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Stimulating Job Search through the Unemployment Insurance System Author(s): Moshe Ben-Horim and Dror Zuckerman Source: Operations Research, Vol. 38, No. 2 (Mar. - Apr., 1990), pp. 359-361 Published by: INFORMS Stable URL: http://www.jstor.org/stable/171346 . Accessed: 26/09/2013 16:25 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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JOB SEARCH THROUGHTHE UNEMPLOYMENT STIMULATING INSURANCESYSTEM MOSHE BEN-HORIM and DROR ZUCKERMAN The Hebrew University, Jerusalem, Israel

(ReceivedJanuary1987;revisionsreceivedOctober1987, March,December1988;acceptedFebruary1989) This note extendsthe model of unemploymentinsurancereportedby D. Zuckermanin 1985. The followingextensions are incorporatedinto the model: The government'sobjective function is now formulatedin a more economically appealingfashion;the searchprocessis allowedto continue beyond the coverageperiod;the time value of money is explicitlyincorporated;and the individual'ssearchstrategyis comparedwiththe sociallyoptimalpolicy.Usingtechniques and conceptsfrom game theory,we investigatethe sociallyand individuallyoptimalstrategiesand derivean unemployment insurancestrategythat best suits the government'sobjective.

Zuckerman

wages determined by the individual, the unemployed individual will search less than is socially optimal.

(1985) and Bernhardt and Gerchak (1986) were the first to develop a theoretical decision model that treats the unemployment insurance (UI) system from the point of view of both the unemployed individual and the government. Building on suggestions in Bernhardt and Gerchak, this note generalizesZuckerman's study in the following ways:

1. THE MODEL In this note, it is assumed that the reader is familiar with the studies by Zuckerman, and Bernhardt and Gerchak. Furthermore, we use the same notation as in Zuckerman. Specifically, assume that an insured unemployed individual is given UI benefits over a time interval of N periods. At the beginning of period n = 1, 2, . . . N the individual receives UI benefits of Un dollars, provided that he is still unemployed. In response to the government's UI policy U = (uI, u2, . .., UN), the individual selects a search expenditure policy C = (c1, c2, . . . ) and a stopping time T, where Cn is the search effort in period n measured in monetary units, and T specifies the period in which the individual stops searchingand accepts an offer. The model further assumes the following:

1. The government's objective function is formulated in a more economically appealing fashion respondingto criticism that has appearedin Bernhardt and Gerchak. 2. The constraint imposed in Zuckerman, forcing a culmination of the search process by accepting a job offer during the insurance period, is eliminated. 3. The time value of money is explicitly introduced by incorporating a discount factor into the decision model. 4. The present model, similar to that of Bernhardt and Gerchak, analyzes and compares the individual's search strategy with the socially optimal strategy. While their comparison focuses on a special case, our study provides a general relationship between the socially and individually optimal search strategies.

1. Search is with recall within periods and without recall across periods. 2. At the end of each period, the unemployed individual makes a decision whether to accept the highest wage offer received during the period or to continue the search effort in the subsequent period. 3. The distribution function of lifetime income associated with the highest wage offer received over one period is denoted by Fc, where c is the periodical search intensity. An increase in c causes a stochastic increase in F,.

Similar to Zuckerman, Bernhardtand Gerchak and most empirical studies, we find that the reservation wages decrease while the search efforts intensify throughout the search period. In addition, as in Zuckerman, the government's optimal UI policy calls for increasing benefits over the eligibility period. By employing a Stackelberg solution concept (see Stackelberg1952) we find that while the socially optimal reservationwages are smallerthan the reservation

Subject classifications: Games/group decisions, bargaining: Stackelberg solution concept. Government, programs: unemployment insurance system. Labor: labor market and job search. Operations Research Vol. 38, No. 2, March-April 1990

359

0030-364X/90/,3802-0359 $01.25 (? 1990 Operations ResearchiSociety of America

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360 /

BEN-HORIM AND ZUCKERMAN

Since the policies U and (C, T) are interdependent, we define an insurance game solved by employing a Stackelbergsolution concept. 1.1. The Objective Functions We will describe the objective function of the two parties. Let Xc(n) for n < T be the present value of the highest offer received in period n, given a search expenditure policy C, and let I be the searcher's expected present value of lifetime earningsfrom future job opportunities in the absence of UI. We assume a time independent non-Bayesian search model, which implies that if T > N, then the individual's expected future lifetime earnings discounted to the beginning of period N + 1 is also equal to I. Given a UI policy U, the searcher selects a search expenditure policy C and a stopping time T, which maximize the expected lifetime income from futurejob opportunities plus UI payments net of search costs. Formally, the individual's objective function is given by f(C, TI U) =

ELITXC(T) +

3

'(U

3-(Un

-

dn

-

Cn)j

T< N

x P(T < N) N

+ E fNI

E

+

Cn)IP(T> N)

(1)

n=1

2. THE SOLUTIONOF THE MODEL As noted, we employ a Stackelbergsolution concept. Specifically,let (C*(U), T*(U)) be the reaction curve of the individual. The government is assumed to know the response function of the searcher,and its policy is determined by maximizing g(U I C*(U)), T*(U)) with respect to U. First we derive the reaction curve of the searcher. Let V"(y) be the individual's optimal expected discounted lifetime income beginning at the end of period n, under the assumptions that T > n, and the highestjob offer received in period n is y. Given that a policy U is imposed by the government, it can be seen that for each period n = 1, 2, . . ., N we have a minimum acceptable offer (reservationwage), say In where (n-1 =

max (U,-CJ -VY) + C

V

+

{(Un

dFn(y)

J

(3)

for n = 2, ... , N and (N = I. Using these relationships we find that V,(x) = maxix, EnJ. Define G,(x) = f0 (y - x) dF.(Y). In what follows, we shall assume that Gc(x) is concave in c for any given x > 0, implying a diminishing return from an increased search effort. By employing similar arguments as in Zuckerman we find that the individual's optimal reaction curve can be derived recursivelyby the following procedure.

where d is a discount factor. Using the same arguments as in Zuckerman, we restrict our attention to search expenditure rates c, < un for n = 1,..., N. Given that a policy (C, T) is employed by the individual, the government's objective is to select a UI strategy U, which maximizes the individual's discounted expected lifetime income (productivity) net of search costs. Accordingly, we define the government's objective function as

2. c* = min('N, UN), where cn is the solution for the functional equation f3&GJ1 J/&c = 1 for n = 1,2, ... , N. 3. For n = N- 1, N- 2,.., 1, the optimal reservation wage and the optimal search cost are obtained by the relations (see 3)

g(UIC, T)

and c*

_

=

~~~T

E f3TXC(T)-

E

f3n-'cCn|T < N P(T < N)

n=1 N

+

3NIJ

E n-

lCnIP(T> N).

(2)

n=1

The payments Un do not explicitly appear in g since they represent transfer payments with zero net effect from the point of view of the society. However, the government's objective function is affected by the policy U through the variablesXc, Cnand T.

1. (N=I.

n =

Unr+1 - Cn+1

=

+

+ 3Gc*+( +1)

d3'n+l

(4)

min(&, Un).

We will derive the socially optimal reservation wages and search intensities. Let Zn(x) be the optimal present value of the social objective at period n, given that the highest offer received during that period is x. The following relationship holds

Zn-1(x) =

max x, max {-Cn

+

F

f

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Zn(y) dF J Y)}}

(5)

TechnicalNotes / for n = 2, ... , Nand ZN(x) = max x, Ii. The socially optimal reservationwages can be expressed as an-

= max 1-c, + O~a,+ fGc(O(a,))

(6)

for n =2, ..., N and aN =I. Letfn be the solution for the functional equation i3Gc((a)/&c = 1, then the socially optimal searchintensity in period n is assumed at the point min(a,, u"). The next theorem compares the individually and socially optimal search strategies.It is found that while the socially optimal reservationwages are smallerthan the reservation wages determined by the individual, the unemployed individual will search less than the socially optimal.

a. an ?, E for n= 1, 2, .. . ,N. b. an ?- C' for n = 1, 2, . . . , N. Part a can be established by induction and the proof of b results from theorem 2 in Zuckerman; see also Ben Horim and Zuckerman (1988). Theorem 2 characterizesthe government's optimal UI policy for a given value of N. The optimal coverage period N* can be obtained using the procedure proposed in Zuckerman, Section 4. Theorem 2. The government's optimal UI strategyfor a given value of N is given by J CA

The proof of this theorem is parallel to the procedure found in Zuckerman. For more details, see also, Ben Horim and Zuckerman. Theorem 3 characterizesthe pattern of socially and individually optimal reservation wages and search intensities. Theorem 3. Under the government's optimal UI policy U*N a. the socially and individually optimal reserva-

tion wagesa,C and i, are decreasingin n for n = 1,2, . .. , N; b. the optimal search intensities e, and ',nare increasingin nforn = 1, 2, ..., N.

The monotonicityof a,, and S,, can be establishedby

Theorem 1

Nr

361

A2

-

,

\

induction. The proof of part b follows directly from part a. REFERENCES BEN-HORIM, M., AND D. ZUCKERMAN. 1988. Job Search

in the Presence of the Unemployment Insurance System.WorkingPaper#88-32, The School of Business, The HebrewUniversity,Jerusalem. BERNHARDT, I., AND Y. GERCHAK. 1986. SociallyOptimal Job Search and Its Inducement. Opns. Res. 34,

844-850. STACKELBERG,V.

H. 1952. The Theoryof MarketEconomy. OxfordUniversityPress,Oxford,England. ZUCKERMAN, D. 1985. Optimal Unemployment Insurance Policy. Opns.Res. 33, 263-276.

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