Search Theory and Unemployment

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iocchini, Francisco J.

Documento de Trabajo N 6

Departamento de Economa de la Facultad de iencias Sociales Econ!micas

Search theor and unemploment

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ontificia Universidad Catlica ArgentinaSanta Mara de los Buenos Aires

Search Theory and

Unemployment

Por

Francisco J. Ciocchini

Facultad de Ciencias Sociales y Econmicas

epartamento de Economa

ocumento de Trabajo N 6

Junio de 2006DOC

UMENTOSDE

TRABAJO


3/21


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Search Theory and Unemployment

Francisco J. Ciocchini

Department of EconomicsUniversidad Catlica Argentina

Abstract

We study a simple search model of the labor market and use it to shed lighton issues like unemployment duration, the determinants of the

unemployment rate, and the potential effects of education

on these two variables.

Resumen

En este artculo se estudia un modelo simple de bsqueda en el mercado de trabajo.Los resultados del modelo permiten echar algo de luz sobre temas tales como la

duracin del desempleo, los determinantes de la tasa de desempleo y losposibles efectos de la educacin sobre estas variables.


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1 Introduction

In this paper we use search theory to analyze some features of the labor market,like unemployment duration, the determinants of the unemployment rate, andthe impact of education on these variables. The importance of understandingthese phenomena does not need too much explanation. What does deservesome explanation is why we will try to understand them by using search theoryinstead of the traditional Walrasian setup. We nd no better way to do thisthan quoting at length from Lucas (1987):

Think, to begin with, about the Walrasian market for a vector ofcommodities, including as one component hours of labor services,that must be at the center of a competitive equilibrium model.In this scenario, households and rms submit supply and demandorders for labor services and other goods at various auctioneer-

determined price vectors and, when the market-clearing price vectoris found, trading is consummated at those prices. Each seller of la-bor sells as much or as little as he pleases at these prices, each isindierent to the identity of the buyer(s) of the labor services hesells, and if this spot market is repeated at later dates, there is noreason to expect any continuity in the relationship between particu-lar buyers and particular sellers. There is no sense in which anyonein this scenario can be said to have a job or to lose, seek or nd a

job.

It seems clear enough that a model in which wages and hours ofemployment are set in this way can, at best, shed light on the deter-mination of these two variables. Whatever success it may enjoy on

these dimensions, it can tell us nothing about the list of labor mar-ket phenomena that have to do with sustained employer-employeerelationships: their formation, their nature, their dissolution. [...][S]uch a model clearly will not provide a useful account of obser-vations on quits, res, lay-os and other phenomena that explicitlyrefer to aspects of the employer-employee relationship. [...]

A theory that does deal succesfully with unemployment needs toaddress two quite distinct problems. One is the fact that job sepa-rations tend to take the form of unilateral decisions - a worker quits,or is laid o or red - in which negotiations over wage rates play noexplicit role. The second is that workers who lose jobs, for whateverreason, typically pass through a period of unemployment insteadof taking temporary work on the spot labor market jobs that are

readily available in any economy. Of these, the second seems to memuch the more important: it does not explain why someone is un-employed to explain why he does not have a job with company X.After all, most employed people do not have jobs with company Xeither. To explain why people allocate time to a particular activity -

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like unemployment - we need to know why they prefer it to al lotheravailable activities... [...]

An analysis of unemployment as an activity was initiated by JohnMcCall in a paper that integrated Stiglers ideas on the economics ofsearch with the sequential analysis of Wald and Bellman.[1 ] McCallscontribution is well-known and justly celebrated, but I would like tocelebrate it a little more, so I need to set out some details.

We set out those details in Section 2, and show that the optimal behaviorof an unemployed worker is a reservation-wage policy. In Section 3 we derivethe expected duration of employment and unemployment spells. In Section4 we obtain the steady-state unemployment rate. In Section 5 we carry outsome interesting comparative statics exercises that are then used in Section 6 toinquiry about the relation between education and unemployment. In Section 7

we show how the theoretical model can be readily mapped into an econometricmodel. In Section 8 we oer some concluding remarks.

2 A Simple Sequential Search Model

Consider an ininite-horizon continuos-time model where individuals occupy,at each instant, one of two states, employment (e) or unemployment (u). Anunemployed individual receives employment oers at rate per unit-time in-terval, independent of the elapsed duration of the unemployment spell.2 Eachemployment oeri2 f1;:::;ng comes with an attached wage oer,wi2 [0; w], soemployment oers and wage oers are interchangeable expressions. Wageoers are realizations of independent random variables Wi with the same cumu-lative distribution function F.3 This distribution is known by the worker, andconstant over time. After receivingnoers the individual has to decide whetherto reject them or to accept one.4 If he accepts an oer, he will choose the onewith the highest wage, w = maxfw1;:::;wng. While employed, layosarrive at

1 See McCall (1970).2 More precisely, employment oers follow a time-homogeneous Poisson process with pa-

rameter . Hence, the probability of receiving n oers during time-interval h is q(n;h) =(h)n

n! eh. It follows that the probability of receiving no oers during h i s q(0; h) = eh,

the probability of receiving one oer isq(1; h) = heh, and the probability of receiving morethan one oer is q(n 2; h) = 1 (1 + h)eh. Applying Mac Laurins formula to eh wecan rewrite these probabilities as follows: q(0; h) = 1 h + o0(h), q(1; h) = h+ o1(h),

and q(n 2; h) = o2(h), where oi() satises limh!0oi(h)h

= 0, i 2 f0; 1; 2g. Notice

that, limh!0q(1;h)h

= and limh!0q(n2;h)

h = 0. Therefore, for small h, q(1; h) = h,

q(0; h)=1 h, and q(n 2; h)=0. Essentially, during a suciently short time-interval, at

most one oer is received.It is not dicult to show that the expected number of oers received during time-intervalh

is h. Therefore, the expected number of oers per unit-time interval is . This makes clearwhy is called the rate of the process.

3 That is, F(w) PrfWi wg.4 The individual cannot decide to accept an oer that was previously rejected.

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rate per unit-time interval, independent of tenure.5 If the individual rejectsthe oers he remains unemployed and obtains an instantaneous net income b c,

until an oer is accepted. The parameter b measures the value of time spentas leisure (assumed constant over time).6 The value of time and out of pocketcosts spent searching for a job are captured byc. The objective of the individualis to maximize the expected discounted lifetime income, U = E

R10 e

ty(t)dt,where E is the expectations operator (conditional on information available attime zero), > 0 is the discount rate, and y(t) is income at instant t (i.e.,y(t) = w if employed at wage w att, and y (t) = b c if unemployed at t).

We start with a discrete-time version of the model in which the length ofa period is h. The continuous-time version will result as we take the limitfor h converging to zero. Since the problem is recursive, we can apply Bell-mans principle of optimality. Stated in words, the principle asserts that thepresent decision in a sequence of decisions maximizes current net return plusthe expected future stream of returns, appropriately discounted, under the pre-sumption that decisions in the future are made optimally where the expectationtaken is conditional on current information. In short, a multi-stage decisionproblem is converted by the principle into a sequence of single-stage decisionproblems (Mortensen (1986)). Let Ve(w;t;h) denote the value of being em-ployed at time t, the beginning of the period that goes from t to t + h, earningwage w. Ve(w;t;h) is the expected value of discounted net income from thepresent moment onward, for a worker who is employed at t, given that he willbehave optimally in the future conditional on current information. Analogously,let Vu(t; h) be the value of unemployment.

7 Consider the situation of an indi-vidual who behaves optimally and is employed at t. At each instant between tandt + hhe earns w. Therefore, total income during the current period is wh.In this period, the worker learns whether he will be red att + h. At timet, the

probability that the layo occurs between t and t + h is heh

. If the workeris red, he obtains Vu(t+h; h) (properly discounted). If he keeps the job, hegets Ve(w; t + h; h)(properly discounted). Putting all these together we get:

Ve(w;t;h) =wh + eh

hehVu(t + h; h) + (1 he

h)Ve(w; t + h; h)

:(1)

Given the assumptions that bc and ware constant, that the future sequenceof best oers is i.i.d. and the distribution Fis constant over time, the problemis stationary. Therefore, Ve(w;t;h) = Ve(w; h) and Vu(t; h) = Vu(h) for all t.Substituting these into (1), rearranging conveniently, dividing by h, and lettinghgo to zero, we get:

5 More precisely, during time-intervalh, the worker is red with probabilityp(h) = heh,

and remains employed with probability1 p(h) = 1 heh. Notice that limh!0p(h)h =.

It follows that, for small h, p(h)=h.6 In some applications, b can be interpreted to include unemployment insurance compensa-

tion.7 Since b c is constant over time, we do not include it as an argument ofVu().

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Ve(w) = w

+ +

+ Vu; (2)

where we have set Ve(w) Ve(w; 0)and Vu Vu(0).8

Equation (2) can be rewritten as follows:

Ve(w) = w+ [Vu Ve(w)]: (3)

This expression has an intuitive asset-pricing interpretation. Consider a risk-neutral investor with required rate of return , who has acces to a risky assetthat pays dividends at rate wper unit time when the worker is employed and bcper unit time when the worker is unemployed. Since the expected present valueof lifetime dividends of this asset (including the current dividend) is the sameas the workers expected discounted value of lifetime income, the assets price

(including the current dividend) must be Ve(w) when the worker is employedand Vu when the worker is unemployed. The (net) rate of return of this assetequals the dividend rate plus any expected capital gain or loss, all divided bythe price of the asset. When the worker is employed, the dividend rate is wand the expected capital gain/loss is [Vu Ve(w)]. The latter holds becausethere is a probability that the worker will be red, which triggers a changein the price of the asset from Ve(w) to Vu (with probability 1 the workerkeeps the job and the price of the asset remains at Ve(w)). It follows that the

rate of return of the risky asset is w+[VuVe(w)]Ve

. To be willingly held by a risk-neutral investor, this return must equal the required (risk-free) rate of return .

Therefore = w+[VuVe(w)]Ve

, which is equivalent to (3).Imagine now the situation of an individual who behaves optimally and is

unemployed at t. At each instant between t and t + hhe gets b c. Therefore,total income during the current period is (b c)h. In this period, the individualreceives oers to start working at t+ h. Both the number of oers and thewage attached to an oer are unknown at t. With probabilityq(0; h) = eh hereceives no oers and remains unemployed, obtaining Vu(h).

9 With probability

q(n; h) = (h)neh

n! he receives n 2 f1; 2;:::g oers and has to decide whetherto reject them or accept (the best) one. The individual rejects a known setof oers whenever the value of being unemployed is higher than the value ofbecoming employed, and accepts the best oer otherwise. Since the wage atwhich the individual can start working at t+h is unknown at t, the value ofthis option is E maxfVe(Xn; h); Vu(h)g, where the expectation runs over Xn maxfW1;:::;Wng.

10 Putting all these together, and discounting properly, we

8

Ve(w; h) and Vu(h) are continuos, so limh!0 Ve(w; h) = Ve(w; 0) and limh!0 Vu(h) =Vu(0).9 We are already imposing sationarity, so we write Vu(h) instead ofVu(t + h;h).

10 Let G(w;n) be the probability that the best of n oers is (weakly) smaller than w.Since wage oers are independent realizations from F, we get: G(w;n) = PrfXn wg =PrfmaxfW1;:::;Wng wg = PrfW1 w;:::;Wn wg = PrfW1 wg:::PrfWn wg =[F(w)]n.

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get:

Vu(h) = (b c)h + eh

"ehVu(h) +

1Xn=1

(h)neh

n! E maxfVe(Xn; h); Vu(h)g

#(4)

Rearranging (4) conveniently, dividing byh, and lettingh go to zero, we get:

Vu= b c

+ +

+ E maxfVe(W); Vug; (5)

where we have set Vu(0) Vu, Ve(W; 0) Ve(W), and the expectation runsover W X1.11

Equation (5) can be rewritten as follows:

Vu= (b c) + E maxfVe(W) Vu; 0g: (6)

Like (3), equation (6) has an asset-pricing interpretation. When the workeris unemployed, the dividend rate is b c. In addition, there is a proba-bility that the unemployed worker will receive a wage oer w. If he ac-cepts the oer, the price of the asset changes from Vu to Ve(w). If he re-

jects the oer, the price of the asset remains at Vu. Therefore, the expectedcapital gain is E maxfVe(W) Vu; 0g. It follows that the (net) rate of re-

turn of the asset is (bc)+EmaxfVe(W)Vu;0gVu

. To be willingly held by a risk-neutral investor, this return must equal the required (risk-free) rate . Therefore

= (bc)+EmaxfVe(W)Vu;0gVu

, which is equivalent to (6).Dene,

! Vu: (7)

Using (2) and (7) we obtain Ve(w) Vu = w!+ . Therefore, Ve(w)> Vu if and

only ifw > !. Hence, an unemployed worker will accept a job oer if and onlyif its associated wage w is higher than !. Because of this property, ! is calledthereservation wage. Notice that, in this version of the model, the reservationwage is independent of the elapsed duration of the unemployment spell.

Substituting (7) and Ve(w) Vu= w!+ into (6) we get:

! (b c) =

+ E maxfW !; 0g: (8)

Equation (6), which determines the value of! , can be rewritten as follows:12

! (b c) =

+

Z w

!

(w !)dF(w): (9)

11 Recall that, when h is small, the probability of receiving more than one oer vanishes.12 We use: E maxfW!; 0g =

Rw0 maxfw!; 0gdF(w) =

R!0 0dF(w) +

Rw! (w!)dF(w) =Rw

!(w !)dF(w).

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The left-hand side of (9) is the marginal cost of search when the wage oerequals !. It is the dierence between what the worker would get by accepting

the oer and what he would get by rejecting and remaining unemployed. Theright-hand side is the (expected) marginal gain from search when the wage oerequals !. It is the expected gain of nding an acceptable oer next instant,properly discounted by taking account of the frequency with which oers arriveand the possibility that the worker is red after accepting the job.

We can represent the determination ofw graphically. Dene the functions

g(x) = x (b c) and h(x) = +

Rwx

(w x)dF(w). Figure 1 displays these

two functions.13 The reservation wage is the value ofx at which both functionscross.

Figure 1. Determination of the reservation wage

g(x), h(x)

x

There is nothing in Figure 1 that prevents ! from being smaller thanb. Thisgives the idea that an individual could accept a job that pays less than whathe would earn by not participating in the labor market.14 This is inconsistentwith a workers rational participation/non-participation decision. For a rationalworker to be part of the labor force as an unemployed searching worker, thevalue of unemployment must be at least as large as the present value of non-participation: Vu >

b

. Therefore: Vu > b. Since ! Vu, we conclude

that the workers participation requires ! > b. And (9) implies: ! > b ,+

Rwb

(w b)dF(w)> c. That is, a rational worker participates as unemployed

(! > b) if and only if the marginal return to search when the reservation wageis equal to the value of leisure is higher than the out-of-pocket cost of search.From now on, we assume this condition holds.

For future reference, we note that (9) can be rewritten as follows:15

13 It is not dicult to show that h(0) = EfWg+

, h(w) = 0, h0(x) = [1F(x)]+

, and

h00(x) = F0(x)

+ .

14 By refusing to participate the worker obtains b without paying the cost c .15 Integrate

Rw!

(w !)dF(w)by parts, using u= w ! a nd v= F(w).

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! (b c) = +

Z w!

[1 F(w)]dw: (10)

It is clear from either (9) or (10) that the value of ! will depend on thedistribution function F and on the values taken by b c, , , and . Anice feature of the model is that we can do comparative statics very easily.Before doing it, however, it will be convenient to analize the determinants ofthe expected duration of employment and unemployment spells, and of thesteady-state unemployment rate.

3 Expected Duration of Employment and Un-

employment Spells

Let Tu denote the (random) duration of completed unemployment spells, andlet Du denote its cumulative distribution function (i.e., Du(t) = PrfTu tg).Consider the following question: Given that the unemployment spell has alreadylasted t units of time, what is the probability that it will end in the next shortinterval h? In other words, what is the value ofPrfTu 2 [t; t+h]jTu tg?

16

It is standard in the literature to characterize this aspect of the distribution

using the hazard function, u(t) limh!0PrfTu2[t;t+h]jTutg

h = du(t)

Su(t), where

du(t) D0u(t) is the density function and Su(t) 1 Du(t) = PrfTu > tg is

the so-calledsurvival function.17 Thus, the hazard function evaluated at t givesthe rate at which unemployment spells are terminated after duration t, giventhat they have already lasted until t.

For the model developed in the previous section, the rate at which a workerescapes unemployment equals the rate at which employment oers arrive mul-tiplied by the probability that an oer is accepted. Since oers arrive at rate and the probability of acceptance is 1 F(!), we get:

u= [1 F(!)]: (11)

Because the reservation wage is stationary, the hazard rate is independent of theelapsed duration of the unemploymen spell. This implies that the distribution

16 Using the law of conditional probabilities we get PrfTu 2 [t; t + h]jTu tg =Du(t+h)Du(t)

1Du(t) .

17 The terms hazard and survivalcome from biostatistics, where it is standard to ask for theprobability that an individual dies during interval [t; t + h] given that it has already survivedup to t. When talking about unemployment spells, these terms ara a bit counterintuitive

because a higher probability of survival means a higher probability of remaining unemployed.

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of completed unemployment spells is exponential with parameter u:18

du(t) = ueut; (12)

Su(t) = eut: (13)

With the distribution of completed unemployment spells at hand we cancalculate their expected duration. We obtain:19

EfTug= 1

u: (14)

We can repeat the calculations for employment spells. Letting Te denotethe duration of completed employment spells, and using analogous notation, we

obtain:

e= ; (15)

de(t) = et; (16)

Se(t) = et; (17)

and

EfTeg= 1

: (18)

4 Unemployment Rate

Consider an economy populated by a continuum of mass L ofex ante identicalworkers. A constant labor force is consistent with our two-state model in whichindividuals always participate in the labor market. Let E(t) and U(t) be themass of employed and unemployed individuals at timet, respectively. It followsthat E(t) + U(t) =L for all t. We want to calculate the instantaneous change

18 Recall that u() = du()Su()

and notice that du()Su()

= d lnSu()

d for all 0. The n

u() = d lnSu()

d for all 0. Integrating between0 and t we get ln Su(t) = Rt0 u()d.

Therefore Su(t) = eRt0 u()d. This, togheter with du(t) = u(t)Su(t), implies du(t) =

u(t)eRt

0u()d. When the hazard rate is independent of the elapsed duration we haveRt

0u()d =

Rt0ud =ut. Hence, (12) and (13) follow.

19 We have EfTug =R10 tdu(t)dt =

R10 tue

utdt. Integrating by parts using u = t and

v= ueut we obtain (14). A similar procedure can be used to show that the variance ofTu is V arfTug=

12u

.

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in the number of unemployed individuals, ddt

U(t). At each instant, the fractionof workers that become unemployed equals the hazard rate for employment, e.

Similarly, the fraction of workers that become employed equals the hazard ratefor unemployment, u. Therefore:

d

dtU(t) = eE(t) uU(t): (19)

Let u UL

be the steady-state unemployment rate, where U is the constant

mass of unemployed workers. Imposing ddt

U(t) = 0in (19), and usingE= LU,we obtain:

u= e

e+ u=

+ [1 F(!)]; (20)

where the second equality follows from (11) and (15). This expression showsthat the steady-state unemployment rate increases with the job-separation rateand decreases with the job-nding rate.

We can rewrite (20) as follows: u=1

u1

u+ 1

e

. Combining this with (14) and

(18) we get:

u= EfTug

EfTug + EfTeg. (21)

The steady-state unemployment rate increases with the expected duration ofcompleted unemployment spells, and decreases with the expected duration of

completed employment spells. Equation (21) is interesting because it shows thatthe unemployment rate, which is an average across workers at each moment,reects the average outcomes experienced by individual workers across time.Thus, there is a link between economy-wide averages at a point in time and thetime-series averages for a representative agent (Ljungqvist and Sargent (2004)).

We can think of each individuals labor-market history as a continuos-timeMarkov chain dened on two states, employment and unemployment. The in-stantaneous transition rate from unemployment to employment is constant overtime and equal to[1 F(!)], while the instantaneous transition rate from em-ployment to unemployment is also constant and equal to . This provides analternative way of deriving the steady-state unemployment rate as the steady-state probability of unemployment for this chain. A well known statistical resultestablishes that the latter is equal to the ratio between and + [1 F(!)].

5 Comparative Statics

We are now ready to study the eects of changes in the primitives of the modelon the reservation wage, the hazard rates, the expected duration of completed

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employment and unemployment spells, and the steady-state unemployment rate.In what follows, we focus on the interior case !2 (b; w).

Consider the eect of an increase in b c. This can happen either becasueb increases (an increase in the value of leisure) or because c decreases (a re-duction in out-of-pocket expenditures while searching for employment). Totallydierentiating (10) we obtain:

@!

@(b c)=

+

+ + u2 (0; 1): (22)

Thus, an increase in b c increases the reservation wage, but less that one forone. This can be obtained from Figure 1, where the upward-sloping line (whichrepresents the marginal cost of search) shifts to the right by a distance equalto the change in b c. This induces an increase in ! which is smaller than theincrease inb cbecause the other curve (which represents the marginal revenue

from continued search) is downward sloping. The increase in ! induces anincrease inF(!), so the probability of accepting a job oer, 1 F(!), decreases.This, in turn, generates a reduction in the hazard rate for unemployment, u=[1 F(!)]. The hazard rate for employment, e = , is not aected. Theexpected duration of unemployment spells, EfTug =

1u

, increases while the

duration of employment spells does not change, EfTeg = 1

. The steady-state

unemployment rate, u= ee+u

, increases.Imagine that the discount rate, , goes up. This reduces the marginal gain

from continued search. In Figure 1, the downward-sloping curve shifts downwhile the upward-sloping curve does not shift. Hence, the reservation wage goesdown. The analytical expression, obtained by totally dierentiating (10), is:

@!

@ =

( + )( + + u)Z w!

[1 F(w)]dw 0: (24)

The eect on the hazard rate for unemployment is a bit more complicated.

The reason is that, unlike the previous cases, has both a direct eect and anindirect eect on u = [1 F(!)]. The direct eect is clearly positive: for aconstant level of! ;an increase in increases u. The indirect eect, however,tends to reduceubecause the increase in! reduces the probability of acceptingan oer,1F(!). The net eect is ambiguous, as can be seen from the following

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expression:

@u@

= [1 F(!)] F0(!) + + u

Z w!

[1 F(w)]dw R 0: (25)

Therefore, the eect on the expected duration of unemployment and on thesteaty-state unemployment rate is also ambiguous. For the intuitive result thatan increase in job availability reduces unemployment duration, the rst termin the right-hand side of (25) must be bigger than the second. Burdett (1981)shows that a sucient condition for this to happen is a log-concave wage-oerdensity funciton.

Consider now an increase in the arrival rate of layos, . This has theobvious eect of increasing the hazard rate for employment, e, and reducingthe expected duration of employment spells, EfTeg =

1

. The impact on thereservation wage can be obtained from Figure 1, where the marginal gain curve

shifts down, implying a reduction in ! . In particular:

@!

@=

( + )( + + u)

Z w!

[1 F(w)]dw


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the area under the new density function and to the right of the new reservationwage is higher than the area under the old density function and to the right of

the old reservation wage. However, the fact that @!@ is smaller than one impliesthat this is actually true. Formally:

@u

@ = (1

@!

@)F0(!)> 0: (28)

It follows from (28) that both EfTug and u fall when the average wage oerincreases.

Mean-preserving spreads have the property of increasing the variance of adistribution without changing its mean. One can show that:

@!

@ =

+ + u

Z !0

F(w;0)]dw 0; (29)

where the non-negative sign follows from the very dention of mean-presearvingspread. The impact on u is, in general, ambiguous:

@u

@ =F0(!)

@!

@ F(!; 0) R 0: (30)

This implies that a mean-preserving spread has an ambiguous eect on EfTugandu.

6 Education and Unemployment

A thorough study of the relationship between education and unemployment is

outside the scope of this paper since it would requiere us to extend the modelby making the decision to invest in education endogenous. That being said, wecan still use the results of the previous section to shed some light on the issue.To this end, we have to think about the likely dierences on the parameters ofthe model across individuals with dierent education levels.21

A lower discount rate is one of the factors that would induce higher investmetin education. Hence, we would expect a negative relation between this parameterand education. If we think ofas an interest rate we obtain a similar conclusion,since higher education is usually associated with better possibilities of nancingsearch.

The relation between b c and education can be negative, both becauseof the greater foregone earnings of more educated people and because of theirtypically lower unemployment benets. This is not guaranteed, however. One

can imagine more educated workers being better connected to other people orinstitutions that may help them search at a lower cost. Also, since b depends onindividuals preferences for leisure, one cannot rule out the possibility of moreeducated people with a higher value for this parameter.

21 The analysis in this section follows the one in Mincer (1991).

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There are reasons to think that - the instantaneous probability of beingred or laid o - is lower for more educated people. On one hand, more educated

people tend to receive more on-the job training, which makes ring them morecostly for the rms. On the other hand, hiring more educated workers may entailhigher xed costs for rms (e.g., screening, hiring, fringe benets independentof hours, etc.), which can also make rms more reluctant to re them. It is alsopossible that the job matching process is easier for more educated people, whichwould also reduce the probability of being red.

The arrival rate of job oers while unemployed, , is likely to be higherfor more educated workers. One reason is that, because of higher foregoneearnings, more educated individuals have incentives to spend more resources onthe search process, either by searching more intensively (more hours per week)or by incurring other kinds of expenditures (advertising, transportation, etc.).It is also likely that more educated workers can gather information about marketopportunities more eciently, and that the intensity of search by rms is higherfor this type of workers. Both phenomena are associated, in the context of ourmodel, with an increase in . Finally, and in accordance with empirical data, itis reasonble to expect that the wage-oer distribution for more educated workershas a higher mean that the one for less educated workers.

Combining the preceding analysis with the comparative-statics results ofthe previous section we conclude that, in all likelihood, more educated workerswill have a higher reservation wage. The impact of more education on thehazard rate for unemployment is ambiguos. Some factors, like and , makethe hazard lower for more educated workers, while others, like the mean ofthe wage distribution, make the hazard higher. Accordingly, the impact ofmore education on the expected duration of completed unemployment spells, isambiguous. The eect of more education on the hazard rate for employment

is likely to be negative. This implies a higher expected duration of completedemployment spells for more educated workers. Finally, the impact of moreeducation on the steady-state unemployment rate is ambiguous.

7 Econometrics

One of the nice features of search theory is that it provides a framework tostructure and analyze labor-market data. This can be shown by means of asimple example.22 Suppose we have a random sample ofn individuals. An ob-servation for individual i 2 f1;:::;ngconsists of the duration of unemployment,ti, and a vector of time-independent explanatory varaibles, xi. For individualsthat were employed at the time of the survey, ti measures the length of the lastcompleted unemployment spell. For individuals that were unemployed,ti mea-sures the elapsed duration of the current unemployment spell, and is thereforeright censored. If the model described in the preceding sections is correct, the

22 Extensions to more complicated setups can be found in Lancaster (1990).

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density and survival functions for unemployment are:

du(ti; i) = uieuiti ; Su(ti; i) = uieuiti ; (31)

where ui= i[1 F(!i; i)]is the individual-specic hazard rate.Since i, F(; i), and ! i are usually unobserved, it is standard to model the

hazard rate directly, as a function of the vector of explanatory variables. Themost common version is the exponential form

ui= u(xi; ) =exi ; (32)

where is a parameter vector to be estimated.Let A, C, and Ube the sets of all, censored, and uncensored observations,

respectively. Combining (31) and (32), and abusing notation, whe can write thelog-likelihood function as follows:

l( j x; t) =Pi2U

ln du(ti; xi; ) +Pi2C

ln Su(ti; xi;)

=Pi2U

ln u(xi; ) Pi2A

u(xi; )ti

= Pi2U

xi Pi2A

exi ti

(33)

where x (x1;:::;xn) and t (t1;:::;tn). The maximum-likelihood estimator,ML , is obtained by maximizing l( j x; t)with respect to .

8 Concluding remarks

We conclude like we started, with a quote from Lucas (1987):Here, then, is a prototype (at least) of a theory of unemployment.Let us take it seriously and criticize it. Indeed, the model explicitnessinvites hard questioning. Why can the worker not work and search atthe same time (the way economists do)? Doesnt he learn anythingabout his job opportunities as he searches? If he is no dierent fromother workers, why does he face a distribution of wage possibilities(as opposed to opportunities to work ad the going wage)?

These are all good questions, and we could think of more. Someof them are hard to deal with, some quite easy. But before turn-ing to some of these questions, note this: in so criticizing McCallsmodel, we are thinkingabout unemployment, really thinking about

what it is like to be unemployed in ways that x-price and othermacroeconomic-level unemployment theories can never lead us todo. Questioning a McCall worker is like having a conversation withan out-of-work friend: Maybe you are setting your sights too high,or Why did you quit your old job before you had a new one linedup? This is real social science: an attempt to model, to understand,

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human behavior by visualizing the situations people nd themselvesin, the options they face and the pros and cons as they themselves

see them.

Many of the extensions listed in Lucas quotation are already available inthe literature, including on-the-job search, on-the-job wage changes, on-the-

job learning, directed search, two-sided search and matching, endogenous wagedetermination, etc. In addition to Mortensen (1986), two excellent surveys ofthis literature are Mortensen and Pissarides (1999), and Rogerson et al. (2004).Numerous papers use search-theoretic models to empirically study labor-marketissues. Eckstein and van den Berg (2006) survey this literature. Search modelshave also been used to study the feedback between labor- and credit-marketimperfections (Wasmer and Weil (2004)). Last but not least, search theory hasbeen used to ask normative questions, especially with regard to the optimaldesign of unemployment insurance schemes (e.g., Shimer and Werning (2006)).

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References

Burdett, K. (1981). A Useful Restriction on the Oer Distribution in JobSearch Models, in G. Eliasson, B. Holmlund and F. Staord (eds.), Studies inLabor Market Behavior: Sweden and the United States, Stockholm, Sweden:I.U.I. Conference Report.

Eckstein, Z. and van den Berg, G. (2006). Empirical Labor Search: A Survey,forthcoming in Journal of Econometrics.

Lancaster, T. (1990). The Econometric Analysis of Transition Data, Cam-bridge University Press.

Lucas, R. (1987). Models of Business Cycles, Yrj Jahnsson Lectures, BasilBlackwell.

Ljungqvist, L. and Sargent, T. (2004). Recursive Macroeconomic Theory,Second Edition, MIT Press.

McCall, J. (1970). Economics of Information and Job Search,The QuarterlyJournal of Economics, Vol. 84, No. 1 (February), pp. 113-126.

Mincer, J. (1991). Education and Unemployment, NBER Working PaperNo. 3838, September.

Mortensen, D. (1986). Job Search and Labor Market Analysis, in O. Ashen-felter and R. Layard (eds.), Handbook of Labor Economics, Vol. 2, Amsterdam:

North Holland, pp. 849-919.

Mortensen, D. and Pissarides, C. (1999). New Developments in Models ofSearch in the Labor Market, in O. Ashenfelter and D. Card (eds.), Handbook ofLabor Economics, Vol. 3, Amsterdam: North Holland, pp. 2567-2627.

Rogerson, R., Shimer, R. and Wright, R. (2004). Search-Theoretic Modelsof the Labor Market: A Survey, NBER Working Paper No. 10655, August.

Shimer, R. and Werning, I. (2006). Reservation Wages and UnemploymentInsurance, mimeo, Massachusetts Institute of Technology.

Wasmer, E. and Weil, P. (2004). The Macroeconomics of Labor and Credit

Market Imperfections,The American Economic Review, Vol. 94, No. 4 (Septem-ber), pp. 944-963.

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Pontificia Universidad Catlica Argentina

Santa Mara de los Buenos Aires

Facultad de Ciencias Sociales y Econmicas

Departamento de Economa

Documentos de Trabajo Publicados:

N 1: Milln Smitmans, Patricio,Panorama del Sector de Transportes en Amrica Latina yel Caribe, Noviembre de 2005.

N 2: Dagnino Pastore, ngeles Servente y Soledad Casares Bledel,La Tendencia y lasFluctuaciones de la Economa Argentina,Diciembre de 2005.

N 3: Gonzlez Fraga, Javier,La Visin del Hombre y del Mundo en John M. Keynes y enRal Prebisch, Marzo de 2006.

N 4: Saporiti de Baldrich, Patricia Alejandra, Turismo y Desarrollo Econmico, Abrilde 2006.

N 5: Kyska, Helga y Marengo, Fernando, Efectos de la devaluacin sobre lospatrimonios sectoriales de la economa argentina,Mayo de 2006.

Documents

Search Theory and Unemployment