search-theoretic models of the labor market: a survey

Journal of Economic LiteratureVol XLIII (December 2005) pp 959ndash988

Search-Theoretic Models of the LaborMarket A Survey

RICHARD ROGERSON ROBERT SHIMER and RANDALL WRIGHTlowast

We survey the literature on search-theoretic models of the labor market We showhow this approach addresses many issues including the following Why do workerssometimes choose to remain unemployed What determines the lengths of employ-ment and unemployment spells How can there simultaneously exist unemployedworkers and unfilled vacancies What determines aggregate unemployment andvacancies How can homogeneous workers earn different wages What are the trade-offs firms face from different wages How do wages and turnover interact Whatdetermines efficient turnover We discuss various modeling choices concerning wagedetermination and the meeting process including recent models of directed search

959

lowast Rogerson Arizona State University Shimer Universityof Chicago Wright University of Pennsylvania We thankDaron Acemoglu Ken Burdett Zvi Eckstein RogerGordon Iourii Manovskii Derek Laing John McMillanDale Mortensen Peter Rupert Martin Schindler Gerardvan den Berg two anonymous referees and many studentsfor their input We also thank the Federal Reserve Bank ofCleveland the National Science Foundation and the SloanFoundation for research support

1 Introduction

The economic fortunes of most individu-als are largely determined by their labor

market experiencesmdashthat is by paths fortheir wages their employers and theirintervening spells of unemployment or non-employment Hence economists are natu-rally interested in documenting theempirical behavior of wages employmentand unemployment and also in buildingmodels to help us understand the forces thatshape these outcomes and using the modelsto assess the consequences of changes inpolicies or institutions

While the usual paradigm of supply anddemand in a frictionless labor market isuseful for discussing some issues manyimportant questions are not easilyaddressed with this approach Why dounemployed workers sometimes choose toremain unemployed say by turning downjob offers What determines the lengths ofemployment and unemployment spellsHow can we simultaneously have unem-ployed workers and unfilled vacanciesWhat factors determine the aggregateunemployment and vacancy rates How canapparently homogeneous workers in similarjobs end up earning different wages Whatare the trade-offs faced by firms in payingdifferent wages How do wages and turnoverinteract What determines the efficientamount of turnover

From its inception search theory hasprovided a rigorous yet tractable frame-work that can be used to address these andrelated questions Central to the approach

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960 Journal of Economic Literature Vol XLIII (December 2005)

is the notion that trading frictions areimportant It takes time and otherresources for a worker to land a job espe-cially a good job at a good wage and for afirm to fill a vacancy There is simply nosuch thing as a centralized market wherebuyers and sellers of labor meet and tradeat a single price as assumed in classicalequilibrium theory Of course economicmodels do not have to be realistic to beuseful and the supply-and-demand para-digm is obviously useful for studying manyissues in labor economics But it is equallyclear that the simple supply-and-demandapproach is ill suited for discussing ques-tions such as those raised in the previousparagraph

We argue that even the earliest searchmodels with their focus on a single workerenhance our ability to organize observa-tions on employment histories We thenexamine more recent research that embedsthe decision-theoretic model into an equi-librium framework Although there areseveral important modeling decisions inequilibrium search theory we argue thattwo questions are paramount First howdo agents meet In particular is searchrandom so that unemployed workers areequally likely to locate any job opening ordirected so that for example firms canattract more applicants by offering higherwages Second exactly how are wagesdetermined Do matched workers andfirms bargain or are wages posted unilat-erally before they meet We consider vari-ous alternative sets of assumptions andindicate how the choice affects predictionsWe also emphasize how a common set ofmethods and ideas are used in all of thedifferent approaches

Before proceeding we mention thatsearch theory constitutes a very large fieldIn addition to labor it has been used inmonetary theory industrial organizationfinance the economics of the marriagemarket and other areas all of which wemust neglect lest this survey becomes

1 Examples in monetary economics include NobuhiroKiyotaki and Randall Wright (1993) Shouyong Shi (1995)and Alberto Trejos and Wright (1995) examples in themarriage literature include Dale T Mortensen (1988)Kenneth Burdett and Melvyn G Coles (1997 1999) andRobert Shimer and Lones Smith (2000) examples in IOinclude Steven C Salop (1977) Boyan Jovanovic (1982)and Jovanovic and Glenn M MacDonald (1994) examplesin finance include Darrell Duffie Nicolae Garleanu andLasse Pedersen (2002) and Pierre-Olivier Weill (2004)

2 Examples of theoretical work studying the question ofwhether frictionless competitive equilibrium is the limit ofsearch equilibrium as the frictions get small include ArielRubinstein and Asher Wolinsky (1985 1990) DouglasGale (1987) and Mortensen and Wright (2002) Theresa JDevine and Nicholas M Kiefer (1991) Kenneth I Wolpin(1995) and Zvi Eckstein and Gerard J van den Berg(forthcoming) survey empirical work

unmanageable1 Search has been used inmuch fairly technical theoretical researchand has also been a workhorse for empiricaleconomics but we can neither delve intopure theory nor pay attention to all of theeconometric issues and empirical findingshere2 Also while we strive to be rigorouswe emphasize issues rather models ormethods per se Hence the presentationrevolves around the ways in which theframework helps us think about substantivequestions like those mentioned above

The logical structure of the paper is as fol-lows We begin in section 2 with the problemof a single agent looking for a job not onlybecause this is the way the literature startedbut because it is a building block for theequilibrium analysis to follow Even thisrudimentary model is consistent with twofacts that do not come out of frictionlessmodels it takes time to find an acceptablejob and what one ends up with is at least par-tially a matter of luck which means similaragents may end up with different wages Insection 3 we describe some generalizationsof this model designed to help understandturnover and labor market transitions Thissection also introduces tools and techniquesneeded for equilibrium search theory

We also spend some time here discussingthe logical transition between decision theo-ry and equilibrium theory We first arguethat one can reinterpret the single-agent


Rogerson Shimer and Wright Search-Theoretic Models of the Labor Market 961

model as an equilibrium of a simple econo-my But in such a model several key vari-ables including the arrival rate anddistribution of wage offers are essentiallyfixed exogenously For some issues one maywant to know how these are determined inequilibrium and in particular how they areaffected by labor market conditions or labormarket policy There is no single way to pro-ceed but any approach requires us to con-front the two questions mentioned abovehow do workers and firms meet and how dothey determine wages

In section 4 we present a class of equilib-rium models built on two main ingredientsthe matching function which determineshow workers and firms get together and thebargaining solution which determineswages once they do The matching functionhelps us in analyzing transitions from unem-ployment to employment as a function of ingeneral the behavior of all the workers andfirms in the market Bargaining is one of themore popular approaches to wage determi-nation in the literature and since it is a keyingredient in many models we take sometime to explain how the generalized Nashsolution works and how to interpret it interms of strategic bargaining theory

In section 5 we consider an environmentwhere wages are posted ex ante rather thanbargained after agents meet and in additionwhere search is directedmdashie workers donot encounter firms completely at randombut try to locate those posting attractiveterms of trade Models with the combinationof wage posting and directed search calledcompetitive search models behave quite dif-ferently from those in section 4 althoughthey can be used to analyze similar issues Insection 6 we consider models where wagesare posted but search is once again purelyrandom This class of models has been wide-ly used in research on wage dispersion andthe distribution of individual employmentunemployment spells

In section 7 we discuss efficiency This isimportant because to the extent that one

3 Earlier surveys of search theory as applied to labormarkets include Steven A Lippman and John J McCall(1976a) Mortensen (1986) and Mortensen andChristopher A Pissarides (1999a 1999b) While there isnaturally some overlap there are important differences inour approach We build equilibrium models up from thedecision problem focusing on the role of random versusdirected search and the role of bargaining versus wageposting In particular previous surveys do not examinedirected search models or explicitly discuss the efficiencyproperties of different models as we do

4 While it is often said that the economics of searchbegan with George Stigler (1961) his formulation was stat-ic The sequential job search model in this section wasdeveloped first by McCall (1970) Mortensen (1970) andReuben Gronau (1971) although others had analyzedrelated problems including Herbert A Simon (1955) whodiscussed the housing market and Samuel Karlin (1962)who discussed asset markets The presentation in this andthe next section is based on some lecture notes that con-tain many more details examples and exercises and can befound at httpwwwsscupennedu~rwrightcoursescourseshtml

wants to analyze policy one would like toknow whether the models rationalize a rolefor intervention in a decentralized economyIt is fine to say for example that a change insome variable reduces unemployment butit is obviously relevant to know whether thisimproves welfare We derive conditionsunder which equilibrium with bargaining isefficient We also show that the combinationof wage posting and directed search gener-ates efficiency under quite general assump-tions providing a version of the first welfaretheorem for economies with frictionsFinally we finish in section 8 with a fewconcluding remarks3

2 Basic Job Search

We begin here with the familiar discrete-time formulation of the basic job searchmodel and then derive its continuous-timeanalogue which is used for the remainder ofthe paper We then use the framework to dis-cuss some issues relating to unemploymentduration and wages

21 Discrete Time

Consider an individual searching for a jobin discrete time taking market conditions asgiven4 He seeks to maximize EE

t=0βtxt



U W w dF wr

U+ minus+

110

max ( ) ( )

5 Our worker is interested in maximizing expected dis-counted income This is the same as maximizing expectedutility if he is risk neutral but also if he is risk averse andconsumption markets are complete since then he canmaximize utility by first maximizing income and thensmoothing consumption The case of a risk averse agentfacing incomplete markets is more difficult Early analysesinclude John P Danforth (1979) and John R HallLippman and McCall (1979) more recent studies includeVictor Valdivia (1996) James Costain (1997) DaronAcemoglu and Shimer (1999b 2000) Martin BrowningThomas F Crossley and Eric Smith (2003) and RasmusLentz and Torben Tranaes (forthcoming)

where β isin (01) is the discount factor xt isincome at t and EE denotes the expectationIncome is x = w if employed at wage w and x = b if unemployed Although we refer to was the wage more generally it could capturesome measure of the desirability of the jobdepending on benefits location prestigeetc and although we refer to b gt 0 as unem-ployment insurance (UI) it can also includethe value of leisure or home production5

We begin with the case where an unem-ployed individual samples one independent-ly and identically distributed (iid) offereach period from a known distribution F(w)If an offer is rejected the agent remainsunemployed that period Assume previouslyrejected offers cannot be recalled althoughthis is actually not restrictive because theproblem is stationary so an offer that is notacceptable today will not be acceptabletomorrow For now we assume that if a job isaccepted the worker keeps it forever Hencewe have the Bellman equations

(1)

(2)

where W(w) is the payoff from accepting awage w (W stands for working) and U is thepayoff from rejecting a wage offer earning band sampling again next period (U stands forunemployed)

Since W(w) w(1 minus β) is strictly increas-ing there is a unique wR called the reserva-tion wage such that W(wR) U with theproperty that the worker should reject

U b U W w dF w= + intβ0

max ( ) ( )

W w w W w( ) ( )= + β

6 Note that in the above analysis as in most of what wedo here it is assumed that the worker knows F If he hasto learn about F while searching the problem gets harderand a reservation strategy may not even be optimal Forexample suppose we know either (a) w = w0 with prob 1or (b) w = w1 with prob π and w = w2 with prob 1 minus π Ifw2 gt w1 gt w0 and π is small it can be optimal to accept w0

but not w1 since an offer of w1 signals that there is a goodchance of getting w2 Michael Rothschild (1974) givesconditions that guarantee a reservation strategy is optimalSee Burdett and Tara Vishwanath (1988a) for additionaldiscussion and references

w lt wR and accept w ge wR (we adopt theconvention that he accepts when indiffer-ent) Substituting U = wR (1 minus β) andW(w) = w(1 minus β) into (2) we have

(3)

The function T is easily shown to be a con-traction so there is a unique solution to wR = T(wR) This implies that if one fixes w0

and recursively defines wN+1 = T(wN) thesequence converges to wR as N rarr If theinitial wage is w0 = b the workerrsquos reserva-tion wage in the final period of a finite hori-zon problem wN has the interpretation ofbeing the reservation wage when N periodsof search remain after which the workerreceives either b or the accepted wage wforever

The optimal search strategy is completelycharacterized by (3) but we also presentsome alternative representations that areoften seen in the literature First subtract-ing βwR from both sides of (3) and simplify-ing gives the standard reservation wageequation

(4)

Using integration by parts we can also writethis as

(5)

which as we shall see is handy in some ofthe applications below6

w b F w dwR wR

= +minus

minusintββ1

1

[ ( )]

w b w w dF wR w RR

= +minus

minusintββ1

( ) ( )

b w w dF wRminus + int( ) max ( ) 10

β β

w T wR R= ( )



7 Alternatively we may assume that the number of wageoffers per period is a Poisson random variable with mean When the time period is short the probability ofreceiving multiple offers within a single period is negligibleSee Mortensen (1986) for details

22 Continuous Time

We now derive continuous-time versionsof the above results First generalize thediscrete-time model to allow the length of aperiod to be ∆ Let β = 1

__1

r__ and assume

that the worker gets a wage offer with prob-ability ∆ in each period7 Then the payoffsto working and unemployment satisfy thefollowing versions of (1) and (2)

(6)

(7)

times

Algebra implies

(8)

(9)

When ∆ rarr 0 we obtain the continuous time

Bellman equations

(10)

(11)

Intuitively while U is the value of beingunemployed rU is the flow (per period)value This equals the sum of the instanta-neous payoff b plus the expected value ofany changes in the value of the workerrsquosstate which in this case is the probability

rU b W w U dF w= + minusint0

0

max ( ) ( )

rW w w( ) =

W w U dF w+ minusint 00

max ( ) ( )

rU r b= +( )1

rW w r w( ) = +( )1

U W w dF wr

U+ minus+int

110

max ( ) ( )

U br

= ++ int

1

W w wr

W w( ) ( )= ++

11

that he gets an offer times the expectedincrease in value associated with the offernoting that the offer can be rejected

The reservation wage wR satisfiesW(wR) = U so equation (10) implies W(w) minusU = (w minus wR)r Substituting this into (11)gives the continuous time reservation wageequation

(12)

Again one can integrate by parts to get

(13)

Although most of the models that we dis-cuss assume fixed search intensity it can beendogenized Suppose a worker can affectthe arrival rate of offers at cost g() whereg gt 0 and g gt 0 Unemployed workerschoose to maximize rUwR where

(14)

The first order condition for an interiorsolution is

(15)

Worker behavior is characterized by a pair(wR ) solving (14) and (15) It easy to showthat an increase in b eg raises wR andreduces

23 Discussion

Traditional frictionless models assume thata worker can costlessly and immediatelychoose to work for as many hours as he wantsat the market wage By relaxing theseextreme assumptions search models allow usto think about unemployment and wages in adifferent light Consider unemploymentduration The probability that the worker hasnot found a job after a spell of length t is eminusHtwhere H = [1 minus F(wR)] is called the hazardrate and equals the product of the contactrate and the probability of accepting

w RR

w w dF w rg

int minus =( ) ( ) ( )

w b gr

w w dF wR w RR

= minus + minusint( ) ( ) ( )

w br

F w dwR wR

= + minusint

[ ( )]1

w br

w w dF wR w RR

= + minusint

( ) ( )



8 Because this simple model is stationary H does notchange with the duration of unemployment Several gen-eralizations would overturn this the simplest being toassume a finite horizon More interestingly Burdett (1979)and Mortensen (1977) allow UI to vary over timeLippman and McCall (1976b) and Lippman and John WMamer (1989) allow wage offers to vary over time Salop(1973) studies systematic search where a worker first looksfor opportunities that are best according to some prior andthen if unsuccessful proceeds to other locations typicallylowering his reservation wage over time The models dis-cussed earlier in which the worker learns about the wagedistribution also predict wR and hence H vary over timeBruce D Meyer (1990) and Wolpin (1987) are examples ofa large body of empirical work on hazard rates

9 We can also study changes in F or As pointed outby Burdett (1981) for some experiments it is useful toassume F is log-concave Suppose we increase every w inF either by a constant or proportionally Perhaps surpris-ingly E[ww ge wR] may actually decrease but it can beguaranteed to increase under log-concavity seeMortensen (1986) or Wright and Janine Loberg (1987)Suppose we increase the arrival rate One might expectthat this must raise the hazard but since it increases wRthe net effect is ambiguous One can show H gt 0 underlog-concavity see Christopher J Flinn and James JHeckman (1983) Burdett and Jan Ondrich (1985) or vanden Berg (1994)

1 minus F(wR)8 The average duration of anunemployment spell is therefore

(16)

Also the observed distribution of wages paidis G(w = F(ww ge wR)

Consider the impact of an increase in bsay more generous UI assuming for simplic-ity here that search intensity and hence arefixed From (12) the immediate effect is toincrease wR which has two secondaryeffects the distribution of observed wagesG(w) is higher in the sense of first order sto-chastic dominance since more low wageoffers are rejected and the hazard rate H islower which increases average unemploy-ment duration Hence even this elementarymodel makes predictions about variablesthat would be difficult to generate using atheory without frictions9

3 Worker Turnover

Although the model in the previous sec-tion is interesting there are important issuesthat it cannot address For instance we

D tHe dtH

Ht= =int minus

0

1

10 An interesting extension of the basic model is to let vary across jobs which implies the reservation strategygenerally depends on the pair (w) see Burdett andMortensen (1980) or Wright (1987)

assumed above that when a worker accepts ajob he keeps it forever Yet according toBruce Fallick and Charles A Fleischman(2004) in the United States from 1994 to2004 66 percent of employment relation-ships ended in a given month (of these fortypercent of workers switched employerswhile the rest either became unemployed orleft the labor force) We now generalize theframework to capture such transitions

31 Transitions to Unemployment

The simplest way of generating transitionsfrom employment into unemployment is toassume that jobs end for some exogenousreason sometimes in the literature this isinterpreted by saying that workers face lay-off risk A tractable formulation is to assumethat this occurs according to a Poissonprocess with parameter λ which for now isan exogenous constant10

Introducing exogenous separations doesnot affect the Bellman equation for Uwhich is still given by (11) but now we haveto generalize (10) to

(17)

The reservation wage still satisfies W(wR)Uand the methods leading to (13) yield

(18)

Notice that λ affects wR only by changing theeffective discount rate to r + λ However aworker now goes through repeated spells ofemployment and unemployment whenunemployed he gets a job at rate H = [1 minus F( wR)] and while an employedhe loses the job at rate λ

A simple way to endogenize transitions tounemployment is to allow w to change at agiven job Suppose that this happens accord-ing to a Poisson process with parameter λand that in the event of a wage change a new

w br

F w dwR wR

= ++

minusint

λ

[ ( )]1

rW w w U W w( ) [ ( )]= + minusλ



11 The on-the-job search model was introduced byBurdett (1978) The presentation here follows Mortensenand George R Neumann (1984)

w is drawn from F(ww) When the wagechanges the worker can stay employed at wor quit to unemployment The exogenouslayoff model discussed above is a special casewhere w= 0 with probability 1 so that atrate λ the job effectively disappears In thismore general model

(19)

A natural assumption is that F(ww2) firstorder stochastically dominates F(ww1)whenever w2 gt w1 This implies W(w) isincreasing and there is a reservation wagewR such that unemployed workers accept ifw ge wR and employed workers quit if theirwage falls to wwR Hence separations aredecreasing in w In the simplest case whereF(w|w) = F(w) (independence) we have

(20)

Notice that λ gt implies wR lt b in this caseworkers accept a job paying less than unem-ployment income and wait for the wage tochange rather than searching while unem-ployed In any case the usual comparativestatic results such as partwR partb gt 0 are similarto what we found earlier

32 Job-to-Job Transitions

To explain how workers change employerswithout an intervening spell of unemploy-ment we need to consider on-the-jobsearch11 Suppose new offers arrive at rate0 while unemployed and 1 whileemployed Each offer is an iid draw fromF Assume that employed workers also losetheir job exogenously at rate λ The Bellmanequations are

(21) rU b W w U dF wwR

= + minusint0

[ ( ) ] ( )

w br

w w dF wR w RR

= + minus+

minusint λλ

( ) ( )

W w U W w dF w w( ) ( ) ( )minus minus |

rW w w W w( ) max ( )= + intλ0

12 As in the basic model we can endogenize searchintensity here Let g0() be the cost of achieving for anunemployed worker and g1() the cost for an employedworker If g0() le g1() for all unemployed workerssearch harder than employed workers In any case searchintensity decreases with w for employed workers

(22)

The second term in (22) represents theevent that an employed worker gets an offerabove his current wage

It is easy to see that W is increasingimplying that unemployed workers use areservation wage satisfying W(wR) = Uand employed workers switch jobs when-ever wgt w Evaluating (22) at w = wR andcombining it with (21) we get

(23)

Observe that wR gt b if and only if 0 gt 1Thus if a worker gets offers more frequent-ly when employed than when unemployedhe is willing to accept wages below b

To eliminate W from (23) use integrationby parts and insert W(w) = r + λ + 1 [1 minusF(w)]minus1 which we get by differentiating(22) to yield

(24)

If 1 = 0 this reduces to the earlier reserva-tion wage equation (13) Many results likepartwR partb gt 0 are similar to what we foundabove but we also have some new predic-tions For instance when wR is higher work-ers are less likely to accept a low w so theyare less likely to experience job-to-job transi-tions Thus an increase in UI reducesturnover12

F wr F w

dwR

+ minus minus ( )+ + minus ( )

⎡⎣⎢

⎤⎦⎥int( )

[ ]

0 11

11

λww

w bR =

W w W w dF ww R

R

+ minus minusint( ) [ ( ) ( )] ( ) 0 1

w bR =

W w dF w U W( ) ( ) [ (minus + minus0 λ ww)]

rW w w W w( ) max ( )= + minusint1 0



13 Other extensions can also deliver similar predictionOne such class of models is the learning models inJovanovic (1979a 1979b) and Louis L Wilde (1979) Inthese models workers have to learn about how good theyare at a job and those with longer tenure have less to learnand hence are less likely to leave Another class of modelsintroduces human capital Since we do not have space todo justice to this topic we refer the reader to recent workby Gueorgui Kambourov and Iourii Manovskii (20042005) that studies human capital within a search frame-work very similar to what is presented here Other search-based models with human capital include Acemoglu(1996) Lars Ljungqvist and Thomas J Sargent (1998)Adrian M Masters (1999) Coles and Masters (2000)Burdett and Smith (2001) and Laing Theodore Palivosand Ping Wang (1995 2003)

33 Discussion

The model of the last subsection is a natu-ral framework in which to analyze individualtransitions between employment and unem-ployment and between employers It makespredictions about the relationships betweenwages tenure and separation rates Forexample workers typically move up thewage distribution during an employmentspell so the time since a worker was lastunemployed is positively correlated with hiswage Also workers who earn higher wagesare less likely to get better opportunitiesgenerating a negative correlation betweenwages and separation rates And the fact thata worker has held a job for a long time typi-cally means that he is unlikely to obtain abetter one generating a negative relation-ship between job tenure and separationrates All of these features are consistentwith the empirical evidence on turnover andwage dynamics summarized in eg HenryFarber (1999)13

With a slight reinterpretation the frame-work can also be used to discuss aggregatevariables Suppose there are many workerseach solving a problem like the one dis-cussed above with the various stochasticevents (like offer arrivals) iid across work-ers Each unemployed worker becomesemployed at rate H = 0[1 minus F(wR)] and eachemployed worker loses his job at rate λ sothe aggregate unemployment rate u evolvesaccording to

u = λ(1 minus u) minus 0[1 minus F(wR)]u

Over time this converges to the steady state

(25)

One can also calculate the cross-sectionaldistribution of observed wages for employedworkers denoted by G(w) given any offerdistribution F(w) For all w ge wR the flow ofworkers into employment at a wage nogreater than w is u0[F(w) minus F(wR)] equal tothe number of unemployed workers times therate at which they find a job paying betweenwR and w The flow of workers out of thisstate is (1 minus u)G(w)λ + 1[1 minus F(w)] equalto the number of workers employed at w orless times the rate at which they leave eitherfor exogenous reasons or because they get anoffer above w In steady state these flows areequal Using (25) and rearranging we have

(26)

We can now compute the steady state job-to-job transition rate

(27)

This type of model has been used in a vari-ety of applications For example Wright(1986) uses a version with learning to discussmacro aggregates showing how the combina-tion of search and learning generates consid-erable persistence in the unemployment rateUnder the interpretation that the learningcomes from a signal-extraction problemwhere workers see nominal wages and have tolearn the real wage the model also generatesa Phillips curve relation between inflation andunemployment Unlike previous signal-extrac-tion models such as Robert E Lucas (1972)unemployment is persistent because searchprovides a natural propagation mechanism

Jovanovic (1987) considers a variation thatallows workers who are not satisfied with

1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]

minus[ ] + minus[ ] λ

λ( ) ( )

( ) ( )1 11

uF wR

lowast =+ minus ( )[ ]

λλ 0 1



their current wage to either search for a bet-ter one or to rest and return to work whenw increases This model can generate pro-cyclical quits and productivity along withcountercyclical unemployment as in thedata Wright and Loberg (1987) use anothervariation to analyze the impact of changes intaxes on unemployment and wages of work-ers at different points in the skill distribu-tion Ljungqvist and Sargent (1998) addhuman capital accumulation during employ-ment and deaccumulation during unemploy-ment and study the effects of policyKambourov and Manovskii (2005) use a ver-sion of the model to study life-cycle earningsprofiles

Although these applications all seeminteresting there is an old critique of theframeworkmdashwhich is that it is partial equi-librium because the distribution F is exoge-nous (Rothschild 1973) From a logical pointof view this critique is not compelling asthere are various ways to embed the modelinto an equilibrium context without chang-ing the results For instance one can imag-ine workers looking for wages as fishermenlooking for lakes Then F is simply the distri-bution of fish across lakes which is some-thing we can logically take as fixed withrespect to most policy interventions

Another approach that involves only asmall change in interpretation is to invokethe island metaphor often used in searchtheory Imagine workers searching acrossislands on each of which there are manyfirms with a constant returns to scale tech-nology using only labor The productivity ofa randomly selected island is distributedaccording to F If the labor market on eachisland is competitive a worker on an islandwith productivity w is paid w This triviallymakes the equilibrium wage distribution thesame as the productivity distribution F Infact this is the Lucas and Edward CPrescott (1974) equilibrium search modelaside from some minor detailsmdasheg theyallow for decreasing returns to scale whichcomplicates the algebra but does not change

14 We do not discuss the LucasndashPrescott model indetail since it can be found in standard textbooks (NancyStokey Lucas and Prescott 1989 chapter 13 Ljungqvistand Sargent 2004 chapter 26) Applications includeJeremy Greenwood MacDonald and Guang-Jia Zhang(1996) Joao Gomes Greenwood and Sergio Rebelo(2001) Fernando Alvarez and Marcelo Veracierto (1999)and Kambourov and Manovskii (2004)

the idea14

In summary we think it is silly to criticizethe framework as being partial equilibriumper se since it is trivial to recast it as an equi-librium model without changing the essenceThe more pertinent question is do we missanything of substance with these storiesabout lakes and islands The models that wepresent below introduce firms and equilibri-um considerations using more economicsand less geography

4 Random Matching and BargainingThis section introduces a popular line of

research emanating from Pissarides (19852000) used to study the determinants ofarrival rates match formation match disso-lution and wages We present a sequence ofmodels that emphasize different marginsincluding entry the decision to consummatematches and the decision to terminatematches Before we begin there are twoissues that need to be addressed how doworkers and firms meet and how are wagesdetermined These models assume meetingsare determined through a matching functionand wages through bargaining

41 MatchingSuppose that at some point in time there

are v vacancies posted by firms looking forworkers and u unemployed workers lookingfor jobs Building on ideas in Peter ADiamond (1981 1982a 1982b) Mortensen(1982a 1982b) Pissarides (1984 1985) andelsewhere assume the flow of contactsbetween firms and workers is given by amatching technology m = m(uv) Assumingall workers are the same and all firms are thesame the arrival rates for unemployed work-ers and employers with vacancies are thengiven by



15 Early empirical work on matching goes back toPissarides (1986) and Olivier J Blanchard and Diamond(1989) see Barbara Petrongolo and Pissarides (2001) for arecent survey

16 An interesting alternative is to assume the number ofmatches depends on both the flows of newly unmatchedworkers and firms and the stocks of existing unemploy-ment and vacancies as in Coles and Smith (1996 1998)See Ricardo Lagos (2000) for another approach

(28)

It is standard to assume the function m iscontinuous nonnegative increasing inboth arguments and concave withm(u0) = m(0v) = 0 for all (uv) It is alsoconvenient to assume m displays constantreturns to scale ie m(uv) m(uv)While alternative assumptions are interest-ingmdasheg Diamond emphasizes thatincreasing returns can generate multipleequilibriamdashconstant returns is consistentwith much empirical work15 Also constantreturns generates a big gain in tractability asit implies w and e depend only on the ratiovu referred to as a measure of market tight-ness Thus w is an increasing and e adecreasing function of vu and there is acontinuous decreasing 1 to 1 relationshipbetween w and e

The matching technology is meant to rep-resent in a simple if reduced-form fashionthe notion that it takes time for workers andfirms to get together Just as a productionfunction maps labor and capital into outputm maps search by workers and firms intomatches There are papers that model thismore deeply some of which we discussbelow but starting with an exogenousmatching function allows us to be agnosticabout the actual mechanics of the process bywhich agents make contact An advantage isthat this is a flexible way to incorporate fea-tures that seem desirablemdasheg more searchby either side of the market yields morematchesmdashand one can regard the exactspecification as an empirical issue Thismight make matching a bit of a black boxbut it is a common and useful approach16

w e

m u vu

m u vv

= =( )and

( )

17 Nash actually showed that the unique outcome con-sistent with his axioms has = 12 Relaxing his symmetryaxiom (R-Nash) with any isin(01) satisfies the otheraxioms and this is what is called the generalized Nashsolution See eg Martin J Osborne and Rubinstein(1990)

42 Bargaining

Consider the situation of a worker and afirm who have met and have an opportunityto produce a flow of output y Suppose thatif the worker gets a wage w his expectedlifetime utility is W(w) while the firm earnsexpected discounted profit J(π) where π = y minus w Again W stands for the value ofworking and now J stands for the value tothe firm of a job that is filled If they fail toreach agreement the workerrsquos payoff falls toU and the firmrsquos to V Again U stands for thevalue of unemployment and now V standsfor the value to the firm of a vacancy We willsoon determine U and V endogenously butfor now take them as given We are of courseinterested in situations where W(w) gt U andJ(y minus w) gt V for some w so that there issomething to bargain over

A standard approach is to assume that w isdetermined by the generalized Nash bar-gaining solution with threat points U and V

(29)

timeswhere isin(01) is the workerrsquos bargainingpower The solution to the maximizationproblem satisfies

(30)

which can be solved for w Since it is animportant building block in this class ofmodels and since wage determination is oneof the main themes of this essay we want todiscuss Nash bargaining carefully

John Nash (1950) did not actually analyzethe bargaining process but took as givenfour simple axioms and showed that his solu-tion is the unique outcome satisfying theseaxioms17 The solution while elegant and

θ( )[ ( ) ] ( )W w U J y w = minus minus minus1

θ[ ) ] ( )J y w V W w( minus minus

J y w Vtimes minus minus minus[ ( ) ] θ1

w W w Uisin minusarg max [ ( ) ]θ



practical is again a black box However onecan provide a game-theoretic description ofthe bargaining process along the lines ofRubinstein (1982) that has a unique sub-game perfect equilibrium with the followingproperty as the time between counteroffersin the game becomes small the equilibriumoutcome converges to the prediction of theNash solution for particular choices of thethreat points and bargaining power thatdepend on details of the underlying gamesee eg Osborne and Rubinstein (1990)For instance suppose that each agent has agiven probability of proposing an offer (asopposed to responding) in each round ofbargaining everything else equal this gamegenerates the same outcome as the Nashsolution in which the bargaining powerequals that probability

Of course this only pushes θ back onelevelmdashwhere does that probability comefrom One position is to say that the natureof bargaining may well differ across indus-tries countries and so on and varying θ isone way to try and capture this Also at leastin simple models as we vary θ between 0and 1 we trace out the set of bilaterally effi-cient and incentive compatible employmentrelationships which would seem to cover thecases of interest Just as the matching func-tion is not the last word on how people meetNash bargaining is not the last word on wagedetermination but it is a useful approach

To proceed suppose as usual that workersand firms are risk neutral infinitely lived anddiscount future payoffs in continuous time atrate r and that matches end exogenously atrate λ Then we have

(31) rW(w) = w + λ[U minus W(w)]

(32) rJ(π) = π + λ[V minus J(π)]

This implies Insertingthese into (30) and rearranging gives

(33) W(w) = U

+ θ[J(y minus w) minus V minus W(w) minus U]

W w J r ( ) ( )= = +π 1λ

18 Notice that S is independent of the wage IntuitivelyS corresponds to the joint payoff available in the matchwhile w simply divides this among the agents

This says that in terms of total lifetimeexpected utility the worker receives histhreat point U plus a share of the surplusdenoted S and defined by

(34)

where the last equality is derived by using(31) and (32)18

From (31) and (32) we have W(w) minus U =w_

r__w

_R and J(π) minus V = 13_

r__13_R where wR and πR

are reservation wage and profit levels for theworker and firm Then (29) reduces to

(35)

which has solution

(36)

Hence in this model the Nash solution alsosplits the surplus in terms of the currentperiod utility Notice that w ge wR if and onlyif y ge yR = πR + wR Similarly π = y minus w ge πR ifand only if y ge yR Hence workers and firmsagree to consummate relationships if andonly if y ge yR

43 Equilibrium

We now combine matching and bargain-ing in a model where a firmrsquos decision to posta vacancy is endogenized using a free entrycondition There is a unit mass of homoge-neous workers and unmatched workerssearch costlessly while matched workerscannot search We focus here on steadystates and let u and v represent unemploy-ment and vacancies The steady-state unem-ployment rate is u = λ (λ + w) wherew = m(uv)u and m is the matching tech-nology As we discussed above assumingconstant returns once we know w we knowe since both are functions of uv

w w y wR R R= + minus minusθ π( )

w w w y wR Risin minus minus minus minusarg max [ ] [ ]θ θπ 1

y rU rVr

=minus minus

+ λ

S J y w V W w U= minus minus + minus( ) ( )



) ( )y y dF yRminus

19 Rather than having entry one can assume a fixed num-ber of firms and then equilibrium determines V endoge-nously Also although we focus on steady states dynamicshere are straightforward The key observation is that thefree entry condition pins down e and therefore w Hencegiven any initial unemployment rate vacancies adjust so thatuv jumps to the steady state level which implies all othervariables are constant along the path as u and v converge totheir steady state levels See Mortensen (1989 1999) egfor related models with more complicated dynamics

The value of posting a vacancy is

(37)

where k is a flow cost (eg recruiting costs)As free entry drives V to 0 we need not keeptrack of V and we can rewrite (37) as

(38)

The value of unemployment satisfies

(39)

while the equations for W and J areunchanged from (31) and (32) Formally anequilibrium includes the value functions(JWU) the wage w and the unemploymentand vacancy rates (uv) satisfying theBellman equations the bargaining solutionfree entry and the steady-state condition19

In terms of solving this model oneapproach would be to try to find the equilib-rium wage Start with some arbitrary wsolve (32) for J(π) and then use (38) to solvefor e and w This determines W and UThis w is an equilibrium if and only if theimplied values for J W and U are such thatthe bargaining condition holds While thisworks here we bypass w by working directlywith the surplus which from (34) is

(40)

Now (33) allows us to rewrite (39) asrU = b + wS and (40) gives

(41)

The next step generally in this method isto obtain expressions that characterize opti-mal choices for each of the decisions made

( )r S y bw+ + = minusλ θ

( )r S y rU+ = minusλ

rU b W w Uw= + minus [ ( ) ]

e J k( )π =

rV k J Ve= minus + minus [ ( ) ]π

outside of a match given S Here the onlysuch decision is whether to post a vacancyUsing (38) and the fact that bargainingimplies J(π) = (1 minus θ)S we have

(42)

Equilibrium is completely characterized by(41) and (42) Indeed we can combinethem as

(43)

Under standard regularity conditions aunique solution for w exists From this wecan recover the wage

(44)

Finally the steady state unemployment ratesatisfies an equation analogous to (25)accounting for the fact that all meetingsresult in matches

A number of results now follow easily Forexample an increase in b reduces the rate atwhich workers contact firms w raises therate at which firms contact workers ereduces S and raises w The conclusion thatunemployment duration and wages increasewith UI is similar to what we found earlierbut here the mechanism is different In thesingle-agent model an increase in b inducedthe worker to raise his reservation wage andto reduce search intensity if it is endoge-nous Now an increase in b raises the bar-gained wage which discourages jobcreation thereby increasing unemploymentduration

44 Match-Specific Productivity

In the above model it takes time for work-ers and firms to get together but every con-tact leads to a match and w is the same inevery match This seems quite special whencompared to what we did in sections 2ndash3 asit corresponds to workers sampling from a

uw

lowast =+λ

λ

w y r S= minus + minus( )( )λ θ1

r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ



degenerate distribution Moreover in appli-cations changes in the probability that acontact leads to a match may be importantHere we extend the model so that not everycontact results in a match and not everymatch has the same w The easiest way toproceed is to assume that when a worker andfirm meet they draw match-specific produc-tivity y from a distribution F where y isobserved by both and constant for the dura-tion of the match From section 42 workersand firms agree to match if and only if y ge yRwhere yR is characterized below

In equilibrium workers in a match withproductivity y earn wage w(y) satisfyingthe Nash bargaining solution Let Wy(w)be the value for an employed worker of amatch with productivity y earning w Jy(y minus w) the value for a firm with a filledjob at productivity y earning profits y minus wand Sy the surplus in a job with productivityy Generalizing (40) gives

(45)

Only the Bellman equations for an unem-ployed worker and the free entry conditionchange appreciably becoming

(46)

(47)

To solve this model combine (46) and (47)to get rU = b + αe

mdashαw(1

kmdashminus) and substitute into (45)

(48)

In particular yR satisfies SyR= 0 or

(49)

Since Sy is linear in y this implies Sy = (y minus yR) (r + λ) so (47) can be written

y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e

+ = minus minusminus

λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ

k J y w y dF ye y yR

= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)

rU b W w y U dF yw y yR

= + minusinfin

int [ ( )] ( )

( )r S y rUy+ = minusλ

20 This section mimics what we did in the single-agentproblem in section 3 Other extensions discussed there canalso be added including on-the-job search (Pissarides1984 1994) and learning (Michael J Pries 2004Giuseppe Moscarini 2005 and Pries and RichardRogerson 2005)

(50)

We can now solve for yR and w from (49)and (50) The first of these equationsdescribes an increasing relationship betweenw and yR (when it is easier for a worker tofind a job he is more willing to turn down apotential match with low productivity) Thesecond gives a decreasing relationshipbetween yR and w (when yR is highermatches are less profitable for firms so theypost fewer vacancies) There exists a uniqueequilibrium under standard conditions Onecan again recover the wage function sincew(yR) = yR and w(y) = υ if y gt yR we havew(y) = yR + θ(y minus yR)

Equilibrium also determines theobserved distribution of productivity acrossexisting relationships or equivalently givenw(y) the observed wage distribution G(w)Since the distribution of match productivityis F(y) truncated at yR the equilibrium wagedistribution G(w) is determined by yR F(y)and w(y)

It is easy to discuss turnover and wagesFor example an increase in b shifts (49) butnot (50) resulting in an increase in yR areduction in w and a reduction inH = w[1 minus F(yR)] From a workerrsquos per-spective this closely resembles the single-agent problem in the sense that he receivesoffers at rate w from a given distributionand needs to decide which to accept exceptnow the arrival rate and distribution areendogenous

45 Endogenous Separations

Mortensen and Pissarides (1994) endoge-nize the separation rate by incorporating on-the-job wage changes20 The resultingframework captures endogenously both the

( ) ( ( ) ( )r k y y dF ye y RR

+ = minus minusintλ θ 1 )



flows into and out of unemployment Giventhat these flows vary a lot across countriesand over time it allows one to begin thinkingformally about factors that may account forthese differences To proceed let y be cur-rent productivity in a match and assumethat at rate λ we get a new draw fromF(y|y) where F(y|y2) first order stochasti-cally dominates F(y|y1) whenever y2 gt y1 Itremains to specify the level of productivity innew matches One can assume they start at arandom y but we assume all new matchesbegin with the same y0

An equilibrium is defined as the naturalextension of the previous model and we canjump directly to the equation for the surplus

(51)

Since rU = b + wSy0 this can be rewritten

(52)

To close the model we again use free entry

(53)

Finding equilibrium amounts to solving (53)and (52) for yR and w

The solution is more complicated herebecause we are now looking for a fixed pointin a system of functional equationsmdash(52)defines both yR and Sy as functions of wNevertheless an increase in w reduces Sy

for all y and hence raises the reservationwage yR Thus (52) describes an increasingrelationship between w and yR At the sametime (53) indicates that when w is higherSy0

must be higher so from (52) yR must belower and this defines a decreasing rela-tionship between w and yR The intersec-tion of these curves gives steady-stateequilibrium which exists uniquely under

k Se y= minusβ θ ( )10

( )S dF y yy

y

yR

+ intλ

|

( )r S y b Sy w y+ = minus minusλ θ0

( )S dF y yy

y

yR

+ intλ

|


standard conditions See Mortensen andPissarides (1994 1999b) for details of theargument

46 Discussion

There are many applications and exten-sions of this framework David Andolfatto(1996) Monika Merz (1995 1999) HaroldL Cole and Rogerson (1999) Wouter J denHaan Gary Ramey and Joel Watson (2000)Costain and Michael Reiter (2003) Shimer(2005) and Robert E Hall (2005) all studyversions of the model quantitatively Thereis a literature that uses versions of themodel to study the behavior of worker andjob flows across countries as well as over thebusiness cycle including Stephen P Millardand Mortensen (1997) Alain Delacroix(2003) Blanchard and Pedro Portugal(2001) and Pries and Rogerson (2005)

There is also a literature that introducesheterogeneous workers and firms includingAcemoglu (1999 2001) James Albrecht andSusan Vroman (2002) Mortensen andPissarides (1999c) Shimer (1999) andShimer and Smith (2000) Ricardo JCaballero and Mohamad L Hammour(1994 1996) and Gadi Barlevy (2002) studywhether recessions are cleansing or sullyingin terms of the distribution of match quality(do they lead to more good jobs or badjobs) Moscarini (2001) also studies thenature of match quality over the businesscycle We do not have space to do justice toall the work but this illustrates that it is anactive and productive area

5 Directed Search and Posting

We now move to models where someagents can post wage offers and otheragents direct their search to the mostattractive alternatives Following Espen RMoen (1997) and Shimer (1996) the com-bination of posting and directed search isreferred to as competitive search Note thatit is the combination of these features thatis important in section 6 we consider wage



21 The idea here is that market makers compete toattract workers and firms to their submarkets since theycan charge them an entrance fee but in equilibrium thisfee is 0 due to free entry into market making

posting with random search which is quitedifferent

The literature has proposed several equiv-alent approaches One posits a group ofagents called market makers who set up sub-markets with the property that any matchconsummated in a submarket must be at theposted wage Within each submarket there isa constant returns matching function m(uv)so that the arrival rates w and e are deter-mined by q = uv which is called the queuelength (the inverse of market tightness)Each unemployed worker and each firmwith a vacancy take as given w and q in everysubmarket and go to the one offering thehighest expected utility In equilibrium q ineach submarket is consistent with agentsrsquoexpectations and no market maker can post adifferent wage and attract both workers andemployers21

Another approach supposes that employ-ers themselves post wages and unemployedworkers direct their search to the mostattractive firms A high posted w attractsmore applicants which reduces workersrsquocontact rate w and raises the employerrsquoscontact rate e In equilibrium workers areindifferent about where to apply at leastamong posted wages that attract some work-ers Firms choose wages to maximize expect-ed profit Still another approach assumesthat workers post wages and firms directtheir search to them As we said theseapproaches are equivalent in the sense thatthey give rise to identical equilibrium condi-tions For brevity we consider only the casewhere firms post wages

51 A One-Shot Model

We first discuss the basic mechanism in astatic setting At the beginning of the periodthere are large numbers u and v of unem-ployed workers and vacancies and qlowast = uv isthe queue length Each firm with a vacancy

must pay cost k and we can either assumefree entry (making v endogenous) or fix thenumber of vacancies Any match within theperiod produces output y which is dividedbetween the worker and firm according tothe posted wage At the end of the periodunmatched workers get b while unmatchedvacancies get 0 Then the model ends

Consider a worker facing a menu of dif-ferent wages Let U denote the highest valuethat he can get by applying for a job at somefirm Then a worker is willing to apply to aparticular job offering a wage w ge b only ifhe believes the queue length q at that jobmdashie the number of workers who applymdashissufficiently small In fact he is willing toapply only if w(q) is sufficiently large in thesense that

(54)

If this inequality is strict all workers wouldwant to apply to this firm reducing the righthand side Therefore in equilibrium if anyworkers apply to a particular job q adjusts tosatisfy (54) with equality

To an employer (54) describes how achange in his wage w affects his queue lengthq Therefore he chooses w to maximize

(55)

taking (54) as a constraint Note that eachemployer assumes he cannot affect Ualthough this is an endogenous variable to bedetermined in equilibrium Eliminating wusing (54) at equality and also using e(q) = qw(q) we get

(56)

The necessary and sufficient first ordercondition is

(57)

In particular (57) implies all employerschoose the same q which in equilibriummust equal the economywide qlowast Hence (57)

e q y b U b( )( )minus = minus

q y b q U be+ minus minus minus( )( ) ( )

V kq

= minusmax

V k q y ww q e= minus + minusmax

( )( )

U q w q bw wle ( ) [ ( )]+ minus1



22 By assumption here firms post wages rather thanmore general mechanisms Coles and Jan Eeckhout (2003)relax this (eg they allow w to be contingent on the num-ber of applicants who turn up) and show it does not affectthe main conclusions

pins down the equilibrium value of U Then(54) at equality determines the market wage

(58)

where ε(qlowast) equiv _qlowast_ee_

(_(_qlowastqlowast

)_) is the elasticity of e(qlowast)

which is in (01) by our assumptions on thematching function m Comparing (58) withthe results in section 42 notice that thiswage rule operates as if the worker and firmbargained over the gains from trade withthe workersrsquo share θ given by the elasticity(qlowast) this has important implications for theefficiency of competitive search as we dis-cuss below

Substituting (58) into (55) pins down

(59)

+ [e(qlowast) minus qlowaste(qlowast)](y minus b)

If the number of vacancies v is fixed thisgives profit or we can use free entry V = 0 toendogenize v and hence qlowast In either casethe model is simple to use Indeed thismodel looks a lot like a one-shot version ofthe random search and bargaining modelThere is a key difference however here thesurplus share is endogenously determined

52 Directed Matching

Before generalizing to a dynamic settingwe digress to discuss some related literatureJames D Montgomery (1991) studies a nas-cent version of the above model (see alsoMichael Peters 1984 1991) He starts withtwo unemployed workers and two firmsFirst firms post wages then each workerapplies to one of them (possibly randomly)A firm receiving at least one applicationhires at the posted w if more than oneapplies the firm selects at random22

Suppose both firms offer the same w gt 0Then there are three equilibria in the appli-cation subgame worker 1 applies to firm 1

V k= minus

w b q y blowast lowast= + minusε( )( )

23 Therefore an increase in the number of undesirabletype 2 workers does not affect the matching rate for type 1workers but an increase in the number of desirable work-ers adversely affects undesirable workers

and worker 2 to firm 2 worker 1 applies tofirm 2 and worker 2 to firm 1 and bothworkers use identical mixed strategiesapplying to each firm with probability 12One can argue that the coordination impliedby the first two equilibria is implausible atleast in large labor markets and so themixed-strategy equilibrium is the naturaloutcome This introduces a coordinationfriction as more than one worker may applyfor the same job

Generalizing this reasoning supposethere are u unemployed workers and vvacancies for any u and v If each workerapplies to each firm with equal probabilityany firm gets a worker with probability 1 minus (1 minus 1_

v)u Taking the limit of this expres-sion as u and v go to infinity with q = uvfixed in a large market a fraction e(q)= 1 minus eq of firms get a worker This is a stan-dard result in statistics Suppose there are uballs independently placed with equal prob-ability into each of v urns Then for large uand v the number of balls per urn is aPoisson random variable with mean uv so afraction euv of the urns do not get any ballsFor this reason this process is often calledan urnndashball matching function

Because the urnndashball matching processprovides an explicit microeconomic story ofboth meetings (a ball is put in a particularurn) and matches (a ball is chosen from thaturn) it is suitable for environments with het-erogeneous workers (not all balls are thesame) For instance suppose there are u1

type 1 workers and u2 inferior type 2 work-ers with u = u1+ u2 Firms hire type 1 work-ers over type 2 workers when both applyhence they hire a type 1 worker with proba-bility 1 minus eu1v and a type 2 worker with prob-ability eu1v(1 minus eu2v) They hire someworker with probability 1 minus euv23

There are several generalizations of thismatching process Burdett Shi and Wright



24 In these models generally one has to be somewhatcareful about strategic interaction between firms In theprevious section in maximizing (55) subject to (54) thefirm takes as given that a workerrsquos market payoff is U Butin general if a firm changes its wage then U will changeBurdett Shi and Wright (2001) solve the model with finitenumbers of agents where each firm must compute theeffect of a change in its w on workersrsquo strategies and on theimplied U In the limit as u and v grow they show that theproblem is equivalent to one where each firm treats Uparametrically as we assumed above

(2001) let some vacancies hire multipleworkers Albrecht Pieter Gautier andVroman (2003) and Albrecht GautierSerene Tan and Vroman (2004) let workersmake multiple applications If workers cansimultaneously apply for every job thiseffectively flips the urnndashball problemaround so a worker is employed if at leastone firm offers him a job see Benoit JulienJohn Kennes and Ian King (2000) Theintermediate case in which workers canapply for a subset of jobs delivers additionalpossibilities In general these directedsearch models provide a way to get insidethe black box of the matching process24

53 A Dynamic Model

To get something like the basic Pissaridesmodel with directed search start with anunemployed worker Suppose he anticipatesan unemploymentndashvacancy ratio q and awage w Then

(60)

(61)It is convenient to combine these into

(62)

Similarly for firms

(63) rV = minus k + e(q)[J(y minus w) minus V]

(64) rJ(y minus w) = y minus w + λ[V minus J(y minus w)]

Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]

Now suppose firms choose w and q to max-imize rV or equivalently by free entry to max-imize e(q)(y minus w) They take (62) as givenEliminating w using this constraint and againusing e(q) = qw(q) this reduces to

(66)

The necessary and sufficient first ordercondition

(67)

has a unique solution so all firms choose thesame q Eliminating U and w from (62) (65)and (67) we get an implicit expression for q

(68)

This pins down the equilibrium q orequivalently the arrival rates w and eUnder standard conditions the solution isunique We can again perform standardexercises such as changing b with similarresults here this raises the uv ratio whichreduces w raises e and increases w Butalthough the conclusions are similar to thosereached in the previous section the mecha-nism is quite different An increase in b inthis model makes workers more willing toaccept an increase in the risk of unemploy-ment in return for an increase in w Firmsrespond by offering workers what theywantmdashfewer jobs at higher wages

54 Discussion

Competitive search equilibrium theoryprovides arguably a more explicit explana-tion of the matching process and of wagedetermination than the bargaining models insection 4 Nash bargaining says w divides thesurplus into exogenous shares here workersface a trade-off between a higher wage and alower probability of getting a job while firmsface a trade-off between profit and the prob-ability of hiring Competition among wagesetters whether these be firms workers or

r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ



25 In section 7 we show that competitive search equi-librium achieves the optimal trade-off between thesefactors

26 The remaining combinationmdashdirected search andbargainingmdashis not so interesting at least if firms are homo-geneous since there is nothing for workers to direct theirsearch toward Lawrence Uren (2004) studies directedsearch and bargaining with heterogeneous firms

market makers leads to a point along theindifference curves of agents trading offwages and arrival rates25

However a potential disadvantage ofthese modelsmdashindeed of any model thatassumes postingmdashis that it is a strongassumption to say that agents commit to theposted terms of trade If the markets is real-ly decentralized what prevents them fromtrying to bargain for a different w after theymeet Again this comes up in any postingmodel In the context of the wage postingmodel in the next section Coles (2001)shows explicitly how to prevent firms fromreneging on their posted wage using reputa-tional considerations but there is more workto be done on the issue

In terms of other extensions Coles andEeckhout (2000) Shi (2001 2002) andShimer (forthcoming) add heterogeneitywhich introduces wage dispersion amongheterogeneous workers and among identicalworkers at different firms Acemoglu andShimer (1999b) allow workers to be risk-averse which means it is no longer possibleto solve the model explicitly They show thatan increase in risk aversion reduces wageswhile an increase in b raises it Building onthis Acemoglu and Shimer (2000) show thatUI can enhance productivity Much moreresearch is currently being done on directedsearch models Again while we cannot dojustice to all of the work in the area we wantto say that it appears to have great potential

6 Random Matching and Posting

We now combine random matching as insection 4 with posting as in section 526

This class of models has been used exten-sively in the literature on the pure theory of

27 As they report van den Berg and Geert Ridder(1998) estimate that up to 25 percent of wage variability isattributable to frictions in the sense that this is what wouldemerge from a posting model that ignored heterogeneitywhile Fabien Postel-Vinay and Jean-Marc Robin (2002)estimate up to 50 percent

wage dispersion which tries to understandhow workers with identical productivity canbe paid different wages Note that the mod-els in the previous two sections generatewage dispersion only if workers are either exante or ex post heterogeneous A pure theo-ry of wage dispersion is of interest for a cou-ple of reasons First the early literaturesuggested that search is relevant only if thedistribution from which you are sampling isnondegenerate so theorists were naturallyled to study models of endogenous disper-sion Second many people see dispersion asa fact of life and for them the issue isempirical rather than theoretical

As Mortensen (2003 p 1) reportsldquoAlthough hundreds if not thousands ofempirical studies that estimate so-calledhuman capital wage equations verify thatworker characteristics that one could view asindicators of labor productivity are positivelyrelated to wages earned the theory is woe-fully incomplete in its explanatory powerObservable worker characteristics that aresupposed to account for productivity differ-ences typically explain no more that 30 per-cent of the variation in compensationrdquoEckstein and van den Berg (forthcoming)argue that ldquoequilibrium search models pro-vide a framework to empirically analyze thesources of wage dispersion (a) workers het-erogeneity (observed and unobserved) (b)firm productivity heterogeneity (observedand unobserved) (c) market frictions Theequilibrium framework can empiricallymeasure the quantitative importance of eachsourcerdquo27

Diamond (1971) is an early attempt toconstruct a model of dispersion andalthough it did not work it is useful tounderstand why Consider an economywhere homogeneous workers each face a



standard search problem We do not give allthe particulars since the model is a specialcase of what is described in detail below butthe key is that the offer distribution F is gen-erated by wage-posting firms each of whichhas a constant returns technology with laboras the only input with productivity y A firmhires any worker it contacts who is willing toaccept its posted w Consider an individualfirm Letting F be the distribution of wagesposted by other firms he wants to maximizeexpected profit taking F as given An equi-librium is defined as a distribution such thatevery wage posted with positive probabilityearns the same profit and no other wageearns greater profit

Diamond provides a rather striking resultthere is a unique equilibrium and in it allemployers set the same wage w = b Theproof is simple Given any F since workersare homogeneous they all choose the samereservation wage wR Clearly no firm postsw lt wR as this would mean it cannot hireand no firm posts w gt wR as it can hire everyworker it contacts at w = wR To see why itturns out that w = b consider an individualfirm when all firms are posting w gt b If itdeviates and offers a wage slightly less thanw it still hires every worker it meets Sincethis is true for any w gt b we must have w = bin equilibrium The model not only fails torationalize wage dispersion it fails to explainwhy workers are searching in the first place

61 Heterogeneous Leisure

Why might one expect to find pure wagedispersion One answer is that search fric-tions produce a natural trade-off for a firmwhile posting higher wages lowers your prof-it per worker it could allow you to hire work-ers faster and so in the long run you getmore of them In Diamondrsquos model thistrade-off is nonexistent since when youincrease your wage above wR there is noincrease in your hiring rate Albrecht and BoAxell (1984) allow for heterogeneity in work-ersmdashnot in productivity but in their value of

28 Albrect and Axell do not actually have firms earningequal profit but allow y to vary across firms and look for acutoff ylowast such that firms with y gt ylowast pay w = w1 and thosewith y gt ylowast pay w = w2 the economic implications are basi-cally the same

leisuremdashwhich leads to heterogeneity inreservation wages and this makes the abovetrade-off operational

Consider two types of workers some withb = b1 and others with b = b2 gt b1 For anywage distribution F there are two reserva-tion wages w1 and w2 gt w1 If Wi(w) is thevalue of a type i worker who is employed atwage w and Ui is the value of an unemploy-ed type i worker these wages satisfyW1(w1) = U1 and W2(w2) = U2 Generalizingour logic from the Diamond model no firmposts a wage other than w1 or w2 It is possi-ble that these two wages could yield equalprofit however since low-wage firms canhire only workers with b = b1 while high-wage firms can hire everyone they contactIn the AlbrechtndashAxell model equal profits attwo different wages can occur in equilibriumfor a large set of parameters28

To see how this works normalize themeasure of firms to 1 and let the measure ofworkers be L = L1+ L2 where Lj is the meas-ure with bj Let be the endogenous frac-tion of firms posting w2 Any candidateequilibrium wage distribution is completelysummarized by w1 w2 and Observe firstthat the reservation wage of type 2 workers isw2 = b2 It is clear that their reservation wagecannot be any lower since otherwise theywould prefer to remain unemployed On theother hand if their reservation wage exceedsb2 an argument like the one we used for theDiamond model ensures that a firm couldreduce w2 and still attract these workers

To determine w1 note that type 1 workersaccept both w = w1 and w = w2 and so giventhe arrival rate w their value functions satisfy

(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]



29 An alternative to having firms maximize expecteddiscounted profit is to maximize the value of posting avacancy which is more in line with sections 4 and 5 seeMortensen (2000) We adopt the criterion in the textbecause it facilitates comparison with the model in thenext subsection

(71) rW1(w2)w2λ[U1W1(w2)]

Note that although type 1 workers acceptw1 they get no capital gain from doing soand suffer no capital loss when laid offsince W1(w1) = U1 Then using w2 = b2 wecan simplify and solve for w1 in terms of

(72)

Unemployment rates for the two types areu1 = αw

mdashλmdash+λ and u2 = αwmdashλmdash+λFor firms the expected value of contact-

ing a worker is the probability he acceptstimes the profit conditional on acceptanceThe acceptance probability is _

L1

_u1

_L1_u1

L2

_u2

__ forfirms paying w1 and 1 for firms paying w2while discounted profit is _y_

r__

_w__i_ So expected

discounted profits from the two wages are

(73)

(74)

where for now we are taking e as given29

Inserting u1 u2 and w1 after some algebrawe see that 2 minus 1 is proportional to

(75)

times

minus

For an equilibrium we need

(76) = 0 and T(0) lt 0 = 1

and T(1) gt 0 or isin(01) and T() = 0

It is easy to show there exists a unique solu-tion to (76) and 0 lt lt 1 if and only ify_ lt y lt

_y where

r L b bw 1 2 1minus ( )

L L y bw( ) ( )+ +[ ] minus minusλ λ λ1 2 1 LL1

T r y bw( ) ( ) ( ) = + + minus timesλ 2

22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ

30 Clearly we get at most two wages here but the argu-ment can be generalized to many types of workers(Eckstein and Wolpin 1990)

31 See Damien Gaumont Martin Schindler and Wright(forthcoming) for details and for several other variationson the general theme of AlbrechtndashAxell models They alsodiscuss a problem with models based on ex ante hetero-geneity Given type 2 workers get no surplus if there is anysearch cost ε gt 0 they drop out of the market leaving onlytype 1 workers Then we are back to Diamond and the dis-tribution collapses (and of course the problem applies withany number of types) Gaumont Schindler and Wrightalso discuss models that avoid this problem

(77)

_y

When productivity is low all firms payw1 = b1 when it is high all firms payw2 b2 and when it is intermediate thereis wage dispersion When isin(01) we cansolve T() = 0 for and use (72) to solvefor w1 and the distribution of paid wagesexplicitly30

Of course all of this is for given arrivalrates and the value of that solves (76)depends on w Using the matching functionwe can endogenize

(78)

where L1u1 + L2u2 is the number of unem-ployed workers and we assume that all firmsare always freely posting a vacancy so thatv = 1 with the idea being that each one willhire as many workers as it can get Since u2

depends on so does w An equilibrium isa pair (w) satisfying (76) and (78)31

62 On-the-Job Search

In the previous section firms may pay higherwages to increase the inflow of workers Therealso exist models where firms may pay higherwages to reduce the outflow of workers (egBurdett Lagos and Wright 2003) In Burdettand Mortensen (1998) both margins are atwork and firms that pay higher wages bothincrease the inflow and reduce the outflow of

w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and



32 Suppose there were a mass point at some cw Then afirm posting cw +ε would be able to hire away any worker itcontacts working from a cw firm increasing its revenue dis-cretely with only an ε increase in cost which means cw doesnot maximize profit Suppose there were a gap in F saybetween cw and tw Then a firm posting tw could lower its wageand reduce cost without reducing its inflow or increasing itsoutflow of workers and again tw could not maximize profit

workers We now present this model in detailwhich is relatively easy here because it isbased on on-the-job search and we can makesubstantial use of results derived in section 3

The arrival rates are 0 and 1 whileunemployed and employed and every offeris a random draw from F(w) For ease ofpresentation we begin with the case0 = 1 = which implies wR = b by (24) andreturn to the general case later Since allunemployed workers use a common reserva-tion wage and clearly no firm posts w lt wRthe unemployed accept all offers and wehave u = λ (λ + ) as a special case of (25)Also the distribution of paid wages is

(79)

a special case of (26)If a firm posts w ge wR a worker he con-

tacts accepts if he is currently unemployedor currently employed but at a lower wagewhich occurs with probability u +(1 minus u)G(w) The employment relationshipthen yields flow profit y minus w until the work-er leaves either due to an exogenous separa-tion or a better offer which occurs at rateλ + [1 minus F(w)] Therefore after simplifica-tion the expected profit from w is

(80)

Again equilibrium requires that any postedwage yields the same profit which is at leastas large as profit from any other wageClearly no firm posts w lt wR = b or w gt y Infact one can show that the support of F is[bw

_] for some w

_ y and there are neither

gaps nor mass points on the support32

( )( )

y wF w r F w

minus+ minus ( )[ ] + + minus[ ]

λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1

We now construct F explicitly The keyobservation is that firms earn equal profitsfrom all posted wages including the lowestw = b (w) = (b) for all w isin [b⎯w] SinceF(b) = 0 (b) = λ(yb) ( + λ)(r + + λ)Combining this and (80) gives an equationthat can easily be solved for F(w) In thesimplest case where r asymp 0 the result is

(81)

We know the lower bound is b and theupper bound⎯w can easily be found by solv-ing F(⎯w) = 1 This yields the unique distri-bution consistent with equal profit for allwages posted

In words the outcome is as follows Allunemployed workers accept the first offerthey receive and move up the wage laddereach time a better offer comes along butalso return to unemployment periodicallydue to exogenous layoffs There is a nonde-generate distribution of wages posted byfirms F given by (81) and of wages earnedby workers G given by inserting F into (79)The model is consistent with many observa-tions concerning worker turnover and alsoconcerning firms including eg the factthat high wage firms are bigger SeeMortensen (2003) for details

There are many interesting extensionsFirst with 0 ne 1 the same methods lead to

(82)

where now wR is endogenous (with 0 = 1

we knew wR = b) To determine wR integrate(24) to get

(83)

One can check that in the limit as 1rarr 0w_

= wR which means there is a single wagew = wR = b This is the Diamond result as aspecial case when there is no on-the-jobsearch Also in the limit as 1 rarr w

_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b

( ) = + minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1



G(w) = 0 for all w lt y hence all workers earnw = y Moreover as 1 rarr clearly u rarr 0Hence the competitive solution also emergesas a special case when 0 and 1 get large

One can let firms be heterogeneous withrespect to y In equilibrium there is a distri-bution of wages paid by each type of firmand all firms with productivity y2 pay morethan all firms with y1 lt y2 Thus higher pro-ductivity firms are more likely to hire andless likely to lose any worker With heteroge-neous firms van den Berg (2003) showsthere may be multiple equilibria Perhapsmore significantly firm heterogeneity isimportant empirically because with homo-geneous firms the equilibrium wage distri-bution (81) has an increasing density whichis not in the data With heterogeneity F canhave a decreasing density Another veryimportant extension is to allow firms whoseworkers contact rivals to make counteroffersas in Postel-Vinay and Robin (2002)

Margaret Stevens (2004) allows firms topost wagendashtenure contracts rather than sim-ply a constant wage She shows firms have anincentive to back load wages to reduceturnover If workers can make an up-frontpayment for a job an optimal contractextracts an initial fee and then pays w = y Iffirms are homogeneous this contract elimi-nates all voluntary quits In equilibrium allfirms demand the same initial fee and thisleaves unemployed workers indifferentabout accepting the position Stevens alsoshows that if initial payments are impossiblesay because of liquidity constraints there isan equilibrium where all firms offer a con-tract that pays the worker 0 for a fixed peri-od and then pays w = y If firms arehomogeneous they extract all of the surplusthere are no job-to-job transitions and allcontracts are identical

However Burdett and Coles (2003) showthat Stevensrsquos results can be overturned ifworkers desire smooth consumption Theyallow firms to commit to wagendashtenure con-tracts but assume workers are risk-averse anddo not have access to financial markets (they

33 We mention some related work Masters (1999) stud-ies a wage-posting model and uses it to analyze changes inthe minimum wage Delacroix and Shi (forthcoming) con-sider directed search in an environment similar toBurdettndashMortensen and show it also gives rise to wage dis-persion although for different reasons Finally some peo-ple consider as an alternative to random search modelswhere workers are more likely to meet large firms as inBurdett and Vishnawath (1988b)

must consume w each period) They provethat all equilibrium wagendashtenure contracts aredescribed by a common baseline salary scalewhich is an increasing continuous relationshipbetween the wage and tenure Firms offer dif-ferent contracts in the sense that they startworkers at different point on the scale Thusconsumption smoothing reintroduces wagedispersion both in the sense that workers getdifferent wages depending on their tenureand in the more fundamental sense that firmsoffer different contracts

63 Discussion

The two models of wage dispersion wehave presented are based on worker hetero-geneity and on-the-job-search respectivelyOf course one can integrate them This isimportant empirically because although on-the-job-search models (with heterogeneousfirms) do a good job accounting for wagesthey do less well accounting for individualemployment histories especially the factthat hazard rates tend to decrease with thelength of unemployment spells Models withworker heterogeneity do better at account-ing for this but less well for wages An inte-grated model can potentially account forboth see Christian Bontemps Robin andvan den Berg (1999) Many other extensionsand applications are possible and this is aproductive area for both theoretical andempirical research33

7 Efficiency

We now move on to efficiency Of coursein an economy with many agents there aremany Pareto optimal allocations Here wefocus on those that maximize the sum ofagentsrsquo utility or equivalently that maximize



34 Because e(q) = qw(q) we have hence the condition can equivalently be stated as firmsrsquobargaining power 1 minus must equal the elasticity of w(q)

q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1

the present discounted value of output netof the disutility of work and search costs

71 A One-Shot Model

Initially u workers are unmatched Firmsdecide how many vacancies v to post each atcost k and then m(uv) matches form Letq = uv and assume constant returns som(uv) = vm(q1) = ve(q) Each match pro-duces y at an opportunity cost b to each work-er Assume provisionally that wages aredetermined by bargaining w = θy + (1 minus θ)bThen the economy ends Firms post vacanciesuntil the free entry condition

(84)

holds This is a static version of the basicPissarides model

Now consider a planner who posts vacan-cies to maximize output net of workersrsquoopportunity cost and the cost of postingvacancies Equivalently he chooses q = uvknowing that each worker will contact avacancy with probability w(q) to maximize

(85)

The necessary and sufficient condition for asolution is(86) [e(q) minus qe(q)](y minus b) = k

Denote the plannerrsquos solution by qlowastThe following is immediate from (84)

and (86) the plannerrsquos solution and thedecentralized solution coincide if and only if

(87)

where ε(qlowast) = qlowastmdashαeα e

mdash(qlowast)mdash(qlowast) was defined in section 5

as the elasticity of e(q) This is the Hosioscondition (Arthur J Hosios 1990) determin-ing the share of the surplus that must go toworkers for bargaining to be efficient34

Alternatively in terms of wages equilibrium is efficient if w = wlowast = θlowasty

θ εlowast lowast= ( )q

uq

q y b ke ( )( )= minus minus[ ]

u q y b vkw ( )( )minus minus

e q y b k( )( )( )1 minus minus =θ

+ (1 minus θlowast)b If is too high for examplethen w gt wlowast and v is too low

Now consider the competitive searchmodel from section 5 From (58) in com-petitive search equilibrium w = wlowast andequilibrium is necessarily efficient That isthe Hosios condition holds endogenouslymdashalthough agents do not bargain the surplusis still being split and the equilibrium splitimplies efficiency To understand why it isuseful to think in terms of competitionbetween market makers as discussed aboveWith free entry market makers effectivelymaximize workersrsquo expected utilityw(q)(w minus b) recognizing that firms mustbreak even to participate e(q)(y minus w) = kUsing the constraint to eliminate w this isidentical to the plannerrsquos problem

Now consider extending the model toendogenize workersrsquo search intensity Thetotal number of matches is m(s

_uv) where s

_

is average intensity Assuming constantreturns a firm hires with probabilitye(s

_q) = m(s

_q1) while a worker with search

intensity s gets hired with probabilitysw(s

_q) = s e(s

_q) s

_q An unemployed work-

er chooses s to maximize

(88)

In equilibrium all workers choose the sames = s

_ where

(89) g

and free entry by firms implies

(90)

The planner solves

(91)

which has necessary and sufficient conditions

(92)

(93) k[e(sndashq)sndashqe(sndashq)](yb)

g s sq y be ( ) ( )( )= minus

max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus

k sq y be= minus minus ( )( )( )1 θ

ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ



Notice something interesting (89) and (92)coincide if the Hosios condition θ = ε (s

_q)

holds while (90) and (93) coincide under thesame condition That is bargaining equilibri-um achieves efficient search intensity andentry under the Hosios condition Thisseems genuinely surprising as there are twovariables to be determined s

_and v and only

one parameter υSince we have already shown that in com-

petitive search equilibrium we get theHosios condition endogenously competitivesearch equilibrium achieves efficient searchintensity and entry As suggested abovemarket makers effectively choose the termsof trade to ensure that both workers andfirms behave optimally There is nothingspecial about endogenous entry decisions orsearch intensity We could consider variousother extensions (eg match-specific pro-ductivity with the reservation match value yR

determined endogenously) and we wouldfind the same result bargaining equilibriumis efficient if and only if the Hosios conditionholds and competitive search equilibriumgenerates this condition endogenouslyRather than go through these exercises wenow move to dynamics

72 A Dynamic Model

Here we formulate the plannerrsquos problemfor the benchmark Pissarides model recur-sively The state variable is the measure ofmatched workers or the employment rate eThe current payoff is y for each of the eemployed workers b for each of the 1 minus eunemployed workers and minus k for each of thev = (1 = e)q vacancies The employment ratefollows a law of motion e= α_e_

q_(q_) (1 minus e) minus λe

Putting this together the plannerrsquos problem is

(94)

One can show that Y(e) is affine ieY(e) = a0 + a1e for some constants a0 and a1

k eq

Y eq ee( )

( )( )(

minus minus +minus1 1

))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1

35 The mapping defined by (94) is a contraction andtakes affine functions into affine functions Since the set ofsuch functions is closed the result follows immediatelyThis method also works and is especially useful when thereis a distribution of productivity across matches

where a0 can be interpreted as the value ofan unemployed worker and a1 the surplusfrom a match35

The first order condition from (94) simpli-fies to

(95)

Using the fact that Y(e) = a0 + a1e and theenvelope theorem we can differentiate bothsides of (94) to get

(96)

Combining (95) and (96) to eliminate a1 wearrive at

(97)

This completely characterizes the optimalpolicy q = q(e) notice that in fact q does notdepend on the state e only on exogenousparameters

How does this compare with equilibriumRecall that equilibrium in section 43 satis-fies (43) It is easy to see that the solutionsare the same and hence bargaining equilib-rium coincides with the plannerrsquos solution ifand only if = ε(q) Now looking at (68)from the competitive search version of themodel we see that competitive search equi-librium is necessarily efficient Hence theresults from the static model generalizedirectly and again carry over to more gen-eral models with endogenous search intensitymatch-specific productivity and so on

73 Discussion

We have demonstrated in several contextsthat competitive search is efficient whilebargaining is efficient if and only if theHosios condition holds Although theseresults are in some sense general it would be

r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe

1 1= minus + minus +⎡⎣⎢

⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )



misleading to suggest that we can get effi-cient equilibria in all search models A gen-eral understanding of the nature ofefficiency in these models is still being devel-oped Mortensen and Wright (2002) providesome results but only for constant returnsmatching functions without this equilibriaare unlikely to be efficient Shimer andSmith (2001) show that in some models withheterogeneity even with constant returnsthe efficient outcome may not even be com-patible with a steady state but exhibitscycles

The efficiency of search equilibria with exante investments is an interesting topic Forexample in Acemoglu (1996) and Masters(1998) agents decide on how much capitalto acquire prior to searching Genericallythere is no θ that makes the equilibrium out-come under bargaining efficientmdashthe valueof θ that provides the right incentive forinvestment in human capital by workers isdifferent from the value of that provides theright incentive for firms (whether firmschoose the number of vacancies the types ofjobs to create or physical capital) HoweverAcemoglu and Shimer (1999a) show com-petitive search equilibrium with ex anteinvestments is efficient Another case wherethere is no θ such that bargaining yields theefficient outcome is discussed by Smith(1999) who assumes firms have concaveproduction functions and hire a large num-ber of workers but bargain with each oneindividually

One can also ask about the efficiency of thewage-posting models in section 6 In generalif firms commit to pay the same w no matterthe circumstances it is unlikely to yield effi-cient outcomes In Albrecht and Axell (1984)eg a planner would want all meetings toresult in matches which does not happen Inthe simplest Burdett and Mortensen (1998)model with 0 = 1 efficiency does resultbecause all meetings involving unemployedworkers result in matches and other meet-ings are irrelevant from the plannerrsquos per-spective In extended versions however

there is no reason to necessarily expect effi-ciency (Mortensen 2000) In general there ismuch more work to be done on this topic

8 Conclusion

In contrast to standard supply-and-demand models search theory emphasizesfrictions inherent in the exchange processAlthough there is no one canonical searchmodel and versions differ in terms of wagedetermination the matching process andother assumptions we have tried to showthat there is a common framework underly-ing all of the specifications We have usedthe different models to discuss severalissues mainly related to worker turnoverand wages We have also used them to dis-cuss some simple policy experiments suchan increase in UI Different models empha-size different margins along which such poli-cies work including the choice ofreservation wage search intensity entry andso on

The approach as we have seen is consis-tent with many interesting observations Fora start it predicts the unemployment rate isnot zero which is perhaps a low hurdle butnot one that can be met by many alternativetheories The reason the model predicts thisis obvious but also obviously correctmdashittakes time for workers to find jobs It alsotakes time for firms to fill vacancies and anice property of these models is that therecan be coexistence of unemployed workersand unfilled vacancies The framework canbe used to organize thinking about individualtransitions between unemployment andemployment as well as job-to-job transitions

Different search-based models can beused to help explain the relationshipsbetween tenure wages and turnoverincluding versions that incorporate on-the-job search learning or human capitalSeveral models can be used to discuss thedistribution of wages and some of the mod-els are consistent with observations such asthe fact that high wage firms tend to have



more workers or high productivity firmstend to pay more We also discussed the wel-fare properties of the models Bargainingmodels achieve efficiency if and only if thebargaining weights satisfy a condition relatedto the matching technology while in com-petitive search models this conditionemerges endogenously

We have only scratched the surface buthopefully we have conveyed the main ideasThe goal was never to claim that search the-orists fully understand all of the issues andindeed there are many interesting directionsfor future research One involves furtherquantifying the models using both microand aggregate data On a more conceptualfront the models we have presented do notoffer a cogent theory of the firm and do notdistinguish between worker and job flowsThe welfare results are incomplete Finallywe have emphasized the importance of bothwage determination and the matching func-tion What is the right model of wages Whatare the mechanics that determine matchingPerhaps this survey will stimulate additionalresearch on these important questions

REFERENCES

Acemoglu Daron 1996 ldquoA Microfoundation for SocialIncreasing Returns in Human Capital AccumulationrdquoQuarterly Journal of Economics 111(3) 779ndash804

Acemoglu Daron 1999 ldquoChanges in Unemploymentand Wage Inequality An Alternative Theory andSome Evidencerdquo American Economic Review 89(5)1259ndash78

Acemoglu Daron 2001 ldquoGood Jobs Versus Bad JobsrdquoJournal of Labor Economics 19(1) 1ndash21

Acemoglu Daron and Robert Shimer 1999aldquoHoldups and Efficiency with Search FrictionsrdquoInternational Economic Review 40(4) 827ndash49

Acemoglu Daron and Robert Shimer 1999bldquoEfficient Unemployment Insurancerdquo Journal ofPolitical Economy 107(5) 893ndash928

Acemoglu Daron and Robert Shimer 2000ldquoProductivity Gains from Unemployment InsurancerdquoEuropean Economic Review 44(7) 1195ndash1224

Albrecht James W and Bo Axell 1984 ldquoAnEquilibrium Model of Search UnemploymentrdquoJournal of Political Economy 92(5) 824ndash40

Albrecht James W and Susan B Vroman 2002ldquoA Matching Model with Endogenous SkillRequirementsrdquo International Economic Review43(1) 283ndash305

Albrecht James W Pieter A Gautier and Susan B

Vroman 2003 ldquoMatching with MultipleApplicationsrdquo Economics Letters 78(1) 67ndash70

Albrecht James W Pieter A Gautier Serene Tan andSusan B Vroman 2004 ldquoMatching with MultipleApplications Revisitedrdquo Economics Letters 84(3)311ndash14

Alvarez Fernando and Marcelo Veracierto 2000ldquoLabor-Market Policies in an Equilibrium SearchModelrdquo in NBER Macroeconomics Annual 1999Volume 14 Ben S Bernanke and Julio J Rotembergeds Cambridge and London MIT Press 265ndash304

Andolfatto David 1996 ldquoBusiness Cycles and Labor-Market Searchrdquo American Economic Review 86(1)112ndash32

Barlevy Gadi 2002 ldquoThe Sullying Effect ofRecessionsrdquo Review of Economic Studies 69(1)65ndash96

van den Berg Gerard J 1994 ldquoThe Effects of Changesof the Job Offer Arrival Rate on the Duration ofUnemploymentrdquo Journal of Labor Economics 12(3)478ndash98

van den Berg Gerard J 2003 ldquoMultiple Equilibria andMinimum Wages in Labor Markets withInformational Frictions and HeterogeneousProduction Technologiesrdquo International EconomicReview 44(4) 1337ndash57

van den Berg Gerard J and Geert Ridder 1998 ldquoAnEmpirical Equilibrium Search Model of the LaborMarketrdquo Econometrica 66(5) 1183ndash1221

Blanchard Olivier and Peter Diamond 1989 ldquoTheBeveridge Curverdquo Brookings Papers on EconomicActivity 1 1ndash60

Blanchard Olivier and Pedro Portugal 2001 ldquoWhatHides Behind an Unemployment Rate ComparingPortuguese and US Labor Marketsrdquo AmericanEconomic Review 91(1) 187ndash207

Bontemps Christian Jean-Marc Robin and Gerard Jvan den Berg 1999 ldquoAn Empirical Equilibrium JobSearch Model with Search on the Job andHeterogeneous Workers and Firmsrdquo InternationalEconomic Review 40(4) 1039ndash74

Browning Martin Thomas Crossley and Eric Smith1999 ldquoAsset Accumulation and Short TermEmploymentrdquo University of Essex Mimeo

Burdett Kenneth 1978 ldquoA Theory of Employee JobSearch and Quit Ratesrdquo American EconomicReview 68(1) 212ndash20

Burdett Kenneth 1979 ldquoUnemployment InsurancePayments as a Search Subsidy A TheoreticalAnalysisrdquo Economic Inquiry 17(3) 333ndash43

Burdett Kenneth 1981 ldquoA Useful Restriction on theOffer Distribution in Job Search Modelsrdquo in GunnarEliasson Bertil Holmlund and Frank P Stafford edsStudies in Labor Market Behavior Sweden and theUnited States Stockholm IUI Conference Report

Burdett Kenneth and Melvyn G Coles 1997ldquoMarriage and Classrdquo Quarterly Journal ofEconomics 112(1) 141ndash68

Burdett Kenneth and Melvyn G Coles 1999 ldquoLong-Term Partnership Formation Marriage and Em-ploymentrdquo Economic Journal 109(456) F307ndash34

Burdett Kenneth and Melvyn G Coles 2003ldquoEquilibrium WagendashTenure Contractsrdquo Economet-rica 71(5) 1377ndash1404



Burdett Kenneth Ryoichi Imai and Randall Wright2004 ldquoUnstable Relationshipsrdquo Frontiers ofMacroeconomics 1(1) Article 1

Burdett Kenneth Ricardo Lagos and Randall Wright2003 ldquoCrime Inequality and UnemploymentrdquoAmerican Economic Review 93(5) 1764ndash77

Burdett Kenneth Ricardo Lagos and Randall Wright2004 ldquoAn On-the-Job Search Model of CrimeInequality and Unemploymentrdquo InternationalEconomic Review 45(3) 681ndash706

Burdett Kenneth and Dale T Mortensen 1980ldquoSearch Layoffs and Labor Market EquilibriumrdquoJournal of Political Economy 88(4) 652ndash72

Burdett Kenneth and Dale T Mortensen 1998 ldquoWageDifferentials Employer Size and UnemploymentrdquoInternational Economic Review 39(2) 257ndash73

Burdett Kenneth and Jan I Ondrich 1985 ldquoHowChanges in Labor Demand Affect UnemployedWorkersrdquo Journal of Labor Economics 3(1) 1ndash10

Burdett Kenneth Shouyong Shi and Randall Wright2001 ldquoPricing and Matching with Frictionsrdquo Journalof Political Economy 109(5) 1060ndash85

Burdett Ken and Eric Smith 2001 ldquoThe Low SkillTraprdquo European Economic Review 46(8) 1439ndash51

Burdett Kenneth and Tara Vishwanath 1988aldquoDeclining Reservation Wages and LearningrdquoReview of Economic Studies 55(4) 655ndash65

Burdett Kenneth and Tara Vishwanath 1988bldquoBalanced Matching and Labor Market EquilibriumrdquoJournal of Political Economy 96(5) 1048ndash65

Caballero Ricardo J and Mohamad L Hammour1994 ldquoThe Cleansing Effect of RecessionsrdquoAmerican Economic Review 84(5) 1350ndash68

Caballero Ricardo J and Mohamad L Hammour1996 ldquoOn the Timing and Efficiency of CreativeDestructionrdquo Quarterly Journal of Economics111(3) 805ndash52

Cole Harold L and Richard Rogerson 1999 ldquoCan theMortensenndashPissarides Matching Model Match theBusiness-Cycle Factsrdquo International EconomicReview 40(4) 933ndash59

Coles Melvyn G 2001 ldquoEquilibrium WageDispersion Firm Size and Growthrdquo Review ofEconomic Dynamics 4(1) 159ndash87

Coles Melvyn G and Jan Eeckhout 2000ldquoHeterogeneity as a Coordination Devicerdquo Mimeo

Coles Melvyn G and Jan Eeckhout 2003ldquoIndeterminacy and Directed Searchrdquo Journal ofEconomic Theory 111(2) 265ndash76

Coles Melvyn G and Adrian Masters 2000ldquoRetraining and Long-Term Unemployment in aModel of Unlearning by Not Doingrdquo EuropeanEconomic Review 44(9) 1801ndash22

Coles Melvyn G and Eric Smith 1996 ldquoCross-SectionEstimation of the Matching Function Evidence fromEngland and Walesrdquo Economica 63(252) 589ndash97

Coles Melvyn G and Eric Smith 1998 ldquoMarketplacesand Matchingrdquo International Economic Review39(1) 239ndash54

Costain James 1997 ldquoUnemployment Insurance in aGeneral Equilibrium Model of Job Search andPrecautionary Savingsrdquo University of Chicago PhDDissertation

Costain James and Michael Reiter 2003 ldquoBusiness

Cycles Unemployment Insurance and theCalibration of Matching Modelsrdquo UniversidadCarlos III Mimeo

Danforth John P 1979 ldquoOn the Role of Consumptionand Decreasing Absolute Risk Aversion in theTheory of Job Searchrdquo in Studies in the Economics ofSearch S Lippman and J McCall eds AmsterdamNorth Holland 109ndash31

Delacroix Alain 1999 ldquoFiring Costs Minimum Wageand the Outside Option Principlerdquo PurdueUniversity Mimeo

Delacroix Alain 2003 ldquoTransitions intoUnemployment and the Nature of Firing CostsrdquoReview of Economic Dynamics 6(3) 651ndash71

Delacroix Alain and Shouyong Shi ForthcomingldquoDirected Search on the Job and the Wage LadderrdquoInternational Economic Review

den Haan Wouter J Garey Ramey and Joel Watson2000 ldquoJob Destruction and Propagation of ShocksrdquoAmerican Economic Review 90(3) 482ndash98

Devine Theresa J and Nicholas M Kiefer 1991Empirical Labor Economics The Search ApproachOxford Oxford University Press

Diamond Peter A 1971 ldquoA Model of PriceAdjustmentrdquo Journal of Economic Theory 3(2)156ndash68

Diamond Peter A 1981 ldquoMobility Costs FrictionalUnemployment and Efficiencyrdquo Journal of PoliticalEconomy 89(4) 798ndash812

Diamond Peter A 1982a ldquoAggregate DemandManagement in Search Equilibriumrdquo Journal ofPolitical Economy 90(5) 881ndash94

Diamond Peter A 1982b ldquoWage Determination andEfficiency in Search Equilibriumrdquo Review ofEconomic Studies 49(2) 217ndash27

Duffie Darrell Nicolae Garleanu and Lasse HejePedersen 2002 ldquoSecurities Lending Shorting andPricingrdquo Journal of Financial Economics 66(2ndash3)307ndash39

Eckstein Zvi and Gerard van den Berg ForthcomingldquoEmpirical Labor Search A Surveyrdquo Journal ofEconometrics

Eckstein Zvi and Kenneth I Wolpin 1990 ldquoEstimatinga Market Equilibrium Search Model from Panel Dataon Individualsrdquo Econometrica 58(4) 783ndash808

Fallick Bruce and Charles A Fleischman 2004ldquoEmployer-to-Employer Flows in the US LaborMarket The Complete Picture of Gross WorkerFlowsrdquo Federal Reserve Board of GovernorsWorking Paper 2004-34

Farber Henry 1999 ldquoMobility and Stability TheDynamics of Job Change in Labor Marketsrdquo in TheHandbook of Labor Economics O Ashenfelter andD Cards eds Amsterdam Elsevier 2439ndash83

Flinn Christopher J and James J Heckman 1983ldquoAre Unemployment and out of the Labor ForceBehaviorally Distinct Labor Force Statesrdquo Journalof Labor Economics 1(1) 28ndash42

Gale Douglas M 1987 ldquoLimit Theorems for Marketswith Sequential Bargainingrdquo Journal of EconomicTheory 43(1) 20ndash54

Gaumont Damien Martin Schindler and RandallWright Forthcoming ldquoAlternative Models of WageDispersionrdquo in Structural Models of Wage and



Employment Dynamics Henning Bunzel Bent JChristensen George R Neumann and Jan-MarcRobin eds Amsterdam Elsevier

Gomes Joao Jeremy Greenwood and Sergio Rebelo2001 ldquoEquilibrium Unemploymentrdquo Journal ofMonetary Economics 48(1) 109ndash52

Greenwood Jeremy Glenn M MacDonald andGuang Jia Zhang 1996 ldquoThe Cyclical Behavior ofJob Creation and Job Destruction A SectoralModelrdquo Economic Theory 7(1) 95ndash112

Gronau Reuben 1971 ldquoInformation and FrictionalUnemploymentrdquo American Economic Review61(3) 290ndash301

Hall Robert E 2005 ldquoEmployment Fluctuations withEquilibrium Wage Stickinessrdquo American EconomicReview 95(1) 50ndash65

Hall John R Steven A Lippman and John J McCall1979 ldquoExpected Utility Maximizing Job Searchrdquo inStudies in the Economics of Search S Lippman andJ McCall eds Amsterdam North-Holland 133ndash56

Hosios Arthur J 1990 ldquoOn the Efficiency of Matchingand Related Models of Search and UnemploymentrdquoReview of Economic Studies 57(2) 279ndash98

Howitt Peter 1988 ldquoBusiness Cycles with CostlySearch and Recruitingrdquo Quarterly Journal ofEconomics 103(1) 147ndash65

Howitt Peter and R Preston McAfee 1987 ldquoCostlySearch and Recruitingrdquo International EconomicReview 28(1) 89ndash107

Jovanovic Boyan 1979a ldquoJob Matching and theTheory of Turnoverrdquo Journal of Political Economy87(5) 972ndash90

Jovanovic Boyan 1979b ldquoFirm-Specific Capital andTurnoverrdquo Journal of Political Economy 87(6)1246ndash60

Jovanovic Boyan 1982 ldquoSelection and the Evolutionof Industryrdquo Econometrica 50(3) 649ndash70

Jovanovic Boyan 1987 ldquoWork Rest and SearchUnemployment Turnover and the Cyclerdquo Journal ofLabor Economics 5(2) 131ndash48

Jovanovic Boyan and Glenn M MacDonald 1994ldquoCompetitive Diffusionrdquo Journal of PoliticalEconomy 102(1) 24ndash52

Julien Benoit John Kennes and Ian King 2000ldquoBidding for Laborrdquo Review of Economic Dynamics3(4) 619ndash49

Kambourov Gueorgui and Iourii Manovskii 2004ldquoOccupational Mobility and Wage Inequalityrdquo PIERWorking Paper 04-026

Kambourov Gueorgui and Iourii Manovskii 2005ldquoAccounting for the Changing Life-Cycle Profile ofEarningsrdquo University of Pennsylvania Mimeo

Karlin Samuel 1962 ldquoStochastic Models and OptimalPolicy for Selling an Assetrdquo in Studies in AppliedProbability and Management Science K J Arrow SKarlin and H Scarf eds Stanford StanfordUniversity Press 148ndash58

Kiyotaki Nobuhiro and Randall Wright 1993 ldquoASearch-Theoretic Approach to Monetary Econom-icsrdquo American Economic Review 83(1) 63ndash77

Lagos Ricardo 2000 ldquoAn Alternative Approach toSearch Frictionsrdquo Journal of Political Economy108(5) 851ndash73

Laing Derek Theodore Palivos and Ping Wang 1995

ldquoLearning Matching and Growthrdquo Review ofEconomic Studies 62(1) 115ndash29

Laing Derek Theodore Palivos and Ping Wang 2003ldquoThe Economics of lsquoNew Bloodrsquordquo Journal ofEconomic Theory 112(1) 106ndash56

Lentz Rasmus and Torben Tranaes ForthcomingldquoJob Search and Savings Wealth Effects andDuration Dependencerdquo Journal of LaborEconomics

Lippman Steven A and John J McCall 1976a ldquoTheEconomics of Job Search A Surveyrdquo EconomicInquiry 14(2) 155ndash89 347ndash68

Lippman Steven A and John J McCall 1976b ldquoJobSearch in a Dynamic Economyrdquo Journal ofEconomic Theory 12(3) 365ndash90

Lippman Steven A and John W Mamer 1989 ldquoASimple Search Model with Procyclical Quitsrdquo Journalof Economic Dynamics and Control 13(2) 247ndash53

Ljungqvist Lars and Thomas J Sargent 1998 ldquoTheEuropean Unemployment Dilemmardquo Journal ofPolitical Economy 106(3) 514ndash50

Ljungqvist Lars and Thomas J Sargent 2004Recursive Macroeconomics Theory CambridgeMIT Press

Lucas Robert E Jr 1972 ldquoExpectations and theNeutrality of Moneyrdquo Journal of Economic Theory4(2) 103ndash24

Lucas Robert E Jr and Edward C Prescott 1974ldquoEquilibrium Search and Unemploymentrdquo Journalof Economic Theory 7(2) 188ndash209

Masters Adrian M 1998 ldquoEfficiency of Investment inHuman and Physical Capital in a Model of BilateralSearch and Bargainingrdquo International EconomicReview 39(2) 477ndash94

Masters Adrian M 1999 ldquoWage Posting in Two-SidedSearch and the Minimum Wagerdquo InternationalEconomic Review 40(4) 809ndash26

McCall John J 1970 ldquoEconomics of Information andJob Searchrdquo Quarterly Journal of Economics 84(1)113ndash26

Merz Monika 1995 ldquoSearch in the Labor Market andthe Real Business Cyclerdquo Journal of MonetaryEconomics 36(2) 269ndash300

Merz Monika 1999 ldquoHeterogeneous Job-Matchesand the Cyclical Behavior of Labor TurnoverrdquoJournal of Monetary Economics 43(1) 91ndash124

Meyer Bruce D 1990 ldquoUnemployment Insurance andUnemployment Spellsrdquo Econometrica 58(4) 757ndash82

Millard Stephen P and Dale T Mortensen 1997ldquoThe Unemployment and Welfare Effects of LabourMarket Policy A Comparison of the USA and theUKrdquo in Unemployment Policy GovernmentOpposition for the Labour Market D J Snower andG de la Dehesa eds New York CambridgeUniversity Press 545ndash72

Moen Espen R 1997 ldquoCompetitive SearchEquilibriumrdquo Journal of Political Economy 105(2)385ndash411

Montgomery James D 1991 ldquoEquilibrium WageDispersion and Interindustry Wage DifferentialsrdquoQuarterly Journal of Economics 106(1) 163ndash79

Mortensen Dale T 1970 ldquoA Theory of Wage andEmployment Dynamicsrdquo in MicroeconomicFoundations of Employment and Inflation Theory



E S Phelps et al eds New York W W Norton124ndash66

Mortensen Dale T 1977 ldquoUnemployment Insuranceand Job Search Decisionsrdquo Industrial and LaborRelations Review 30(4) 505ndash17

Mortensen Dale T 1982a ldquoThe Matching Process as aNoncooperative Bargaining Gamerdquo in TheEconomics of Information and Uncertainty John JMcCall ed Chicago University of Chicago Press233ndash54

Mortensen Dale T 1982b ldquoProperty Rights andEfficiency in Mating Racing and Related GamesrdquoAmerican Economic Review 72(5) 968ndash79

Mortensen Dale T 1986 ldquoJob Search and LaborMarket Analysisrdquo in Handbook of Labor EconomicsO Ashenfelter and R Layard eds AmsterdamNorth Holland 849ndash920

Mortensen Dale T 1988 ldquoMatching Finding aPartner for Life or Otherwiserdquo American Journal ofSociology 94(Supplement) S215ndash40

Mortensen Dale T 1989 ldquoThe Persistence andIndeterminancy of Unemployment in SearchEquilibriumrdquo Scandinavian Journal of Economics91(2) 347ndash70

Mortensen Dale T 1999 ldquoEquilibrium Unemploy-ment Dynamicsrdquo International Economic Review40(4) 889ndash914

Mortensen Dale T 2000 ldquoEquilibrium Unemploy-ment with Wage Posting BurdettndashMortensen MeetsPissaridesrdquo in Panel Data and Structural LaborMarket Models H Bunzel B J Christianesen NM Keifer and D T Mortensen eds AmsterdamElsevier 281ndash92

Mortensen Dale T 2003 Wage Dispersion Cam-bridge MIT Press

Mortensen Dale T and George R Neumann 1984ldquoChoice or Chance A Structural Interpretation ofIndividual Labor Market Historiesrdquo in Studies inLabor Market Dynamics G R Neumann and WWestergaard-Neilsen eds Heidelberg Springer-Verlag 98ndash131

Mortensen Dale T and Christopher A Pissarides1994 ldquoJob Creation and Job Destruction in theTheory of Unemploymentrdquo Review of EconomicStudies 61(3) 397ndash415

Mortensen Dale T and Christopher A Pissarides1999a ldquoNew Developments in Models of Search inthe Labor Marketrdquo in Handbook of LaborEconomics O Ashenfelter and D Card edsAmsterdam North Holland 2567ndash2627

Mortensen Dale T and Christopher A Pissarides1999b ldquoJob Reallocation Employment Fluctuationsand Unemployment Differencesrdquo in Handbook ofMacroeconomics M Woodford and J Taylor edsAmsterdam North Holland 1171ndash1228

Mortensen Dale T and Christopher A Pissarides1999c ldquoUnemployment Responses to lsquoSkill-BiasedrsquoTechnology Shocks The Role of Labour MarketPolicyrdquo Economic Journal 109(455) 242ndash65

Mortensen Dale T and Randall Wright 2002ldquoCompetitive Pricing and Efficiency in SearchEquilibriumrdquo International Economic Review43(1) 1ndash20

Moscarini Giuseppe 2001 ldquoExcess Worker

Reallocationrdquo Review of Economic Studies 68(3)593ndash612

Moscarini Giuseppe 2005 ldquoJob Matching and theWage Distributionrdquo Econometrica 73(2) 481ndash516

Nash John 1950 ldquoThe Bargaining ProblemrdquoEconometrica 18(1) 155ndash62

Osborne Martin J and Ariel Rubinstein 1990Bargaining and Markets London Academic Press

Peters Michael 1984 ldquoBertrand Equilibrium withCapacity Constraints and Restricted MobilityrdquoEconometrica 52(5) 1117ndash27

Peters Michael 1991 ldquoEx Ante Price Offers inMatching Games Non-Steady Statesrdquo Econometrica59(5) 1425ndash54

Petrongolo Barbara and Christopher A Pissarides2001 ldquoLooking into the Black Box A Survey of theMatching Functionrdquo Journal of EconomicLiterature 39(2) 390ndash431

Pissarides Christopher A 1984 ldquoSearch Intensity JobAdvertising and Efficiencyrdquo Journal of LaborEconomics 2(1) 128ndash43

Pissarides Christopher A 1985 ldquoShort-RunEquilibrium Dynamics of Unemployment Vacanciesand Real Wagesrdquo American Economic Review 75(4)676ndash90

Pissarides Christopher A 1986 ldquoUnemployment andVacancies in Britainrdquo Economic Policy 3(3) 499ndash559

Pissarides Christopher A 1994 ldquoSearchUnemployment with On-the-Job Searchrdquo Review ofEconomic Studies 61(3) 457ndash75

Pissarides Christopher A 2000 EquilibriumUnemployment Theory Cambridge MIT Press

Postel-Vinay Fabien and Jean-Marc Robin 2002ldquoThe Distribution of Earnings in an EquilibriumSearch Model with State-Dependent Offers andCounteroffersrdquo International Economic Review43(4) 989ndash1016

Postel-Vinay Fabien and Jean-Marc Robin 2002ldquoEquilibrium Wage Dispersion with Worker andEmployer Heterogeneityrdquo Econometrica 70(6)2295ndash2350

Pries Michael J 2004 ldquoPersistence of EmploymentFluctuations A Model of Recurring Job LossrdquoReview of Economic Studies 71(1) 193ndash215

Pries Michael J and Richard Rogerson 2005 ldquoHiringPolicies Labor Market Institutions and LaborMarket Flowsrdquo Journal of Political Economy 113(4)811ndash39

Robin Jean-Marc and Sebastien Roux 1998ldquoEquilibrium Search with Decreasing Returns toScale and Hiring Costs Theory and EstimationrdquoMimeo

Rothschild Michael 1973 ldquoModels of MarketOrganization with Imperfect Information A SurveyrdquoJournal of Political Economy 81(6) 1283ndash1308

Rothschild Michael 1974 ldquoSearching for the LowestPrice When the Distribution of Prices Is UnknownrdquoJournal of Political Economy 82(4) 689ndash711

Rubinstein Ariel 1982 ldquoPerfect Equilibrium in aBargaining Modelrdquo Econometrica 50(1) 97ndash109

Rubinstein Ariel and Asher Wolinsky 1985ldquoEquilibrium in a Market with SequentialBargainingrdquo Econometrica 53(5) 1133ndash50

Rubinstein Ariel and Asher Wolinsky 1990



ldquoDecentralized Trading Strategic Behaviour and theWalrasian Outcomerdquo Review of Economic Studies57(1) 63ndash78

Salop Steven C 1973 ldquoWage Differentials in aDynamic Theory of the Firmrdquo Journal of EconomicTheory 6(4) 321ndash44

Salop Steven C 1977 ldquoThe Noisy MonopolistImperfect Information Price Dispersion and PriceDiscriminationrdquo Review of Economic Studies 44(3)393ndash406

Shi Shouyong 1995 ldquoMoney and Prices A Model ofSearch and Bargainingrdquo Journal of EconomicTheory 67(2) 467ndash96

Shi Shouyong 2001 ldquoFrictional Assignment IEfficiencyrdquo Journal of Economic Theory 98(2)232ndash60

Shi Shouyong 2002 ldquoA Directed Search Model ofInequality with Heterogeneous Skills and Skill-Based Technologyrdquo Review of Economic Studies69(2) 467ndash91

Shimer Robert 1996 ldquoContracts in Frictional LaborMarketsrdquo MIT Mimeo

Shimer Robert 1999 ldquoJob Auctionsrdquo PrincetonUniversity Mimeo

Shimer Robert 2005 ldquoThe Cyclical Behavior ofEquilibrium Unemployment and VacanciesrdquoAmerican Economic Review 95(1) 25ndash49

Shimer Robert Forthcoming ldquoThe Assignment ofWorkers to Jobs in an Economy with CoordinationFrictionsrdquo Journal of Political Economy

Shimer Robert and Lones Smith 2000 ldquoAssortativeMatching and Searchrdquo Econometrica 68(2) 343ndash69

Shimer Robert and Lones Smith 2001ldquoNonstationary Searchrdquo Princeton UniversityMimeo

Simon Herbert A 1955 ldquoA Behavioral Model ofRational Choicerdquo Quarterly Journal of Economics69(1) 99ndash118

Smith Eric 1999 ldquoSearch Concave Production andOptimal Firm Sizerdquo Review of Economic Dynamics

2(2) 456ndash71Stevens Margaret 2004 ldquoWagendashTenure Contracts in a

Frictional Labour Market Firmsrsquo Strategies forRecruitment and Retentionrdquo Review of EconomicStudies 71(2) 535ndash51

Stigler George 1961 ldquoThe Economics ofInformationrdquo Journal of Political Economy 69(3)213ndash25

Stokey Nancy Robert E Lucas Jr and EdwardPrescott 1989 Recursive Methods in EconomicDynamics Cambridge Harvard University Press

Trejos Alberto and Randall Wright 1995 ldquoSearchBargaining Money and Pricesrdquo Journal of PoliticalEconomy 103(1) 118ndash41

Uren Lawrence 2004 ldquoThe Allocation of Labor andEndogenous Search Decisionsrdquo Mimeo

Valdivia Victor 1995 ldquoEvaluating the Welfare Benefitsof Unemployment Insurancerdquo Mimeo

Weill Pierre-Olivier 2003 ldquoLeaning Against theWindrdquo Stanford University Mimeo

Wilde Louis L 1979 ldquoAn Information-TheoreticApproach to Job Quitsrdquo in Studies in the Economicsof Search S A Lippman and J J McCall edsAmsterdam North Holland 35ndash52

Wolpin Kenneth I 1987 ldquoEstimating a StructuralSearch Model The Transition from School to WorkrdquoEconometrica 55(4) 801ndash17

Wolpin Kenneth I 1995 Empirical Methods for theStudy of Labor Force Dynamics LuxembourgHarwood Academic Publishers

Wright Randall D 1986 ldquoJob Search and CyclicalUnemploymentrdquo Journal of Political Economy94(1) 38ndash55

Wright Randall D 1987 ldquoSearch Layoffs andReservation Wagesrdquo Journal of Labor Economics5(3) 354ndash65

Wright Randall D and Janine Loberg 1987ldquoUnemployment Insurance Taxes andUnemploymentrdquo Canadian Journal of Economics20(1) 36ndash54



is the notion that trading frictions areimportant It takes time and otherresources for a worker to land a job espe-cially a good job at a good wage and for afirm to fill a vacancy There is simply nosuch thing as a centralized market wherebuyers and sellers of labor meet and tradeat a single price as assumed in classicalequilibrium theory Of course economicmodels do not have to be realistic to beuseful and the supply-and-demand para-digm is obviously useful for studying manyissues in labor economics But it is equallyclear that the simple supply-and-demandapproach is ill suited for discussing ques-tions such as those raised in the previousparagraph

We argue that even the earliest searchmodels with their focus on a single workerenhance our ability to organize observa-tions on employment histories We thenexamine more recent research that embedsthe decision-theoretic model into an equi-librium framework Although there areseveral important modeling decisions inequilibrium search theory we argue thattwo questions are paramount First howdo agents meet In particular is searchrandom so that unemployed workers areequally likely to locate any job opening ordirected so that for example firms canattract more applicants by offering higherwages Second exactly how are wagesdetermined Do matched workers andfirms bargain or are wages posted unilat-erally before they meet We consider vari-ous alternative sets of assumptions andindicate how the choice affects predictionsWe also emphasize how a common set ofmethods and ideas are used in all of thedifferent approaches

Before proceeding we mention thatsearch theory constitutes a very large fieldIn addition to labor it has been used inmonetary theory industrial organizationfinance the economics of the marriagemarket and other areas all of which wemust neglect lest this survey becomes

1 Examples in monetary economics include NobuhiroKiyotaki and Randall Wright (1993) Shouyong Shi (1995)and Alberto Trejos and Wright (1995) examples in themarriage literature include Dale T Mortensen (1988)Kenneth Burdett and Melvyn G Coles (1997 1999) andRobert Shimer and Lones Smith (2000) examples in IOinclude Steven C Salop (1977) Boyan Jovanovic (1982)and Jovanovic and Glenn M MacDonald (1994) examplesin finance include Darrell Duffie Nicolae Garleanu andLasse Pedersen (2002) and Pierre-Olivier Weill (2004)

2 Examples of theoretical work studying the question ofwhether frictionless competitive equilibrium is the limit ofsearch equilibrium as the frictions get small include ArielRubinstein and Asher Wolinsky (1985 1990) DouglasGale (1987) and Mortensen and Wright (2002) Theresa JDevine and Nicholas M Kiefer (1991) Kenneth I Wolpin(1995) and Zvi Eckstein and Gerard J van den Berg(forthcoming) survey empirical work

unmanageable1 Search has been used inmuch fairly technical theoretical researchand has also been a workhorse for empiricaleconomics but we can neither delve intopure theory nor pay attention to all of theeconometric issues and empirical findingshere2 Also while we strive to be rigorouswe emphasize issues rather models ormethods per se Hence the presentationrevolves around the ways in which theframework helps us think about substantivequestions like those mentioned above

The logical structure of the paper is as fol-lows We begin in section 2 with the problemof a single agent looking for a job not onlybecause this is the way the literature startedbut because it is a building block for theequilibrium analysis to follow Even thisrudimentary model is consistent with twofacts that do not come out of frictionlessmodels it takes time to find an acceptablejob and what one ends up with is at least par-tially a matter of luck which means similaragents may end up with different wages Insection 3 we describe some generalizationsof this model designed to help understandturnover and labor market transitions Thissection also introduces tools and techniquesneeded for equilibrium search theory

We also spend some time here discussingthe logical transition between decision theo-ry and equilibrium theory We first arguethat one can reinterpret the single-agent










2 Basic Job Search


21 Discrete Time


t=0βtxt



U W w dF wr

U+ minus+

110

max ( ) ( )




(1)

(2)




max ( ) ( )

W w w W w( ) ( )= + β




(3)




(4)


(5)


w b F w dwR wR

= +minus

minusintββ1

1

[ ( )]

w b w w dF wR w RR

= +minus

minusintββ1

( ) ( )


β β

w T wR R= ( )




22 Continuous Time


__1

r__ and assume


(6)

(7)

times

Algebra implies

(8)

(9)


Bellman equations

(10)

(11)



0

max ( ) ( )

rW w w( ) =


max ( ) ( )

rU r b= +( )1

rW w r w( ) = +( )1

U W w dF wr

U+ minus+int

110

max ( ) ( )

U br

= ++ int

1

W w wr

W w( ) ( )= ++

11



(12)


(13)


(14)


(15)


23 Discussion


w RR

w w dF w rg

int minus =( ) ( ) ( )

w b gr

w w dF wR w RR


w br

F w dwR wR

= + minusint

[ ( )]1

w br

w w dF wR w RR

= + minusint

( ) ( )






(16)



3 Worker Turnover


D tHe dtH

Ht= =int minus

0

1






(17)


(18)



w br

F w dwR wR

= ++

minusint

λ

[ ( )]1






(19)


(20)





= + minusint0

[ ( ) ] ( )

w br

w w dF wR w RR

= + minus+

minusint λλ

( ) ( )




(22)



(23)



(24)


F wr F w

dwR


⎡⎣⎢

⎤⎦⎥int( )

[ ]

0 11

11

λww

w bR =

W w W w dF ww R

R

+ minus minusint( ) [ ( ) ( )] ( ) 0 1

w bR =






33 Discussion





(25)


(26)


(27)



1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]


λ( ) ( )

( ) ( )1 11

uF wR


λλ 0 1







the idea14










(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES

































































































































































































2 Basic Job Search


21 Discrete Time


t=0βtxt



U W w dF wr

U+ minus+

110

max ( ) ( )




(1)

(2)




max ( ) ( )

W w w W w( ) ( )= + β




(3)




(4)


(5)


w b F w dwR wR

= +minus

minusintββ1

1

[ ( )]

w b w w dF wR w RR

= +minus

minusintββ1

( ) ( )


β β

w T wR R= ( )




22 Continuous Time


__1

r__ and assume


(6)

(7)

times

Algebra implies

(8)

(9)


Bellman equations

(10)

(11)



0

max ( ) ( )

rW w w( ) =


max ( ) ( )

rU r b= +( )1

rW w r w( ) = +( )1

U W w dF wr

U+ minus+int

110

max ( ) ( )

U br

= ++ int

1

W w wr

W w( ) ( )= ++

11



(12)


(13)


(14)


(15)


23 Discussion


w RR

w w dF w rg

int minus =( ) ( ) ( )

w b gr

w w dF wR w RR


w br

F w dwR wR

= + minusint

[ ( )]1

w br

w w dF wR w RR

= + minusint

( ) ( )






(16)



3 Worker Turnover


D tHe dtH

Ht= =int minus

0

1






(17)


(18)



w br

F w dwR wR

= ++

minusint

λ

[ ( )]1






(19)


(20)





= + minusint0

[ ( ) ] ( )

w br

w w dF wR w RR

= + minus+

minusint λλ

( ) ( )




(22)



(23)



(24)


F wr F w

dwR


⎡⎣⎢

⎤⎦⎥int( )

[ ]

0 11

11

λww

w bR =

W w W w dF ww R

R

+ minus minusint( ) [ ( ) ( )] ( ) 0 1

w bR =






33 Discussion





(25)


(26)


(27)



1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]


λ( ) ( )

( ) ( )1 11

uF wR


λλ 0 1







the idea14










(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES


























































































































































































U W w dF wr

U+ minus+

110

max ( ) ( )




(1)

(2)




max ( ) ( )

W w w W w( ) ( )= + β




(3)




(4)


(5)


w b F w dwR wR

= +minus

minusintββ1

1

[ ( )]

w b w w dF wR w RR

= +minus

minusintββ1

( ) ( )


β β

w T wR R= ( )




22 Continuous Time


__1

r__ and assume


(6)

(7)

times

Algebra implies

(8)

(9)


Bellman equations

(10)

(11)



0

max ( ) ( )

rW w w( ) =


max ( ) ( )

rU r b= +( )1

rW w r w( ) = +( )1

U W w dF wr

U+ minus+int

110

max ( ) ( )

U br

= ++ int

1

W w wr

W w( ) ( )= ++

11



(12)


(13)


(14)


(15)


23 Discussion


w RR

w w dF w rg

int minus =( ) ( ) ( )

w b gr

w w dF wR w RR


w br

F w dwR wR

= + minusint

[ ( )]1

w br

w w dF wR w RR

= + minusint

( ) ( )






(16)



3 Worker Turnover


D tHe dtH

Ht= =int minus

0

1






(17)


(18)



w br

F w dwR wR

= ++

minusint

λ

[ ( )]1






(19)


(20)





= + minusint0

[ ( ) ] ( )

w br

w w dF wR w RR

= + minus+

minusint λλ

( ) ( )




(22)



(23)



(24)


F wr F w

dwR


⎡⎣⎢

⎤⎦⎥int( )

[ ]

0 11

11

λww

w bR =

W w W w dF ww R

R

+ minus minusint( ) [ ( ) ( )] ( ) 0 1

w bR =






33 Discussion





(25)


(26)


(27)



1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]


λ( ) ( )

( ) ( )1 11

uF wR


λλ 0 1







the idea14










(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES



























































































































































































22 Continuous Time


__1

r__ and assume


(6)

(7)

times

Algebra implies

(8)

(9)


Bellman equations

(10)

(11)



0

max ( ) ( )

rW w w( ) =


max ( ) ( )

rU r b= +( )1

rW w r w( ) = +( )1

U W w dF wr

U+ minus+int

110

max ( ) ( )

U br

= ++ int

1

W w wr

W w( ) ( )= ++

11



(12)


(13)


(14)


(15)


23 Discussion


w RR

w w dF w rg

int minus =( ) ( ) ( )

w b gr

w w dF wR w RR


w br

F w dwR wR

= + minusint

[ ( )]1

w br

w w dF wR w RR

= + minusint

( ) ( )






(16)



3 Worker Turnover


D tHe dtH

Ht= =int minus

0

1






(17)


(18)



w br

F w dwR wR

= ++

minusint

λ

[ ( )]1






(19)


(20)





= + minusint0

[ ( ) ] ( )

w br

w w dF wR w RR

= + minus+

minusint λλ

( ) ( )




(22)



(23)



(24)


F wr F w

dwR


⎡⎣⎢

⎤⎦⎥int( )

[ ]

0 11

11

λww

w bR =

W w W w dF ww R

R

+ minus minusint( ) [ ( ) ( )] ( ) 0 1

w bR =






33 Discussion





(25)


(26)


(27)



1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]


λ( ) ( )

( ) ( )1 11

uF wR


λλ 0 1







the idea14










(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES





























































































































































































(16)



3 Worker Turnover


D tHe dtH

Ht= =int minus

0

1






(17)


(18)



w br

F w dwR wR

= ++

minusint

λ

[ ( )]1






(19)


(20)





= + minusint0

[ ( ) ] ( )

w br

w w dF wR w RR

= + minus+

minusint λλ

( ) ( )




(22)



(23)



(24)


F wr F w

dwR


⎡⎣⎢

⎤⎦⎥int( )

[ ]

0 11

11

λww

w bR =

W w W w dF ww R

R

+ minus minusint( ) [ ( ) ( )] ( ) 0 1

w bR =






33 Discussion





(25)


(26)


(27)



1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]


λ( ) ( )

( ) ( )1 11

uF wR


λλ 0 1







the idea14










(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES




























































































































































































(19)


(20)





= + minusint0

[ ( ) ] ( )

w br

w w dF wR w RR

= + minus+

minusint λλ

( ) ( )




(22)



(23)



(24)


F wr F w

dwR


⎡⎣⎢

⎤⎦⎥int( )

[ ]

0 11

11

λww

w bR =

W w W w dF ww R

R

+ minus minusint( ) [ ( ) ( )] ( ) 0 1

w bR =






33 Discussion





(25)


(26)


(27)



1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]


λ( ) ( )

( ) ( )1 11

uF wR


λλ 0 1







the idea14










(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES



























































































































































































33 Discussion





(25)


(26)


(27)



1 1wR

F w dG w

int minus[ ( ( ))]

G wF w F w

F w F wR

R

( ) =minus[ ]


λ( ) ( )

( ) ( )1 11

uF wR


λλ 0 1







the idea14










(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES






























































































































































































the idea14






de05_Article1 111605 353 PM 67




(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES




























































































































































































(28)



w e

m u vu

m u vv

= =( )and

( )


42 Bargaining



(29)


(30)















(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES
































































































































































































(33) W(w) = U


W w J r ( ) ( )= = +π 1λ



(34)



r__w




(35)

which has solution

(36)


43 Equilibrium




y rU rVr

=minus minus

+ λ




) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES


























































































































































































) ( )y y dF yRminus



(37)


(38)


(39)



(40)


(41)





e J k( )π =



(42)


(43)


(44)





uw

lowast =+λ

λ


r y bk

w

e

+ +minus

=minusλ θ

θ

( )1

k Se= minus ( )1 θ





(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES




























































































































































































(45)


(46)

(47)


mdashαw(1


(48)


(49)


y bk

Rw

e

= +minus

θθ( )1

( )( )

r S y bk

yw

e


λθ

θ

1

S dF ye y yR

= minusinfin

int ( ) ( )1 θ


= minusinfin

int [ ( )] ( )

b S dFw y yR

= +infin

int (θ yy)


= + minusinfin

int [ ( )] ( )



(50)






( ) ( ( ) ( )r k y y dF ye y RR






(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES




























































































































































































(51)


(52)


(53)






( )S dF y yy

y

yR

+ intλ

|


( )S dF y yy

y

yR

+ intλ

|



46 Discussion











51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES






























































































































































































51 A One-Shot Model




(54)



(55)


(56)


(57)




V kq

= minusmax


( )( )






(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES




























































































































































































(58)


(_(_qlowastqlowast




(59)






V k= minus













53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES




























































































































































































53 A Dynamic Model


(60)


(62)

Similarly for firms



Free entry yields

(65) kq y wr

e=minus

+ ( )( )

λ

rU bq w rUr

w= +minus

+ ( )( )

λ


rU b q W w Uw= + minus ( )[ ( ) ]


(66)


(67)


(68)


54 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )( ) ( )

e qy rUr

rU b( )minus+

= minusλ

max q e q

y rUr

q rU b( ) ( )minus+

minus minusλ


























(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES

















































































































































































































(69) rU1b1w(1)[W1(w1)U1]

w[W1(w2)U1]

(70) rW1(w1)w1λ[U1W1(w1)]




(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES



























































































































































































(71) rW1(w2)w2λ[U1W1(w2)]


(72)




L1

_u1

_L1_u1

L2

_u2


r__

_w__i_ So expected


(73)

(74)



(75)

times

minus


(76) = 0 and T(0) lt 0 = 1



_y where




22=

minus+

e

y br λ

11 1

1 1 2 2

1=+

minus+

e

L uL u L u

y wr λ

wr b b

rw

w1

1 2=+ +

+ +( )λ

λ



(77)

_y



(78)





w

m L u L uL u L u

=++

( )1 1 2 2

1 1 2 2

1

yr L b bw= +

minus1 2 1( ))( )( )r Lw w+ + + λ λ 2

y bL b b

Lw

= +minus

+21 2 1

2

λλ

( )( )

and






(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES





























































































































































































(79)



(80)


_] for some w



( )( )

y wF w r F w


λλ λ 1 1

( )w =

G wF w

F w( )

( )( )

=+ minus[ ]

λλ 1


(81)




(82)



(83)



_= y and

wb y

R =+( ) + minus( )

+( ) + minus( )λ

λ

1

2

0 1 1

1

2

0 1 1

F wy wy wR

( ) =+

minusminusminus

⎛⎝⎜

⎞⎠⎟

λ

1

1

1

F wy wy b


⎛⎝⎜

⎞⎠⎟

λ

1









63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES
































































































































































































63 Discussion


7 Efficiency





q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES



























































































































































































q q

q

q q

qe

e

w

w

αα

αα

prime prime

+ =( )( )

( )( ) 1


71 A One-Shot Model


(84)



(85)




(87)






uq







_uv) where s

_


_q) = m(s



_q) = s e(s

_q) s



(88)


_ where

(89) g


(90)

The planner solves

(91)


(92)



max q s

eb g ssq y b k

qu( )

( )( )minus +

minus minus


ssq y b

sqe( )( ) ( )

=minus θ

b g ss sq y b

sqeminus +

minus( )

( ) ( ) θ




_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES



























































































































































































_q)


_and v and only




72 A Dynamic Model




(94)


k eq

Y eq ee( )

( )( )(


))

qeminus⎡

⎣⎢⎤⎦⎥

λ

rY e ye b eq

( ) ( )= + minusmax 1




(95)


(96)


(97)



73 Discussion


r q

q q qy b

ke

e e

+ +minus

=minusλ

( )

( ) ( )

ra y bkq

aq

qe


⎤⎦⎥

( )λ

k a q q qe e= minus[ ]1 ( ) ( )







8 Conclusion








REFERENCES






























































































































































































8 Conclusion








REFERENCES




























































































































































































REFERENCES

























































































































































































search-theoretic models of the labor market: a survey

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