Search models and applied labor economics
Search models and applied labor economics NICHOLAS M. KIEFER Cornell Univers...

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Search models and applied labor economics

Search models and applied labor economics NICHOLAS M. KIEFER Cornell University

GEORGE R. NEUMANN University of Iowa

The right of the University of Cambridge to print and sell all manner of books was granted by Henry VIII in 1534. The University has printed and published continuously since 1584.

CAMBRIDGE UNIVERSITY PRESS Cambridge New York New Rochelle Melbourne Sydney

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521360531 © Cambridge University Press 1989 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 This digitally printed first paperback version 2006 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication data Kiefer, Nicholas M. Search models and applied labor economics/Nicholas M. Kiefer, George R. Neumann. p.

cm.

ISBN 0 521 36053 6 I. Job hunting. 2. Unemployment. II. Title. HF5382.7.K43 1989 331.12'5-dcl9

3. Wages.

ISBN-13 978-0-521-36053-1 hardback ISBN-10 0-521-36053-6 hardback ISBN-13 978-0-521-02464-8 paperback ISBN-10 0-521-02464-1 paperback

I. Neumann, George R.

88-15302 CIP

To Charlotte and Margaret

Contents

Preface 1

page ix

Introduction

Part I Wages, reservation wages, and duration 2 Estimation of wage offer distributions and reservation wages 3 An empirical job-search model, with a test of the constant reservation-wage hypothesis 4 Individual effects in a nonlinear model: explicit treatment of heterogeneity in the empirical jobsearch model Part II Continuous-time models of duration 5 Earnings, unemployment, and the allocation of time over time 6 Choice or chance? A structural interpretation of individual labor market histories 7

Layoffs and duration dependence in a model of turnover

Part III 8 9 10

23 42

63

83 109 139

Applications

Structural and reduced form approaches to analyzing unemployment durations Wages and the structure of unemployment rates

163 179

How long is a spell of unemployment? Illusions and biases in the use of CPS data

203

Part IV 11

1

Mobility and contracting

Employment risk and labor market diversification

vn

229

viii 12 13 14

Contents A proposition and an example in the theory of job search with hours constraints Interfirm mobility and earnings Methods for analyzing employment contracts and other agreements

Index

236 247 284 293

Preface

The search approach to labor economics allows a productive interaction between economic theory and applied work because it incorporates uncertainty about the economic environment from the outset of the analysis. Optimal decisions derived in the search framework take account of market opportunities and market uncertainties, as well as considerations of costs of search or the "value of nonmarket time." In this sense, search models represent a sharp break with conventional practice in empirical modeling of employment and labor force participation. Because these models are explicitly stochastic, the distinction in tone between theoretical arguments and econometric and empirical arguments is much less pronounced than is usually the case. Indeed, theory and econometrics can be made to blend quite smoothly. The chapters in this volume represent our work in labor economics, in the search framework, over roughly the decade 1977-87. We began our work at the University of Chicago, and the effort continued at the Center for Operations Research and Econometrics (CORE), University of Louvain, and Cornell (NMK), and at Northwestern and Iowa (GRN). Over the years we have received many useful comments and suggestions from teachers, colleagues, and students, including Gary Becker, Ken Burdett, Hyun Joon Chang, Theresa Devine, Ron Ehrenberg, James Heckman, Tony Lancaster, Ed Lazear, Steve Lippman, Bob Lucas, Shelly Lundberg, John McCall, Lars Muus, Melvin Reder, Geert Ridder, Sherwin Rosen, Sunil Sharma, Robert Topel, and Neils Westergaard-Neilsen. We are grateful to all of those named and especially to Dale T. Mortensen, who has, as this volume shows, influenced us as teacher, coauthor, and critic. Many others, including participants at numerous seminars, made additional helpful comments. We are grateful for all of them. This work has been supported over the years by the National Science Foundation and by the Department of Labor. We appreciate their support. IX

CHAPTER 1

Introduction

The labor market is a fascinating and important market in modern economies. Imperfect and asymmetric information, heterogeneity among workers and firms, labor unions and bargaining, implicit and explicit long-term employment arrangements are all present in the labor market. Thus, this market has a much more complicated and diverse structure than, say, spot markets for homogeneous goods or financial markets. Yet understanding the labor market is crucial for understanding movements in macroeconomic aggregates - business cycles - as well as for evaluating the important welfare questions associated with the presence of unemployment. Payments to labor make up about two-thirds of gross domestic product in the United States, in the United Kingdom, and across the European Economic Community. How well do economists understand this important component of the economy? Perhaps it will suffice to point out that a wage equation, run on a sample of individual workers, is considered to have a "good fit" if it explains about 25 percent of the variance in wages. The literature on job search represents a breakthrough in modeling the labor market. In the search framework, uncertainty is explicitly handled in the theoretical treatment of the worker's behavior. Employment is determined as workers sample wage offers from a distribution and accept jobs that have acceptable wages. In this model, "lucky" workers may earn more than identical workers who were not so lucky in their draws from the wage distribution. Thus, at least a portion of the unexplained variance in a wage regression is part of the model; there is no presumption that the equation should fit perfectly apart from specification and measurement errors. From a macroeconomic theory point of view, the search framework is attractive because it admits the existence of unemployment. Indeed, it is possible to begin to ask questions about the efficient level of unemployment - the level that allows workers time to find suitable jobs without enduring hardship due to lack of offers.

2

Introduction

The search literature began with Stigler's pioneering articles (1961, 1962) on information in the labor market. Stigler formulated the job-shopping problem facing a worker as an optimal sample size problem. The worker chooses a sample of offers from a wage distribution at some cost per offer. The worker then accepts the job at the highest wage sampled. The decision problem for the worker is to choose the sample size so as to maximize the expected net return to search. In this framework, the worker applies for a number of jobs at once. The next round of theoretical work on job search introduced a sequential approach based on the literature on optimal stopping in statistical decision theory. In this work, the worker samples a wage in one period and decides whether to accept employment at that wage or to wait, sampling another wage in the next period. In most of the models the worker's optimal policy is a "reservation wage" policy — the worker chooses a reservation wage and accepts the first offer above the reservation wage. In this approach the sample size, that is, the number of offers received, is random, rather than determined by the worker in advance of sampling, as in Stigler's model. The sample size has an interpretation in terms of duration of search since each observation corresponds to a period of time. Thus, this framework has implications for lengths of spells of unemployment as well as for reemployment wages. The relevant literature includes Gronau (1971), McCall (1970), and Mortensen (1970). A key insight of this literature is that the reservation wage depends not only on the value of "nonmarket production," but on market opportunities. This insight represented a sharp break with standard approaches to labor force participation and labor supply. The search paradigm was the basis for many of the contributions in the celebrated "Phelps volume" (Phelps et al. 1970), which announced a new and rigorous approach to macroeconomics. Since then the theoretical development has been continuous and search ideas have continued to be influential in macroeconomics, demography, and other areas as well as in labor economics. Theoretical developments, adding to the applicability of the search framework, include continuous-time models (avoiding the theoretically clean but empirically awkward problem of defining a "period"), utility-maximizing models, models allowing layoffs and on-thejob search, models explicitly treating unemployment insurance variables, models in which the offer distribution must be learned, models with capital or asset constraints, and many others. The literature is carefully unified and reviewed in the insightful survey by Mortensen (1986).

Introduction

3

At least one empirical study using search ideas appeared even before the formal sequential search models of the 1970s. Kasper (1967) studied data on a sample of 3,000 unemployed workers in Minnesota. These workers were asked for their reservation wages - "What wage are you seeking?" - as well as for their previous wages and other information. Kasper addressed two questions: (1) Are unemployed workers willing to accept a lower wage than that received in their previous job? and (2) does the reservation wage decline with the duration of a spell of unemployment? Kasper used the term "asking wage" rather than the now conventional "reservation wage." Although he presented no formal model, the reasoning Kasper used in forming these equations was exactly that suggested by the search models developed later. The 1970s saw an expansion of interest in search models, following the publication of the papers by Gronau (1971), McCall (1970), and Mortensen (1970). This interest was due in part to the fact that an important policy question, namely, the effect of unemployment insurance benefits on unemployment, appeared to be amenable to analysis within the search framework. Unemployment insurance benefits subsidize search costs and therefore allow workers to be more choosy about jobs. This leads to longer unemployment spells and therefore possibly to higher unemployment rates. Of course, workers whose search costs are subsidized can be expected to find better jobs, and this effect must be considered as well in evaluating the welfare effects of an unemployment insurance system. The effects of benefits on the duration of unemployment spells and on reemployment wages were investigated by Ehrenberg and Oaxaca (1976). This study examined the regressions of durations and reemployment wages (both in logarithms) on a variety of demographic variables and the "replacement rate" - the ratio of benefits to previous wages. Coefficients were interpreted within the search framework in terms of costs of search and so on. No formal search model was presented in the published paper, although the appendix, regrettably unpublished, was an advance in linking empirical and theoretical search models. Estimation of an empirical model closely tied to a theoretical job search model was the aim of the paper "Estimation of Wage Offer Distributions and Reservation Wages," which appears here as Chapter 2. Before turning to a discussion of this and the following chapters, we present a simple, stylized job search model and discuss some issues involved in estimating such models. A full review of the empirical literature on job search models, loosely interpreted, is given by Devine and Kiefer (1987).

4

Introduction

1.1

A simple job search model

This section sets out the basic job search model in continuous time. The discrete-time version can be obtained simply from the continuous-time model, as indicated at the end of the section. For simplicity, we focus here on one transition, that from unemployment to employment. This is sufficient to introduce the central ideas of search theory. Assume that (1) workers maximize expected lifetime income, discounted over an infinite horizon at constant rate r; (2) the income flow while an individual is unemployed, net of search costs, is a constant b, fixed over the duration of a given spell (thus, the income received over a short interval of unemployment of length h is bh)\ (3) the probability that a worker receives an offer during a short interval of length h is bh, where 8 is constant over the duration of a spell; (4) a job offer is characterized by a wage rate w, which will be received continuously over time if the job is accepted; and (5) job offers are independent draws from a fixed distribution F(w) known to the worker. These assumptions are obviously unrealistic. The interesting questions are whether they can form the basis of a model that increases our understanding of the working of the labor market, and whether that model is helpful in organizing and interpreting labor market data. Assumption (1) involves an infinite horizon. This assumption simplifies calculations substantially, and examples seem to show that an infinite horizon is not much different from a "long" horizon as long as there is discounting. Thus, the model may have to be modified for workers nearing retirement, as in Chang (1985). Assumption (1) also restricts attention to incomemaximizing workers. Generalizing to utility maximization is straightforward. The assumption of a constant discount rate r does not seem objectionable. Assumption (2) abstracts from the possibilities that workers draw down assets significantly when unemployed or that unemployment insurance benefits vary over the course of a spell. The model has been modified to study the effects of benefit exhaustion (Burdett 1979; this is also an example of a utility-maximization model). Generally, data on workers' financial status through the course of a spell of unemployment are either unavailable or unreliable, with the occasional exception of records of unemployment insurance benefits paid. Assumption (3) implies that offers arrive according to a Poisson process; that is, the probability of receiving an offer in an interval is proportional to the length of the interval. This assumption eliminates the possibility that a worker will vary

Introduction

5

the effort devoted to search over the course of an unemployment spell or that firms will use elapsed duration as an indicator of a worker's suitability. This assumption is probably too restrictive. More work should be done on the determination of offer arrivals in labor markets. Assumption (4), that a job once accepted is held forever, is obviously unrealistic but is easily modified to allow future layoffs or quits. The basic implications for the unemployment-to-employment transition remain unchanged. Assumption (5), that draws from the offer distribution are independent, means that firms cannot take duration into account when determining the size of an offer, for example. It also requires that the offer distribution remain fixed over the course of an unemployment spell, abstracting from business cycle effects or deterioration of human capital. We are now in a position to construct the value function for a worker in the labor market. First, consider the maximum expected discounted income for a worker who becomes employed at wage w. Under our assumptions no expectations are involved, and that value is Ve(w) = w/r. This value is a function of the wage w, but not of the offer distribution due to the simplifying assumption that the job is held forever. The value, or discounted expected income, for a worker who is unemployed is a little harder to calculate. It is defined by the equation bh bh 1 - 8/i Vu = — — + — — E max{Ve(w), V j + — — Vu +

Search models and applied labor economics NICHOLAS M. KIEFER Cornell University

GEORGE R. NEUMANN University of Iowa

The right of the University of Cambridge to print and sell all manner of books was granted by Henry VIII in 1534. The University has printed and published continuously since 1584.

CAMBRIDGE UNIVERSITY PRESS Cambridge New York New Rochelle Melbourne Sydney

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521360531 © Cambridge University Press 1989 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 This digitally printed first paperback version 2006 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication data Kiefer, Nicholas M. Search models and applied labor economics/Nicholas M. Kiefer, George R. Neumann. p.

cm.

ISBN 0 521 36053 6 I. Job hunting. 2. Unemployment. II. Title. HF5382.7.K43 1989 331.12'5-dcl9

3. Wages.

ISBN-13 978-0-521-36053-1 hardback ISBN-10 0-521-36053-6 hardback ISBN-13 978-0-521-02464-8 paperback ISBN-10 0-521-02464-1 paperback

I. Neumann, George R.

88-15302 CIP

To Charlotte and Margaret

Contents

Preface 1

page ix

Introduction

Part I Wages, reservation wages, and duration 2 Estimation of wage offer distributions and reservation wages 3 An empirical job-search model, with a test of the constant reservation-wage hypothesis 4 Individual effects in a nonlinear model: explicit treatment of heterogeneity in the empirical jobsearch model Part II Continuous-time models of duration 5 Earnings, unemployment, and the allocation of time over time 6 Choice or chance? A structural interpretation of individual labor market histories 7

Layoffs and duration dependence in a model of turnover

Part III 8 9 10

23 42

63

83 109 139

Applications

Structural and reduced form approaches to analyzing unemployment durations Wages and the structure of unemployment rates

163 179

How long is a spell of unemployment? Illusions and biases in the use of CPS data

203

Part IV 11

1

Mobility and contracting

Employment risk and labor market diversification

vn

229

viii 12 13 14

Contents A proposition and an example in the theory of job search with hours constraints Interfirm mobility and earnings Methods for analyzing employment contracts and other agreements

Index

236 247 284 293

Preface

The search approach to labor economics allows a productive interaction between economic theory and applied work because it incorporates uncertainty about the economic environment from the outset of the analysis. Optimal decisions derived in the search framework take account of market opportunities and market uncertainties, as well as considerations of costs of search or the "value of nonmarket time." In this sense, search models represent a sharp break with conventional practice in empirical modeling of employment and labor force participation. Because these models are explicitly stochastic, the distinction in tone between theoretical arguments and econometric and empirical arguments is much less pronounced than is usually the case. Indeed, theory and econometrics can be made to blend quite smoothly. The chapters in this volume represent our work in labor economics, in the search framework, over roughly the decade 1977-87. We began our work at the University of Chicago, and the effort continued at the Center for Operations Research and Econometrics (CORE), University of Louvain, and Cornell (NMK), and at Northwestern and Iowa (GRN). Over the years we have received many useful comments and suggestions from teachers, colleagues, and students, including Gary Becker, Ken Burdett, Hyun Joon Chang, Theresa Devine, Ron Ehrenberg, James Heckman, Tony Lancaster, Ed Lazear, Steve Lippman, Bob Lucas, Shelly Lundberg, John McCall, Lars Muus, Melvin Reder, Geert Ridder, Sherwin Rosen, Sunil Sharma, Robert Topel, and Neils Westergaard-Neilsen. We are grateful to all of those named and especially to Dale T. Mortensen, who has, as this volume shows, influenced us as teacher, coauthor, and critic. Many others, including participants at numerous seminars, made additional helpful comments. We are grateful for all of them. This work has been supported over the years by the National Science Foundation and by the Department of Labor. We appreciate their support. IX

CHAPTER 1

Introduction

The labor market is a fascinating and important market in modern economies. Imperfect and asymmetric information, heterogeneity among workers and firms, labor unions and bargaining, implicit and explicit long-term employment arrangements are all present in the labor market. Thus, this market has a much more complicated and diverse structure than, say, spot markets for homogeneous goods or financial markets. Yet understanding the labor market is crucial for understanding movements in macroeconomic aggregates - business cycles - as well as for evaluating the important welfare questions associated with the presence of unemployment. Payments to labor make up about two-thirds of gross domestic product in the United States, in the United Kingdom, and across the European Economic Community. How well do economists understand this important component of the economy? Perhaps it will suffice to point out that a wage equation, run on a sample of individual workers, is considered to have a "good fit" if it explains about 25 percent of the variance in wages. The literature on job search represents a breakthrough in modeling the labor market. In the search framework, uncertainty is explicitly handled in the theoretical treatment of the worker's behavior. Employment is determined as workers sample wage offers from a distribution and accept jobs that have acceptable wages. In this model, "lucky" workers may earn more than identical workers who were not so lucky in their draws from the wage distribution. Thus, at least a portion of the unexplained variance in a wage regression is part of the model; there is no presumption that the equation should fit perfectly apart from specification and measurement errors. From a macroeconomic theory point of view, the search framework is attractive because it admits the existence of unemployment. Indeed, it is possible to begin to ask questions about the efficient level of unemployment - the level that allows workers time to find suitable jobs without enduring hardship due to lack of offers.

2

Introduction

The search literature began with Stigler's pioneering articles (1961, 1962) on information in the labor market. Stigler formulated the job-shopping problem facing a worker as an optimal sample size problem. The worker chooses a sample of offers from a wage distribution at some cost per offer. The worker then accepts the job at the highest wage sampled. The decision problem for the worker is to choose the sample size so as to maximize the expected net return to search. In this framework, the worker applies for a number of jobs at once. The next round of theoretical work on job search introduced a sequential approach based on the literature on optimal stopping in statistical decision theory. In this work, the worker samples a wage in one period and decides whether to accept employment at that wage or to wait, sampling another wage in the next period. In most of the models the worker's optimal policy is a "reservation wage" policy — the worker chooses a reservation wage and accepts the first offer above the reservation wage. In this approach the sample size, that is, the number of offers received, is random, rather than determined by the worker in advance of sampling, as in Stigler's model. The sample size has an interpretation in terms of duration of search since each observation corresponds to a period of time. Thus, this framework has implications for lengths of spells of unemployment as well as for reemployment wages. The relevant literature includes Gronau (1971), McCall (1970), and Mortensen (1970). A key insight of this literature is that the reservation wage depends not only on the value of "nonmarket production," but on market opportunities. This insight represented a sharp break with standard approaches to labor force participation and labor supply. The search paradigm was the basis for many of the contributions in the celebrated "Phelps volume" (Phelps et al. 1970), which announced a new and rigorous approach to macroeconomics. Since then the theoretical development has been continuous and search ideas have continued to be influential in macroeconomics, demography, and other areas as well as in labor economics. Theoretical developments, adding to the applicability of the search framework, include continuous-time models (avoiding the theoretically clean but empirically awkward problem of defining a "period"), utility-maximizing models, models allowing layoffs and on-thejob search, models explicitly treating unemployment insurance variables, models in which the offer distribution must be learned, models with capital or asset constraints, and many others. The literature is carefully unified and reviewed in the insightful survey by Mortensen (1986).

Introduction

3

At least one empirical study using search ideas appeared even before the formal sequential search models of the 1970s. Kasper (1967) studied data on a sample of 3,000 unemployed workers in Minnesota. These workers were asked for their reservation wages - "What wage are you seeking?" - as well as for their previous wages and other information. Kasper addressed two questions: (1) Are unemployed workers willing to accept a lower wage than that received in their previous job? and (2) does the reservation wage decline with the duration of a spell of unemployment? Kasper used the term "asking wage" rather than the now conventional "reservation wage." Although he presented no formal model, the reasoning Kasper used in forming these equations was exactly that suggested by the search models developed later. The 1970s saw an expansion of interest in search models, following the publication of the papers by Gronau (1971), McCall (1970), and Mortensen (1970). This interest was due in part to the fact that an important policy question, namely, the effect of unemployment insurance benefits on unemployment, appeared to be amenable to analysis within the search framework. Unemployment insurance benefits subsidize search costs and therefore allow workers to be more choosy about jobs. This leads to longer unemployment spells and therefore possibly to higher unemployment rates. Of course, workers whose search costs are subsidized can be expected to find better jobs, and this effect must be considered as well in evaluating the welfare effects of an unemployment insurance system. The effects of benefits on the duration of unemployment spells and on reemployment wages were investigated by Ehrenberg and Oaxaca (1976). This study examined the regressions of durations and reemployment wages (both in logarithms) on a variety of demographic variables and the "replacement rate" - the ratio of benefits to previous wages. Coefficients were interpreted within the search framework in terms of costs of search and so on. No formal search model was presented in the published paper, although the appendix, regrettably unpublished, was an advance in linking empirical and theoretical search models. Estimation of an empirical model closely tied to a theoretical job search model was the aim of the paper "Estimation of Wage Offer Distributions and Reservation Wages," which appears here as Chapter 2. Before turning to a discussion of this and the following chapters, we present a simple, stylized job search model and discuss some issues involved in estimating such models. A full review of the empirical literature on job search models, loosely interpreted, is given by Devine and Kiefer (1987).

4

Introduction

1.1

A simple job search model

This section sets out the basic job search model in continuous time. The discrete-time version can be obtained simply from the continuous-time model, as indicated at the end of the section. For simplicity, we focus here on one transition, that from unemployment to employment. This is sufficient to introduce the central ideas of search theory. Assume that (1) workers maximize expected lifetime income, discounted over an infinite horizon at constant rate r; (2) the income flow while an individual is unemployed, net of search costs, is a constant b, fixed over the duration of a given spell (thus, the income received over a short interval of unemployment of length h is bh)\ (3) the probability that a worker receives an offer during a short interval of length h is bh, where 8 is constant over the duration of a spell; (4) a job offer is characterized by a wage rate w, which will be received continuously over time if the job is accepted; and (5) job offers are independent draws from a fixed distribution F(w) known to the worker. These assumptions are obviously unrealistic. The interesting questions are whether they can form the basis of a model that increases our understanding of the working of the labor market, and whether that model is helpful in organizing and interpreting labor market data. Assumption (1) involves an infinite horizon. This assumption simplifies calculations substantially, and examples seem to show that an infinite horizon is not much different from a "long" horizon as long as there is discounting. Thus, the model may have to be modified for workers nearing retirement, as in Chang (1985). Assumption (1) also restricts attention to incomemaximizing workers. Generalizing to utility maximization is straightforward. The assumption of a constant discount rate r does not seem objectionable. Assumption (2) abstracts from the possibilities that workers draw down assets significantly when unemployed or that unemployment insurance benefits vary over the course of a spell. The model has been modified to study the effects of benefit exhaustion (Burdett 1979; this is also an example of a utility-maximization model). Generally, data on workers' financial status through the course of a spell of unemployment are either unavailable or unreliable, with the occasional exception of records of unemployment insurance benefits paid. Assumption (3) implies that offers arrive according to a Poisson process; that is, the probability of receiving an offer in an interval is proportional to the length of the interval. This assumption eliminates the possibility that a worker will vary

Introduction

5

the effort devoted to search over the course of an unemployment spell or that firms will use elapsed duration as an indicator of a worker's suitability. This assumption is probably too restrictive. More work should be done on the determination of offer arrivals in labor markets. Assumption (4), that a job once accepted is held forever, is obviously unrealistic but is easily modified to allow future layoffs or quits. The basic implications for the unemployment-to-employment transition remain unchanged. Assumption (5), that draws from the offer distribution are independent, means that firms cannot take duration into account when determining the size of an offer, for example. It also requires that the offer distribution remain fixed over the course of an unemployment spell, abstracting from business cycle effects or deterioration of human capital. We are now in a position to construct the value function for a worker in the labor market. First, consider the maximum expected discounted income for a worker who becomes employed at wage w. Under our assumptions no expectations are involved, and that value is Ve(w) = w/r. This value is a function of the wage w, but not of the offer distribution due to the simplifying assumption that the job is held forever. The value, or discounted expected income, for a worker who is unemployed is a little harder to calculate. It is defined by the equation bh bh 1 - 8/i Vu = — — + — — E max{Ve(w), V j + — — Vu +

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