BurdettMortensenIER1998

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Economics Department of the University of Pennsylvania

Institute of Social and Economic Research -- Osaka University

Wage Differentials, Employer Size, and UnemploymentAuthor(s): Kenneth Burdett and Dale T. MortensenSource: International Economic Review, Vol. 39, No. 2 (May, 1998), pp. 257-273Published by: Blackwell Publishing for the Economics Department of the University ofPennsylvania and Institute of Social and Economic Research -- Osaka University

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INTERNATIONALECONOMIC May 1998REVIEW Vol. 39, No. 2

WAGE DIFFERENTIALS, EMPLOYER SIZE, AND UNEMPLOYMENT*BY KENNETH BURDETr AND DALE T. MORTENSEN1

University f Essex, U.KNorthwesternUniversity,U.S.A.

The unique equilibriumsolution to a game in which a continuum ofindividual mployers hoosepermanentwageoffers anda continuum f work-ers search by sequentially ampling rom the set of offers is characterized.Wage dispersionis a robust outcome provided that workers search whileemployedas well as whenunemployed.The uniquenondegenerate quilibriumdistribution f wage offers is constructed or threecases:(i) identicalworkersandemployers, ii) identicalemployersandan atomlessdistribution f workersupply prices, and (iii) identicalworkersand an atomlessdistribution f jobproductivities.

1. INTRODUCTIONEmpiricalresearch has documented that inter-industryand cross-employerwagedifferentials exist, are stable, and cannot be explained by observabledifferences inworkeror job characteristics hat might requirecompensation.Whyshouldworkers

of apparentlyequal abilitybe paid differentlyon similar obs? Manyhave attemptedto provide an explanation.Some writers have arguedthat workerssort on nonobservableabilityin ways thatexplainthe data without contradicting irstprinciplesof competitivemarketanalysis,for example, Murphy and Topel (1987). Others appeal to alternativetheory with'efficiency' and 'fair' wage-type arguments, for example, Kreuger and Summers(1987a, 1987b). Utilizing equilibrium sequential search theory, several differentauthors have provided insights into how a dispersed wage equilibriumcan exist, ormore precisely, how difficult it is to generate dispersed wages as an equilibriumphenomena. (See, for example, Diamond 1971, Albrecht and Axell 1984, andBurdett and Judd 1983.) Here we show that persistent wage differentials are

* Manuscript received January 1995.E-mail: [email protected];[email protected]

257


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258 BURDETT AND MORTENSENconsistent with strategic wage formation in an environmentcharacterizedby marketfriction with and without observable heterogeneity across workers and jobs.1In the environment studied, workers randomlysearch employers for a job thatpays a higher wage while employed and an acceptable wage when unemployed,whereas each employer posts a wage conditional on the search behaviorof workersand the wages offered by other firms. Given the wages offered by all others and thedistributionof workerreservationwage rates, the labor force available to a specificemployerevolves in response to the employer'swage. The higherthe wage the largerthe steady-state labor force, because higher wage firms attractmore workers fromand lose fewer workers to other employers. The resulting labor supply relationdetermines the profit of each employer conditional on the wages offered by otheremployers and the reservationwages demanded by workers.This profit function isthe payoff in a 'wage posting' game played by employers.We show that the uniquenoncooperative steady-state equilibrium to the game can be characterized by anondegenerate distribution of wage offers even when all workers and jobs arerespectivelyidentical if a relativelynatural condition holds-the arrivalrate of joboffers experienced by all workers is strictly positive but finite. As the arrivalrate ofjob offers tends to infinite, the competitive equilibriumresultsin the limit. However,if employed workersdo not receive job offers but unemployedworkers face a strictlypositive but finite arrival rate of offers, all employers offer the monopsony wage,that is, the monopsony equilibriumobtains.Wage dispersion exists in equilibriumeven when workers are equallyproductivein all jobs. Three furtherstrong predictionsalso follow from this simplestversion ofthe model. First, more experienced workersand those with more tenure are morelikely to be found in higher paying jobs. Second, there is a positive associationbetween the labor force size and the wage paid. Finally, there is also a negativerelationship between wage offers and quit rates acrossemployers.Several extensions of the basic framework are explored. The presence of searchfrictions in the form of lags in the arrival of information about the availabilityandterms of job offers unifies the labor marketmodels analyzed.Each model presentedis related to a perfectly competitive counterpart n a naturalwayin the sense that itssolution converges to the competitive equilibriumas frictions vanish. However, thecharacteristics of equilibriumwhen frictions are present provide novel theoreticalinsights and new empirical predictions.

The firstmodel, the pure wage dispersioncase analyzed n Section 2, is too simpleto addressall of the framework's mpiricaland policy implicationsof interest. In thefirst extension, introduced and studied in Section 3, all jobs are equallyproductivebut workers differ with respect to opportunitycosts of employment. In this case,equilibriumunemploymentexceeds that associated with the operationof an efficientjob-worker matching process as some matches that yield gains from trade fail toform. The unemployment inefficiency arisingin this case is attributableto monop-sony power, which accrues to wage-setting employers when frictions are present.

1The origin of this paper is Burdett and Mortensen (1989). We also extend arguments made inBurdett (1990) and Mortensen (1990).


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WAGE DIFFERENTIALS 259Hence, a mandatedminimumwage reduces inefficientunemploymentand increasesthe wage earned by all workers.In Section 4 employer heterogeneity is introduced. First, we consider the casewhere there are two types of employer; high productivityand low productivityones.Any worker employed by a high productivity employer generates more marginalrevenue flow than a worker employed by a low productivityemployer, by assump-tion. In this case, more productive employersoffer higher wage rates and hire moreworkers in equilibrium,a result that reinforces the implicationthat wages offeredincreasewith labor force size in equilibrium.In the second extension,we consider acontinuum of employer productivity types. The relationship between the givendistributionof productivity ypes and the resulting equilibriumdistributionof wageoffers is explored.

2. PURE WAGE DISPERSIONSuppose a large fixed number of both employers and workers participate in alabor market, formallya continuumof each. The measure of workersis indicated bym, whereas the measure of employersis normalized to equal the number 1. In theinitial model considered, all workers and all firms are respectivelyidentical.The decision problem faced by a workeris considered first. At a moment in time,each worker is either unemployed (state 0) or employed (state 1). At randomtimeintervals, however, a worker receives information about a new or alternative job

opening. Allowing the arrival rate to depend on a worker's current state, let A1,i E {0, 1), representingthe parameterof a Poisson arrivalprocess, denote the arrivalrate of offers while a worker is currentlyoccupyingstate i. As workers are assumedto randomlysearch among employers,an offer is assumedto be the realizationof arandom draw from F, the distribution of wage offers across employers. Workersmust respond to offers as soon as they arrive;there is no recall.As jobs are identical apart fromthe wage associatedwith them, employed workersmove from lower to higher paying jobs as the opportunityarises.Workers also movefrom employment to unemployment as well as from job to job. In particular,job-worker matchesare destroyedat an exogenous positive rate 8. Any unemployedworker receives utility flow b per instant. All agents discount future income atrate r.Given the framework briefly outlined above, the expected discounted lifetimeincome when a worker is unemployed, V0,can be expressed as the solution to theasset pricing equation

(1) rVo= b + AO[max(Vo, V(xc)1 dF(G) - /o].

In other words, the opportunitycost of searching while unemployed, the interest onits asset value, is equal to income while unemployed plus the expected capital gainattributable o searchingfor an acceptable job where acceptance occurs only if thevalue of employment, V1(w), exceeds that of continued search. Similarly, the


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260 BURDETT AND MORTENSENexpected lifetime income of a workercurrentlyemployed at wage rate w solves

(2) rWl(w) = w + Al lmax V1(w), V1( ) I - V1(w) dF i) + 8[Vo-V1( w)] .As V1() is increasingin w whereas VO s independentof it, a reservationwage, R,exists such that

(3) V1(w) tVO asw;Rwhere V1(R)= V. The above,plus equations(1) and (2), and an integrationby partsimply.2

(4) R-b = [AO-A1]l [V1(x)-VO] dF(x)

= [AO- A1J VZl(x)[1 -F(x)] dx

- r ~~~1F(x) 1LAo A [ r + 8 + A(1 -F(x))Keeping the analysisas simple as possible, we consider the special case where r issmall relative to the offer arrival rate when unemployed. In particular, lettingr/AO -O 0 implies that (4) can be written as

(5) R -b = [ko-k1]f [] 1 4 F()]] ]where(6) ko= Ao/8 and k1= A1/8represents the ratios of state-dependent arrivalrates to the job separationrate.3Given the reservationwage, the flows of workersinto and out of unemploymentcan be easily specified. Let u denote the steady-state number of workers unem-ployed. In the steady state, the flow of workers into employment, AO[1F(R)]u,equalsthe flow from employmentto unemployment,8(m - u), and therefore

m(7) U 1 +kO[1-F(R)]2 The detailsof the derivation anbe foundin MortensenandNeuman 1987).3 Foreverystrictlypositiver/AO,no matterhowsmall, (4) holdsand therefore 5) holds as thelimit.Ofcourse,at r = 0, there aremanyotheroptimizingtrategies,noneof whichare of particularinterest.


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WAGE DIFFERENTIALS 261Given an initial allocation of workers to firms, the number of employedworkersreceivinga wage no greater than w at time t, G(w, t)(m - u(t)), can be calculated,where G(w, 0 is the proportion of employed workers at t receiving a wage no

greater than w, and u(t) is the measure of unemployedat t. Its time derivative canbe written as(8) dG(w, t)(m - u(t))/dt = AOmax{F(w) - F(R), 0}u(t)

- [8+ A1(1 - F(w))]G(w, t)(m - u(t)).The first term on the right-hand-side of (8) describes the flow at time t ofunemployed workers into firms offering a wage no greater than w, whereas thesecond term representsthe flow out into unemploymentand into higherpayingjobs,respectively. The unique steady-state distribution of wages earned by employedworkers can be written as

(9) G(w) = [F(w) -F(R)]/[1 -F(R)](9) G~~~~w)= 1 +kl(1- F(w))by virtue of (7) and (8) for all w ? R.In what follows, attention is focused on steady-statebehavior. The steady-statenumber of workers earning a wage in the interval [w - e, w] is represented by[G(w) - G(w - 0)](1 - u), while F(w) - F(w - e) is the measure of firmsofferingawage in the same interval.Thus, the measure of workersper firm earninga wage wcan be expressed as

1(wJR )= imG(w) - G(w - e) ((Wl, F)= lio F(W) - F(w - e)Therefore,(10) 1(wJR,F) = mkO[1+ k1(l - F(R))] /[1 + ko(1 - F(R))]

(10) (wRF [1 + k1(l -F(w))] [1 + k1(l -F(w-))]if w ? R and 1(w IR,F) = 0, if w < R, where

F(w) =F(w-) +v(w)given v(w) is the fraction,or mass, of firmsofferingwage w. Thus, (10) specifies thesteady-state number of workers available to a firm offering any particularwage,conditional on the wages offered by other firms, represented by the distributionfunction F, and the workers reservationwage R. From (10) it follows immediatelythat 1( IR,F) (for w 2 R) is (i) increasing n w; (ii) continuousexceptwhere F has amass point; and (iii) strictlyincreasingon the support of F and a constant on anyconnected interval off the supportof F.Firmbehavior is now considered. Let p denote the flow of revenuegeneratedperemployedworker. Hence, an employer'ssteady-stateprofit given the wage offer w


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262 BURDETT AND MORTENSENcan be written as (p - w)l(w IR,F).4 Conditional on R and F, each employer isassumed to post a wage that maximizes its steady-state profit flow, that is, anoptimal wage offer solves the following problem:(11) Tr=max(p-w)l(wIR,F).w

An equilibriumolutionto the searchand wage-posting ame outlined above can bedescribed by a triple (RF, IT), such that R, the common reservation wage ofunemployed workers,satisfies (5), iT satisfies (11), and F is such that(12) (p - w)l(wlR,F) = Tr for all won support of F

(p - w)l(wlR,F) < ir otherwise.Our first task is to establish the existence of a unique equilibriumsolution. To ruleout the trivial, assume oc >p > b > 0 that is, the productivityof workers is greaterthan the common opportunity cost of employment. Further, and this is a morecritical restriction, assume oo> ki > 0, i = 0, 1. The role this restriction plays isdiscussed later.Let w andw denote the infimum and supremumof the supportof an equilibriumF (given one exists).The first thing to note is that no employerwill offer a wage lessthan R in an equilibriumas any employer offering such a wage would have noemployees. Hence, without loss of generality,we consider only those distributionfunctionswhich have w ? R.Before establishingexistence of a unique equilibrium,noncontinuouswage offerdistributionsare first ruled out as possibilities.5 As stated previously, l(w IR,F) isdiscontinuous at w = wi if and only if wiis a mass point of F and w' ? R. This impliesany employer offeringa wage slightly greaterthan wi, a masspoint where R < wi< p,has a significantly larger steady-state labor force and only a slightlysmallerprofitper worker than an employer offering wias (p - w) is continuous in w. Hence, anywage just above wiyields a greater profit. If there were a mass of F at wi p, allfirms offering such a wage make a nonpositive profit. However, any firm offering awage slightly lower than p will make a strictly positive profit as it still attracts apositive steady-state labor force. In short, offering a wage equal to a mass point wcannot be profit maximizingin the sense of (12). Note, this conclusionrules out asinglemarketwageas an equilibriumossibility.As noncontinuous offer distributionshave been ruled out, (10) implies that(13) 1( wIR, F) = mko/(1 + ko)(1 + k1)independent of the wage offered as long as w 2 R. This, of course, implies theemployer offering the lowest wage in the marketwill maximizeits profit flow if andonly if(14) w=R.

4Steady-state profit is the appropriatecriterion in the limitingcase as r/AO> 0, which isassumed or simplicity.See Cole (1997)for an analysisof the discounted ase.5For a more detailedversionof the argument, ee BurdettandMortensen 1989).


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WAGE DIFFERENTIALS 263At any equilibriumeveryoffer mustyield the same steady-stateprofit,which equals(15) T= (p - R)mko/(l + ko)(l + kj) = (p - w)l(wlR, F)for all w in the support of F by equation (13). As w = R, equations (10) and (15)implythat the unique candidate for F is

(16) F(W)= [1k ][1(- )p /]

Substituting(16) into (5) yields

[rOk1[]-R 1+(1 -F(2X) )_][l R (dx l)[

1k1] j 1 -kF \x p -R J[k, ]| 1+kl ((p-R )

However, recognizingthat F(G) = 1, manipulationof equation (16) yields(17) p-W = ( p-R)/(1 + kj)2so that

R(b +kl )Equivalently,(18) R= (1+k1)2b+(ko-kl)klp(1 + k1)2 + (ko - kj)k,Equations(17) and (18) implythe supportof the only equilibriumcandidate,[R, w],is nondegenerate and lies strictlybelow p. Therefore, profit on the support, 7r, isstrictlypositive.6To complete the proof that equations (14), (16), (17), and (18) characterizetheunique equilibrium, we need only show that no wage off the support of thecandidate F yields higher profits. Profits from offers less those on the support

6At thisstage,one can endogenize he relativemeasureof firmsas reflected n m, the measureof workersper firm,byassuminghe existenceof a positive ixedcost c > 0 andinvokingree entry,i.e., 7T= (p - R)mko/(l + k0Xl + kj) = c. All relationships derived above hold in this case as well.Specifically,R is independentof m andtherefore s independent f c givenfree entry.


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264 BURDETT AND MORTENSENattract no workers and therefore yield zero profits, whereas a wage offer greaterthanw, the supremumof the support,attracts no more workers than 1(-wR,F), andhence yields a lower profit. Hence, the claim is established.

The equilibriumwage offer distribution derived here contrasts sharplywith boththe competitive Bertrand solution and Diamond's(1971) monopsonysolution to thewage-posting game. Still, both are limitingcases. Specifically,as k1 tends to zero,(17) implies that the highest wage in the marketgoes to R. Further, (18) implies anunemployed worker's reservation wage, R, converges to the opportunity cost ofemployment, b. Hence, Diamond's solution is obtained in this limit. Allowing thepossibilitythat employedworkers receive alternative offers and move from lower tohigher paying jobs resolves the paradoxical nature of Diamond's (1971) solution.Finally, the equilibriumwage distribution G limits to a mass point at p, the value ofrevenue product, as frictions vanish in the sense that k1 tends to infinity. As kotends to infinity as well, the steady-state unemploymentrate tends to zero. Hence,the competitive equilibrium is the limiting solution when frictions vanish in thesense that offer arrivalrates tend to infinitywhen employedand not.A criticalfeature of the model is the positive relationshipbetween the wage offerand employer labor force size it implies. As the voluntary quit rate, AF(w),decreases with the wage offer, larger firms experience lower quit rates. Becauseworkers only switch employers in response to a higher wage offer, workers witheither more experience or tenure are more likely to be earninga higher wage.

3. HETEROGENEOUS WORKERSHere we extend the basic model by allowingworkersto value leisure differently.Assume all job-worker matches are equally productive as before. Workers,however,

differ in how they value nonemployment. In particular, let H(b) denote theproportion of workers whose opportunity cost of employment is no greater than b.Assume H is continuousand let b and b indicate the infimum and supremumof itssupport. To simplify the analysis, assume that the arrivalrate is independent ofemployment status, that is, AO A1= A, so that the reservation wage of a type bworker is R = b. In this case, the steady-state measure of unemployed workerswilling to accept a wage offer less than or equal to x conditional on the wage offerdistributionF is(19) u(xIF) = j 5 + A[i-F(b)] dH(b)since the unemployment rate of worker of type b is 8/(0 + A[1 F(b)]) and thedensityof type b is mdH(b).Let the steady-statenumber of workersemployed by employers offering a wageno greater than w be given by G(w)(m - u), where u = u(bIF) is total unemploy-ment. In a steady-state the flow of workers leaving employers offering a wage nogreaterthan w equals the flow of workers entering such employers,(20) (8+ A[1-F(w)])(m - u(b IF))G(w) = A [F(w)-F(x)] du(xIF),


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WAGE DIFFERENTIALS 265where du(b IF) is the measure of unemployedworkerswith reservationwage b andF(w) - F(b) is the probability that an offer received by a type b worker isacceptable and less than or equal to w. Solving (20) yields

k [F(w) -F(x)]du(xIF)(21) G(w) (m - u(b IF)) = 1 +k[1 -F(w)]where k = A/8. As [1 + k(1 - F(x))] du(xlF) = mdH(x) from (19), the steady-statenumber of workers available to an employer offering wage w given the offerdistributionF can be written as

(22) l(w, F) (m - u(7 IF)) dG(w) kmH(w)dF(w) [ + k[l -F(w) ]]2at least when F is continuous.7In this case, an equilibriumolution to the searchand wagepostinggame is a triple(R, F, ir) composed of a reservationwage function R(b) = b from (5), a maximalprofit 7ras defined by (11), and an offer distributionF that satisfies(23) (p - w)l(w,F) =r for all w on the supportof F

(p - w) l(w, F) < 7r otherwise.Equations (23) and (22) imply(24) 7r= (p - w)kmH(w)/[1 + k]2

= (p - w)kmH(w)/[1 + k[1 - F(w)]]2where w is the infimum of the supportof F. Thus, all candidatesfor F must satisfy

(25) F( ) [ ~1 k'][ (p - w)H(w)] |25) F(w)=[k1[l-[ ;4]for all w on its support. Of course, in the case where workers have the sameopportunitycost of employment(25) reduces to (16).Let F represent a candidate for an equilibriumoffer distributionand let w, andwh indicate the infimum and supremumof its support.We have established that (25)describes F on its support. We now characterize the support of an equilibriumdistributionof offers. For this purpose, define(26) 4b(w,H) = (p - w)H(w).

7Noncontinuous wageofferdistributions an be ruledoutas an equilibrium henomenautilizingarguments ssentially he same as those used in the previous ection.


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266 BURDETT AND MORTENSENNote, 4(, F), which is equal to monopsony profit were there only one employer, iscontinuous in w as HO is continuous.Assume that H(b) > 0 for some b w where w < w < wi andO(w,H) < O(w', H), then w is not profit maximizing, hat is,

ko(w, F) ko(w', F)(p -w)l(w, F)


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WAGE DIFFERENTIALS 267When workers are heterogenous with respect to the opportunitycost of employ-ment, wage dispersion implies that some efficient matches do not form in the sense

that the wage offered is less than the reservation wage, even though the worker'scontributionto employerrevenue exceeds the worker'sopportunitycost of employ-ment. For the sake of the argument,let frictionalunemploymentbe defined as thelevel that prevails if every employerwere to offer marginalrevenue product, p, andevery worker with opportunity cost of employment less or equal to that of thecompetitive wage, b


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268 BURDETT AND MORTENSENwageeven though atomisticwage competition,not classic monopsonyin the formalsense of one buyer, characterizes the market structure.8The result is the conse-quence of monopsonistic competition conditions generated by the existence ofsearch friction.

4. JOB PRODUCTIVITY DIFFERENTIALSConsider the case of identical workers and two types of employers, one moreproductive than the other. In particular, et pi denote a type i employer'srevenueflow per worker, i E {1, 2), and assume P2 > Pi*The fraction of employersthat havea high productivityis indicated by o-. Given all other aspects of the model are aspresented in the previous section, an equilibriumin this case can be described by(F1, F2, R, iiT, ir2), where R, is the (common to all workers) reservation wage

satisfying(5), Fi, an offer distributionof type i employers,i = 1,2, and(32) (pi - w)l(wIR, F) = 7ri, on supportof F

(pi - w)l(wIR, F) < 7ri, otherwisewhere the marketdistribution,F, is the followingmixture:(33) F(w) = (1 - o)Fl(w) + uF2(w).Existence of such an equilibriumcan be establishedusing argumentssimilarto thoseused in Section 2.A critical characteristic of an equilibriumin this case is that more productiveemployers offer higher wages. Formally, we claim that w2? w1 if wi is on thesupportof Fi, i =1, 2. This follows as(34) T2= P2-W2)l(w2IRF) 2 (P2-w1)l(w1IR,F)

> (p, - wj1)twj JR,F) = 7r 2 (P1- W2)1(W21R,F)where the first inequality and last inequality are implied by (32). Comparing thedifference between the first and last terms of (34) with the difference between themiddle two yields the inequality (P2-p1)l(w2IR,F)2(P2-pl)l(wuIR,F). Thisinequalityand the fact that 1( IR,F) is increasing n w imply W2 w1 as claimed. Itfollows that F1 and F2 can be written as(35) Fi(w) = [l +k Pi-) ]on its support [wi, ii3),i = 1,2.

wy= R, where R satisfies (5)w1= w2, where p, - V1= (p1 - w1)/(1 + k1)2

)2P2-W2=(P22-W2)/(1+k1)8Still, he result sglobal nly ntheconstant eturnsase.


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WAGE DIFFERENTIALS 269There are several relativelyobviousimplicationsof the aboveresult. As the moreproductive employers offer higher wages, they have larger workforces,make moreprofit, and keep workers longer than less productivefirms. Thus, the existence of

productivitydifferences and matchingfriction together provide an explanationforthe persistence in cross-firmand inter-industrywage and profitdifferentials that hasrecentlybeen highlightedin the empiricalliterature.Note, in the above example the equilibriumdistribution of wages offered is amixture of the two underlyingdistributionsof wages across employersof the sametype and the proportion of high productivity employers. As shown in Mortensen(1990), this result holds for any finite numberof productivity ypes. In other words,when there are n productivity ypes among employers,the equilibriumdistributionof wage offers is a mixture of n distributions,one for each type, with weights equalto the relativefrequencyof types. An employerwith productivitypj will offer a wageat least as great as one with productivityPj-i (pj >pj-): that is, the supportof thedistributionof wage offers associated with employerswith productivitypj, [w1,w1) issuch that wj_1= wj. Hence, when there are n productivity types, the marketdistributionof wage offers has n - 1 'kinks.' Between these 'kinks,'the distributionsatisfies (35).Below we briefly consider the case of a continuum of productivity ypes. In thiscase there is a unique wage associated with each productivity ype. This fact impliesthe market distributionof wage offers is a transformationof the underlyingdistribu-tion of productivity ypes.9Let J(p) denote the proportionof employerswith productivityno greater than p.Assume J is continuous and differentiablewith support [p, p]. Keeping things as

simple as possible, we again assume A1= AO= A, which implies R = b from (5). Asall wage offers must be at least as great as the common workerreservationwage, b,only employerswith productivityp ? b can make a profit and hence will participate.Hence, without loss of generality assume that p ? b.As the argumentmade above in the case of two employer types applies here aswell, the employermakingthe lowest offer will be the least productive.Because allunemployed workers accept if the offer exceeds R = b and all the employer'sworkers will leave in response to an outside offer, the lowest offer is profitmaximizing f and only if it equals b, that is,(36) w(p)=b.

For any employer of productivitytype p > p, steady-state profit is determined by awage choice. Hence,

(37) 7r(p) = max{ (p - w)l(wlb, F)}.web

9Bontemps et al., (1997) independently derive and characterize the same equilibrium solution.


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270 BURDETT AND MORTENSENThe first-order onditionfor an interiorsolution is

(38) (p-w)V( 1, 2kF'(w) 11(38) (P - W)[l'(wlbF) =(p - W) 1+k(1-Fw))Il(wlIb,) ] J~)from (10). Finally,the sufficient econd-order ondition is(39) F"(w)[1 + k(1 - F(w))] - 2k[F (w)]2 < 0.

A wage offer distributionF is an equilibriumolutionto the wagepostinggame ifand only if it satisfies (38) and (39) for all p E (b,p] with lower support equal to bfrom (36). Below we constructa unique distribution unction with this property.Theprincipal insightused is that all employerswith the same productivitymustoffer thesame wage in the case of a continuum of productivity types. The wage offercorrespondence that maps productivityto the wage offer is a function w(p) thatsatisfies the first-orderprofit maximizationcondition (38), so that(40) F(w(p)) =J(p) for all p E [b,p].Substituting rom (40) into (38), the first-orderordinarydifferentialequationfollows

_[p - w(p)]2kJ'(p)(41) w'(P)- 1 +k[l -J(p)]As (36) provides a boundary condition, its solution w(p) is unique and, conse-quently, the equilibriumoffer distributioncandidate is unique.To complete the existence proof, it is sufficient to show that the candidatesatisfies the second-order profit maximization condition (39). That it does followsfrom a more explicitrepresentationof the solution to (41). For this purpose, define(42) T(p) -21og(1 + k(1-J(p))).It follows that(43) T'(p)= 2kJ'(p)[1 +k(1-J(p))]Substituting(43) into (41) it follows that(44) w'(p) + T'(p)w(p) = T'(p)p.Any solution to this differentialequation must satisfy(45) w(p)eT(P) = J xT/(x)eT(x) de +A


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WAGE DIFFERENTIALS 271where A is the constant of integration. However, deT(x)/dX = TU(x)eT(x) andtherefore(46) fxT'xT(x) d =pe(p - beT(b) JPeT(x) XHence, (45) can be written as

w(p) =p + e T(P)[A beT(b)] - eT(P)JPeT(x) dx.As we have alreadyestablished that w(b) = b, condition (36), it follows that

A = beT(b)and

w(p) =p -e-T(P)JPeT(x) dX.Finally, as

eT(p) = [1 + k(1 - J(p))]2from definition (42), the wage-productivityprofile is(47) W(p) =p7 - fb( 1 + k(1 -J(x)) dx.

By differentiation(48) w'(p) = 2kJe(p\rP( 1+k(l-J(p)) dx>0 forallp>b.bP [1 + k(1 -J(x)) ]2Consequently,equations (40) and (48) imply

1(49) F(w(p)) = P( l+k(1-J(p)) >0 forallp>b.2kfd(b [1 + k(1 - J(x))]By completely differentiatingthe first-ordercondition for profit maximization(38)with respect to p, equations (48) and (49) imply(50) F"(w( p)) [1 + k(l -F(w(p)))] - 2k[F'(w( p))]2

((2kF(w(p)))


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272 BURDETT AND MORTENSENIn other words, the second-order condition for profit maximization holds on thesupportof J(p) and consequentlyon the supportof F(w).

When jobs have the same productivity, he equilibriumwage distributionhas anincreasing density, a fact implied by equation (16) in the case of a degeneratedistribution of reservation wage rates, and equations (25) and (29) in the moregeneral case. Indeed, for largevalues of the offer arrivalrates, mostwage offers arebunched in a neighborhoodto the left of the common value of marginalrevenue, p.This property,one sharedwith the perfectly competitivecase, implies that the longflat right tail observed in the earnings distributions must be explained by hetero-geneity in workerproductivityor job productivity.The model in this section impliesthat job productivitydifferences can indeed generate a right skew even withoutskewing in the distribution of productivities.For example, if the distribution ofproductivityJ(p) is uniform on the support [b, P], where, without loss of generalityfor the point at issue b = 0 and j = 1, then

kp2W(P)= k

by equation (47). By differentiating (40) and the equation above twice and thensubstituting appropriately,one finds that the offer density declines throughout itssupport. Namely,J"(p) -F'(w(p))w"(p) F'(w(p))

W'(p) P

5. CONCLUSIONThe unique equilibriumsolution to a wage posting game in which a continuumofindividualemployers choose permanent wage offers and a continuum of workerssearchby randomlyand sequentially samplingfrom the set of offers is characterized.In contrast to Diamond's(1971) conclusion, the principalresult of the paper is thatwage dispersionis a robustoutcome when informationabout the individualoffers isincomplete, provided that workers search while employed as well as when unem-ployed. The unique solution for the equilibrium distribution of wage offers is

constructed for three cases: (i) identical workers and employers, (ii) identicalemployers and an atomless distribution of worker supplyprices, and (iii) identicalworkers and an atomless distributionof employer productivities.A question arises whether the solution derived in the paper is also an equilibriumof a more general repeated game in which employerscannot precommit to a wageoffer once and for all. Coles (1997) providesan affirmativeanswer to the question byshowingthat our solution is the limit of a particularsequential equilibrium of themore general game as the common rate of time discount tends to zero. Cole alsoshows that this equilibriumsolution is supported by a 'reputation'for not cuttingwage rates. However, other equilibria also exist once the precommitment require-ment is relaxed.


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WAGE DIFFERENTIALS 273REFERENCES

ALBRECHT,.W. ANDB. AXELL,"An Equilibrium Model of Search Unemployment," Journal ofPolitical Economy 92 (1984), 824-840.

BONTEMPS,, J-M. ROBIN,ANDG. VANDENBERG,"Equilibrium Search with Productivity Disper-sion: Theory and Estimation," Paper presented at the 1997 meeting of the Society forEconomic Dynamics, Oxford, U.K.

BURDETI, K., "A New Framework for Labor Market Policy" in J. Hartog, G. Ridder, andJ. Theeuwes, eds., Panel Data and Labor Market Studies (Amsterdam: North Holland, 1990).

ANDK. JUDD, "Equilibrium Price Distributions," Econometrica 51 (1983), 955-970.AND D.T. MORTENSEN,Equilibrium Wage Differentials and Employer Size," Discussion

Paper no. 860, Northwestern University, 1989.COLES,M.G., "Equilibrium Wage Dispersion, Firm Size, and Growth,"Working paper, Department

of Economics, University of Essex, 1997.DIAMOND, ., "A Model of Price Adjustment,"Journalof Economic Theory3 (1971), 156-168.KRUEGER,. ANDL. SUMMERS,Efficiency Wages and the Inter-Industry Wage Structure,"Econo-metrica 56 (1987a), 259-293.AND ,"Reflections on the Inter-Industry Wage Structure," n K. Lang and J. Leonard,

eds., Unemploymentand the Structureof the LaborMarket (London: Basil Blackwell, 1987b).MORTENSEN,.T., "Equilibrium Wage Distributions: A Synthesis," in J. Hartog, G. Ridder, andJ. Theeuwes, eds., Panel Data and Labor Market Studies (Amsterdam: North Holland, 1990).

ANDG.R. NEUMANN, Estimating Structural Models of Unemployment and Job Duration,"in W.A. Barnett, E.R. Bernt, and H. White, eds., Dynamic Econometric Modelling, Proceedingsof the Third International Symposium in Economic Theoty and Econometrica (Cambridge:Cambridge University Press, 1988).

MURPHY,K. AND R. TOPEL,"Unemployment, Risk, and Earnings: Testing for Equalizing WageDifferences in the Labor Market," in K. Lang and J. Leonard, eds., Unemployment and theStructureof the Labor Market (London: Basil Blackwell, 1987).

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