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Optimal Unemployment Insurance in Search EquilibriumAuthor(s): Peter Fredriksson and Bertil HolmlundReviewed work(s):Source: Journal of Labor Economics, Vol. 19, No. 2 (April 2001), pp. 370-399Published by: The University of Chicago Press on behalf of the Society of Labor Economists and theNORC at the University of ChicagoStable URL: http://www.jstor.org/stable/10.1086/319565 .Accessed: 30/07/2012 17:32

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370

[ Journal of Labor Economics, 2001, vol. 19, no. 2]q 2001 by The University of Chicago. All rights reserved.0734-306X/2001/1902-0005$02.50

Optimal Unemployment Insurance inSearch Equilibrium

Peter Fredriksson, Uppsala University

Bertil Holmlund, Uppsala University

Should unemployment benefits be paid indefinitely at a fixed rate orshould the rate decline (or increase) over a worker’s unemploymentspell? We examine these issues using an equilibrium model of searchunemployment. The model features worker-firm bargaining overwages, free entry of new jobs, and endogenous search effort amongthe unemployed. The main result is that an optimal insurance pro-gram implies a declining benefit sequence over the spell of unem-ployment. Numerical calibrations of the model suggest that theremay be nontrivial welfare gains associated with switching from anoptimal uniform benefit structure to an optimally differentiatedsystem.

I. Introduction

The economics of unemployment insurance (UI) has attracted consid-erable attention over the past couple of decades. The research has primarilybeen concerned with positive analysis, such as the effects of UI benefits

We gratefully acknowledge comments from an anonymous referee and fromGerhard van den Berg, Carl Davidson, Nils Gottfries, Torsten Persson, and AsaRosen, as well as seminar and session participants at Australian National Uni-versity, Curtin University, Institute for International Economic Studies, UppsalaUniversity, Tinbergen Institute, and the 1998 meetings of European Associationof Labour Economists, the European Economic Association, and the InternationalInstitute of Public Finance.

Optimal Unemployment Insurance 371

on the duration of unemployment.1 Much less attention has been devotedto the normative issues: What does an optimal UI system look like? Theultimate rationale for public UI is, after all, to provide income insurancefor risk-averse workers. A welfare analysis of UI policies thus requires aunified treatment of the insurance benefits provided by UI as well as ofthe adverse incentive effects induced by the usual moral hazard problems.The purpose of this article is to contribute to the normative analysis ofUI by means of an equilibrium model of search unemployment.

The seminal papers on optimal UI appeared in the late 1970s (Baily1978; Flemming 1978; Shavell and Weiss 1979). These papers analyzedthe problem of UI design in an optimal taxation framework; more gen-erous benefits caused lower search intensity and, hence, longer spells ofunemployment. Shavell and Weiss (1979) focused in particular on theoptimal sequencing of benefits. Their analysis, based on a model of theindividual worker’s search behavior, suggested that benefits should declineover the spell of unemployment, provided that the unemployed can in-fluence their job-finding probabilities. Baily’s (1978) two-period analysisanalogously suggested a case for a redundancy payment, that is, a lump-sum transfer to the worker at the start of the unemployment spell.

Recently, a number of papers have extended the analysis of Shavell andWeiss (1979). One strand of the literature stays within the Shavell andWeiss (1979) framework in the sense that it focuses solely on the behaviorof the worker. Hopenhayn and Nicolini (1997) enlarge the set of policyinstruments by considering a wage tax after reemployment in conjunctionwith the sequence of benefit payments. According to their analysis, ben-efits should decrease throughout the unemployment spell, and the tax atreemployment should increase with the length of the spell. The result thatbenefits should fall monotonically over the unemployment spell is, how-ever, called into question by Wang and Williamson (1996). They addanother source of moral hazard by examining an environment where aworker’s employment status depends on the choice of effort. The tran-sition rate from unemployment to employment is increasing in searcheffort; analogously, the probability of remaining employed is increasingin work effort. The optimal UI in this setting involves a large drop inconsumption during the first period of unemployment (so as to discourageshirking), and a large reemployment bonus (so as to encourage searcheffort). The implied time profile of UI compensation is nonmonotonic;compensation increases initially and then falls throughout the spell.

Another strand of the recent literature on benefit sequencing has ap-proached the issue by incorporating some aspects of firm behavior. Da-vidson and Woodbury (1997) examine whether benefits should be paid

1 See Atkinson and Micklewright (1991) and Holmlund (1998) for two recentsurveys and assessments of the literature on UI and unemployment.

372 Fredriksson and Holmlund

indefinitely or for a fixed number of weeks. The analysis is cast in a searchand matching framework, albeit with a fixed number of jobs and exog-enous wages. They conclude that the optimal UI program should offerrisk-averse workers indefinite benefit payments, a conclusion that seemsto suggest that most existing UI programs with finite benefit periods aresuboptimal.2 Cahuc and Lehmann (1997) ignore job search but allow forendogenous wage determination through union-firm bargaining. Theyfind that a constant time sequence yields a lower unemployment rate thana program with a declining time profile; the reason is that a decreasingbenefit schedule increases the welfare of the short-term unemployed atthe expense of the long-term unemployed, and, therefore, wage pressurerises. Based on a “Rawlsian” welfare criterion, they argue that a constantbenefit sequence is optimal.

Our article reexamines the question of the optimal sequencing of ben-efits using an equilibrium model of search unemployment along the linesof Pissarides (1990). We allow for endogenous search effort among un-employed workers. In contrast to Shavell and Weiss (1979), Wang andWilliamson (1996), Davidson and Woodbury (1997), and Hopenhayn andNicolini (1997), we incorporate endogenous wage determination and freeentry of new jobs. Search effort as well as the wage bargains are affectedby the parameters of the UI program. In a search equilibrium framework,as well as in other models of equilibrium unemployment, there is a linkbetween benefits and wages, which in turn implies a relationship betweenbenefits and job creation.3 The endogenous response of wages to benefitsmay potentially prove to be an important channel that affects the optimaldesign of UI, a point stressed by Mortensen (1996). Indeed, the result ofCahuc and Lehmann (1997) suggests that the sequencing of benefit pay-ments matters for wage setting and equilibrium unemployment.

Our analysis ignores the possibility of smoothing consumption throughborrowing and saving. The optimal UI system would presumably offerlower replacement rates if workers had access to the capital market. Afew recent papers have addressed the welfare implications of UI usinggeneral equilibrium search models that allow for capital markets. Forinstance, Costain (1997) develops a model with endogenous search effortand precautionary savings. The setup of the wage bargain is greatly sim-

2 Davidson and Woodbury (1997) do not offer a formal proof of their conclu-sion. On the basis of a numerical experiment, where the compensation after benefitexhaustion is arbitrarily set, they argue that the constant benefit sequence is pref-erable, since it reduces risk as it is defined by Rothschild and Stiglitz (1970).However, we would argue that their result does not hold if the compensationafter the expiration of UI benefits is determined optimally as well.

3 This aspect of UI has been widely discussed in research on European un-employment over the past 10–15 years; see, e.g., Layard, Nickell, and Jackman(1991).


plified by ignoring that the wage in general will depend on outside op-portunities; hence, the benefit level will not directly affect the outcomeof the bargain. Despite this simplification, the model is much too com-plicated to solve analytically, so the results are based on numerical cali-brations. According to Costain (1997), optimal replacement ratios in therange of 30%–40% arise under plausible assumptions. Valdivia (1996)reports similar results with a somewhat different wage-setting rule thanthe one adopted in Costain’s paper. Neither of these papers, however,consider the optimal sequencing of benefits.

For ease of exposition, and without loss of generality, we mainly focuson a two-tiered UI system, that is, a program with two benefit levels.Workers who lose their jobs are entitled to UI benefits. UI benefits maynot be paid indefinitely, however; some workers lose their benefits andare thereafter entitled to what we refer to as “social assistance.” Socialassistance payments have infinite duration but are potentially lower thanregular benefits. We ask whether a two-tiered system dominates, in welfareterms, a program with indefinite payments of a constant wage replacementrate. The answer to this question turns out to be an unambiguous yes,provided that we ignore discounting. We also show that the result gen-eralizes to the case with a multitiered benefit structure; unemploymentbenefits should decline monotonically over the spell of unemployment.

Section II presents the basic model and some of its comparative staticsproperties. Some of the results from this positive analysis are well knownfrom the search literature, whereas others are new. Section III turns tothe normative analysis and shows that search effort is too low in marketequilibrium; the reason is that workers do not internalize the tax burdenthey impose on others by reducing their search effort. We also characterizethe optimal sequencing of benefits. A declining sequence always domi-nates a system with indefinite payment of a constant replacement rate inthe limiting case with zero discounting. The result is driven by a featureknown from models of individual worker search, which implies that theeffect of higher benefits on the individual worker’s search behavior de-pends on whether he is presently qualified for UI or not. A rise in benefitswill in general increase search effort among those not insured, as this willbring them more quickly to employment, which in turn results in eligi-bility for future UI payments. A two-tiered UI system exploits this “en-titlement effect” by providing incentives for active search among workersnot currently entitled to benefits.

With discounting, the optimality of a declining benefit sequence cannotbe established analytically. The reason for the ambiguity lies in the factthat a declining sequence increases the welfare of the short-term unem-ployed at the expense of the long-term unemployed, which in turn inducesstronger wage pressure than a flat (or increasing) sequence (see Cahucand Lehmann 1997). According to our numerical calibrations, however,


this “wage pressure effect” is dominated by the case for exploiting theentitlement effect. Moreover, the numerical experiments suggest that theoptimal degree of differentiation should be substantial. The welfare gainsof switching from a uniform benefit structure to a two-tiered one appearto be nontrivial.

II. The Model

A. Job Matching and Labor Market Flows

Consider an economy with a fixed labor force, without loss of generalitynormalized to unity. Workers are either employed or unemployed, in-dividuals have infinite horizons, and time is continuous. Employed work-ers are separated from their jobs at the exogenous rate f. When they enterunemployment, they are immediately eligible for UI benefits; the un-employed worker is insured as long as he receives UI benefits.4 Benefitsare time-limited, however. We assume for simplicity that benefits expireat a rate l that is exogenous to the individual (although endogenouslydetermined as part of the optimal UI program). The expected potentialduration of benefit receipt is thus 1/l. An unemployed worker whosebenefits have expired is referred to as “noninsured.” The insured workerescapes unemployment and enters employment at the rate aI, whereas thenoninsured worker enters employment at the rate aN. Figure 1 illustratesthe labor market flows.

The assumption that benefits have a stochastic rather than a fixed du-ration is made for tractability.5 It is not restrictive, however, as the analysisgeneralizes to the case with an arbitrary number of insured unemploymentstates, each characterized by a given benefit level (see app. D). Searcheffort varies by unemployment state but is constant within a state.6 Onemight also argue that the practical implementation of the work test inexisting UI systems is bound to involve a degree of randomness in benefitreceipt from the job searcher’s perspective, since the authorities cannotmonitor the job acceptance behavior of all unemployed workers.

Unemployed individuals can affect the rate at which they enter em-ployment. Let sI denote the search intensity of a representative insuredworker and sN the corresponding intensity for a noninsured worker. The

4 The assumption of immediate eligibility for UI compensation simplifies theanalysis considerably, as it implies that all employed workers will receive the samewage.

5 If benefit duration is fixed and perfectly predictable, one would have to dealwith the issue of how search effort changes over the spell of insured unemploy-ment. Such time dependence results from the fact that the value of insured un-employment declines as the worker approaches the date of benefit exhaustion(Mortensen 1977).

6 Thus, as the number of steps in the benefit ladder becomes sufficiently large,we mimic the search behavior arising in a setup with fixed duration.


Fig. 1.—Labor market flows

effective number of searchers in the economy is then given as S {, where uI and uN, respectively, denote the number of un-I I N Ns u 1 s u

employed in the two categories. The matching process is summarized byan aggregate matching function that relates new hires (H) to the numberof effective searchers and the number of vacancies (v): . TheH p H(S, v)probability per unit time that individual i gets an acceptable offer is givenby , , assuming constant returns to scale;j j ja { s H(S, v)/S p s a(v) j p I, Ni i i

is a measure of labor market tightness, and .v { v/S a(v) p H(S, v)/SFirms fill vacancies at the rate . Clearly, .q(v) p H(S, v)/v a(v) p vq(v)Differentiating a(v) with respect to v, we have ,′a (v) p q(v)(1 2 h) 1 0where is the elasticity of the expected duration of a vacancyh P (0, 1)with respect to v. Further, ; thus, the tighter the labor market,′q (v) ! 0the more difficult it is to fill a vacancy.

The flow equilibrium conditions for this economy are as follows:

I I N Nfe p a u 1 a u (1)N N Ia u p lu . (2)

The first condition pertains to employment (e) and the second to non-insured unemployment. Equations (1) and (2) imply a flow equilibriumcondition for insured unemployment as well. The solution for the em-ployment rate takes the form:


I I N Nm a 1 m ae p , (3)I I N Nf 1 m a 1 m a

where we define mj as the number of unemployed in state j relative tototal unemployment: .j j I Nm { u /(u 1 u )

B. Worker Behavior

Workers do not have access to a capital market, so individuals consumeall of their income at each instant. The employed worker’s income is givenby his wage, w, and the insured unemployed worker’s income by UIbenefits, B. Noninsured unemployed workers receive a transfer, Z, fromthe government; Z can be thought of as “social assistance” that is availablefor workers who have run out of benefits. We proceed under the as-sumption that ; an objective of the normative analysis will be toB ≥ Zdetermine whether this inequality is socially optimal.

The utility of unemployed workers is decreasing in search effort, assearch reduces available leisure time. The utility functions can accordinglybe written as and . An employed individual works a fixedI Nu(B, s ) u(Z, s )amount of hours ( ) and does not search; hence, we write his utilityhfunction as . The utility function is assumed to have the¯u(w,h) p u(w)following specific form:

d j[(cø ) 2 1], j ≤ 1,j ( 0,d P (0,1)

jv p (4){lnc 1 dlnø, j p 0,d P (0,1)

where c denotes consumption and ø leisure. If T denotes the availabletime, then consumption and leisure in the three states are given by c pw and if employed; and if insured un-Iø p T 2 h c p bw ø p T 2 semployed; and and if noninsured unemployed.7 TheNc p zw ø p T 2 svariables B and Z are thus proportional to the aggregate wage, where band z are the wage replacement rates in the two states.

Let UI, UN, and E denote the expected present values of being in insuredunemployment, noninsured unemployment, and employment, respec-tively. The relevant value functions for worker i can be written as assetequations of the form:

7 If individuals choose their working time, taking the hourly wage as given,hours during employment will be independent of the wage and given as h p

. The specific utility function is commonly used in the growth literature,T/(1 1 d)as it is consistent with a balanced growth rate assumption (King, Plosser, andRebelo 1988).


I I I I I NrU p u(B, s ) 1 a (E 2 U ) 2 l(U 2 U ), (5)i i i i i

N N N NrU p u(Z, s ) 1 a (E 2 U ), (6)i i i i

IrE p u(w ) 2 f(E 2 U ), (7)i i i

where r is the subjective rate of time preference. These present values canbe solved in terms of the utilities pertaining to each state and the transitionrates. For many purposes it is the differences in present values betweenemployment and unemployment that matter; these are given in appendixA.

An unemployed worker i in state j chooses search intensity, , to max-jsi

imize the value of unemployment, . Using and the factj j jU a /s p a(v)i i i

that all workers in state j choose an identical search intensity, the first-order conditions have the following structure:

j j jL { u 1 a(v)(E 2 U ) p 0, j p I, N, (8)s s

where . Thus, in optimum, the marginal cost of increasingj j ju p u(7, s ) sZs

search effort is equated to the expected marginal gain of doing so. As-suming that the second-order condition holds ( ), imposing sym-u ! 0ss

metry and taking wages and labor market tightness as givens, we can statethe following results concerning individual search behavior:

Lemma 1. (i) an increase in UI benefits (B) reduces sI (if Iu psB

, i.e., ) but increases sN; (ii) an increase in the potential2 I u/Bs ≤ 0 j ≥ 0duration of UI benefit receipt (1/l) reduces sI but increases sN; (iii) anincrease in social assistance (Z) reduces sN (if , i.e., ) and sI.Nu ≤ 0 j ≥ 0sZ

Proof. Differentiate equation (8) implicitly with respect to B, l, andZ, using equation (A1): (if , i.e., ); ; ;I I N IL ! 0 u ≤ 0 j ≥ 0 L 1 0 L 1 0sB sB sB sl

; ; and (if , i.e., ). Note that the “if-state-N I N NL ! 0 L ! 0 L ! 0 u ≤ 0 j ≥ 0sl sZ sZ sZ

ments” are ones of sufficiency. Q.E.D.The result that noninsured search rises when the benefit level is in-

creased is known in the literature as an “entitlement effect” (Mortensen1977). The entitlement effect arises because higher benefits make em-ployment more attractive relative to noninsured unemployment, as a spellof employment is a prerequisite for benefit eligibility, that is, isNE 2 Uincreasing in B. For the exact same reason, noninsured search is increasingin benefit duration.

According to lemma 1, the effect on insured search of increasing UIbenefits, for example, depends on whether consumption and leisure aresubstitutes or complements in the production sense. With our specificutility function, this provision is translated into a condition on the degree


of relative risk aversion ( ).8 We will mostly discuss the positive results1 2 j

in terms of ; the sign of j will not matter for our normative results.j ≥ 0

C. Firm Behavior and Wage Determination

The modeling of firms and wage determination follows Pissarides (1990)closely. Let J and V denote the expected present values of an occupiedjob and a vacant job, respectively. Labor productivity is constant anddenoted y. The cost of holding a vacancy open is ky, where . Thek 1 0assumption that the cost of holding a vacancy is proportional to laborproductivity is akin to the conventional idea that vacancy costs are pro-portional to real wage costs (see Pissarides 1990).9 Wages are taxed at theproportional rate t. The flow values of having an occupied and a vacantjob are given by

rJ p y 2 q 2 f( J 2 V), (9)

rV p 2ky 1 q(v)( J 2 V), (10)

where is the real labor cost. For simplicity we assume thatq { w(1 1 t)firms discount the future at the same rate as workers.10 There are no costsassociated with opening and closing a vacancy, so free entry ensures that

. From equations (9) and (10), then, we get:V p 0

y 2 q kyJ p p . (11)

r 1 f q(v)

We refer to equation (11) as a zero-profit condition. It gives the wage

8 The importance of the cross-derivative of utility with respect to leisure andconsumption for search behavior is common in the literature; see, e.g., Mortensen(1977). Risk aversion will influence whether two goods are complements or sub-stitutes; as Samuelson (1974, p. 1277) notes, “[risk aversion] pushes all comple-mentarity coefficients towards substitutability.” Empirical work on search inten-sity is scarce, but Jones (1989) finds some support for the hypothesis that higherbenefits reduce the search effort among benefit recipients. By contrast, Blau andRobins (1990) and Wadsworth (1991) find that unemployed workers eligible forunemployment compensation search more actively than do those not eligible. Assuggested by Wadsworth (1992), the latter results may reflect that benefit claimantsenter into an environment favorable to search, involving, e.g., regular contactswith officials at employment exchange offices. Our analysis ignores these aspectsof the search process.

9 To rationalize this assumption, think of a world where firms allocate theirworkforce between production and recruitment activities. In such a setup the costof recruiting—the vacancy cost—consists of the alternative cost, i.e., the marginalproduct of labor.

10 We think of firms as owned by a group of risk-neutral “rentiers” who donot work. Our welfare calculations take their income from profits into account(although this is only relevant as long as r 1 0; see Sec. III).


cost as proportional to the marginal product of labor, that is, to q p, . According to equation (11), firms′[1 2 (r 1 f)k/q(v)]y p q(v) q (v) ! 0

react by posting fewer vacancies (v falls) as q increases.Wages are set in decentralized Nash-bargains between workers and

firms. Wages can be renegotiated at any time. Hence, the relevant fallbackposition for the worker is the state of insured unemployment, irrespectiveof whether he entered employment from insured or noninsured unem-ployment. The Nash-bargain thus solves the problem:

I b 12bmax [E (w ) 2 U ] [ J (w ) 2 V ] , b P (0, 1),i i i iwi

where the definition of Ji is analogous to equation (9).11 The first-ordercondition for the maximization of the Nash-product can be written as:

IE 2 U b Jp , (12)

u w 1 2 bqw

where and symmetry across bargaining units has been imposed.V p 0Note that depends on labor market tightness, the various transitionIE 2 Urates, and compensation in the three states (see app. A).

D. Equilibrium

To complete the characterization of equilibrium, we need to specifythe government’s budget constraint, which simply states that taxes on thetotal wage bill are used to finance UI benefits and social assistance trans-fers. Since and , the budget restriction can be written as:Z p zw B p bw

I Nte p bu 1 zu . (13)

The equilibrium of this model may in principle be very complex. How-ever, the model has a convenient recursive structure, which simplifies theanalysis considerably.

Lemma 2. The zero-profit condition (11) and the wage-setting equa-tion (12) determine q and v, independently of the remaining endogenousvariables. With v determined, we get search behavior from equation (8).Having determined v and sj, we get e and uj from equations (1)–(3). Finally,the tax rate is given by equation (13) and the wage by .w { q/(1 1 t)

Proof. See appendix B.The assumptions of constant wage replacement rates and the specific

utility function thus render search and unemployment independent of the

11 Rubinstein and Wolinsky (1985) show that surplus sharing of the kind de-picted in eq. (12) and b p 1/2 is the outcome of a strategic bargaining game ifthe bargaining parties can costlessly search while negotiating.


payroll tax; they also imply that changes in labor productivity have noeffect on search and unemployment.

We arrive at a single equation determining v by combining the free-entry condition ( ) with the solution to the wage bargain:J p ky/q(v)

j j1 2O m rjW(v, l, b, z) {

j

b k/q(v)j j2 r 1 f 1 m a p 0, (14)( O )j 1 2 b 1 2 (r 1 f)k/q(v)

where , , andI N N N Im p (r 1 a ) (r 1 a 1 l) m p 1 2 mZdj

jT 2 sj jr p x , x P {b if j p I, z if j p N}.( )¯T 2 h

Equation (14) defines . The properties of this relationshipv p v(l, b, z)are as follows.

Lemma 3. Equilibrium labor market tightness increases if benefitduration, the benefit level, or social assistance is reduced.

Proof. By implicit differentiation: , , and . Q.E.D.v 1 0 v ! 0 v ! 0l b z

The intuition for lemma 3 is straightforward: every change that reducesworkers’ threat point in the wage bargain produces more moderate wage-setting behavior on the part of workers and, consequently, increases equi-librium market tightness.

As a first step toward deriving the equations for equilibrium searchintensity, we substitute the Nash-bargaining solution and the free-entrycondition into the first-order conditions for optimal search.12

Ir d b vkp (15a)IT 2 s 1 2 b 1 2 (r 1 f)k/q(v)

N I Nr d a(v) r 2 rp (15b)N NT 2 s r 1 l 1 a j

Ir 1 l 1 a b vk1 .Nr 1 l 1 a 1 2 b 1 2 (r 1 f)k/q(v)

Equations (15a) and (15b) define the following “semireduced forms”:and . Lemma 4 summarizes the propertiesI I N Ns p s (v, b) s p s (v, b, z, l)

of these relationships.Lemma 4. When the outcome of the wage bargain is taken into ac-

12 Equation (15b) is derived as follows: first decompose (E 2 UN) into E 2 UN

p (E 2 UI) 1 (UI 2 UN); then use UI 2 UN p (r 1 l 1 aN)21 (u I 2 uN 1 (aI

2 aN)(E 2 UI)), and eliminate (E 2 UI) using eq. (12) and eq. (11).


count, search effort has the following properties: , , ,I N Ns 1 0 s 1 0 s 1 0v v b

and ; and if ; ; and if .N I N I Ns ! 0 Gj s ≤ 0 s ! 0 j ≥ 0 s 1 0 s _ 0 j ! 0l b z b z

Proof. Implicit differentiation of equations (15a) and (15b), recog-nizing that the right-hand side of equation (15b) is independent of sj bythe envelope theorem. Q.E.D.

According to lemma 4, the entitlement effect remains when wage ad-justments are taken into account, but v is held constant. To find the generalequilibrium effects on search behavior of changing b, z, or l, we mustalso invoke .v p v(l, b, z)

E. Employment Effects of Introducing Benefit Differentiation

From the point of view of positive analysis, we are ultimately interestedin the employment effects of changing the parameters of the two-tieredbenefit system. In general, these effects are ambiguous, the reason beingthat the parameters of the UI system may have opposite effects on thesearch behavior of the insured and noninsured. However, the case wherewe introduce a two-tiered benefit system yields some conclusive results;we summarize these in proposition 1.

Proposition 1. A reform that introduces marginally different benefitlevels, b and z, while holding unemployment expenditure per unemployedconstant will:

a) decrease employment if the wage cost and, hence, v adjust but searchis constant,

b) increase employment if search adjusts but the wage cost is constant,c) increase employment if .r r 0Proof. Differentiate employment with respect to b, assuming log util-

ity for convenience.13

de dz dzI N N Np e 1 e s 1 e s v 1 v 1 e s 1 s ,I N N( )v s v s v b z s b z( ) ( )db db db

where , . We evaluate this derivative at the initialI Ne p e/x x p {v, s , s }x

point, that is, . The reform we consider thus satisfies Ib p z m db 1. At , and . HenceN N I N Nm dz p 0 b p z v p (m /m )v s p 2sz b z b

I Nde m m e NsI N Np e 1 e s 1 e s 1 2 v 1 s .I N( )v s v s v b bN I N( )db m m m

a) If search is exogenous then

13 This is only for convenience. All results hold for a more general utilityfunction.


I Nde m mp e 1 2 v ! 0,v bN I( )db m m

since , , and .I N N Ie 1 0 1 2 (m /m )(m /m ) p r/[r 1 sa(v)] 1 0 v ! 0v b

b) If wage costs are fixed, then v is constant by equation (11). In thiscase, we must use the partial equilibrium relationships for search: equation(8). To facilitate comparison, we write the employment effect in terms of

, as defined in lemma 4.14 We derive:Nsb

de e f 1 sa(v)Ns Np s 1 0.bNdb m r 1 f 1 sa(v)

c) If , then , and soI N N Ir r 0 1 2 (m /m )(m /m ) p 0

de e Ns Np s 1 0.bNdb m

Q.E.D.Part a of proposition 1 is the pure “wage pressure effect” highlighted

by Cahuc and Lehmann (1997). When search is exogenous, “front-load-ing” the benefit system has adverse employment effects; shifting benefitpayments in favor of the insured implies that wage claims rise, as it is thewelfare of the insured that is of direct relevance for wage setting. Part bof proposition 1 provides the essence of the result in Shavell and Weiss(1979). They maximize the expected utility of a newly unemployed in-dividual holding the expected size of the UI budget constant. When wagecosts are exogenous, front-loading the benefit system restores search in-centives and, hence, reduces the expected cost of providing benefits. Formarginal differences between b and z, the effects on expected utility areof the second order, but, since employment increases, compensation inall states can be raised, and so individuals in all labor market states gain.Part c of proposition 1 will be crucial for our welfare analysis. When

, the timing of benefits does not matter for wage setting. Thus ar r 0reform that holds average compensation constant is beneficial for em-ployment since search incentives are restored.

III. Optimal Unemployment Insurance

A. The Optimal Policy

We are now ready to address the welfare economics of unemploymentinsurance. What does a socially optimal UI system look like? Determiningthe welfare optimizing policy involves an optimal choice of the policy

14 This is just to say that the expressions are algebraically equivalent. Lemma1 still defines behavior.


variables (b, z, l). We take the welfare objective to be utilitarian, that is,

I I N NW p erE 1 u rU 1 u rU 1 erJ 1 vrV. (16)

Substituting the explicit expression for the value functions into equation(16), invoking the flow equilibrium conditions and taking the limit of theresulting expression as , yields the following simple expression:r r 0

I I N NW p eu(w) 1 u u(bw, s ) 1 u u(zw, s ). (17)

Thus, the flow of steady state welfare simplifies to a weighted average ofworkers’ instantaneous utilities. We ignore discounting in order to be ableto compare alternative steady states without considering the adjustmentprocess. As we illustrate in Section III.B, the general case of a positiverate of discount is more complicated, and we are restricted to numericalanalysis on this account.

Before we characterize the optimal policy, let us consider the derivativechanges of equation (17) with respect to sj. This yields the following result.

Lemma 5. Search intensity is too low in market equilibrium.Proof. Differentiate equation (17) with respect to sj, while recognizing

that , , ,I N I N Nw { q(v)/(1 1 t) e 1 u 1 u { 1 e p e(v, l, s , s ) u p, and :N I N Nu (v, l, s , s ) t p t(b, z, e, u )

W dtj j j j j 21p u [u 1 a(v)(E 2 U )] 2 u w e 1 u r (1 1 t) .( O )s w jj js ds

According to the first-order conditions for optimal search, equation (8),we get:

W dtj j 21p 2u w e 1 u r (1 1 t) 1 0, j p I, N,( O )w jj js ds

as . Q.E.D.j j N N jdt/ds p (t/e)(e/s ) 1 (t/u )(u s ) ! 0ZIncreases in search reduce equilibrium unemployment and, hence, re-

quired unemployment expenditures. Since these gains are external to theunemployed, search intensity is too low in market equilibrium.

Now, let us proceed to the optimal choice of the parameters of thebenefit system: b, z, and l. Maximizing equation (17), taking into accountthat , , , and the rest of the con-I I N Nv p v(b, z, l) s p s (v, b) s p s (v, b, z, l)straints listed above, we have:


dW W W dWjp 1 s 1 v p 0, (18a)Oj b bjdb b s dv

dW W W dWNp 1 s 1 v p 0, (18b)z zNdz z s dv

dW W W dWNp 1 s 1 v p 0, (18c)l lNdl l s dv

where . We can establish the followingj jdW dv p W/v 1O (W/s )sZ j v

result.Proposition 2. The optimal benefit system involves , providedb 1 z

that an interior solution to equation (18b) exists.Proof (sketch; see app. C for details): pick an arbitrary and0l P (0, `)

consider the trial solution . A uniform benefit structure cannot beb p zoptimal if at . Making use of equation (14), equationdW/db 1 0 b p z(18a), and equation (18b), we derive

dW 1 WNp s 1 0.bN Ndb m s

According to lemma 4, we have ( ). So, at , .Ns 1 0 Gj b p z dW db 1 0Zb

Q.E.D.Corollary. The potential duration of UI benefit receipt is finite

and positive: .l P (0,`)Proof. By construction, the policies ( ), zero potential du-0 0 0l , b p z

ration ( ), and infinite potential duration of UI benefit receipt0l r `, z( ) all yield identical welfare, since they effectively pay the same0l p 0, buniform wage replacement rate. According to proposition 2, however,

; therefore, an interior l is optimal. Q.E.D.0 0 0 0 0W(l , b 1 z ) 1 W(l , b p z )The gist of the proof of proposition 2 is the following. By invoking

equation (18b) on equation (18a), we evaluate equation (18a) at the optimaluniform insurance scheme. Hence, the costs associated with an unequalflow of income are of the second order for marginal differences betweenb and z. The relevant first-order effects are the effects on wage settingand search, but as the timing of benefits does not matter for wage settingwhen , the only remaining first-order effect is that of restoring searchr p 0incentives. Therefore, the entitlement effect, , in conjunction withNs 1 0b

the fact that the search effort is too low in equilibrium, , drivesNW/s 1 0the result that a two-tiered benefit structure dominates a uniform one.15

15 The provision of the proposition translates into a restriction on j. If agentsare sufficiently risk averse, they will demand some insurance, and then it willalways be optimal to differentiate the benefit system.


B. Discounting and Optimal Differentiationof the Benefit System

In this section we consider the case where . Discounting compli-r 1 0cates the analysis for two reasons: first, we must take the adjustmentprocess toward the new steady state into account; second, front-loadingthe benefit system ( ) has adverse employment effects when searchb 1 zeffort is exogenous.

With discounting, the flow of steady state utility equals

kyI I N NW p eu(w) 1 u u(bw, s ) 1 u u(zw, s ) 1 er . (19)

q(v)

If , this expression simplifies to equation (17). Equation (19) dependsr r 0on time (t) because of the matching technology; the relevant equationsof motion are:

I I N Ne p a u (t) 1 a u (t) 2 fe(t), (20a)N I N Nu p lu (t) 2 a u (t). (20b)

The proper objective is`

ˆ ˆ ˆQ(t) p exp [2r(t 2 t)]W(t)dt,Et

as the timing of gains and losses matters when . Since Q(t) satisfiesr 1 0the standard equation of asset equilibrium,

˙rQ(t) p W(t) 1 Q. (21)

In order to find the optimal benefit system, we calculate the effects ofpermanent and unexpected differential changes in b, z and , starting froml

an initial steady state, defined by equations (1) and (2), while taking intoaccount that the economy must obey the laws of motion (eqs. [20a] and[20b]). We do this using the methodology in Diamond (1980).16 Budgetbalance is assumed to hold in every period.

We focus on the case of a logarithmic utility function. Log utility impliesadditive separability between consumption and leisure, which simplifiesthe analysis somewhat; nevertheless, the fundamental properties of themore general problem are preserved. With log utility: andI Is p s (v)

; thus, the relative compensation in the two unemploy-N Ns p s (v, b/z, l)ment states determines noninsured search.

Evaluating at , we have that impliesdrQ/db drQ/dz p 0 drQ/db 1 0

16 Notice that if we take the policy experiment to be unexpected and permanent,p 0 and eq. (14) always holds. The reason for this result is that it is free tov

open and close vacancies (see Pissarides 1990).


and vice versa. The time-sequencing of benefits is therefore deter-b 1 zmined by

drQ zsign p sign 2r 1 2 (22)( ) { [ ]db (1 1 t)e

Nb z rQ sN N N2a 2 1 1 « r 1 l 1 a ,( )bN N( ) }z (1 1 t)e s u

where , with , isN N N N N 21 N N 2« p a g(s )/(r 1 l 1 a ) 1 0 g(s ) p d [(T 2 s )/s ]b

the elasticity of noninsured search with respect to b. The first term inequation (22) captures the wage pressure effect of front-loading the benefitsystem. The second term is the direct “tax cost” of having . The lastb 1 zterm is a consequence of the externalities associated with search. Differ-entiating welfare with respect to noninsured search, we get

NrQ au ky zIp r 1 l 1 a er 1 (23)( )N { [ ]s eA q(v) (1 1 t)e

b zI1ru 2 1 1 0,( ) }z (1 1 t)e

where . In this case, the searchN I I N NA p (r 1 l 1 a )(r 1 f 1 m a 1 m a )externality not only reflects the fact that taxes can be lowered when searchis higher, it also includes the gain for capital owners ( ) whenJ p ky/q(v)employment increases. For given l and benefit parameters, the externalityhas increased in size in comparison to the case that has no discounting.

It is ambiguous whether a declining benefit sequence improves welfare.Evaluating equations (22) and (23) at , we deduce that the virtueb p zof having is determined by the size of (the negative) wage-pressureb 1 zeffect vis-a-vis (the positive) search externality. In contrast, if , wer r 0can derive an explicit expression for benefit differentiation. From equa-tions (22) and (23), we have

bI N I Nm 2 1 p « p m g(s ) 1 0, (24)b( )z

implying .b 1 zBefore proceeding to the numerical analysis, we wish to emphasize two

things related to the special cases we analyzed in proposition 1. First, ifwe ignore search effort, a uniform or increasing benefit structure wouldbe optimal, since only the effect on wage setting matters. Second, if wefix wage costs and ignore discounting, we can in fact derive a conditionfor differentiation that is identical to equation (24). This does not implythat the extent of benefit differentiation will be identical in the partialand general equilibrium setting. The reason is that the case for exploiting


the entitlement effect is more potent in a situation when overall benefitgenerosity is high. When wages are endogenous, the optimal policy hasto take an additional source of moral hazard into consideration; this wouldtend to reduce benefit generosity. Consequently, we should expect thatour general equilibrium analysis delivers less benefit differentiation thanthe partial equilibrium analysis. However, as we show numerically inSection III.C, the case for differentiation will still be valid, even if wetake discounting into account.

C. Numerical Results

To provide some rough indications of the numbers involved, we havecalibrated the model numerically. We begin with the simpler case of nodiscounting. Then we proceed to analyze the case with discounting; when

distributional issues are relevant, and, hence, we can also addressr 1 0the political economy of UI.

The matching function is taken to be Cobb-Douglas, that is, H p. The day is the basic time unit. We impose the “Hosios-condition,”h 12haS v

, implying that equilibrium labor market tightness is efficient absentb p h

policy interventions.17 The rationale for doing so is that we do not wantour results to be influenced by particular assumptions about whethermarket tightness is efficient or not; moreover, Moen (1997) finds that

is the endogenous outcome in a framework in which employersb p h

can credibly announce the pay associated with vacant jobs. We set h p, which is in the upper range of the estimates according to Blanchard0.5

and Diamond (1989). Other parameters imposed are anda p 0.023. Hours of work are set to , which is the optimal¯y p 1 h p T/(1 1 d)

working time if workers were free to choose.For the remaining parameters, T, d, f, and k, we calibrated the model

assuming that utility is logarithmic, the benefit structure is uniform, and. A wage replacement rate of 30% approximately correspondsz p b p 0.3

to a uniform characterization of the present generosity of UI and welfare

17 As in all models of search and matching equilibrium, there are externalitiesassociated with firm and worker entry into the market; hence, tightness (v) is notnecessarily efficient. Provided that the matching function is constant returns toscale, however, there exists a rule for sharing the total surplus of a match, whichyields the efficient v (see Hosios 1990). The efficient outcome occurs if workers’share of the total surplus (b) equals the elasticity of the expected vacancy durationwith respect to tightness (h). When there are no policy interventions, we alsoarrive at the conclusion that v is efficient if and only if b p h.


benefits in the United States.18 Also, it is reasonably close to the Organ-ization for Economic Cooperation and Development average replacementratio in 1995 (see Martin 1996). The variables T, d, and k were then chosensuch that we obtained an unemployment duration of 12 weeks, Is p

, and a partial equilibrium elasticity of unemployment durationNs p s p 1with respect to benefits of 0.5—which is in the middle range of the avail-able estimates (see Layard, Nickell, and Jackman 1991). This procedureresulted in values of , , and .19 The value of kT p 1.60 d p 0.72 k p 4.13implies that the expected vacancy cost ( ) amounts to around 14ky/q(v)weeks of employers’ labor cost (q).20 The separation rate, finally, was setat , which implies an annual separation rate of around 30%f p 0.000828and an unemployment rate of 6.5% (given the above value of unem-ployment duration).21

Table 1 presents the outcome in terms of some key variables. We mea-sure the welfare gain associated with a particular policy in the followingway. Let the welfare associated with two kinds of policies be denoted byW1 and W0. A measure of the welfare gain of policy 1 relative to policy0 is, then, given by the value of y that solves

1 d 0W [(1 2 y)cø , 7] p W . (25)

Equation (25) gives the consumption tax (y) that would make a repre-sentative individual indifferent between living in policy regime 1 and 0,

18 A uniform characterization of the U.S. benefit system would be: gb[1 2, where g denotes the fraction of unemployed workersexp(2sa/l)] 1 (1 2 g)z

eligible for UI. According to Blank and Card (1991), g roughly equals 0.5. Thevalue of UI (the first term) is corrected to take its finite nature into account.Setting the expected duration of unemployment to 12 weeks, the potential durationof UI to 26 weeks, g p 0.5, b p 0.5, and z p 0.17 (as suggested by Wang andWilliamson 1996) yields a replacement rate of around 0.3.

19 The careful reader will note that our baseline calibration has the unrealisticfeature that the unemployed search more than the employed work. We can changethe relationship between search and working time by introducing a “death risk.”However, we do not pursue this extension since we think that it will not changethe essentials of our results.

20 Admittedly, expected vacancy costs are on the high side compared to the fewestimates available. One could argue that vacancy costs also reflect training costs,but these costs are distinct and should probably not be treated as synonymous.Note, however, that studies similar in spirit to ours (e.g., Valdivia 1996; Costain1997) encounter the same problem; an extreme case is Valdivia (1996), whereexpected vacancy costs equal 4.5 times the quarterly producer wage.

21 The average rate of unemployment in the United States during 1983–96 was6.5%; see OECD (1997). The inflow rate into unemployment averaged 30.8%per year and the average duration of completed unemployment spells was 11.4weeks during 1984–89 (see Layard, Nickell, and Jackman 1991). Our calibrationcorresponds to the one in Mortensen (1994). We have also conducted calculationsfor low values of f (an annual separation rate of around 10%) and for high values(an annual rate of 50%); the results are only marginally changed.


Table 1Optimal UI and Risk Aversion (No Discounting)

j p 0 jp21

bp z p 0.3 bp z {b, z, l} b p z p 0.3 bp z {b, z, l}

(Base Run) Optimal Optimal (Base Run) Optimal Optimal

Policy variables:Social assistance

(z) (%) 30 37.91 33.84 30 41.81 36.16Benefits/social

assistance(b/z) 1 1 1.69 1 1 2.04

Benefit duration(1/l) (weeks) … … 8.4 … … 17.4

Market variables:Tightness (v) .268 .216 .199 .489 .263 .203Employment (e)

(%) 93.50 91.64 91.61 93.35 88.61 89.54Fraction insured

(mI) (%) … … 36.27 … … 45.37Noninsured

search(sN/T) (%) 63 53 59 45 34 41

Insured search(sI/T) (%) sN/T sN/T 49 sN/T sN/T 45

Welfare and spread:Welfare gain, %

of consumption(ybase) … .43 .65 … 2.91 4.08

Welfare gain,% ofconsumption,relative toU.S. system(yUS) … … .38 … … 3.55

Coefficient ofvariation(CV) .503 .402 .415 .439 .279 .304

Note.—CV p std(cød)/E(cød) denotes the coefficient of variation of the utility index, cød, where c isconsumption, ø is leisure, and d a positive parameter. The parameters are set as follows: the matchingfunction a p .023, h p .5, the worker’s bargaining power b p .5, the separation rate f p .000827604,time endowment T p 1.5998, the utility function d p .719514, labor productivity y p 1, the vacancycost k p 4.13297.

respectively. We let ybase denote the welfare gain relative to the base run,which has a replacement rate of 30%. We have conducted two types ofexperiments: first, choosing the optimal uniform benefit system; and sec-ond, choosing the optimally differentiated benefit system. The welfaregain of moving from an optimal uniform to an optimally differentiatedsystem is given by the difference between the two entries for ybase.

The simulations reveal, unsurprisingly, that optimal benefit generosityincreases in the degree of relative risk aversion ( ).22 With a uniform1 2 j

22 Note that we apply a broad definition of “consumption,” since it refers to


benefit structure, the optimal wage replacement rate varies between 38%and 42%. These numbers are of the same order of magnitude as resultsobtained by others using related models (e.g., Valdivia 1996). Althoughthe computed replacement rates may seem low compared to most real-world UI systems, it is important to remember that effective replacementrates typically are much lower than the statutory ones because of variousrestrictions on eligibility (Martin 1996). We also note that the impliedemployment rates in the uniform system decline from 91.6% to 88.6%as risk aversion increases, an implication of the fact that higher risk aver-sion calls for a more generous unemployment compensation.

The potential duration of unemployment benefits is increasing in riskaversion, as insurance arguments would suggest. More surprising, perhaps,is another result. It turns out that the optimal degree of benefit differ-entiation increases in risk aversion. With “low” risk aversion, we have

, and with “high” risk aversion ; these numbers sug-b/z p 1.69 b/z p 2.04gest that the degree of differentiation should be substantial. The key tounderstanding this result is to note that the time devoted to search declineswith risk aversion, partly because of risk aversion per se but also becauseof the adjustments of the benefit system induced by risk aversion.23 Whenrisk aversion is high, the optimal uniform system calls for increasingbenefit generosity, which has adverse effects on search incentives. Whenthe unemployed devote a small fraction of their time to search, there isgreat scope for increasing noninsured search via the entitlement effect (seeeq. [24]). Thus, increasing the wedge between b and z is an efficient wayof restoring search incentives when risk aversion is high; agents are willingto pay the price of increased dispersion in order to increase search andreduce taxes.

The welfare gains implied by moving from an optimal uniform benefitsystem to an optimal two-tiered system appear to be fairly substantial:agents would be willing to pay between 0.2% ( ) and 1.2% (j p 0 j p

) of consumption to live in the optimal two-tiered system as opposed21to the optimal uniform one.24 The gains of designing the benefit systemoptimally per se are of course even greater. The effects on employmentof moving from the optimal uniform to the differentiated system, bycontrast, seem to be negligible.

goods as well as leisure; hence, risk aversion implies that individuals would liketo smooth the flow of the consumption index: cød. See Hansen and Imrohoroglu(1992) for a similar procedure.

23 To see this, hold v constant and differentiate eq. (15a) and eq. (15b) withrespect to j.

24 Empirical studies–based union wage-setting models usually find that the co-efficient of relative risk aversion (1 2 j) ranges between 1 and 4 (see, e.g., Farber1978; Carruth and Oswald 1985). It should be noted, though, that risk aversionrefers to income only in these studies.


Table 2Optimal UI and Discounting (Log Utility)

Annual Discount Rate

0 .05 .10 .20

z (%) 33.84 33.74 33.65 33.49b/z 1.69 1.70 1.71 1.741/l (weeks) 8.45 8.27 8.09 7.79ye (%) .22 .27 .32 .44yu (%) .22 .15 .08 2.06

Note.—See table 1 for parameter values and definitions of variables. Variables yk,k p e, u, are measures of the welfare gain, analogous to eq. (25), of a move from theoptimal uniform to the optimal two-tiered benefit system. Let “*” and “0” index theoptimal two-tiered and uniform system, respectively. We then have ye p 1 2

, for the employed, and yu p , for the unemployed,—

exp(rE 2 rE*) 1 2 exp(rU 2 rU*)0 0

where is a weighted average of and . The calculations of and— I NrU* rU * rU * rE*

take the transition path into account.—

rU*

We have also asked the following question: What are the welfare effectsof moving from the current U.S. benefit system to the optimal two-tieredone? The U.S. unemployment insurance is crudely characterized by

and a potential duration of 26 weeks; finally, we set ,b p 0.5 z p 0.17and the average social assistance payments per recipient amounted to 17%of average earnings in 1991 (see Wang and Williamson 1996). The welfaregains (yUS) range from 0.4% to 3.6%; these gains are obtained throughmore generous levels of compensation and shorter potential duration ofregular benefits.

Finally, we have calculated the benefit parameters that would be theoutcome if v (and hence the wage cost) was constant. To facilitate com-parison, we fix v at the solution of the optimal two-tiered benefit systemwhen utility is logarithmic. As expected, this partial equilibrium analysisyields more benefit differentiation; in particular, the parameters of thebenefit system would be set as follows: z would equal 36.82%, the po-tential duration of UI benefit receipt would be 7.6 weeks, and b/z p

.1.91

D. Discounting and the Political Economy of UI

We conclude Section III by analyzing the case with discounting. Wedo this for two reasons. First, we want to resolve the ambiguity illustratedin Section III.B. Second, when distributional issues are relevant; thus,r 1 0we can ask whether a proposal to move to a differentiated benefit systemis politically viable, in the sense that it pleases the (presumably) employedmajority.

Table 2 presents the results of varying the annual discount rate from0% to 20%. We look only at the case of log utility and use the sameparameter values as in table 1.

For a given benefit system, market tightness and aggregate search effort


are decreasing in the discount rate. To see this, first note that the valueof employment relative to unemployment decreases with r, so workersopt for higher wages in the bargain; second, the present value of an oc-cupied job falls, given the wage cost. These two effects imply that v andaggregate search effort decrease (since v falls). The reduction in searchtogether with the fact that the search externality now also involves thevalue of operating firms, in addition to the taxation externality, suggeststhat the case for exploiting the entitlement effect is strengthened. Asshown in table 2, the benefit system becomes more differentiated whenthe discount rate increases. It seems that the adverse effects on tightnessof front-loading the benefit system are compensated by reducing the pe-riod of UI benefit receipt.

We have also calculated the consumption taxes that would make dif-ferent categories of individuals indifferent between the optimal uniformsystem and the optimal two-tiered one. The implied welfare gain for theemployed is given by ye, and yu gives the welfare change for the unem-ployed. The value of yu is the consumption tax that would make anunemployed individual indifferent between, on the one hand, a weightedaverage of the values of insured and noninsured unemployment in thetwo-tiered system and, on the other hand, the value of unemployment inthe uniform system. These welfare measures are evaluated at the policythat maximizes welfare for society as a whole. The move to a differentiatedbenefit system always represents a gain for the employed. Thus, a proposalto introduce an optimal two-tiered (or multitiered) benefit system is apolitically viable one. The political support received from the unemployed,by contrast, varies with the discount rate: with low rates of time preferencethey are in favor of the policy, but when they discount the future heavily(20% annually), they actually oppose the proposal.

IV. Concluding Remarks

The main result of this article is that the socially optimal UI programis characterized by a declining time profile of benefit payments over thejob searcher’s spell of unemployment. A two-tiered benefit structure isoptimal because it exploits the differential impact of higher benefits onsearch incentives among insured and noninsured unemployed workers;raising the compensation offered to the insured induces additional searcheffort among the noninsured (the so-called entitlement effect). The nu-merical calibration of the model suggests that the welfare gains obtainedfrom optimal differentiation may well be substantial if individuals aresufficiently risk averse. The numerical results also indicate that a proposalto move from a uniform to a differentiated benefit structure has the virtueof being politically viable, assuming that the employed comprise the ma-jority of the electorate.


The absence of savings is an unrealistic feature of our analysis, althoughit should be kept in mind that an analytical treatment of equilibrium searchmodels with endogenous savings has proven to be extremely difficult.Would the optimality of the two-tiered benefit structure carry over to amodel where workers were allowed to save? It is not obvious why itshould not. The driving forces, in particular the entitlement effect, wouldstill be present and motivate some differentiation of benefits.

A complete welfare analysis of UI policies would also have to considerthe eligibility rules and how these affect behavior on both sides of thelabor market. Existing systems require that workers have demonstratedsome attachment to the labor force, for example, in the form of a minimumnumber of weeks in employment over the past year. A normative analysisof these issues in an equilibrium framework remains a topic for futureresearch.

Appendix A

Present Value Differences

In a symmetric equilibrium, the differences between the expected pre-sent values are:

I 21 N I NE 2 U p A {(r 1 a )[u(w) 2 u ] 1 l[u(w) 2 u ]}N 21 I N I NE 2 U p A {(r 1 l 1 a )[u(w) 2 u ] 1 f(u 2 u )} (A1)

I N 21 I I N I N IU 2 U p A {(r 1 f 1 a )(u 2 u ) 1 (a 2 a )[u(w) 2 u ]},

where , withN I I N N I NA { (r 1 l 1 a )(r 1 f 1 m a 1 m a ) m p (r 1 a )/and . Moreover, andN N I I I N(r 1 a 1 l) m p 1 2 m u p (B, s ) u p

.Nu(Z, s )

Appendix B

Proof of Lemma 2

From equation (11), the right-hand side of equation (12) depends onq and v but not on sj and w. Here, we establish that the left-hand sideof equation (12) is also independent of sj and w.

We begin by showing that is independent of sj. Imposing sym-IE 2 Umetry in equation (7), we have . Hence,I(r 1 f)E p u(w) 1 fU

, . By the envelope theorem it followsj I jE/s p [f/(r 1 f)]U /s j p I, Nimmediately that . From equations (5) and (7) (and symmetry),IE/s p 0we get . There-I I I I N[r 1 l 1 ra /(r 1 f)]U p u(B, s ) 1 [a /(r 1 v)]u(w) 1 lUfore, and , since .I N N N NU /s p 0 E/s p 0 U /s p 0

Next, we show that the left-hand side of equation (12) does not dependon w given our utility function, , and . According toB p bw Z p zwequation (A1), , wherej j[u(w) 2 u (xw, s )]/u w x P {b if j p I, z if j pw

, should not depend on w if is to be independent of w.IN} (E 2 U )/u ww

From equation (4), we have , given that andj dj¯u w p w (T 2 h) c p ww


for an employed person. The difference between instantaneous¯ø p T 2 hutilities is given by

dj a j dj¯w(T 2 h) 2 (xw) (T 2 s )j ju(w) 2 u (xw, s ) p .

j

Hence,

j ju(w) 2 u (xw) 1 T 2 sjp 1 2 x ,¯[ ( )]u w j T 2 hw

which demonstrates that is independent of the wage. Q.E.D.I(E 2 U ) u wZ w

Appendix C

Proof of Proposition 2

The proof is by contradiction. A uniform benefit structure cannot beoptimal if at . Pick an arbitrary and consider0dW/db 1 0 b p z l P (0, `)the trial solution . At , equation (18c) is irrelevant since thereb p z b p zis no difference between insured and noninsured unemployment. For thetrial solution to be well defined, the value of l must be interior, otherwiseeither b or z is not determined. First, we want to eliminate the term

in equation (18a). From equation (14): andI Iv dW/dv v p (m r )/(bW )b b v

, where . Consequently,N Nv p (m r ) (zW ) W p W/v ! 0 v dW/dv pZz v v b

. Suppose that equation (18b) holds at .I I N N[(m r z)/(m r b)]v dW/dv b p zz

Solving equation (18b) for and substituting into the expressionv dW/dvz

for ,dW/dbI IdW W m W W m W W

I N Np 2 1 s 2 s 1 s ,b z bN I N N Ndb b m z s m s s

where we have used at . Also, , andI N I Nr /r p 1 b p z W/b p (m /m )W/z. We then haveI I N NW/s p (m /m )W/s

IdW W mI N Np s 2 s 1 s .( )b z bN N[ ]db s m

To determine the sign of , we make use of equations (15a) andI N(s 2 s )b z

(15b). Implicit differentiation yields:Nz b r b

N I Ns p s 2 s .z b bN I I NT 2 s T 2 s r T 2 s

So,

I N Ns F 2 s F p s ,b bpz z bpz b

and we obtain the desired result:


dW 1 WNp s 1 0,bN Ndb m s

since , according to lemma 4. Hence, at , .Ns 1 0(Gj) b p z dW/db 1 0b

Q.E.D.

Appendix D

Several Unemployment States

Let us introduce the following notation. Let X p x w, j p 1, … , Jj j

denote compensation in state j; sj denote search effort; denotea p s a(v)j j

the outflow rate from unemployment; uj denote unemployment; anddenote the fraction of the unemployed in state j.m p u /(1 2 e)j j

We restrict attention to the case where l is identical across unemploy-ment states. The flow equilibrium conditions then take the form

fe p a(v) s uOj j j,

(l 1 a )u p lu j p 2, ..., J 2 1 (A2)j j j21,

a u p lu .j j J21

The flow values of being in each of the possible labor market states aregiven by

rE p u(w) 2 f(E 2 U ),1

rU p u(X , s ) 1 a (v)(E 2 U ) 2 l(U 2 U ), j p 1, … , J 2 1,j j j j j j j11

rU p u(X , s ) 1 a (v)(E 2 U ).J J J J J

The equilibrium conditions are straightforward generalizations of thecase of two unemployment states. Equilibrium labor market tightness isdetermined by

1 2O m rj j j…W(v, x , , x , l) { (A3)1 Jj

b k/q(v)2 f 1 m a p 0,( O )j j j 1 2 b 1 2 fk/q(v)

where and for simplicity. From equationj dj¯r p x [(T 2 s )/(T 2 h)] r p 0j j j

(A3) we have that market tightness is decreasing in xj:

v m rj jp ! 0 (as W ! 0). (A4)v

x W xj v j

Taking wage bargaining into account, search is given by


r d b vk1p

T 2 s 1 2 b 1 2 fk/q(v)1

Jr d D b vkj jp m (l 1 a ) m (A5)O1 1 l[T 2 s m 1 2 b 1 2 fk/q(v) lpjj j

j21 J r 2 rk l1a(v) m m , j p 2, … , J,O Ok l( )]jkp1 lpj

where Dj denotes the duration of state j: D p 1/(l 1 a ), j pj j

.1, … , J 2 1; D p 1/aJ J

Lemma D1. Search intensity in state is increasing in , de-j ≥ 2 xj21

creasing in xj (sufficient condition: ), and decreasing in .j ≥ 0 xj11

Proof. Implicit differentiation of equation (A5), recognizing that theright-hand side of equation (A5) is independent of sj. Q.E.D.

The welfare objective and the budget constraint are given by W pand , respectively. Consider the deriv-eu(w) 1O u u(x w, s ) te p O x uj j j j j j j

ative of W with respect to sj, taking equation (A5) and the budget con-straint into account:

W dt21p 2u w e 1 u r (1 1 t) 1 0, j p 1, … , J. (A6)( O )w j j j

s dsj j

Thus, search intensity is too low in market equilibrium, because of thetaxation externality alluded to in the main text.

Consider the fully optimal policy, that is, a choice of .(x , … , x , l)1 J

To simplify the exposition, we only consider the case of log utility. Thisdoes not change any of the fundamental properties of the problem. Thus,maximizing W(7) subject to equation (A2), the budget constraint, v p

, , and , , yields thev(x , … , x , l) s p s (v) s p s (x , … , x , l, v) j ≥ 21 J 1 1 j j 1 J

optimality conditions:

JdW W W s dW vkp 1 1 p 0, j p 1, … , J, (A7)O

dx x s x dv xkp2j j k j j

JdW W W s dW vkp 1 1 p 0, (A8)O

dl l s l dv lkp2 k

where .dW dv p W/v 1O (W/s )(s /v)Z j j j

Proposition D1. Given , the optimal benefit policy in-l P (0, `)volves , provided that .x 1 x 1 … 1 x x 1 01 2 J J

Proof. Suppose that the Jth condition in equation (A7) holds withequality. Let us consider the th. Using equation (A4), we can re-( J 2 1)write the latter condition as follows:


dW W m WJ21x p x 2 xJ21 J21 Jdx x m xJ21 J21 J J

J W s m sk J21 k1 x 2 x p 0.O J21 J( )s x m xkp2 k J21 J J

Now, for .(s /x )x 2 (m /m )(s /x )x p 0 k p 2, … , J 2 2k J21 J21 J21 J k J J

Equation (A6) and , , yields:(W/x )x p u {1 2 x /[e(1 1 t)]} Gjj j j j

dW uJ2121x p (1 1 t) (x 2 x )j J J21[dx ej

J dt s m sk J21 k2 x 2 x p 0.O J21 J( )]ds x m xkpJ21 k J21 J J

Since , and sJ is increasing in but decreasing in xJ, wedt/ds ! 0, Gj xj J21

know that . There is left todt/ds [(s x )x 2 (m /m )(s x )x ] ! 0Z ZJ J J21 J21 J21 J J J J

show that raising xJ has more adverse effects on than raising .s xJ21 J21

Differentiation of equation (A5) gives:

2s m s (T 2 s )J21 J21 J21 J21x 2 x p D a(v) m 1 0.J21 J J21 J21x m x dJ21 J J

Therefore, . Proceeding analogously one can establish thatx 1 xJ21 J

; indeed, for any , we have . Q.E.D.x 1 x j p 2, … , J x 1 xJ22 J21 j21 j

References

Atkinson, Anthony, and Micklewright, John. “UnemploymentCompensation and Labor Market Transitions: A Critical Review.”Journal of Economic Literature 29 (December 1991): 1679–1727.

Baily, Martin Neil. “Some Aspects of Optimal UnemploymentInsurance.” Journal of Public Economics 10 (December 1978): 379–402.

Blanchard, Olivier Jean, and Diamond, Peter. “The Beveridge Curve.”Brookings Papers on Economic Activity, no. 1 (1989): 1–60.

Blank, Rebecca, and Card, David. “Recent Trends in Insured andUninsured Employment: Is There an Explanation?” Quarterly Journalof Economics 106 (November 1991): 1157–89.

Blau, David, and Robins, Philip. “Job Search Outcomes for the Employedand Unemployed.” Journal of Political Economy 98 (June 1990): 637–55.

Cahuc, Pierre, and Lehmann, Etienne. “Equilibrium Unemployment andthe Time Sequence of Unemployment Benefits.” Working Paper no.97.49. Paris: Universite de Paris I, 1997.

Carruth, Alan, and Oswald, Andrew. “Miners’ Wages in Post-War Britain:An Application of a Model of Trade Union Behaviour.” EconomicJournal 86 (December 1985): 1003–20.

Costain, James. “Unemployment Insurance with Endogenous Search


Intensity and Precautionary Savings.” Working Paper no. 243.Universitat Pompeu Fabra, 1997.

Davidson, Carl, and Woodbury, Stephen. “Optimal UnemploymentInsurance.” Journal of Public Economics 64 (June 1997): 359–87.

Diamond, Peter. “An Alternative to Steady State Comparisons.”Economics Letters 5 (1980): 7–9.

Farber, Henry. “Individual Preferences and Union Wage Determination:The Case of the United Mine Worker.” Journal of Political Economy86 (October 1978): 923–42.

Flemming, John. “Aspects of Optimal Unemployment Insurance: Search,Leisure, Savings and Capital Market Imperfections.” Journal of PublicEconomics 10 (December 1978): 403–25.

Hansen, Gary, and Imrohoroglu, Ayse. “The Role of UnemploymentInsurance in an Economy with Liquidity Constraints and MoralHazard.” Journal of Political Economy 100 (February 1992): 118–42.

Holmlund, Bertil. “Unemployment Insurance in Theory and Practice.”Scandinavian Journal of Economics 100, no. 1 (1998): 113–41.

Hopenhayn, Hugo, and Nicolini, Juan Pablo. “Optimal UnemploymentInsurance.” Journal of Political Economy 105 (April 1997): 412–38.

Hosios, Arthur. “On the Efficiency of Matching and Related Models ofSearch and Unemployment.” Review of Economic Studies 57 (April1990): 279–98.

Jones, Stephen. “Job Search Methods, Intensity and Effects.” OxfordBulletin of Economics and Statistics 51 (August 1989): 277–96.

King, Robert; Plosser, Charles; and Rebelo, Sergio. “Production, Growthand Business Cycles I: The Basic Neoclassical Model.” Journal ofMonetary Economics 21 (March/May 1988): 195–232.

Layard, Richard, Nickell, Stephen, and Jackman, Richard. Unemploy-ment: Macroeconomic Performance and the Labour Market. Oxford:Oxford University Press, 1991.

Martin, John. “Measures of Replacement Rates for the Purpose ofInternational Comparisons: A Note.” OECD Economic Studies 26(1996): 99–115.

Moen, Espen. “Competitive Search Equilibrium.” Journal of PoliticalEconomy 105 (April 1997): 385–411.

Mortensen, Dale. “Unemployment Insurance and Job Search Decisions.”Industrial and Labor Relations Review 30 (July 1977): 505–17.

———. “Reducing Supply-Side Disincentives to Job Creation.” InReducing Unemployment: Current Issues and Policy Issues. Kansas City,MO.: Federal Reserve Bank of Kansas City, 1994.

———. “Unemployment Insurance, Labor Market Dynamics, and SocialWelfare: A Comment.” Carnegie-Rochester Conference Series on PublicPolicy 44 (June 1996): 83–86.

Organization for Economic Cooperation and Development. OECDEmployment Outlook. Paris: Organization for Economic Cooperationand Development, 1997.

Pissarides, Christopher. Equilibrium Unemployment Theory. Oxford:Blackwell, 1990.


Rothschild, Michael, and Stiglitz, Joseph. “Increasing Risk: I. ADefinition.” Journal of Economic Theory 2 (September 1970): 225–43.

Rubinstein, Ariel, and Wolinsky, Asher. “Equilibrium in a Market withSequential Bargaining.” Econometrica 53 (September 1985): 1133–50.

Samuelson, Paul. “Complementarity: An Essay on the 40th Anniversaryof the Hicks-Allen Revolution in Demand Theory.” Journal ofEconomic Literature 12 (December 1974): 1255–89.

Shavell, Stephen, and Weiss, Laurence. “The Optimal Payment ofUnemployment Insurance Benefits over Time.” Journal of PoliticalEconomy 87 (December 1979): 1347–62.

Valdivia, Victor. “Evaluating the Welfare Benefits of UnemploymentInsurance.” Unpublished manuscript. Evanston, IL: NorthwesternUniversity, 1996.

Wadsworth, Jonathan. “Unemployment Benefits and Search Effort in theUK Labour Market.” Economica 58 (February 1991 ): 17–34.

———. “Unemployment Benefits and Labour Market Transitions inBritain.” Discussion Paper no. 73. London: London School ofEconomics, Centre for Economic Performance, 1992.

Wang, Cheng, and Williamson, Stephen. “Unemployment Insurance withMoral Hazard in a Dynamic Economy.” Carnegie-RochesterConference Series on Public Policy 44 (June 1996): 1–41.

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