical studies by Jaramillo et al. (1993), Davis and Haltiwanger (1990) and Summers (1981) show that labor adjustment costs, although small as a percentage of output, are important in altering the production process of the firm. To match these two facts, we consider quadratic labor adjustment costs following Cogley and Nason (1995) and Fairise and Langot (1994). In our case, we obtain average labor adjustment costs that are less then one hundred of one percent of total output. Cogley and Nason (1995) consider capital adjustment costs as well, however their findings indicate that capital adjustment costs do not help to propagate shocks and hence we do not incorporate capital adjustment costs in our model.2 We consider two types of shocks, a standard technology shock and a government spending shock. Christiano and Eichenbaum (1992) analyze the role of government spending shocks and find these shocks to substantially improve the performance of the RBC model. Specifically, the labor market side of the model is improved upon, as government spending shocks allow for shocks to labor supply. However, the model is such that the resulting impulse response functions do not quantitatively match the data. Interestingly, Benhabib and Wen (2004) and Wen (2001) obtain impulse responses of output to consumption and government spending shocks by substantially departing from the standard RBC model.3 In contrast, we are able to match the data quite well by incorporating small labor adjustment costs. We assess the importance of labor adjustment costs under both constant capital utilization as well as variable capital utilization.4 We find that the inclusion of labor adjustment costs under both specifications leads to impulse response functions of output to technology, as well as government shocks, that are persistent and hump-shaped. As firms face costs to adjusting labor, they have an incentive to increase labor slowly over several periods to minimize the costs of adjustment. Further, we find that without labor adjustment costs, variable capital utilization leads to quantitative differences only. It does not lead to impulse responses that are hump-shaped. Only with labor adjustment costs does changing the utilization rate increase persistence in output.

2 Janko

(2005) finds that capital adjustment costs worsen the propagation mechanism of shocks in a Calvo-type wage rigidity model. 3 They find that by including externalities the model is able to propagate shocks well. 4 Neiss and Pappa (2005) find that variable capital utilization increases the propagation mechanism of monetary shocks. Specifically, variable capital utilization magnifies the impulse response functions, however it does not alter the qualitative results. In their model substantial price rigidity leads to qualitatively different impulses.

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We obtain business cycle statistics for the economy with and without variable capital utilization considering three alternative labor adjustment cost values.5 With labor adjustment costs equal to zero, the economy is quite volatile, given both supply and demand shocks. However, incorporating small labor adjustment costs leads to volatilities that match the economy well, when it comes to output, hours and wages.6 Labor adjustment costs do not help to improve the volatility of consumption and labor productivity, which we find to be lower then in the data.7 Next, we find that the contemporaneous correlations of output with other macroeconomic variables are improved when capital utilization is constant. Specifically, with LAC and constant capital utilization we obtain a positive correlation between output and consumption, and more closely match the correlation between output and hours as well as wages.8 Interestingly, overall we find that it is the model with constant capital utilization and labor adjustment costs that does best in matching the business cycle statistics of the data. This chapter is organized as follows. In Section 2 we construct a model with labor adjustment costs and variable capital utilization. The solution method is briefly discussed in Section 3, while in Section 4 we calibrate the model and discuss the results. Section 5 concludes the chapter.

2. Model The economy consists of identical, infinitely-lived households who choose labor by maximizing expected discounted lifetime utility given in (1) ∞ 1+χ Ant t β log(ct ) − , A > 0, Et (1) 1+χ t=0

s.t. ct + τt = wt nt + πt ,

(2)

where ct is consumptions, nt is hours worked, β is the discount factor, χ is the inverse elasticity of labor supply with χ 0, τt is lump-sum tax, wt is the real wage and πt are profits paid out by the firm. Optimization leads to the following first-order condition χ

Ant ct = wt .

5 Specifically, we consider three

(3)

alternative values for the labor adjustment costs parameter in our quadratic representation. 6 These results hold under both types of utilization rates. 7 We find that the volatilities of consumption and productivity to be much lower with variable capital utilization than with constant capital utilization. 8 We are unable to closely match the correlation between hours and productivity under any specification.

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Equation (3) equates the real wage with the marginal rate of substitution between consumption and labor. The economy also consists of a large number of competitive firms who produce output according to the Cobb–Douglas production function9 yt = ezt (ut kt )α n1−α , t

0 < α < 1.

(4)

The inputs used for production are labor and physical capital stock kt , where capital is utilized at the rate ut . The production function is subject to a technology shock zt that alters total factor productivity and evolves according to zt = ρz zt−1 + εz ,

0 < ρz < 1.

(5)

The law of motion for capital stock is given by kt+1 = it + (1 − δt )kt ,

(6)

where it is investment in new capital and δt is endogenous capital depreciation rate that increases as firms increase the rate at which they utilize capital.10 Specifically, 1 θ u , θ > 1. (7) θ t In addition to input costs, firms face costs as labor is adjusted between periods. We follow Fairise and Langot (1994) and Cogley and Nason (1995) by adopting quadratic labor adjustment costs: η nt − nt−1 2 , η > 1. LACt = (8) 2 nt−1 δt =

The firm’s objective is to maximize discounted stream of expected lifetime profits given by ∞ η nt − nt−1 2 β t Λt yt − wt nt − it − , Et (9) 2 nt−1 t=0

subject to the production function (4), laws of motion for capital (6) and the depreciation technology (7). The firm discounts future profits at the household’s market rate of discount, where Λt is the expected marginal utility of consumption. The first-order conditions for the firm over (nt , kt+1 , ut ) are: ut kt α nt − nt−1 nt Λt wt = Λt (1 − α)ezt − ηΛt nt nt−1 nt−1

9 When

capital is fully utilized the production function is a function of output, capital and the technology shock only not ut . 10 In a model with full capital utilization depreciation rate is constant and hence Equation (7) is eliminated from the problem.

Labor Adjustment Costs, Shocks and the Real Business Cycle

nt+1 − nt n2t+1 + Et ηΛt+1 , nt n2t αyt+1 + 1 − δt+1 , Λt = Et βΛt+1 kt+1 yt α = θ δt . kt

299

(10) (11) (12)

Equation (10) shows that real wage is obtained by setting it equal to the marginal product of labor, taking into account the effect on next period’s labor adjustment costs from today’s change in labor (shown by the last term). Equation (11) is the standard Euler equation for the intertemporal consumption choice and Equation (12) is used to equate the marginal benefit to the marginal costs of a change in the capital utilization rate.11 The government raises funds through lump-sum taxation and uses these funds to purchase goods and service. Thus, the government’s budgets constrained is gt = τt with government spending evolving according to gt = (1 − ρg )g¯ + ρg gt−1 + εt,g ,

0 < ρg < 1,

(13)

where g¯ is the level of government expenditure at steady state and the error terms are normally distributed with mean zero and a standard deviation of σεg .

3. Solution Method We use the method of undetermined coefficient (Cambell, 1994) to solve for the equilibrium laws of motion. Since no analytical solution exists to the households’ and firms’ maximization problems we start by postulating a linear relationship between the decision variables and the state variables. A general representation is given by Xt = AXt−1 + Bwt , Vt = CXt−1 + Dwt ,

(14)

where Xt is an endogenous state vector, Vt is a vector of other endogenous variables and wt is a vector of stochastic processes. The stochastic processes considered here are the technology shock and the government spending shock. The objective is to solve for the matrices A, B, C and D in order to obtain equilibrium decision rules for Xt , that include Kt+1 and Ht as well as equilibrium decision rules for Vt , that include Yt , It and Ct .

11 In

only.

the case of full capital utilization the F.O.C. of the firm are Equations (10) and (11)

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Since the equations in (14) are postulated recursive laws of motion, the coefficients in matrices A, B, C and D are undetermined. We solve for these matrices by log-linearizing the equations characterizing the equilibrium around the steady state. Specifically, we log-linearize the first-order conditions of households and firms, the budget constraint, the output function, and the utilization rate equation. Together, the postulated laws of motion and the log-linearized equilibrium equations from the model are used to solve for the undetermined coefficients in (14).12

4. Calibration and Results The model is calibrated to post-war quarterly US data. The capital share in total income, α, is set at 0.36, β equals 0.989 corresponding to a one percent quarterly real interest rate, δ (=0.025) is set such that annual depreciation rate is at 10%, A (=1.95) is chosen to match the average total hours at steady state, and χ is set equal to 0 corresponding to indivisible labor. In the model with variable capital utilization we use Equations (11) and (12) to obtain θ (=1.384). The parameter values of the AR(1) technology and government shock processes are both highly persistent with ρz = ρg = 0.95. The standard deviation for the technology shock following Prescott (1986) are 0.0072 and 0.0012 given constant and variable capital utilization respectively. We obtain the value of g¯ by setting the steady state ratio of government expenditure to output equal to 0.2 and the standard deviation of the technology shock is set at 0.022 – obtained by running and ordinary least square regression of (13). Lastly, for the parameter value η we consider two values 0.38 and 1.0.13 Both imply average adjustment costs of less than one hundred of one percent of output. Figure 1 and 2 show the impulse response of output to a technology shock with variable capital utilization and constant capital utilization respectively.14 In Figure 1, when labor adjustment costs are zero output increases dramatically and stays above its steady state value for a number of periods, however it begins to fall immediately after the initial sharp rise. In contrast, with positive labor adjustment costs output continues to increase for several periods following its initial increase, as firms have an incentive to increase hours worked slowly over time in order to reduce the costs incurred by adjusting labor. Consequently, labor adjustment costs introduce an endogenous mechanism whereby the behavior of output is hump-shaped in response to a technology shock. Quantitatively, we find the impulses to be quite strong with moderate labor adjustment costs (η = 0.38). Note 12 We

use a modified algorithm formulated by Uhlig (1999) to obtain the undetermined coefficients. 13 Shapiro (1986) suggest a value of 0.38 for the labor adjustment costs parameter. We consider the value of 1.0 for sensitivity purposes. 14 Note that a period is equivalent to one quarter.

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Figure 1: Impulse Response of Output to a Technology Shock Given Variable Capital Utilization.

Figure 2: Impulse Response of Output to a Technology Shock Given Constant Capital Utilization.

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that increasing the size of the adjustment costs parameter (η = 1) dampens the responses, yet it takes slightly longer for output to reach its peak before slowly falling down. Once we consider constant capital utilization (Figure 2) the dynamics are slightly different. Labor adjustment costs continue to lead to hump-shape dynamics of output, however without variable capital utilization the impulse responses of output after six quarters (given η = 0.38) are identical to the case without labor adjustment costs. Hence, labor adjustment costs together with variable capital utilization lead to a more persistent hump-shaped response of output to a technology shock. We observe similar output dynamics when the economy is subject to a government expenditure shock. Figure 3 shows the response of output given variable capital utilization. Without adjustment costs output rises sharply in the initial period followed by a fall towards steady state. With labor adjustment costs it takes slightly over a year for output to reach its highest levels before adjusting towards the steady state given η = 0.38 and close to two years for η = 1. Again, with labor adjustment costs the firm has an incentive to slowly increase labor resulting in a hump-shape response of output to a government shock. This result is not dependent on whether capital utilization is variable or not as seen in Figure 4. As in Neiss and Pappa (2005), we find variable capital utilization to amplify the responses of output to the shocks. Table 1 gives the statistical properties of a model with labor adjustment costs when capital utilization is variable (model 1) and for a model with constant

Figure 3: Impulse Response of Output to a Government Spending Shock Given Variable Capital Utilization.

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Figure 4: Impulse Response of Output to a Government Spending Shock Given Constant Capital Utilization.

capital utilization (model 2). Both models are subjected to technology and government spending shocks. We find the economy to be much too volatile without labor adjustment costs. However, once labor adjustment costs are introduced, the volatilities of output and hours fall sharply. We find the volatilities of output and hours to be similar in both models, however the volatilities of productivity, consumption and wages are higher in model 1, thus more closely matching the data. With respect to the correlations, labor adjustment costs together with constant capital utilization (model 2) do better in matching the correlation of output with hours, productivity and consumption (given small adjustment costs). We find the overall correlation between output and productivity to be positive and rising as labor adjustment costs rise. Note that while technology shocks alone lead to a positive correlation between output and productivity, government spending shock result in a negative correlation due to it shifting labor supply. Furthermore, the correlation between output and consumption is high and positive in model 2, while being negative in model 1. This correlation is best matched to the data when labor adjustment costs are positive. Next, we find the correlation between output and wages to be well matched by a model with variable capital utilization when labor adjustment costs are zero. Note that introducing adjustment costs to this model leads to negative correlations. This correlation is positive, but high in model 2. Lastly, we are unable to match the correlations between hours and productivity under any specification.

304

Table 1: Business Cycle Statistics Standard deviations

Correlations

H

Y /H

C

W

corr(Y, W )

corr(Y, C)

1.65

1.51

1.03

1.29

0.67

0.86

0.49

0.13

0.86

Model 1: Variable capital utilization η=0 2.334 2.275 η = 0.38 1.837 1.696 η=1 1.578 1.404

0.435 0.437 0.452

0.371 0.301 0.275

0.435 0.479 0.609

0.983 0.973 0.961

0.228 0.431 0.509

0.228 −0.155 −0.257

−0.094 −0.056 −0.016

Model 2: Constant capital utilization η=0 2.04 1.88 η = 0.38 1.85 1.56 η=1 1.73 1.36

0.93 0.91 0.91

0.93 0.92 0.94

0.69 0.66 0.65

0.8899 0.8721 0.8524

0.3988 0.5466 0.6279

Data

corr(Y, H )

corr(Y, Y /H )

0.3988 0.4659 0.4812

0.6305 0.7554 0.8136

corr(H, Y /H ) −0.22 0.0426 0.2091 0.2498 −0.0634 0.0668 0.1283

Z. Janko

Y

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5. Conclusion We analyze the impact of labor adjustment costs in a model with variable capital utilization and in a model without variable capital utilization. Two shocks are introduced, a technology shock and a government spending shock. Labor adjustment costs introduce a propagation mechanism whereby the impulse response functions are hump-shaped and persistent. We look at standard deviations and correlations of key macroeconomic variables when both shocks are present. Labor adjustment costs decrease the volatilities, and a model that includes both labor adjustment costs and constant capital utilization then best matches the model. With respect to correlations, we again find the labor adjustment costs model with constant capital utilization to best matched the model, with the exception of the correlation between hours and productivity, which we are unable to match under any specification. Overall, we find labor adjustment costs to be an important propagation mechanism for shocks.

Acknowledgement The chapter substantially benefited from comments made by anonymous referees.

References Benhabib, J. and Wen, Y. (2004). Indeterminacy, aggregate demand and the real business cycle. Journal of Monetary Economics 51, 503–530. Burnside, C. and Eichenbaum, M. (1996). Factor-hoarding and the propagation of businesscycle shocks. The American Economic Review 86, 1154–1174. Cambell, J.Y. (1994). Inspecting the mechanism, an analytical approach to the stochastic growth model. Journal of Monetary Economics 33, 463–506. Christiano, L.J. and Eichenbaum, M. (1992). Current real-business-cycle theories and aggregate labor-market fluctuations. The American Economic Review 82, 430–450. Cogley, T. and Nason, J.M. (1995). Output dynamics in real-business-cycle models. American Economic Review 85, 492–511. Davis, S.J. and Haltiwanger, J. (1990). Gross job creation and destruction: Microeconomic evidence and macroeconomic implications. In Blanchard, O.J. and Fisher, S. (Eds), NBER Macroeconomics Annual 1990. Cambridge: MIT Press, pp. 123–168. Fairise, X. and Langot, F. (1994). Labor productivity and business cycle: Can RBC models be saved? European Economic Review 38, 1581–1594. Janko, Z. (2005). The dynamic effects of adjustment costs in a model with stochastic wage staggering. Manuscript, University of Calgary. Jaramillo, F., Schiantarelli, J.F. and Sembenelli, A. (1993). Are adjustment costs for labor asymmetric? An econometric test on panel data for Italy. Review of Economics and Statistics 75, 640–648.

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Neiss, K.S. and Pappa, E. (2005). Persistence without too much price stickiness: The role of variable factor utilization. Review of Economic Dynamics 8, 231–255. Prescott, E.C. (1986). Theory ahead of business cycle measurement. Federal Reserve Bank of Minneapolis Quarterly Review 10, 9–22. Shapiro, M.D. (1986). Dynamic demand for capital and labor. Quarterly Journal of Economics 101, 523–542. Summers, L.H. (1981). Taxation and corporate investment: A “q” theory approach. Brookings Papers on Economic Activity 1, 67–127. Uhlig, H. (1999). A toolkit for analyzing nonlinear dynamic stochastic models easily. In Marimon, R. and Scott, A. (Eds), Computational Methods for the Study of Dynamic Economies. Oxford: Oxford University Press, pp. 30–61. Wen, Y. (2001). Demand-driven business cycle: Explaining domestic and international comovements. Manuscript, Cornell University.

Chapter 15

Optimal Fiscal Policy in an Asymmetric Small Open Economy Constantine Angyridis Department of Economics, Ryerson University, 350 Victoria Street, Toronto, ON, Canada M5B 2K3, e-mail: [email protected]

Abstract The objective of this chapter is to study the optimal fiscal policy in a dynamic stochastic small open economy with a financial asymmetry between the government and the representative household. The government can borrow and lend at a variable nonstate-contingent interest rate, while the representative household has access to complete financial markets. Uncertainty consists of shocks to asset prices and government expenditures. In this environment, we solve numerically for the optimal fiscal policy and discuss its predictions in relation to the existing literature. Keywords: fiscal policy, complete markets, incomplete markets, Ramsey problem JEL: E62, H21

1. Introduction What is the optimal response of the income tax rate to stochastic innovations in government expenditures and interest rates? How should the level of public debt evolve over time? Should the government be constrained in terms of the size of the budget deficit it can incur? These are important questions than can be addressed by taking advantage of recent theoretical advancements in solving stochastic intertemporal optimization problems in the presence of implementability constraints.1 This chapter applies these techniques to study the optimal fiscal policy in a small open economy with a government facing stochastic expenditures and interest rates. 1 Standard

dynamic programming techniques are applicable under the condition that the set of feasible current actions available to a social planner depend only on past variables. Implementability constraints imposed in contracting and optimal fiscal policy problems usually depend on plans for future variables, thus constraining the set of current feasible actions available to the social planner. Extending the work of Kydland and Prescott

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18015-7

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Based on the idea that when taxation is distortionary random fluctuations in the tax rate are welfare-reducing, Barro (1979) uses a partial equilibrium model to show that the optimal fiscal policy is that of “tax smoothing”. That is, budget deficits should vary in order to maintain expected constancy in the tax rate, regardless of the serial correlation properties of the exogenous government expenditures. Barro also provided empirical evidence that the behavior of public debt in the US and United Kingdom was consistent with the predictions of his theory during the period 1916–1976.2 The tax smoothing result has been reexamined by various authors in the context of a closed economy under different assumptions regarding the availability of financial markets. Lucas and Stokey (1983) considered the issue in a general equilibrium model with complete markets. Their framework emphasized the role of state-contingent debt as “insurance” against bad times purchased by the government from the households, which allows the former to smooth tax distortions across both time and states of nature. Thus, in contrast to Barro’s prediction, temporary increases in government expenditures lead to debt issue falling instead of rising. In addition, the authors show that the serial correlations of optimal tax rates are closely related to those for government expenditures. In their environment, tax rates are smooth, not in the sense of following a random walk, but in exhibiting a smaller variance than a balanced budget would imply. Marcet and Scott (2001) argue that, unlike in a model like Lucas and Stokey’s, the behavior of US public debt can be accounted for by a model in which the government issues only one-period risk-free bonds. Allowing for shocks to labor productivity in Lucas and Stokey’s model, the authors show that when the government issues a full set of state-contingent claims, public debt exhibits similar persistence compared to other variables in the economy and declines in response to an adverse shock in government expenditures. This prediction is in contrast to the US data, where public debt is not only substantially more persistent than other macroeconomic variables, but actually increases in response to an innovation in government expenditures.

(1980) for a general class of contracting problems involving incentive constraints, Marcet and Marimon (1999) formally show how one can compensate for this lack of recursivity by expanding the state space to include a new variable that depends on past Lagrange multipliers. 2 Sargent and Velde (1995) argue that the behavior of British debt during the 18th century conforms with Barro’s theory, since it closely resembles a martingale with a drift. During the latter period, the British government accumulated large deficits during wars and small, but sufficient, surpluses during peacetime. These features can be readily contrasted to the case of France with its recurrent debt defaults, which can be interpreted as occasionally low state-contingent payoffs. Extending Barro’s model to allow for stochastic variation in interest and growth rates, Lloyd-Ellis et al. (2001) show that the US debt policy is also in agreement with the theory of tax smoothing during the transition period from the large budget deficits of the 1980s to the high surpluses in the mid-1990s.

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Motivated by the empirical evidence in favor of tax smoothing, researchers have tried to develop general equilibrium models which give rise to Barro’s result. Aiyagari et al. (2002) revisit Lucas and Stokey’s closed economy environment, but restrict the government to borrow and lend at an endogenous risk-free rate. The authors show that the presence of the newly introduced market incompleteness imposes additional constraints with respect to equilibrium allocations on the Ramsey planner beyond a single implementability constraint imposed under complete markets. In particular, these constraints require the allocation to be such that at each date the present value of the budget surplus evaluated at current period Arrow prices is known one period ahead. Under the condition that an ad hoc limit is imposed on the government’s asset holdings, the authors show that the Ramsey equilibrium exhibits features shared by both paradigms of optimal debt policy, but the dynamics of debt and taxes actually resemble more closely Barro’s “taxsmoothing” result. An interesting question from a policy perspective is whether the result of Aiyagari et al. extends to the case of a small open economy, such as Canada, which faces exogenous interest rates. In this chapter, we analyze the tax-smoothing problem in this environment. Specifically, we consider an asymmetric small open economy with respect to accessibility to financial markets: households are allowed to perform transactions involving state-contingent financial claims, while the government is restricted to borrow and lend only at a risk-free interest rate.3 In this setting, the government is required to determine the optimal fiscal policy plan as of time zero in order to maximize the welfare of the representative household, given a stream of stochastic government expenditures which are financed by taxing labor income and issuing debt. However, the nonstate-contingent rates of return of the government bonds are also assumed to be exogenous and stochastic. Solving the Ramsey problem numerically for given specifications of government expenditure and asset prices processes yields series of equilibrium tax rates and public debt issue that exhibit more persistence than the underlying fiscal shocks. In addition, it is shown that allowing the government to borrow and lend at a constant nonstate-contingent interest rate results in equilibrium outcomes that resemble Barro’s tax smoothing result. A possible justification for the assumed asymmetry between the two sectors of the economy with respect to their accessibility to complete financial markets might be the inherent “moral hazard” that is present if the government is allowed to issue state-contingent debt. The return on these bonds would have been

3 Note

that the existence of a market for riskless government bonds is redundant from the point of view of the households, since it does not alter their trading opportunities: households can always adjust their portfolio of state-contingent securities in such a way to obtain the same return as that of a government bond.

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linked to the level of macroeconomic variables, such as the inflation rate or the debt/GDP ratio. However, these variables are under the strong influence of the government’s actions, thus damaging the marketability of these bonds among consumers. Furthermore, as Sleet (2004) and Sleet and Yeltekin (2006) demonstrate, if the government has access to a complete set of contingent claims markets but is unable to either commit to future debt repayments or truthfully reveal private information regarding its spending needs, then the market for public debt becomes endogenously incomplete. The coexistence of complete and incomplete markets in a small open economy is new in the literature on optimal taxation.4 Furthermore, if one would like to maintain the representative agent framework, the simplest environment to discuss the implications of this asymmetry is that of a small open economy.5 Finally, the distinction between the private and public sectors of the economy in terms of the nature of assets that they can trade captures an aspect of reality: private agents purchase and sell financial assets that have state-contingent payoffs, such as stocks, forwards, options etc., while governments issue exclusively nonstate contingent bonds.6 The rest of the chapter is organized as follows. Section 2 describes the theoretical model for the small open economy and states the optimal implementable competitive allocations that maximize the welfare of the representative household. Section 3 discusses the computational scheme employed in solving the Ramsey problem numerically. Section 4 provides details regarding the parameterization of

4 The

literature on optimal fiscal policy for a small open economy focuses exclusively on factor income taxation, incorporates capital stock accumulation and assumes away uncertainty. Recent studies include Atkeson et al. (1999), Chari and Kehoe (1999), Correia (1996) and Razin and Sadka (1995). 5 It should be noted, however, that the literature has considered different degrees of market completeness between the public and private sectors of a closed economy. For instance, Aiyagari et al. (2002) assume that the government faces ad hoc debt and asset limits that are more stringent than those faced by the representative household. On the other hand, Schmitt-Grohé and Uribe (2004) consider a stochastic production economy with sticky product prices in which households can acquire two types of final assets: fiat money and one-period state-contingent nominal assets. In contrast, the government can print money and issue nominal nonstate contingent bonds. 6 Canada is an example of a country, in which the federal government has the legal ability, at least in principle, to engage in transactions that involve financial assets with statecontingent payoffs, as of 1991. The legal restrictions that govern the issue of public debt are outlined in Part IV of the Financial Administration Act (also known as Chapter F-11). Section 45.1 states that: “The Governor in Council may authorize the Minister, subject to any terms and conditions that the Governor in Council may specify, to enter into any contract or agreement of a financial nature, including options, derivatives, swaps and forwards, on such terms and conditions as the Minister considers necessary”. This document is available at: http://laws.justice.gc.ca/en/F-11/53712.html

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311

the model that is used in the examples that follow. Section 5 discusses the quantitative predictions of the model for the debt and tax rate dynamics. Section 6 concludes the chapter.

2. The Model Consider a small open production economy populated by a large number of identical and infinitely lived individuals, in which there is no population growth. There is a single final good, which is nonstorable and the technology used in producing it is assumed to be such, that one unit of labor yields one unit of output. The endowment of labor at each point in time is unity. In any given period, an event st is assumed to take place. Let S denote a finite set whose elements are all the possible events that might occur. At any point in time, the state of this economy can be described by the state variable s t = (s0 , s1 , . . . , st ). This vector can also be thought of as representing the history of events up to and including period t. As a consequence, the history at any period can be described by s t = (s t−1 , st ). Finally, the probability as of period 0 of any specific history s t taking place is denoted by π(s t ). The representative household’s income consists of the sum of its net-of-tax labor earnings [1 − τ (s t )]n(s t ) and matured state-contingent securities bh (st |s t−1 ). The latter are one-period forward Arrow securities that pay one unit of output if state st is realized and nothing otherwise. Total income is allocated between consumption expenditures c(s t ) and the purchase of financial claims bh (st+1 |s t ) at the exogenously determined prices p h (st+1 |s t ) for each possible event st+1 . The asset prices are measured in units of time-t output and are assumed to be determined by the international financial markets. The objective of the household is to maximize expected lifetime utility, while taking the fiscal policy determined by the government as of time 0 as given. Formally,

max

{c(s t ),l(s t ),n(s t ),bh (st+1 |s t )}+∞ t=0,s t

W =

+∞

βt π st u c st , l st

(1)

t=0 s t

subject to: h c st + p st+1 |s t bh st+1 |s t st+1

1 − τ s t n s t + bh st |s t−1 ∀t 0, st+1 ∈ S, l s t + n s t = 100 ∀t 0

(2) (3)

312

C. Angyridis

and b0h given, where 0 < β < 1. The period utility u(·) is strictly increasing in consumption c(s t ) and leisure l(s t ), strictly concave and satisfying the Inada conditions.7 The first-order conditions for this intertemporal optimization problem are the following: ul (s t ) ∀t 0, = 1 − τ st t uc (s ) uc (s t+1 ) p h st+1 |s t = βπ st+1 |s t uc (s t )

(4) ∀t 0, st+1 ∈ S

(5)

and the household’s flow budget constraint (2) satisfied with equality. Condition (4) equates the marginal rate of substitution between leisure and consumption with the net return to the household from supplying one extra unit of labor. Expression (5) is the one-period asset pricing kernel for a state-contingent financial security purchased at time t and maturing in the following period. On the other hand, the government is faced with an exogenous stream of stochastic and wasteful expenditures g(s t ), which can be financed by either taxing labor income through a time-varying flat tax rate τ (s t ) or by issuing debt. In contrast to the financial claims purchased by the private sector, government bonds are assumed to be nonstate-contingent: in other words, they yield a rate of return that is independent of the state of nature in the economy. Formally, the government’s flow budget constraint can be described as: ∀t 0, p g s t bg s t = bg s t−1 + g s t − τ s t n s t (6) where bg (s t ) denotes the stock of bonds issued at time t and maturing at time t + 1, while p g (s t ) represents their price measured in units of time-t output.8 ,9 In

7 We follow Aiyagari et al. (2002) and adopt a rescaled version of the time endowment con-

straint most commonly used in the literature: l(s t ) + n(s t ) = 1. This is done for technical reasons, since it facilitates the computational scheme to be discussed later. 8 The existence of a market of riskless government bonds has no effect on the trading opportunities of the representative household. To see this, consider the simultaneous purchase of bg (s t ) units of output of Arrow securities corresponding to each possible state of nature. This transaction will yield next period a payoff of bg (s t ) units of output regardless of st+1 . However, this payoff is identical to that of a government bond. 9 As it is mentioned in the main text, the state- and nonstate-contingent (i.e. “riskless”) financial claims are both assumed to be one-period forward ones. In this sense, we are not dealing with the issue of the optimal maturity structure of public debt. Angeletos (2002) and Buera and Nicolini (2001) show how the maturity structure of riskless debt can be manipulated in such a way that the Ramsey allocations generated under complete markets are replicated, as long as there is a finite number of random outcomes and enough long-term bonds exist. Intuitively, the social planner takes advantage of the endogenous dependence of the term structure of interest rates on the state of the economy. Thus, if the maturity

Optimal Fiscal Policy in an Asymmetric Small Open Economy

addition, we assume that: h pg s t = p st+1 |s t

∀t 0, st+1 ∈ S

313

(7)

st+1

which ensures that both riskless and state-contingent bonds are traded and there is no arbitrage. Finally, in order to ensure a sustainable and meaningful optimal fiscal policy, the following asset and debt constraints are imposed on the government: ∀t 0, B p g s t bg s t B (8) and B denote the government’s maximum level of indebtedness and maxwhere B imum level of asset holdings, respectively. These limits correspond to Aiyagari’s (1994) notions of the “natural” debt and asset limits for the consumer’s problem. represents the maximum level of indebtedness for which In the present context, B debt can be paid back with probability 1, given the stochastic processes for government expenditures and interest rates. On the other hand, B represents the level of assets beyond which the government has no incentive to accumulate additional assets, since it can finance all of its expenditures solely through the interest on its assets, even if g(s t ) and p g (s t ) take their highest possible values in all periods. To define a competitive equilibrium for this economy, let: x s t = c s t , l s t , n s t , bh st+1 |s t ∀s ∈S t+1

denote an allocation for a given history at a given period t, and X = denote an allocation for every possible s t and t. Define ω(s t ) = {x(s t )}+∞ t=0,∀s t (τ (s t ), bg (s t )) as the government’s policy at time t given history s t , and Ω = {ω(s t )}+∞ t=0,∀s t the infinite sequence of policies implemented by the government. A competitive equilibrium for this economy is a policy Ω and an allocation X such that, given: (i) the sequence of asset prices P h = {p h (st+1 |s t )}+∞ t=0,∀st+1 ∈S , t (ii) the stochastic process for government expenditures g(s ) and prices of nonstate contingent government bonds p g (s t ), and (iii) the initial indebtedness of the g government b0 and the household’s b0h , (1) is maximized subject to (2) and (3), while the government’s budget constraints (6) and (8) are satisfied. The government’s or the Ramsey problem is to maximize (1) over competitive equilibria. Our formulation of this problem follows the primal approach in st ,

structure is chosen carefully ex ante, the ex post variation in the market value of outstanding long-term debt may offset the contemporaneous fluctuation in government expenditure: while in “good” times interest rates are low and the market value of long-term debt high, in “bad” times the market value of long-term debt falls and the capital gain enjoyed by the government compensates for the increase in government expenditures. However, this argument does not go through in the context of a small open economy, where interest rates are independent of domestic shocks.

314

C. Angyridis

characterizing the competitive equilibrium with distortionary taxes of the current model, as it is described in Atkinson and Stiglitz (1980). The basic idea is to recast the problem of choosing the optimal fiscal policy as a problem of choosing allocations subject to constraints which capture the restrictions on the type of allocations that can be supported as a competitive equilibrium for some choice of taxes and bond issue. Then, combining the first-order conditions for the government’s problem with those of the household’s one can determine the optimal tax rates. Working towards this end, we combine conditions (4), (5) with (2) and use forward substitution along with a transversality condition on bh (st+1 |s t ) to derive the household’s present value budget constraint as of time t. In addition, we substitute (4) into the government’s flow budget constraint (6). In the present context, the Ramsey problem can be formally described as follows: max

{c(s t ),l(s t ),n(s t ),bh (st+1 |s t )}+∞ t=0,s t

W =

+∞

βt π st u c st , l st

(9)

t=0 s t

subject to: bh st |s t−1 =

uc (s t+j )c(s t+j ) − ul (s t+j )n(s t+j ) β j π s t+j |s t uc (s t ) t+j

+∞ j =0 s

∀t 0, ∀st ∈ S, ul (s t ) t p g s t bg s t = bg s t−1 + g s t − 1 − ∀t 0, n s uc (s t ) uc (s t ) ∀t 1, ∀st ∈ S, p h st |s t−1 = βπ st |s t−1 uc (s t−1 ) ∀t 0, B p g s t bg s t B t t l s + n s = 100 ∀t 0

(10) (11) (12) (13) (14)

g b0 , b0h

and given. The Ramsey allocation can be characterized by composing a Lagrangian for the Ramsey problem. Multipliers β t π(s t )φ(s t ), β t π(s t )λ(s t ), β t π(s t )θ1 (s t ) and β t π(s t )θ2 (s t ) are attached to constraints (10), (11) and the left-hand and righthand sides of (13). In addition, constraint (12) is rewritten as: p h st |s t−1 uc s t−1 − βπ st |s t−1 uc s t = 0 and multiplier β t μ(s t ) is attached to it. Then, in the Lagrangian for the Ramsey problem (9)–(14) described above, the term: ··· + β t μ s t p h st |s t−1 uc s t−1 − βπ st |s t−1 uc s t + · · · s t−1 st

Optimal Fiscal Policy in an Asymmetric Small Open Economy

315

can be rewritten as: ··· + β t π s t μˆ s t p h st |s t−1 uc s t−1 − βuc s t + · · · , st

where pˆ h (st |s t−1 ) ≡ p h (st |s t−1 )/π(st |s t−1 ) and μ(s ˆ t ) ≡ μ(s t )/π(s t−1 ). t Next, define a pseudo-multiplier γ (s ) as: γ s t = φ s t + γ s t−1 ∀t 0

(15)

with γ−1 = 0. Using a similar argument as in Aiyagari et al. (2002) by applying the Law of Iterated Expectations and appropriately rearranging the sums involved, the original nonrecursive Lagrangian for the Ramsey problem is transformed to the following recursive one: L=

+∞

β t π s t u c s t , 100 − n s t

t=0 s t

+ γ s t − γ s t−1 bh st |s t−1 uc (s t )c(s t ) − ul (s t )n(s t ) − γ st uc (s t ) + λ s t p g s t bg s t − bg s t−1 t ul (s t ) t n s −g s + 1− uc (s t ) t − p g s t bg s t + θ1 s B + + θ2 s t p g s t b g s t − B

+

+∞

β t π s t μˆ s t pˆ h st |s t−1 uc s t−1 − βuc s t .

(16)

t=1 s t

3. Solving the Ramsey Problem This section provides a description of the computational scheme involved in solving the Ramsey problem numerically. The decision variables at the initial period t = 0 are: y s 0 = c s 0 , n s 0 , l s 0 , bg s 0 , bh s1 |s 0 ,

γ s 0 , φ s 0 , λ s 0 , θ1 s 0 , θ2 s 0 ,

316

C. Angyridis

while at t 1: y s t = c s t , n s t , l s t , bg s t , bh st+1 |s t ,

γ s t , φ s t , μˆ s t , λ s t , θ1 s t , θ2 s t , where bh (st+1 |s t ) is a vector, ∀t 0, with each element corresponding to a possible realization of st+1 . Letting St = (pˆ h (st |s t−1 ), p g (s t ), g(s t )), the state variables at t = 0 are: g

x s 0 = b0 , b0h , γ−1 , S0 , where, by definition, γ−1 = 0, while at t 1:

x s t = bg s t−1 , bh st |s t−1 , c s t−1 , γ s t−1 , St , with bh (st |s t−1 ) being now a scalar, ∀t 0. Next, we argue that bh (st |s t−1 ) should not be included in the vector of state variables, x(s t ). Constraint (10) implies that: ∀t 1, st ∈ S. bh st |s t−1 = D bg s t−1 , c s t−1 , γ s t−1 , St In other words, the stock of state-contingent financial claims held by the household at time t 1 depends on the same set of state variables as y(s t ). Therefore, it is not necessary to include it as an extra state variable when solving for y(s t ). Based on that, the implementation of the algorithm does not require solving for bh (st+1 |s t ).10 To summarize, the decision and state variables of interest at this point are: y s t = c s t , n s t , l s t , bg s t , γ s t ,

φ s t , μˆ s t , λ s t , θ1 s t , θ2 s t and

x s t = bg s t−1 , c s t−1 , γ s t−1 , St

∀t 0, st ∈ S,

where St = (pˆ h (st |s t−1 ), p g (s t ), g(s t )) and μ(s ˆ 0 ) = c−1 = γ−1 = 0. For the quantitative component of the chapter, we assume that the instantaneous utility function of the representative household is additively separable in

10 This

statement does not imply that the portfolio of state-contingent financial claims purchased by the household in each period is indeterminate: once the Ramsey problem in its current form is solved and the policy rules for all endogenous variables as a function of the state variables become known, we can use (10) to derive bh (st |s t−1 ) for each period. However, it should be noted that this is computationally a very expensive task to perform.

Optimal Fiscal Policy in an Asymmetric Small Open Economy

consumption and leisure:11 u c st , l st = v c st + H l st

317

∀t 0.

Given this assumption, the first-order conditions for the Ramsey problem with respect to c(s t ), n(s t ), bg (s t ), μ(s ˆ t ), λ(s t ) and γ (s t ) can be shown to be the following: 0 0 0 0 0 v

(c(s 0 ))H (100 − n(s 0 ))

v c s −γ s − γ s −λ s n s [v (c(s 0 ))]2 + βv

c s 0 (17) π s1 |s 0 μˆ s 1 pˆ h s1 |s 0 = 0, s1

v

(c(s t ))H (100 − n(s t )) v c s t − γ s t − γ s t − λ s t n s t [v (c(s t ))]2 − βv

c s t μˆ s t + βv

c s t π st+1 |s t μˆ s t+1 pˆ h st+1 |s t = 0 ∀t 1, × (18)

st+1

−H 100 − n s t + λ s t t H (100 − n(s t )) − H

(100 − n(s t ))n(s t ) t =0 + γ s −λ s v (c(s t )) ∀t 0, (19) −1 t t t t+1 g t t p s β λ s − θ1 s + θ2 s = π st+1 |s λ s

st+1

∀t 0, h pˆ st |s t−1 v c s t−1 = βv c s t ∀t 1, st ∈ S, H (100 − n(s t )) t n s p g s t bg s t = bg s t−1 + g s t − 1 − v (c(s t )) ∀t 0, H (100 − n(s t )) t bh st |s t−1 = c s t − n s v (c(s t )) ∀t 1, st+1 ∈ S. π st+1 |s t bh st+1 |s t +β

(20) (21)

(22)

(23)

st+1

11 Introducing nonseparability in the utility function between consumption and leisure adds

to the complexity of the calculations involved, but leaves the general logic of the proposed numerical scheme, which will be discussed later, intact. Whether this specification would alter the results obtained in the present context is a question that can only be answered once the actual simulations are performed.

318

C. Angyridis

In addition, the standard Kuhn–Tucker conditions apply: − p g s t bg s t = 0, θ1 s t 0 ∀t 0, θ1 s t B θ2 s t p g s t bg s t − B = 0, θ2 s t 0 ∀t 0.

(24) (25)

A useful result can be derived from the first-order condition with respect to the state-contingent bonds purchased by the household, bh (st+1 |s t ), which is given by: β t+1 π st+1 |s t φ s t+1 = 0 ∀t 0, st+1 ∈ S. (26) This, in turn, implies: φ s t+1 = 0 ∀t 0, β t+1 π s t+1 γ s t+1 − γ s t = 0

(27) ∀t 0

and, in addition, by making use of definition (15), we obtain: γ s t+1 = γ s t = γ¯ ∀t 0.

(28)

(29)

γ (s t )

remains constant in all periods. For this reason, it can be In other words, safely removed from being an element of x(s t ), ∀t 0. Using a more compact notation than previously, the state and decision variables that will be the focus of the computational component in the chapter can be summarized as follows: y s t = c s t , n s t , l s t , bg s t , γ s t ,

φ s t , μˆ s t , λ s t , θ1 s t , θ2 s t ∀t 0 and

x s t = bg s t−1 , c s t−1 , St

∀t 0,

where ∀t 0, St = pˆ h st |s t−1 , p g s t , g s t t γ s = γ¯ ∀t 0, φ s 0 = γ¯ , φ s t = 0 ∀t 1, c−1 = γ−1 = μˆ s 0 = 0. The proposed algorithm used in solving the Ramsey problem numerically is an appropriate adaptation to the present context of the Parameterized Expectations Algorithm (PEA), as it is described in den Haan and Marcet (1990). According to this algorithm the agent’s conditional expectations about functions of future variables are replaced by an approximating function involving the state variables of

Optimal Fiscal Policy in an Asymmetric Small Open Economy

319

the system and coefficients on each of these variables. The approximating function is used to generate T realizations of the endogenous variables of the model, where T is a very large number. These realizations are used as observations in a series of nonlinear least squares regressions which are used to reestimate the coefficients of the approximating function. The new set of obtained coefficients are then used to generate a new series of length T for the endogenous variables. Iterations continue until the regression coefficients obtained from the use of successive sets of coefficients for the approximating function converge up to a prespecified tolerance level. The steps involved in implementing the algorithm in the present context are described in the Appendix.

4. Parameterization and Computational Details The parameterization scheme resembles that of Aiyagari et al. (2002). In particular, we assume that the instantaneous utility function of the representative household is separable in consumption and leisure: t 1−σ2 t t c(s t )1−σ1 − 1 −1 l(s ) +η . u c s ,l s = 1 − σ1 1 − σ2 The preference parameters have been assigned the following values: σ1 = 0.5, g σ2 = 2 and η = 1. Regarding initial indebtedness, we set b0 = b0h = 0. Government expenditures are assumed to obey the following autoregressive stochastic process: φ(s t+1 ) g s t+1 = (1 − ρ)g¯ + ρg s t + , sg

(30)

where φ(s t+1 ) is an independently and identically distributed sequence of random numbers distributed N (0, 1), and sg is a scale factor. The long run mean of g(s t ), g, ¯ is set equal to 30, while sg = 0.40. With respect to parameter ρ, we consider the case of serially uncorrelated government expenditures. In other words, we set ρ = 0. Furthermore, g(s t ) is constrained to lie in the interval [20, 40] for all periods. This specification is the same as the one adopted by Aiyagari et al. in one of their numerical examples. The assumed stochastic process for the price of government bonds is given by: ε(s t ) pg s t = β 1 + (31) , sr where β = 0.95 and sr = 800, while ε(s t+1 ) is an independently and identically distributed sequence of random numbers distributed N (0, 1). In addition, p g (s t ) is constrained to lie in the interval [0.94, 0.96] for all periods. Finally, pˆ h (st+1 |s t )

320

C. Angyridis

is assumed to be generated from the following stochastic process: u(s t+1 ) h t g t pˆ st+1 |s = p s 1 + , sp

(32)

where u(s t+1 ) is also an independently and identically error term drawn from a N(0, 1) distribution and the scale factor sp is set equal to 800. Note that the specification in (32) ensures that the no-arbitrage condition (7) is satisfied. The conditional expectations in the system of first-order conditions of the Ramsey problem were parameterized using the family of approximating functions mapping 5+ into + : ψ(αi , xt ) = exp Pn (xt ) , where Pn denotes a polynomial of degree n and the parameters αi are the coefficients in the polynomial. In addition, to ensure that all the variables involved in the nonlinear least squares regressions performed were of similar orders of magni¯ → (−1, 1) tudes, we followed Aiyagari’s et al. in applying the function ϕ : (k, k) ¯ to each state variable separately, where k and k are prespecified lower and upper bounds for the argument of ϕ, k. This function is defined as: ϕ(k) = 2

k−k − 1, k¯ − k

which implies that the ψ(·) functions used in the nonlinear least squares regressions component of the algorithm were: ψ(αi , xt ) = exp Pn ϕ(xt ) . The lower and upper bounds used in transforming g(s t ), pˆ h (st+1 |s t ) and p g (s t ) before they were used as inputs in the parameterized expectations correspond to the intervals stated previously. For bg (s t ) the bounds were B/p g (s t ) and g (s t ), while the ones for c(s t−1 ) were set to be 40 and 100.12 B/p

5. Quantitative Results The objective of the above quantitative exercise is to determine whether the structure of optimal taxation policy and the dynamics of public debt issue for this artificial economy conform to either of the two main paradigms of optimal debt determination. We consider two cases: the first case was outlined in the previous

12 These

bounds are arbitrary. Once the program has been executed, we can check the generated equilibrium allocation to see whether consumption exceeds the prescribed bounds and adjust them accordingly.

Optimal Fiscal Policy in an Asymmetric Small Open Economy

321

Figure 1a: Various Variables for the Stochastic Nonstate-Contingent Interest Rate Case. section and involves a stochastic nonstate-contingent interest rate. In the second case, we assume that government can borrow and lend at a constant risk-free rate equal to β. In other words, we replace (31) by: p g s t = β ∀t 0. The results for each case are displayed in groups of graphs in Figures 1 and 2. Each figure consists of 3 groups of graphs. The first group, group (a), plots the entire allocation series for consumption, leisure, price of state-contingent financial claims, debt, tax rate and government expenditures. Group (b) is simply a truncated version of the graphs in group (a), respectively, depicting only the last 150 periods. Finally, group (c) displays the impulse response functions of leisure, tax rate, tax revenue, budget deficit and debt with respect to the innovation in government expenditures g(s t ).13 13 First-order

condition (21) in conjunction with the assumption of separable preferences in consumption and leisure implies that, once c0 is determined, ct evolves as a function

322

C. Angyridis

Figure 1b: Various Variables for the Stochastic Nonstate-Contingent Interest Rate Case (T = 150). The first case presented is that of the stochastic nonstate-contingent interest rate and appears in Figure 1. The asset and debt limit were set equal to −200 and 200, respectively. The graph of debt issue makes apparent that the government holds assets more often than it owes debt. Overall, the government is a lender for roughly 82% of the time. The graphs for the last 150 periods clearly demonstrate a pattern of co-movement between leisure and the tax rate. This is anticipated, since that after the initial period, consumption is determined from the stochastic process for pˆ h (st+1 |s t ) from (21). However, the way this process is specified does not allow asset prices to move too far away from β. As a consequence, the consumption series is relatively smooth. Since the optimal tax rate is determined from (4), changes in leisure are

of the ratio (β/pˆ h (st |s t−1 ))−1/σ1 ∀t 1, which is parametric to both the representative household and the government. This is the reason why no impulse response functions for consumption are displayed.

Optimal Fiscal Policy in an Asymmetric Small Open Economy

323

Figure 1c: Impulse Response Functions for the Stochastic Nonstate-Contingent Interest Rate Case. mirrored to a large extent in changes in the tax rate. Finally, debt issue appears to be positively correlated with government expenditures: as g(s t ) increases, the level of public debt rises. This is consistent with the prediction of Barro’s model. The impulse response functions displayed in Figure 1c differ from the ones reported in Aiyagari et al. In their case, a positive innovation in government expenditures causes both leisure and consumption to fall. The overall effect is an increase in the tax rate. In our case, leisure falls as well. However, since consumption is purely determined by the stochastic process governing the prices of the state-contingent financial claims and, as a consequence, is fairly smooth, the overall effect is a decrease in the tax rate. The case of the constant nonstate-contingent interest rate is shown in Figure 2. The upper and lower bounds of debt were set to be 50 and −50, respectively. The observation made earlier regarding the government’s tendency to hold assets rather than owe debt carries over to the present context: the government now borrows almost 25% of the time.

324

C. Angyridis

Figure 2a: Various Variables for the Constant Nonstate-Contingent Interest Rate Case. As it is shown in Figure 2b, the co-movement pattern between leisure and the tax rate observed earlier is again present. In addition, compared to Figure 1b, allowing the government to borrow and lend at a constant risk-free interest rate results in all endogenous variables being smoother. The impulse response functions displayed in Figure 2c differ from the ones in Figure 1c both quantitatively and qualitatively. The effect of a positive innovation in government expenditures on the endogenous variables of the model appears to be smaller in magnitude. Furthermore, leisure now increases instead of falling, causing the tax rate to rise instead of falling. However, the dynamics of the budget deficit in the two cases considered are quite similar. As mentioned earlier, the main implication of Barro’s model is that the optimal tax rate should follow a random walk, regardless of the stochastic process for government expenditures. In contrast, Lucas and Stokey suggest that the tax rate inherits the serial correlation properties of government expenditures and should be smooth, not in the sense of following a random walk, but in having a smaller variance than a balanced budget would imply. In the current context, due to the

Optimal Fiscal Policy in an Asymmetric Small Open Economy

325

Figure 2b: Various Variables for the Constant Nonstate-Contingent Interest Rate Case (T = 150). coexistence of complete and incomplete markets, intuition suggests that the tax rate should behave in a manner that lies between these two extremes, but is closer to Barro’s proposition: it is the government that is imperfectly insured against future contingencies, but it is also the agent who sets the tax rates in this economy. A first step in the direction towards verifying this conjecture is to perform the following autoregression using the generated series for the tax rate: τt = a + bτt−1 + ut .

(33)

Table 1 presents the first two unconditional moments of the tax rates, as well as the results from the above least squares regressions. The first two columns refer to the two cases discussed in the present context, while the third one provides the corresponding results reported by Aiyagari et al. for their incomplete markets case with ρ = 0.14 In the case of the stochastic nonstate-contingent interest rate

14 The

set by these authors was (−1000, 1000). range (B, B)

326

C. Angyridis

Figure 2c: Impulse Response Functions for the Constant Nonstate-Contingent Interest Rate Case. Table 1: Autoregressions of the Tax Rate

E[τ ] std[τ ] a b R2

Case 1

Case 2

Aiyagari et al.

0.2852 0.0462 0.1310 0.5408 0.9819

0.3048 0.0137 0.0050 0.9835 0.9675

0.2776 0.0191 0.0031 0.9888 0.9944

(case 1), the average tax rate is lower compared to both the constant risk-free interest rate case (case 2) and the one obtained by Aiyagari et al. Furthermore, the tax rate in case 1 appears to be considerably more volatile relative to the one in the other two cases. In contrast to the prediction of Lucas and Stokey’s model, the estimated coefficient of the lagged tax rate b is significantly different from zero and implies that the tax rate in case 1 exhibits high persistence. However, the

Optimal Fiscal Policy in an Asymmetric Small Open Economy

327

degree of persistence (0.5408) is not close enough to 1, which is the value of b implied by Barro’s model. Providing the government with the ability to borrow and lend at a constant riskfree interest (case 2), allows it to achieve “tax smoothing”. Although the asset and debt limits are four times smaller than their counterparts in case 1, the elimination of the interest rate effect on public debt issue results in the tax rate becoming highly persistent. The estimated coefficient b is close to 1 and similar in value with the one obtained by Aiyagari et al. in the context of a closed economy. In other words, the tax rate in case 2 tends to resemble a martingale.

6. Conclusion This chapter studies an artificial small open economy in which only households have access to complete financial markets. Constraining the government to borrow and lend at a stochastic risk-free interest rate we solve for a Ramsey equilibrium numerically given a stochastic process of serially uncorrelated government expenditures. The Ramsey equilibrium exhibits behavior that lies between the two main theories of public debt determination, but is closer to Barro’s proposition that the tax rate should follow a random walk. Eliminating the interest rate effect on public debt issue, results in debt and tax rate dynamics that are consistent with the “tax smoothing” result. The basic model of this chapter can be extended by endogenizing the choice of government expenditures and adding a maturity structure in the issue of public debt. These modifications will not only add more realistic features to the present setup, but will also facilitate a quantitative exercise in which the model is calibrated to an actual economy. We intend to undertake these extensions in future work.

Acknowledgements I am indebted to Eric Kam, Leo Michelis and Xiaodong Zhu for their helpful comments and suggestions. All possible errors are mine alone.

Appendix This appendix discusses the numerical scheme adopted in order to solve the Ramsey problem described in Section 3. The steps involved are the following: Step 1: Choose appropriate values for the parameters of the model. Assuming that the exogenous state variables of the model, g(s t ), p g (s t ) and

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pˆ h (st |s t−1 ), follow particular stochastic processes, generate a series of realizations for each one of them. These series have length T , where T is a large number, and are drawn only once. Step 2: Make a guess for the value of the pseudo-multiplier γ¯ , γ¯j , where subscript j is an integer used to denote the current outer-loop iteration step. Step 3: Replace the conditional expectations included in first-order conditions (17), (18) and (20) of the Ramsey problem by approximating functions ψ(αik , x(s t )), where αik denotes a fixed vector of parameters, subscript i represents the current inner-loop iteration step, and k = 1, 2 corresponds to the particular conditional expectation replaced by a function ψ(·, x(s t )). Formally, expressions (17), (18) and (20) can be compactly rewritten as: v c s t − γ s t v

(c(s t ))H (1 − n(s t )) − γ st − λ st n st [v (c(s t ))]2 − βv

c s t μˆ s t + βv

c s t ψ αi1 , x s t = 0 ∀t 0

(34)

and p g s t β −1 λ s t − θ1 s t + θ2 s t = ψ αi2 , x s t ∀t 0,

(35)

where μ(s ˆ 0 ) = 0. Step 4: Letting a˜ i = [αi1 , αi2 ] , obtain a long series of the endogenous variables of the model {y(s t , a˜ i ; γ¯j )}Tt=0 , that solves the system of equations (14), (19), (21), (22), (24), (25), (34) and (35). Towards this end, we have to distinguish between four possible cases, depending on whether: (i) constraint (13) is binding, and (ii) the allocation derived corresponds to the initial period. (a) Unconstrained Case for t = 0: In this case, B < p g (s 0 ) × and from the Kuhn–Tucker conditions (24) and (25) bg (s 0 ) < B it follows that θ1 (s 0 ) = θ2 (s 0 ) = 0. From (35), we obtain λ(s 0 ). First-order conditions (19) and (34) consist a system of two (nonlinear) equations in two unknowns: c(s 0 ) and n(s 0 ). Once n(s 0 ) has been calculated, the time endowment constraint (14) provides us with l(s 0 ). Finally, making use of c(s 0 ) and n(s 0 ), we compute bg (s 0 ) from (22). (b) Constrained Case for t = 0: In this case, either bg (s 0 ) = g (s 0 ) or B/p g (s 0 ), which by making use of (24) and (25) B/p

Optimal Fiscal Policy in an Asymmetric Small Open Economy

329

implies that θ1 (s 0 ) > 0 and θ2 (s 0 ) = 0 or θ2 (s 0 ) > 0 and θ1 (s 0 ) = 0, respectively. Equations (19), (22) and (34) consist a system of three equations in three unknowns: c(s 0 ), n(s 0 ) and λ(s 0 ). Once n(s 0 ) has been calculated, the time endowment constraint (14) yields l(s 0 ). Finally, given the multiplier λ(s 0 ), first-order condition (35) provides us with either θ1 (s 0 ) or θ2 (s 0 ). (c) Unconstrained Case for t 1: In this case, B < p g (s t )bg (s t ) < and from the Kuhn–Tucker conditions (24) and (25) it folB lows that θ1 (s t ) = θ2 (s t ) = 0. From (35), we obtain λ(s t ). Given c(s t−1 ) calculated from the previous period, first-order condition (21) provides us with c(s t ). Equations (19) and (34) then consist a system of two equations in two unknowns: n(s t ) and μ(s ˆ t ). Given the optimal supply of labor, the time endowment constraint (14) yields l(s t ). Finally, we compute bg (s t ) by substituting c(s t ) and n(s t ) into (22). (d) Constrained Case for t 1: In this case, either bg (s t ) = g (s t ) or B/p g (s t ), which by making use of (24) and (25) B/p implies that θ1 (s t ) > 0 and θ2 (s t ) = 0 or θ2 (s t ) > 0 and θ1 (s t ) = 0, respectively. As in the previous case, given c(s t−1 ) calculated from the previous period, c(s t ) can be derived from first-order condition (21). Equations (19), (22) and (34) consist then a system of three equations in three unknowns: μ(s ˆ t ), n(s t ) and λ(s t ). Once n(s t ) has been calculated, we obtain l(s t ) from the time endowment constraint (14). Finally, given the multiplier λ(s t ), first-order condition (35) provides us with either θ1 (s t ) or θ2 (s t ). Step 5: Given the series of the endogenous variables of the model calculated in the previous step, compute the variables that appear inside the conditional expectations in expressions (17), (18) and (20): that is, μ(s ˆ t+1 )pˆ h (st+1 |s t ) and λ(s t+1 ), ∀t 0. Next, perform nonlinear least squares regressions of these variables on the corresponding ψ(·, x(s t )) functions. Define the result of these regressions as S(a˜ i ). Step 6: Using a predetermined relaxation parameter δ ∈ (0, 1], update a˜ i according to the following scheme: a˜ i+1 = (1 − δ)a˜ i + δS(a˜ i ). Step 7: Iterate until a˜ i+1 − a˜ i < ε, where ε is a small positive number. Denote the fixed point obtained by a˜ 0 . Step 8: Consider H histories, each history being P periods long. Generate corresponding series for the exogenous state variables of the model (g, pg and pˆ h ) and, using a˜ 0 , repeat steps 3 and 4 to derive the allocations and

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corresponding tax rates. Next, compute the following expression:

P P H 1 g g h,j −1 p s d j = b0 − H h=1 p=0 j =1 × τ s h,p ; γ¯j n s h,p ; γ¯j − g s h,p , which is an approximation to the government’s present value budget constraint.15 Keep iterating on γ¯j until |dj | < ε, where ε is a small positive number.

References Aiyagari, S.R. (1994). Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109(3), 659–684. Aiyagari, S.R., Marcet, A., Sargent, T.J. and Seppala, J. (2002). Optimal taxation without state-contingent debt. Journal of Political Economy 110, 1220–1254. Angeletos, G.M. (2002). Fiscal policy with non-contingent debt and the optimal maturity structure. Quarterly Journal of Economics 117(3), 1105–1131. Atkeson, A., Chari, V.V. and Kehoe, P.J. (1999). Taxing capital income: A bad idea. Federal Reserve Bank of Minneapolis Quarterly Review 23, 3–17. Atkinson, A.B. and Stiglitz, J. (1980). Lectures on Public Economics. New York: McGrawHill. Barro, R.J. (1979). On the determination of public debt. Journal of Political Economy 87, 940–971. Buera, F. and Nicolini, J.P. (2001). Optimal maturity of government debt without state contingent bonds. Mimeo, University of Chicago. Chari, V.V. and Kehoe, P.J. (1999). Optimal fiscal and monetary policy. In Taylor, J.B. and Woodford, M. (Eds), Handbook of Macroeconomics. Amsterdam: North-Holland, pp. 1671–1745. Correia, I.H. (1996). Dynamic optimal taxation in small open economies. Journal of Economic Dynamics and Control 20, 691–708. den Haan, W.J. and Marcet, A. (1990). Solving the stochastic growth model by parameterizing expectations. Journal of Business and Economic Statistics 8, 31–34. Kydland, F.E. and Prescott, E.C. (1980). Dynamic optimal taxation, rational expectations and optimal control. Journal of Economic Dynamics and Control 2, 79–91.

15 The

latter can be obtained by using forward substitution on (6) and imposing a transversality condition on the government’s debt as T → +∞ to obtain:

t +∞ j −1 t t g t g π s p s τ s n s − g st . b0 = t=0 s t

j =1

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331

Lloyd-Ellis, H., Zhan, S. and Zhu, X. (2001). Tax smoothing with stochastic interest rates: A re-assessment of Clinton’s fiscal legacy. Working Paper No. 125, Center for Research on Economic Fluctuations and Employment, UQAM. Lucas Jr, F.E. and Stokey, N.L. (1983). Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 55–93. Marcet, A. and Marimon, R. (1999). Recursive contracts. Mimeo, Universitat Pompeu Fabra, Barcelona. Marcet, A. and Scott A. (2001). Debt and deficit fluctuations and the structure of bond markets. Mimeo, Universitat Pompeu Fabra, CREI and CEPR. Razin, A. and Sadka, E. (1995). The status of capital income taxation in the open economy. FinanzArchiv 52, 21–32. Sargent, T.J. and Velde, F. (1995). Macroeconomic features of the french revolution. Journal of Political Economy 103(3), 474–518. Schmitt-Grohé, S. and Uribe, M. (2004). Optimal fiscal and monetary policy under sticky prices. Journal of Economic Theory 114, 198–230. Sleet, C. (2004). Optimal taxation with private government information. Review of Economic Studies 71, 1217–1239. Sleet, C. and Yeltekin, S. (2006). Optimal taxation with endogenously incomplete debt markets. Journal of Economic Theory 127, 36–73.

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Part VII Monetary Policy and Capital Accumulation

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Chapter 16

Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect Eric Kam Department of Economics, Ryerson University, Toronto, ON, Canada M5B 2K3, e-mail: [email protected]

Abstract This chapter describes the effects of monetary growth on real sector variables using optimizing models with an endogenous rate of time preference. Results depend on whether money is introduced in a utility function or a cash-in-advance constraint. With money in the utility function, time preference wealth effects endogenize the real interest rate so that inflation raises capital and consumption through the Tobin effect. If money is introduced by a cash-in-advance constraint on consumption and capital goods, inflation lowers the demand for both goods and reverses the Tobin effect. However, if the constraint applies to consumption goods, monetary growth is superneutral.

Keywords: time preference, Tobin effect, cash-in-advance JEL: E40, E50, E52

1. Introduction The controversial Uzawa (1968) function, in which the rate of time preference is modeled as an increasing function of instantaneous utility has produced significant criticism. Many theorists argue that the underlying assumptions that are necessary for steady state stability are ad-hoc, counterintuitive, and lack the necessary empirical underpinnings (Persson and Svensson, 1985; Blanchard and Fischer, 1989; Turnovsky, 2000; Smithin, 2004). This chapter describes a restatement of the Tobin effect and stability by emphasizing parallels between Uzawa and accepted aggregative models (Mundell, 1963; Tobin, 1965; Laidler, 1969; Begg, 1980). The rate of time preference and the discount factor applied to future utility streams are modeled as an increasing function of real wealth to produce nonsuperneutrality and a stable Tobin effect using the optimizing foundations of the infinitely lived, representative agent. However, these results do not rely on the International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18016-9

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counterintuitive preference criticisms that bring the applicability of the Uzawa time preference specification into dispute. This chapter further demonstrates that if a cash-in-advance constraint is applied to purchases of capital and consumption goods or consumption goods alone, this new motive either dominates or accommodates the endogenous time preference wealth effects. When the cashin-advance constraint applies to the purchase of both capital and consumption goods, inflation lowers the steady state production and demand for each commodity. Alternatively, if the cash-in-advance constraint applies solely to consumption goods, monetary growth is superneutral. Following decades of research, economists have failed to reach a consensus regarding how changes in the monetary growth rate affect the “real” economy in general and steady state capital in particular. In the modern literature, this debate started with Tobin’s (1965) extension of Solow (1956) and Swan’s (1956) one-sector, neoclassical growth model that assumes money is an asset. Using descriptive, nonoptimizing models, the “Tobin effect” demonstrates how monetary growth yields inflation and reduces the relative return on real balance holdings so that agents convert their assets from money to capital. More recently, the optimizing underpinnings of the infinitely-lived representative agent model have been employed to determine the real sector implications of inflation and monetary growth. One seminal technique for modeling money in the Walrasian tradition is Stockman (1981) and Abel’s (1985) cash-in-advance constraint models where liquidity constraints are levied on the purchase of consumption and capital goods or consumption goods independently. The results of these models contradict Tobin,1 but this ambiguity is supported by the empirical evidence that fails to confirm any stylized relationship between inflation, monetary policy and steady state, real sector variables such as consumption and the capital stock.2 The Tobin effect has largely been forgotten because it tends not to arise in optimizing models. In neoclassical models with optimizing agents and competition in production, the return to capital is determined by its marginal product. The household’s Euler equation in turn links the return to capital to the growth rate of consumption. Given fixed preferences and no productivity growth, these conditions hold constant the amount of physical capital and its return. Monetary growth has no affect on real variables because the savings rate is endogenous. Households are not concerned with the amount of savings because decisions rely on the return to investment relative to the intertemporal marginal rate of substitution. Monetary

1 Inflation

lowers consumption and capital if the cash-in-advance constraint applies to the purchase of both commodities but is superneutral if the constraint applies only to the purchase of consumption goods. 2 The Tobin effect is verified in Rapach (2003), Ahmad and Rogers (2000), Shrestha et al. (2002) and Woodward (1992) and is reversed in Bae and Ratti (2000). Serletis and Koustas (1998, 2001) and Serletis and Krause (1996) reject superneutrality using recent econometric techniques.

Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect 337

policy has no effect on the household Euler equation, which implies that steady state capital is unaffected and monetary growth is superneutral (Sidrauski, 1967). Uzawa’s utility function, and specifically his endogenous rate of time preference, was first created to break the monetary superneutrality result in Sidrauski’s (1967) model with real balances in the utility function. Sidrauski showed that if the rate of time preference is fixed, monetary policy has no affect on steady state capital as it is determined by equality across the rate of time preference and the marginal product of capital. However, using the Uzawa function, monetary growth increases the opportunity cost of holding real balances, which leaves the initial steady state equilibrium too costly. As a result, there is a decrease in real balance holdings and an increase in savings, which is then converted through a fall in the rate of time preference to steady state capital and consumption. However, this nonsuperneutrality result depends critically on the assumption that the rate of time preference is an increasing function of instantaneous utility, which is itself, an increasing function of consumption. Uzawa argues correctly that this assumption is necessary for stability, but it is also the reason why his model is criticized and often dismissed. Savings are modeled as an increasing function of real wealth, which implies that agents are increasingly impatient as consumption rises. Blanchard and Fischer (1989), Lawrance (1991), Turnovsky (2000) and Persson and Svensson (1985, p. 45) refer to this modeling assumption as “arbitrary, and even counterintuitive”,3 as it contradicts the empirical and theoretical evidence describing savings as a decreasing function of wealth.4 The Uzawa function also received significant attention in the open-economy literature with respect to the Harberger–Laursen–Metzler effect. Using Uzawa preferences, Obstfeld (1982) demonstrates that a deterioration in the terms of trade raises savings and generates a current account surplus for a small open economy that is constrained by an exogenously fixed world interest rate. However, his results contradict the accepted Harberger–Laursen–Metzler intuition, which generated a large literature (including Svensson and Razin, 1983; Persson and Svensson, 1985; Matsuyama, 1987, 1988; Sen and Turnovsky, 1989) which examine the effect of a terms of trade deterioration using the optimizing underpinnings of the infinitely-lived, representative agent. This paper instead models the rate of time preference as an increasing function of real wealth and generates the Tobin effect and steady state stability. How-

3 Turnovsky (2000, p. 357) argues “. . .the requirement that the rate of time discount . . . must increase with the level of utility and therefore consumption, is not particularly appealing. It implies that, as agents become richer and increase their consumption levels, their preference for current consumption over future consumption increases, whereas intuitively, one would expect the opposite to be more likely”. 4 See Edwards (1996), Perotti (1996) and Marchante et al. (2001) for empirical verification; Epstein (1983), Epstein and Hynes (1983), Alesina and Rodrik (1994), Persson and Tabellini (1994) and Alesina and Perotti (1996) for theoretical underpinnings.

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ever, these results do not rely on the counterintuitive behavior assumptions that challenge Uzawa’s (1968) time preference specification. Inflation yields substitution effects that lower the initial value of real wealth and raise the opportunity cost of holding real balances. The corresponding fall in steady state real balance holdings lowers the rate of time preference and raises savings. Endogenous time preference wealth effects consequently reinforce the substitution effects and connect the monetary and real sectors to convert additional savings from real balance holdings to steady state capital and consumption. Therefore, this model also generates the missing theoretical foundations for empirical verification of the Tobin effect (Woodward, 1992; Ahmad and Rogers, 2000; Shrestha et al., 2002; Rapach, 2003). This time preference specification has two important advantages. First, it is consistent with the traditional literature including Mundell (1963), Tobin (1965), Laidler (1969) and Begg (1980) that integrate reinforcing wealth effects into aggregative ISLM models using ad-hoc saving and consumption functions. Allowing the rate of time preference to depend positively on real wealth implies that optimizing behavior, not ad-hoc specification yields wealth effects that endogenize the real interest rate and link the monetary and real sectors to generate a Tobin effect. Second, this time preference specification produces optimizing foundations for modeling savings as a decreasing function of real wealth using theoretical methods consistent with life cycle hypothesis predictions of consumption as an increasing function of real wealth (Dornbusch and Frankel, 1973; Orphanides and Solow, 1990). This chapter also demonstrates the monetary growth effects of modeling an endogenous rate of time preference and a cash-in-advance constraint on the purchases of consumption and capital goods or consumption goods alone. First, when the cash-in-advance constraint is levied on the purchase of both consumption and capital goods, substitution effects raise the opportunity cost of holding real balances, which in turn raises savings by reducing the level of real balance holdings, real wealth and the rate of time preference. The question is whether the additional savings raises or lowers steady state consumption and capital? The answer rests on the relative magnitude of two opposing effects. First, the cash-in-advance constraint effect generates an investment tax that converts added savings into real balance holdings and lowers the production and demand for steady state consumption and capital. However, as the substitution effect lowers the initial level of real wealth, it also generates endogenous time preference wealth effects that convert the new savings from real balance holdings into steady state consumption and capital through the Tobin effect. The cash-in-advance constraint effects strictly dominate endogenous time preference wealth effects so that the new steady state is characterized by lower consumption and capital, reversing the real sector implications of the Tobin effect. Second, if the cash-in-advance constraint is imposed on the purchases of consumption goods, monetary growth raises the opportunity cost of holding real bal-

Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect 339

ances and the costs of current consumption. The demand for real balance holdings is residually determined so that substitution effects increase savings and permanent real wealth by decreasing current consumption. However, this generates time preference wealth effects where the rise in the rate of time preference lowers savings and increases the demand for current consumption. When an endogenous rate of time preference is coupled with a cash-in-advance constraint imposed solely on consumption purchases, it creates a contradictory effect on steady state capital and consumption. However, it is the equal magnitudes of the opposing effects that yields a perfectly accommodating neutrality on permanent real wealth and savings so that monetary growth has no effect on any real sector variables and is superneutral. Section 2 demonstrates the representative agent problem and the steady state effect of monetary growth. Section 3 derives the real sector implications of combining endogenous time preference and a cash-in-advance constraint on consumption goods. The next section expands the cash-in-advance constraint so that it applies to purchases of consumption and capital goods. Section 5 demonstrates an alternative explanation for the results in Sections 3 and 4. The final section offers a summary.

2. The Representative Agent Model with Endogenous Time Preference Consider an economy that is characterized by infinitely-lived representative agents where the discount factor applied to future streams of utility depends positively on real wealth t θv (av ) dv, β(v) = (1) 0

where v is a chronological time index and θ is the rate of time preference, assumed to be an increasing function of real wealth a. Representative agents maximize utility ∞ (2) u(ct , mt )e−β(v) dt, 0

where c is consumption, m is real balance holdings, uc , um , ucm > 0 and ucc , umm < 0. Two flow budget constraints describe the evolution and composition of real wealth a˙ t = f (kt ) + x − ct − πt mt − δt kt ,

(3)

at = kt + mt

(4)

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and there is one stock budget constraint lim at e−

t

0 rv

dv

t→∞

0,

(5)

where f is a constant returns to scale production function, x is the value of public sector transfers to hold real balances constant following inflation, k is the capital stock, r is the real interest rate, π is the inflation rate and δ is the depreciation rate on capital. Maximizing (2) subject to (3)–(5) yields first-order optimality conditions uc (c, m) − λ = 0, um (c, m) − λ f (k) − δ + π = 0, λ˙ = −λ f (k) − δ − θ (k + m) and the transversality condition limt→∞ at λt e− state variable. In the steady state, combine (6) and (7) um (m, c) = f (k) − δ + π. uc (m, c)

(6) (7) (8) t 0 θv

dv

= 0, where λ is the co-

(9)

The resource constraint is given by k˙ = f (k) − δk − c.

(10)

From (10) with k˙ = 0 c = f (k) − δk.

(11)

From (8) with λ˙ = 0 θ (k + m) = f (k) − δ.

(12)

From (3) and (11) with a˙ = 0 π ∗ = μ,

(13)

where μ is the monetary growth rate. The steady state is characterized by (9), (11) and (12), which is next linearized around steady state levels of consumption, real balances and capital (c∗ , m∗ , k ∗ ) using (13)

∗ −1 0 θ dc 0 (14) 0 θ

α dm∗ = 0 dμ, dk ∗ φ γ −f

uc uc where φ = umc − ucc ( uumc ) > 0, γ = umm − ucm ( uumc ) < 0 and α = θ − f

> 0.

Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect 341

The determinant of the coefficient matrix in (14) is |Δ| = (uc f

θ + γ α) − θ θ φ < 0

(15)

and the effect of monetary growth is determined for steady state capital −θ uc dk ∗ = >0 dμ |Δ|

(16)

consumption dc∗ −θ θ uc = >0 dμ |Δ|

(17)

and real balance holdings dm∗ uc α = < 0. dμ |Δ|

(18)

Substitution effects lower the initial level of real wealth and raise the opportunity cost of holding real balances. This raises savings and lowers the rate of time preference, which generates endogenous time preference wealth effects that reinforce the substitution effect, link the real and monetary sectors and convert added savings from real balances to steady state consumption and capital. The increase in steady state capital reduces the real interest rate to replicate the key inverse relationship between the real interest rate and steady state inflation in Tobin (1965). Optimizing behavior creates wealth effects that in turn generate a Tobin effect and saddle-point stability but neither result depends on any ad-hoc savings or consumption functions or Uzawa’s (1968) counterintuitive preference assumptions.5

3. A Cash-in-Advance Constraint on Consumption Goods Stockman (1981) models a cash-in-advance constraint that applies strictly to the purchase of consumption goods. In continuous time, this constraint becomes6 ct mt .

(19)

Substituting (19) and (4) into (3) obtains7 a˙ t = f (at − ct ) + x − ct (1 + πt ).

5 Kam

(20)

(2005) demonstrates the steady state stability of the dynamic system. does not yield utility so that the nominal interest rate is positive and (19) holds with equality. 7 Capital depreciation has been removed with no loss of generality. 6 Money

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Unlike the previous section, money must be held in advance of consumption purchases so that the need to model real balances as an asset in the utility function is eliminated. Thus, representative agents now maximize the intertemporal utility function ∞ t (21) u(ct )e− 0 β(v) dt 0

subject to (5) and (20) using (1) to get the first-order optimality conditions uc (c) − ξ f (a − c) + (1 + π) = 0, ξ θ (k + c) − ξf (a − c) = ξ˙ , lim at ξt e

−

t 0 θv

t→∞

dv

= 0,

(22) (23) (24)

where ξ is the co-state variable. In the steady state, from (23) with ξ˙ = 0 θ (k + c) = f (a − c). Linearizing around steady state levels of consumption and capital acterized by (22), (25) and (11) using (13), obtains

∗ ucc −ξf

−α dc ξ ϕ 0 dk ∗ = 0 dμ, θ

0 dξ 0 −1 f

(25) (c∗ , k ∗ ),

char-

(26)

where ϕ = f + 1 + μ > 0. The determinant of the coefficient matrix in (26) is |Ψ | = −α(θ f + ϕ) < 0.

(27)

It is possible to determine the effect of monetary growth on steady state consumption dc∗ 0 = dμ |Ψ |

(28)

and the capital stock 0 dk ∗ = . dμ |Ψ |

(29)

Monetary growth raises the opportunity cost of real balance holdings and the real cost of consumption. The demand for money is residually determined and equal to the demand for consumption (since the cash-in-advance constraint is not imposed on capital goods) so that the substitution effect lowers the demand for consumption goods by raising the initial level of saving and real wealth. However, this also generates endogenous time preference wealth effects that simultaneously reduces saving and raises the demand for consumption. Combining a cash-inadvance constraint that is imposed solely on consumption purchases with this time preference specification generates offsetting substitution and wealth effects

Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect 343

on consumption and capital. Equal magnitudes of these two effects ensures that monetary growth has no effect on the steady state real sector variables and is therefore superneutral.

4. Cash-in-Advance Constraint on Consumption and Capital Goods Stockman models a second cash-in-advance constraint that applies to purchases of capital and consumption. In continuous time, this constraint becomes mt ct + k˙t .

(30)

Substituting (30) into (4) and (3) yields two flow budget constraints k˙t = (at − kt ) − ct ,

(31)

a˙ t = f (kt ) + x − ct − πt (at − kt ).

(32)

The two stock budget constraints are given by (5) and lim kt e−

t 0 rv

dv

t→∞

0.

(33)

Maximize (22) subject to (31)–(33) and (5) to obtain the first-order conditions uc (c) − (ψ + ρ) = 0, ˙ ρθ (k + m) − ψ f (k) + π + ρ = ρ, ˙ ψθ (k + m) + ψπ − ρ = ψ, lim kt ρt e−

t 0 θv

lim at ψt e−

t→∞

(35) (36)

dv

= 0,

(37)

dv

= 0,

(38)

t→∞

t 0 θv

(34)

where ρ and ψ are co-state variables. In the steady state, from (35) with ρ˙ = 0 ρθ (k + m) − ψ f (k) + π + ρ = 0. (39) From (36) with ψ˙ = 0 ψθ (k + m) + ψπ − ρ = 0. From (4) and (31) with k˙ = 0

(40)

c∗ = m∗ .

(41) (c∗ , m∗ , k ∗ )

Linearizing (11), (34), (39)–(41) around using (14) obtains ⎡ ⎤⎡ ∗ ⎤ ⎡ ⎤ ucc 0 0 −1 −1 dc 0

∗ σ ρθ (1 + θ ) −(f + μ) ⎥ ⎢ dk ⎥ ⎢ ψ ⎥ ⎢ 0 ⎢ ⎥⎢ ⎥ ⎢ ⎥ −1 (θ + μ) ⎥ ⎢ dm∗ ⎥ = ⎢ −ψ ⎥ dμ, ⎢ 0 ψθ ψθ

⎣ ⎦⎣ ⎦ ⎣ ⎦

0 0 0 dρ −1 f 0 −1 0 1 0 0 dψ 0 (42)

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E. Kam

where σ = θ ρ − f

ψ > 0. The determinant of the coefficient matrix in (42) is |Φ| = σ (θ + μ) + ψθ (f + μ) + σ + ψθ (1 + θ ) + f ρθ (θ + μ) + ψθ (f + μ) + ρθ + ψθ (1 + θ ) > 0. (43) Therefore, the effect of monetary growth is determined for steady state consumption −ψ(f )2 dc∗ = < 0, dμ |Φ|

(44)

real balance holdings dm∗ −ψ(f )2 = <0 dμ |Φ|

(45)

and the capital stock dk ∗ −ψ(f ) = < 0. dμ |Φ|

(46)

Substitution effects increase the opportunity cost of holding real balances and savings while decreasing the initial level of real wealth, real balance holdings and the rate of time preference. Determining whether the additional savings raises or lowers consumption and capital relies on the relative magnitude of two opposing effects. First, the cash-in-advance constraint acts functions as an investment tax, which redirects new savings away from the production and demand for steady state real sector variables. However, the decline in real wealth generates endogenous time preference wealth effects, which convert the additional savings from real balance holdings to capital and consumption via a Tobin effect process. Thus, the cash-in-advance constraint effect dominates endogenous time preference wealth effects so that monetary growth reduces steady state capital and consumption to invert the real sector implications of the Tobin effect.

5. One Complementary Explanation Reversing the Tobin effect can also be shown using (11) and (41), which implies f (k ∗ ) = c = m. Substituting (47) into (1) yields t θv kv + f (kv ) dv. β(v) =

(47)

(48)

0

Comparing (1) and (48) demonstrates that the rate of time preference, first modeled as an increasing function of real wealth is transformed into a function that

Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect 345

depends solely on capital. As the rate of time preference is no longer an increasing function of real wealth, it no longer generates time preference wealth effects. Therefore, in terms of the first model, substitution effects raise savings and the cash-in-advance constraint acts independently to generate superneutrality. With respect to the second model, inflation yields an investment tax that lowers the effective return and demand for investment goods. Substitution effects again increase savings, but the cash-in-advance constraint directs the added savings away from the accumulation of steady state consumption and capital.

6. Conclusions This chapter demonstrates a restatement of the Tobin effect, and steady state stability using a model that incorporates the microeconomic foundations of an optimizing and infinitely-lived, representative agent. However, neither result relies on the counterintuitive behavior assumptions that discredit Uzawa’s time preference specification. Instead, the rate of time preference is modeled as an increasing function of real wealth. Therefore, the substitution effect raises savings and lowers real balances but as the rate of time preference falls, time preference wealth effects convert savings into steady state consumption and capital. If the model is then altered by a cash-in-advance constraint, the effect of inflation on real sector variables depends on the relative magnitude of contradictory effects. First, when the cash-in-advance constraint is imposed on capital and consumption purchases, endogenous time preference wealth effects are dominated by the constraint so that monetary growth lowers steady state consumption and capital to reverse the Tobin effect. However, if the cash-in-advance constraint applies strictly to consumption purchases, the contradictory effects are of equal magnitude. The cash-in-advance constraint effect is strengthened relative to time preference wealth effects so that inflation, while lowering the demand to hold money, has no effect on any steady state real sector variables. The demand functions for consumption and real balance holdings are equal and unaffected by inflation so that the same monetary policy is superneutral.

Acknowledgements The author wishes to thank Arman Mansoorian and John Smithin without any implication for errors and omissions.

References Abel, A. (1985). Dynamic behavior of capital accumulation in a cash-in-advance model. Journal of Monetary Economics 16, 55–71.

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Ahmad, S. and Rogers, J. (2000). Inflation and the great ratios: Long term evidence from the U.S. Journal of Monetary Economics 45, 3–35. Alesina, A. and Perotti, R. (1996). Income distribution, political instability, and investment. European Economic Review 40, 1203–1228. Alesina, A. and Rodrik, D. (1994). Distributive politics and economic growth. Quarterly Journal of Economics 109, 465–490. Bae, S.-K. and Ratti, R. (2000). Long run neutrality, high inflation, and bank insolvencies in Argentina and Brazil. Journal of Monetary Economics 46, 581–604. Begg, D. (1980). Rational expectations and the non-neutrality of systematic monetary policy. Review of Economic Studies 47, 293–303. Blanchard, O. and Fischer, S. (1989). Lectures on Macroeconomics. Cambridge: MIT Press. Dornbusch, R. and Frankel, J. (1973). Inflation and growth: Alternative approaches. Journal of Money, Credit and Banking 50, 141–156. Edwards, S. (1996). Why are Latin America’s saving rates so low? An international comparative analysis. Journal of Development Economics 51, 5–44. Epstein, L. (1983). Stationary cardinal utility and optimal growth under uncertainty. Journal of Economic Theory 31, 133–152. Epstein, L. and Hynes, J. (1983). The rate of time preference and dynamic economic analysis. Journal of Political Economy 91, 611–635. Laidler, D. (1969). Money, wealth and time preference in a stationary economy. Canadian Journal of Economics 2, 526–535. Lawrance, E. (1991). Poverty and the rate of time preference: Evidence from panel data. Journal of Political Economy 99, 54–77. Kam, E. (2005). A note on time preference and the Tobin effect. Economics Letters 89, 127–132. Marchante, A., Ortega, B. and Trujillo, F. (2001). Regional differences in personal savings rates in Spain. Papers in Regional Science 80, 465–482. Matsuyama, K. (1987). Current account dynamics in a finite horizon model. Journal of International Economics 23, 299–313. Matsuyama, K. (1988). Terms-of-trade, factor intensities and the current account in a lifecycle model. Review of Economic Studies 55, 247–252. Mundell, R. (1963). Inflation and real interest rates. Journal of Political Economy 71, 280– 283. Obstfeld, M. (1982). Aggregate spending and the terms of trade: Is there a Harberger– Laursen–Metzler effect? Quarterly Journal of Economics 97, 251–270. Orphanides, A. and Solow, R. (1990). Money, inflation and growth. In Friedman, B. and Hahn, F. (Eds), Handbook of Monetary Economics. Amsterdam: North-Holland, pp. 223–259. Perotti, R. (1996). Growth, income distribution, and democracy: What the data say. Journal of Economic Growth 1, 149–187. Persson, T. and Svensson, L. (1985). Current account dynamics and the terms of trade: Harberger–Laursen–Metzler two generations later. Journal of Political Economy 93, 43–65. Persson, T. and Tabellini, G. (1994). Is inequality harmful for growth? Theory and evidence. American Economic Review 84, 600–621.

Endogenous Time Preference, Cash-in-Advance Constraints and the Tobin Effect 347 Rapach, D. (2003). International evidence on the long-impact of inflation. Journal of Money, Credit, and Banking 35, 23–48. Sen, P. and Turnovsky, S. (1989). Deterioration of the terms of trade and capital accumulation: A reexamination of the Laursen–Metzler effect. Journal of International Economics 26, 227–250. Serletis, A. and Koustas, Z. (1998). International evidence on the neutrality of money. Journal of Money, Credit, and Banking 30, 1–25. Serletis, A. and Koustas, Z. (2001). Monetary aggregation and the neutrality of money. Economic Inquiry 39, 124–138. Serletis, A. and Krause, D. (1996). Empirical evidence on the long run neutrality hypothesis using low frequency international data. Economics Letters 50, 323–327. Shrestha, K., Chen, S.-S. and Lee, C.-F. (2002). Are expected inflation and expected real rates negatively correlated? A long run test of the Mundell–Tobin hypothesis. Journal of Financial Research 25, 305–320. Sidrauski, M. (1967). Rational choice and patterns of growth in a monetary economy. American Economic Review 30, 534–544, Papers and Proceedings. Smithin, J. (2004). Controversies in Monetary Economics. Cheltenham: Edward Elgar Publishing. Solow, R. (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics 70, 65–94. Stockman, A. (1981). Anticipated inflation and the capital stock in a cash-in-advance economy. Journal of Monetary Economics 8, 387–393. Svensson, L. and Razin, A. (1983). The terms of trade and the current account: The Harberger–Laursen–Metzler effect. Journal of Political Economy 91, 97–125. Swan, T. (1956). Economic growth and capital accumulation. Economic Records 32, 334– 361. Tobin, J. (1965). Money and economic growth. Econometrica 33, 671–684. Turnovsky, S. (2000). Methods of Macroeconomic Dynamics. Cambridge: MIT Press. Uzawa, H. (1968). Time preference, the consumption function, and optimal asset holdings. In Wolfe, J. (Ed.), Value, Capital and Growth: Papers in Honour of Sir John Hicks. Chicago: Aldine Publishing Company. Woodward, G. (1992). Evidence of the Fisher effect from U.K. indexed bonds. Review of Economics and Statistics 74, 315–320.

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Chapter 17

On the Implications of Different Cash-in-Advance Constraints with Endogenous Labor Arman Mansooriana,∗ and Mohammed Mohsinb a Department b Department

of Economics, York University, North York, ON, Canada M3J 1P3 of Economics, University of Tennessee, Knoxville, TN 37996, USA

Abstract This chapter compares the effects of inflations on cash-in-advance models with endogenous labor. The steady state effects of an increase in the inflation rate on welfare, consumption and capital are significantly different, depending on whether there are cash-in-advance constraints on consumption alone, or on both consumption and investment. The steady state effects on employment are not significantly different for the two models. The impact effects and the dynamics of the two models are also compared and contrasted.

Keywords: monetary policy, cash-in-advance, employment, capital accumulation JEL: E22, E52, E58

1. Introduction One of the long-standing issues in Monetary Economics is the debate on the effects of changes in the rate of growth of money on investment and the capital stock. The two prominent models that have been used to discuss this issue are the money-in-utility model (Sidrauski, 1967) and the cash-in-advance model (CIA) (Stockman, 1981; Abel, 1985). These two alternative models give rise to sharp and contrasting results. With Sidrauski’s money-in-utility formulation an increase in the rate of growth of money will have no steady state effects, because the equality of the rate of time preference and the marginal productivity of capital dictates the level of the capital stock which must be maintained in the steady state. The steady state capital stock

∗ Corresponding author; e-mail: [email protected]

International Symposia in Economic Theory and Econometrics, Vol. 18 W.A. Barnett and A. Serletis (Editors) Copyright © 2007 Elsevier B.V. All rights reserved ISSN: 1571-0386/DOI: 10.1016/S1571-0386(07)18017-0

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is, therefore, unaffected by changes in the rate of growth of money (money is super-neutral in the steady state).1 With the CIA formulation the results are sensitive to the assumptions made regarding the CIA constraints. If the CIA constraints are on consumption alone, then changes in the rate of growth of money will have no steady state effects. Higher inflation will increase the cost of current consumption. Higher inflation will also increase the cost of future consumption arising from the dividend payment on titles to capital. On net, the trade-off between current and future consumption is unaffected and, therefore, there are no effects on steady state capital.2 On the other hand, if there are CIA constraints on all transactions, including transactions involving assets, then an increase in the rate of growth of money will reduce the steady state capital stock. The reason is that, then, the higher inflation will act as a tax on the purchases of assets, reducing the real rate of return on assets and the steady state capital.3 It is important to establish how robust these results are, by relaxing some of the assumptions made in their derivations. One important assumption is the abstraction from labor–leisure choice. Turnovsky (2000, pp. 264–268) considers the Sidrauski model with labor–leisure choice. His analysis shows that the long run super-neutrality of Sidrauski holds if the instantaneous utility function has the common form of Cobb–Douglas utility, or strong separability between consumption and leisure on the one hand, and real balances on the other. To date, the effects of labor–leisure choice on the Stockman and Abel results have not been fully worked out.4 Ahmed and Rogers (2000) consider the model with labor–leisure choice, but confine the analysis to the steady state effects only, while Dotsey and Sarte (2000) consider the effects of inflation uncertainly with different CIA models without labor–leisure choice.

1 Nevertheless, Fischer (1979) has shown that there are significant real effects along the adjustment path to the steady state. 2 As with money-in-utility, there will be some off steady state effects. 3 As pointed out by Dotsey and Sarte (2000, p. 635) the fraction of investment that is subject to CIA constraints “may be substantial in developing economies that do not have well functioning credit markets”. One can, therefore, view the model with CIA on investment as well as consumption expenditures as applying to less developed countries. 4 The reason could be that the CIA constraints have traditionally been employed in discrete time frameworks. In that setting, the analysis tends to be very cumbersome, which has given rise to the general view that CIA constraints are relatively hard to work with. Thus, according to Turnovsky (1997, p. 20), “One difficulty with this [CIA] approach is that the introduction of the various constraints, embodying the role played by money in transactions, can very quickly become intractable”. (See also Turnovsky, 2000, p. 264). Similarly, Blanchard and Fischer (1989, p. 155) state that “Models based explicitly on (CIA) constraints . . . can quickly become analytically cumbersome. Much of the research on the effects of money has taken a different shortcut, that of . . . putting real money services directly in the utility function”.

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 351

In the present chapter we work out the implications of labor–leisure choice on the Stockman and Abel results. We make two assumptions in order to simplify the analysis, and derive the full dynamics of the model. First, we consider the CIA constraints in a continuous time setting. Second, we assume that the central bank targets the inflation rate (not the rate of growth of money per se).5 This precludes complicated, yet no so crucial, off steady state effects, similar to those analyzed by Fischer (1979). It, thus, reduces the dimensions of the dynamic system corresponding to the model.6 We show that the steady state effects of the an increase in the inflation rate on the steady state levels of the important macroeconomic variables are qualitatively the same, regardless of whether the CIA constraint is on consumption alone, or on both consumption and investment. In both cases, the steady state levels of these variables fall. Hence, with endogenous labor supply the discussion of the qualitative effects of policy changes on the steady state equilibrium, which was central in the CIA models with fixed labor, is a rather mute point. The focus now should be on the dynamics, and the quantitative analysis of the steady state effects. In both models, the increase in inflation, by increasing the relative price of consumption, will lead to a substitution of leisure for consumption. With CIA constraint on investment as well as on consumption, there is an additional effect, because the higher inflation acts as a tax on investment. This tends to reduce the return on investment, tilting the consumption and leisure profiles towards the present. Hence, an increase in the inflation rate leads to a larger fall in the steady state levels of capital, employment and consumption when the CIA constraint is on consumption as well as on investment. The dynamic adjustments of the important macroeconomic variables are also qualitatively very similar. On impact, there will be a substantial fall in employment in both models, as the representative agent substitutes consumption for leisure. The fall in employment is substantially more with CIA on consumption and investment (a fall of 1.5% versus 1% due to a 1% increase in inflation), as then the increase in the inflation rate directly reduces the return on investment, tilting the leisure profile to the present. The fall in employment would reduce the

5 This

assumption is consistent with the assumptions in the literature concerned with the time consistency of monetary policy (e.g., Kydland and Prescott, 1977; Backus and Driffill, 1985; Walsh, 1995), where it is also assumed that the central bank targets the inflation rate (not the rate of growth of money per se). Recently, Mishkin (2000) also argued that the central banks of most developed as well as emerging countries do indeed target the inflation rate rather than the rate of growth of money. 6 When the central bank targets the rate of growth of money, instead, the inflation rate will be endogenous, and variable off the steady state. This then increases the dimension of the dynamic system. In any case, it is well known that the steady state policy effects are the same, regardless of whether the central bank fixes the rate of growth of money or the inflation rate.

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marginal productivity of capital, reducing capital accumulation. Hence, capital will be falling along the adjustment path to the new steady state. With capital falling along the adjustment path, there will be two competing effects on employment along the adjustment path. First, with capital falling the marginal productivity of labor will be falling, which tends to reduce employment along the adjustment path. Second, with capital falling, the marginal productivity of capital will be rising along the adjustment path. This tends to increase employment over time, through the intertemporal substitution of leisure. In both cases, the second effect dominates, and employment rises over time along the adjustment path. Consumption is the only important macroeconomic variable whose adjustment along the optimum path is different for the two models. Immediately after the increase in the inflation rate, in both models there is a fall in consumption, because the representative agent substitutes leisure for consumption. The impact fall in consumption is larger with CIA on both consumption and investment (a fall of over 10 percent versus a third of one percent due to a 1 percent increase in inflation). Along the adjustment path, with falling capital, there are two forces impinging on the adjustment of consumption. First, with falling capital, there will be falls in the wage rate, inducing the representative agent to substitute consumption for leisure. Second, with falling capital, there will be increases in the marginal productivity of capital, which would tend to tilt the consumption profile towards the future. With CIA constraint on consumption alone the second effect dominates; and consumption falls along the adjustment path. With CIA constraint on investment as well as consumption, the first effect dominates; and consumption rises along the adjustment path. Quantitatively, a 1% increase in the inflation rate will result in approximately 2% fall in the steady state capital stock when both investment and consumption are subject to CIA constraints, while it results in an approximately 0.66% fall in the steady state capital when only consumption is subject to CIA constraints. A 1% increase in the inflation rate will result in an approximately 1.06% fall in steady state consumption when both investment and consumption are subject to CIA constraints, while it results in an approximately 0.67% fall in steady state consumption when only consumption is subject to CIA constraints. A 1% increase in the inflation rate will result in approximately 0.67% fall in steady state employment. Surprisingly, the steady state effects on employment are the same for both models. This is so for different parameter values, as long as logarithmic utility is used.7 Finally, the steady state welfare falls by 0.23% if only consumption is sub-

7 This

is due to the functional forms used in the calibration process. One would expect steady state employment to fall by a larger amount when there are CIA constraints on both consumption and investment, because in that case the steady state wage rate will be lower, with lower capital.

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 353

ject to a CIA constraint, while it falls by 0.6% if both consumption and investment are subject to CIA constraints. The chapter is organized as follows. The model with CIA on both consumption and investment is presented in Section 2. The model with CIA on consumption alone is presented in Section 3. The effects of an increase in the inflation rate on the two models are compared and contrasted in Section 4. Some concluding remarks are made in Section 5.

2. The Model with CIA on Consumption and Investment The economy is modeled as a representative worker–entrepreneur with an infinite planning horizon. He chooses his consumption, c, labor supply, l and holding of capital, K, in order to maximize ∞ u(ct , lt )e−βt dt, (1) 0

where u(ct , lt ) = U (ct ) + V (lt ), with U (ct ) > 0, U

(ct ) < 0, V (lt ) < 0 and V

(lt ) < 0. Output is produced with a standard neoclassical constant return to scale production function Yt = F (Kt , lt ) with the following properties: FK > 0, Fl > 0, 2 = 0. The agent also receives FKK < 0, Fll < 0, FKl > 0 and Fll FKK − FKl monetary transfer with real values of τ from the government. There are two kinds of assets in the model, money balances (m) and capital (K). The total real value of the assets held by the representative agent is at : at = Kt + mt .

(2)

His flow budget constraint is K˙ t + m ˙ t + εt mt = F (Kt , lt ) + τt − ct , where εt is the inflation rate, and it is assumed that capital does not depreciate. Using (2), we can re-write this equation as a˙ t = F (Kt , lt ) + τt − εt mt − ct .

(3)

The agent also has the No Ponzi game condition lim e

t→∞

t 0 rv

dv

at 0.

(4)

Money is introduced through a cash-in-advance (CIA) constraint. In this section we assume that the agent will demand money for both his consumption and investment expenditures: mt ct + K˙ t

∀t.

(5)

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As money does not yield utility directly and as the return on bonds completely dominates the return on money, Equation (5) will always hold with strict equality: K˙ t = mt − ct .

(6)

Therefore, the representative agent’s problem is to maximize (1) subject to (2)–(4), (6) and the initial conditions a0 and K0 . The Hamiltonian for the representative agent’s problem is H = U (c) + V (l) + λ F (K, l) + τ − c − ε(a − K) + μ[a − K − c]. The optimality conditions are8 U (c) = λ + μ,

V (l) = −λFl (K, l), λ˙ = λ(β + ε) − μ, μ˙ = μ(1 + β) − λ[FK + ε]

(7) (8) (9) (10)

and the standard transversality conditions lim at λt e−βt = 0,

t→∞

lim Kt μt e−βt = 0.

t→∞

Now note that, from (7) and (8), the equilibrium levels of c and l can be represented by the following equations: −

−

ct = c(λt , μt ), + + lt = l λ t , K t

(11) (12)

with cλ < 0, cμ < 0, lλ > 0 and lK > 0.9 The government side of the model is kept as simple as possible. As mentioned in the Introduction, we assume that the government chooses the lump sum transfers τt in order to maintain the inflation rate εt at a constant level ε, according to its flow constrain m ˙ t + εmt = τt .

(13)

The right-hand side of Equation (13) is total government expenditures, while the left-hand side is total government revenue from seigniorage.

8 Clearly,

in making these optimal decisions the representative agent takes the values of τt and ε as given to him exogenously. 1 ∂c 1 ∂l 9 These partial derivatives are as follows: ∂c = ∂λ U

(c) < 0, ∂μ = U

(c) < 0, ∂λ = −Fl −λFlK ∂l V

(l)+λFll > 0 and ∂K = V

(l)+λFll > 0.

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 355

We are now in a position to work out the dynamics of the model with rational expectations. To this end, first note that from Equations (3) and (13) we obtain the product market clearing condition: F (Kt , lt ) = ct + K˙ t .

(14)

The dynamics of the economy are obtained by substituting for c and l from (11) and (12) into (9), (10) and (14). This gives λ˙ = λ(β + ε) − μ, μ˙ = μ(1 + β) − λ FK K, l(K, λ) + ε , K˙ = F K, l(K, λ) − c(λ, μ).

(15) (16) (17)

To study the transitional dynamics linearize (15)–(17) around the steady state to obtain

˙

λ − λ˜ Φ11 −1 0 λ (18) μ − μ˜ , μ˙ = Φ21 Φ22 Φ23 K −K Φ31 Φ32 Φ33 K˙ where tildes denote steady state values, Φ11 = β + ε (>0), Φ21 = −[FK + ε + λFKl lλ ] (<0), Φ22 = (1 + β) (>0), Φ23 = −[FKK + FKl lλ ] (>0), Φ31 = Fl lλ − cλ (>0), Φ32 = −cμ (>0) and Φ33 = FK + Fl lK (>0). The stable path of the system is given by the following equations:10 = (K0 − K)e ξt , Kt − K Φ23 ξt , (K0 − K)e λt − λ˜ = − Φ21 + (Φ22 − ξ )(Φ11 − ξ ) Φ23 (Φ11 − ξ ) ξt , μt − μ˜ = − (K0 − K)e Φ21 + (Φ22 − ξ )(Φ11 − ξ )

(19) (20) (21)

where ξ is the negative eigenvalue of the coefficient matrix in (18). Now, linearizing (11) and (12) and using (19)–(21), we obtain the adjustments of consumption and employment along the optimum path: ct − c˜ = −

cμ Φ23 (Φ11 − ξ ) + cλ Φ23 ξt , (K0 − K)e Φ21 + (Φ22 − ξ )(Φ11 − ξ )

(22)

10 Since the above system has two jump variables and one predetermined variable, the stability condition requires that the system have one negative and two positive eigenvalues. The determinant of the above matrix is negative and so the product of three eigenvalues is negative. This indicates that either one or all the three roots have negative real parts. Now consider the trace of the matrix. Trace = Φ11 + Φ22 + Φ33 > 0. Since the trace is the sum of the characteristic roots of the system; it being positive implies that at least one of the roots must be positive. Therefore, only one of the roots has a negative real part; and the steady state of the system exhibits saddle point stability.

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˜ lt − l = lK −

lλ Φ23 ξt . (K0 − K)e Φ21 + (Φ22 − ξ )(Φ11 − ξ )

(23)

In the following section we will derive the adjustment paths of the important macroeconomic variables for the case in which the CIA constraint is on consumption expenditures alone. After that we will compare the effects of an increase in the inflation rate in the two models.

3. The Model with CIA on Consumption Alone In this section we consider the model with the alternative assumption that only consumption expenditures are subject to CIA constraints.11 The CIA constraint in this case will be mt ct

∀t.

(24)

The agent’s problem is to maximize (1) subject to (2)–(4), (24) and the initial condition a0 . Again, as money does not yield utility directly and as the return on bonds completely dominates the return on money, Equation (24) will always hold with strict equality. Hence, mt is residually determined once ct is chosen. Thus, setting mt = ct and also using (2), we can write the Hamiltonian for the representative agent’s problem as H = U (c) + V (l) + λ F (a − c, l) + τ − c − εc . The optimality conditions for this problem are: U (c) = λ 1 + ε + FK (K, l) ,

V (l) = −λFl (K, l), λ˙ = λ β − FK (K, l)

(25) (26) (27)

and the transversality condition lim at λt e−βt = 0.

t→∞

Next, note that from (25) and (26), the equilibrium c and l will be represented by the following equations: + − − ct = c Kt , λt , ε t , + + lt = l Kt , λt ,

(28) (29)

11 This case is also considered in Mansoorian and Mohsin (2004). In that paper the effects of monetary policy on the term structure of interest rates are also discussed.

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 357

where cK > 0, cλ < 0, cε < 0, lK > 0 and lλ > 0. As in the previous section, we assume that the government chooses the real lump sum transfers τ according to (13) in order to maintain the inflation rate ε at a constant level. Hence, from Equations (3) and (13), we obtain the product market clearing condition: F (Kt , lt ) = ct + K˙ t .

(30)

Next, substituting for ct and lt from (28) and (29) into (27) and (30), we obtain K˙ t = F Kt , l(Kt , λt ) − c(Kt , λt , ε), (31) ˙λt = λ β − FK K, l(Kt , λt ) . (32) These two equations jointly determine the dynamics of K and λ. Linearizing them around the steady state, we obtain Λ11 Λ12 (Kt − K) K˙ t (33) = , Λ21 Λ22 λ˙ t (λt − λ˜ ) where, Λ11 = FK + Fl lK − cK (>, =, < 0), Λ12 = Fl lλ − cλ (>0), Λ21 = −λ[FKK + FKl lK ] (>0) and Λ22 = −λFKl lλ (<0). As K is predetermined while λ is a jump variable, for saddlepoint stability the determinant of the coefficient matrix in (33) should be negative. This condition will be satisfied with very mild assumptions. The equation of the saddle path is given by the following equations: = (K0 − K)e ςt , Kt − K (34) ς − Λ 11 ςt , λt − λ˜ = (35) (K0 − K)e Λ12 where ς is the negative eigenvalue of the coefficient matrix in (33). Linearizing (28) and (29), and using (34) and (35), we obtain the adjustments of consumption and employment along the optimum path: ς − Λ11 ςt , ct − c˜ = cK + cλ (36) (K0 − K)e Λ12 ς − Λ11 ςt . (K0 − K)e lt − l˜ = lK + lλ (37) Λ12 We are now in a position to compare and contrast the effects of an increase in the inflation rate on the two models.

4. The Effects of an Increase in the Inflation Rate To obtain the steady state effects of an increase in the inflation rate on capital, consumption, and employment for the model with CIA constraint on both consumption and investment totally differentiate Equations (6)–(10) at the steady

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state, with μ˙ = λ˙ = K˙ = 0. This gives dK λβ{Φ32 Φ11 + Φ31 } + λΦ32 Fkl lλ (38) = < 0, dε D dc˜ λ{Φ23 Φ32 − βΦ33 } = cλ dε D λ{Φ11 Φ33 + Φ21 Φ33 − Φ31 Φ23 } + cμ (39) < 0, D λ{Φ23 Φ32 − βΦ33 } dl˜ = lλ dε D λβ{Φ32 Φ11 + Φ31 } + λΦ32 Fkl lλ + lK (40) < 0, D where D (<0) is the determinant of the coefficient matrix in (18). Similarly, in order to obtain the steady state effects on capital, consumption, and employment for the model with CIA constraint on consumption alone totally differentiate Equations (25)–(27) and (30) at the steady state, with λ˙ = K˙ = 0. This gives cε Λ22 dK < 0, = dε Λ11 Λ22 − Λ21 Λ12 cε Λ22 −cε Λ21 dc˜ + cλ + cε < 0, = cK dε Λ11 Λ22 − Λ21 Λ12 Λ11 Λ22 − Λ21 Λ12 dl˜ cε Λ22 −cε Λ21 + lλ < 0. = lK dε Λ11 Λ22 − Λ21 Λ12 Λ11 Λ22 − Λ21 Λ12

(41) (42) (43)

From (38)–(43) it is clear that an increase in the inflation rate will have qualitatively the same effect on the steady state levels of capital, employment and consumption. In both models the increase in the inflation rate will increase the cost of consumption relative to leisure. This will reduce the steady state consumption and employment, as the representative agent substitutes leisure for consumption. The fall in employment will reduce the marginal productivity of capital, which in turn will reduce capital accumulation and the steady state capital stock. In the case with CIA constraint on both consumption and investment there will be an additional effect on capital that was also present in the Stockman–Abel setting without labor–leisure choice. The higher inflation will, in that model, act as a tax on the return on capital, which tends to reduce capital accumulation, and the steady state capital. Hence, with endogenous labor supply the discussion of the qualitative effects of policy changes on the steady state equilibrium, which was central to the CIA models with fixed labor, is a rather mute point. The focus now should be on the quantitative analysis of the steady state effects, and the dynamics of the two models.

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 359

Table 1: % Change in Steady State Capital due to 1% Increase in Inflation Rate Increase in inflation rate From 0% to 1% From 1% to 2% From 2% to 3% From 3% to 4% From 4% to 5% From 5% to 6% From 6% to 7% From 7% to 8% From 8% to 9% From 9% to 10%

˙ Model I (CIA on c and K)

Model II (CIA on c only)

−2.0374 −2.0201 −2.0031 −1.9863 −1.9699 −1.9537 −1.9378 −1.9221 −1.9067 −1.8916

−0.6916 −0.6869 −0.6822 −0.6776 −0.6730 −0.6685 −0.6641 −0.6597 −0.6554 −0.6511

To execute a quantitative analysis of the model we follow the Real Business Cycle (RBC) literature and assume that instantaneous utility is logarithmic, U (ct , lt ) = (1 − α) ln ct + α ln(1 − lt ), and the production function is Cobb– Douglas, Yt = Ktθ lt1−θ . The values of the coefficients we choose are also taken from the RBC literature (Cooley, 1995, Chapter 1). Hence, we set α = 0.64, θ = 0.3 and β = 0.042. In Table 1 we compare the effects of a 1% increase in the inflation rate on the steady state capital stock for the two models. For different initial inflation rates in the range 0% until 10%, a 1% increase in the inflation rate will result in approximately 2% fall in the steady state capital stock when both investment and consumption are subject to CIA constraints, while it results in an approximately 0.66% fall in the steady state capital when only consumption is subject to CIA constraints. The reason for such discrepancy is the fact that with CIA constraint on both consumption and investment the increase in the inflation rate acts as a tax on the return on capital. This reinforces the steady state fall in capital that results from the substitution of leisure for consumption. In Table 2 we compare the effects of a 1% increase in the inflation rate on steady state consumption for the two models. For different initial inflation rates in the range 0% until 10%, a 1% increase in the inflation rate will result in approximately 1.06% fall in steady state consumption when both investment and consumption are subject to CIA constraints, while it results in an approximately 0.67% fall in steady state consumption when only consumption is subject to CIA constraints. The reason for this difference between the two models is that with CIA constraint on both consumption and investment an increase in the inflation rate directly reduces the return on capital. This reduces savings during the adjustment path to the steady state, which reinforces the steady state fall in consumption resulting from the substitution of leisure for consumption. In Table 3 we compare the effects of a 1% increase in the inflation rate on steady state employment for the two models. For different initial inflation rates

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Table 2: % Change in Steady State Consumption due to 1% Increase in Inflation Rate Increase in inflation rate From 0% to 1% From 1% to 2% From 2% to 3% From 3% to 4% From 4% to 5% From 5% to 6% From 6% to 7% From 7% to 8% From 8% to 9% From 9% to 10%

˙ Model I (CIA on c and K)

Model II (CIA on c only)

−1.0973 −1.0887 −1.0803 −1.0720 −1.0639 −1.0558 −1.0479 −1.0401 −1.0324 −1.0249

−0.6916 −0.6869 −0.6822 −0.6776 −0.6730 −0.6685 −0.6641 −0.6597 −0.6554 −0.6511

Table 3: % Change in Steady State Employment due to 1% Increase in Inflation Rate Increase in inflation rate From 0% to 1% From 1% to 2% From 2% to 3% From 3% to 4% From 4% to 5% From 5% to 6% From 6% to 7% From 7% to 8% From 8% to 9% From 9% to 10%

˙ Model I (CIA on c and K)

Model II (CIA on c only)

−0.6916 −0.6869 −0.6822 −0.6776 −0.6730 −0.6685 −0.6641 −0.6597 −0.6554 −0.6511

−0.6916 −0.6869 −0.6822 −0.6776 −0.6730 −0.6685 −0.6641 −0.6597 −0.6554 −0.6511

in the range 0% until 10%, a 1% increase in the inflation rate will result in approximately 0.67% fall in steady state employment. Surprisingly, the steady state effects on employment are the same for both models. This is so for different parameter values, as long as logarithmic utility is used. This is due to the functional forms used in the calibration process. One would expect steady state employment to fall by a larger amount when there are CIA constraints on both consumption and investment, because in that case steady state wage rate will be lower, with lower capital. In Table 4 we compare the effects of a 1% increase in the inflation rate on steady state welfare for the two models. For different initial inflation rates in the range 0% until 10%, a 1% increase in the inflation rate will result in approximately 0.6% fall in steady state welfare if both consumption and investment are subject to CIA constraints, while the fall in steady state welfare is 0.23% if only consumption is

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 361

Table 4: % Change in Steady State Welfare due to 1% Increase in Inflation Rate Increase in inflation rate From 0% to 1% From 1% to 2% From 2% to 3% From 3% to 4% From 4% to 5% From 5% to 6% From 6% to 7% From 7% to 8% From 8% to 9% From 9% to 10%

˙ Model I (CIA on c and K)

Model II (CIA on c only)

−0.6156 −0.6108 −0.6060 −0.6012 −0.5964 −0.5917 −0.5870 −0.5824 −0.5777 −0.5731

−0.2253 −0.2275 −0.2296 −0.2315 −0.2334 −0.2351 −0.2367 −0.2381 −0.2395 −0.2408

subject to CIA constraint. Compared with the welfare effects reported by Lucas (2000), the welfare effects are significant regardless of whether there is a CIA constraint on consumption and investment, or on consumption alone. Figures 1 and 2 describe the adjustments of c, l and K for the two models, assuming an increase in the inflation rate by 1% from an initial inflation rate of 4%.12 On impact, there will be a substantial fall in employment in both models, as the representative agent substitutes consumption for leisure (see Table 5). The fall in employment is substantially more with CIA on consumption and investment (a fall of 1.4231% versus 0.9663%), as then the increase in the inflation rate directly reduces the return on investment, tilting the leisure profile to the present. The fall in employment would reduce the marginal productivity of capital, reducing capital accumulation. Hence, capital will be falling along the adjustment path to the new steady state. With capital falling, there will be two competing effects on employment along the adjustment path. First, with capital falling the marginal productivity of labor will be falling, which tends to reduce employment along the adjustment path. Second, with capital falling, the marginal productivity of capital will be rising along the adjustment path. This would tend to increase employment over time, through the intertemporal substitution of leisure. In both cases, the second effect dominates, and employment rises over time along the adjustment path. Consumption is the only important macroeconomic variable whose adjustment along the optimum path is qualitatively, as well as quantitatively, different for

12 With CIA constraint on both consumption and investment the initial steady state level of consumption, capital stock and labor supply are 0.5990789, 3.9548438 and 0.2668134, respectively. With CIA constraint on consumption alone the initial steady state level of consumption, capital stock and labor supply are 0.6196592, 4.4261371 and 0.26681348, respectively.

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The path of K during the adjustment period:

The path of c during the adjustment period (after initial jump at time 0):

The path of l during the adjustment period (after initial jump at time 0):

Figure 1: Adjustments of K, c and l with CIA on Consumption and Investment.

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 363

The path of K during the adjustment period:

The path of c during the adjustment period (after initial jump at time 0):

The path of l during the adjustment period (after initial jump at time 0):

Figure 2: Adjustments of K, c and l with CIA on Consumption Alone.

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Table 5: % Change in Employment on Impact due to 1% Increase in Inflation Rate Increase in inflation rate From 0% to 1% From 1% to 2% From 2% to 3% From 3% to 4% From 4% to 5% From 5% to 6% From 6% to 7% From 7% to 8% From 8% to 9% From 9% to 10%

˙ Model I (CIA on c and K)

Model II (CIA on c only)

−1.4802 −1.4656 −1.4512 −1.4371 −1.4231 −1.4094 −1.3960 −1.3827 −1.3697 −1.3568

−0.9895 −0.9836 −0.9778 −0.9720 −0.9663 −0.9607 −0.9552 −0.9497 −0.9443 −0.9389

Table 6: % Change in Consumption on Impact due to 1% Increase in Inflation Rate Increase in inflation rate From 0% to 1% From 1% to 2% From 2% to 3% From 3% to 4% From 4% to 5% From 5% to 6% From 6% to 7% From 7% to 8% From 8% to 9% From 9% to 10%

˙ Model I (CIA on c and K)

Model II (CIA on c only)

−14.8912 −14.0820 −13.2975 −12.5368 −11.7989 −11.0832 −10.3888 −9.7150 −9.0612 −8.4265

−0.3498 −0.4372 −0.3446 −0.3421 −0.3397 −0.3373 −0.3349 −0.3325 −0.3302 −0.3279

the two models. Immediately after the increase in the inflation rate in both models there is a fall in consumption, because the representative agent substitutes leisure for consumption. The impact fall in consumption is substantially larger when there are CIA constraints on consumption as well as on investment; a fall of 11.7989% versus 0.3397% (see Table 6). Along the adjustment path, with falling capital, there are two forces impinging on the adjustment of consumption. First, with falling capital, there will be falls in the wage rate, inducing the representative agent to substitute consumption for leisure. Second, with falling capital, there will be increases in the marginal productivity of capital, which would tend to tilt the consumption profile towards the future. With CIA constraint on consumption alone the first effect dominates; and consumption falls along the adjustment path. With CIA constraint on investment as well as consumption, the second effect dominates; and consumption rises along the adjustment path.

Implications of Different Cash-in-Advance Constraints with Endogenous Labor 365

Finally, comparing the results reported in Tables 1–6, it is clear that when there is a CIA constraint on both consumption and investment the impact effect on consumption of an increase in the inflation rate is quite sensitive to the initial value of the inflation rate. Such sensitivity is not present for any other variable either on impact or in the steady state.

5. Conclusion In this chapter we have considered the effects of monetary policies in economies with endogenous labor in which money is introduced through CIA constraints. We compared and contrasted the effects of monetary policies, involving inflation rate targeting, in a model where the CIA constraint was on both investment and consumption, with the case in which the CIA constraint was on consumption alone. Although an increase in the inflation rate had qualitatively similar effects on the steady state levels of capital, employment and consumption, numerical evaluations of the two models revealed substantial quantitative differences between them. The dynamic adjustments of these variables for the two models were also compared and contrasted.

Acknowledgement Financial support from the Social Sciences and Humanities Research Council of Canada (grant # 410-97-1212) is gratefully acknowledged.

References Abel, A.B. (1985). Dynamic behaviour of capital accumulation in a cash-in-advance model. Journal of Monetary Economics 16, 55–71. Ahmed, S. and Rogers, J.H. (2000). Inflation and the great ratios: Long term evidence from the U.S. Journal of Monetary Economics 45, 3–35. Backus, D. and Driffill, J. (1985). Inflation and reputation. American Economic Review 75, 530–538. Blanchard, O. and Fischer, S. (1989). Lectures on Macroeconomics. Cambridge, MA: MIT Press. Cooley, T. (1995). Frontiers of Business Cycle Research. Princeton, NJ: Princeton University Press. Dotsey, M. and Sarte, P.D. (2000). Inflation uncertainty and growth in a cash-in-advance economy. Journal of Monetary Economics 45, 631–655. Fischer, S. (1979). Capital accumulation on the transition path in a monetary optimizing model. Econometrica 47, 1433–1439.

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Kydland, F. and Prescott, E. (1977). Rules rather than discretion: The inconsistency of optimal plans. Journal of Political Economy 85, 473–492. Lucas, R.E. (2000). Inflation and welfare. Econometrica 68, 247–274. Mansoorian, A. and Mohsin, M. (2004). Monetary policy in a cash-in-advance economy: Employment, capital accumulation and the term structure of interest rates. Canadian Journal of Economics 37, 336–352. Mishkin, F.S. (2000). Inflation targeting in emerging-market countries. American Economic Review 90, 105–109, Papers and Proceedings. Sidrauski, M. (1967). Rational choice and patterns of growth in a monetary economy. American Economic Review 30, 534–544, Papers and Proceedings. Stockman, A.C. (1981). Anticipated inflation and the capital stock in a cash-in-advance economy. Journal of Monetary Economics 8, 387–393. Turnovsky, S.J. (1997). International Macroeconomic Dynamics. Cambridge, MA: MIT Press. Turnovsky, S.J. (2000). Methods of Macroeconomic Dynamics. Cambridge, MA: MIT Press. Walsh, C. (1995). Optimal contracts for central bankers. American Economic Review 85(1), 150–167.

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