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Journal of Economic Literature Vol. XLIII (December 2005), pp. 959–988

Search-Theoretic Models of the Labor Market: A Survey

∗ RICHARD ROGERSON, ROBERT SHIMER, and RANDALL WRIGHT

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

1. Introduction While the usual paradigm of supply and demand in a frictionless labor market is he economic fortunes of most individu- useful for discussing some issues, many als are largely determined by their labor T important questions are not easily market experiences—that is, by paths for addressed with this approach. Why do their wages, their employers, and their unemployed workers sometimes choose to intervening spells of unemployment or non- remain unemployed, say by turning down employment. Hence, economists are natu- job offers? What determines the lengths of rally interested in documenting the employment and unemployment spells? empirical behavior of wages, employment, How can we simultaneously have unem- and unemployment, and also in building ployed workers and unfilled vacancies? models to help us understand the forces that What factors determine the aggregate shape these outcomes and using the models unemployment and vacancy rates? How can to assess the consequences of changes in apparently homogeneous workers in similar policies or institutions. jobs end up earning different wages? What are the trade-offs faced by firms in paying ∗ Rogerson: Arizona State University. Shimer: University different wages? How do wages and turnover of Chicago. Wright: University of Pennsylvania. We thank interact? What determines the efficient Daron Acemoglu, Ken Burdett, Zvi Eckstein, Roger Gordon, Iourii Manovskii, Derek Laing, John McMillan, amount of turnover? Dale Mortensen, Peter Rupert, Martin Schindler, Gerard From its inception, search theory has van den Berg, two anonymous referees, and many students provided a rigorous yet tractable frame- for their input. We also thank the Federal Reserve Bank of Cleveland, the National Science Foundation, and the Sloan work that can be used to address these and Foundation for research support. related questions. Central to the approach

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is the notion that trading frictions are unmanageable.1 Search has been used in important. It takes time and other much fairly technical theoretical research, resources for a worker to land a job, espe- and has also been a workhorse for empirical cially a good job at a good wage, and for a economics, but we can neither delve into firm to fill a vacancy. There is simply no pure theory nor pay attention to all of the such thing as a centralized market where econometric issues and empirical findings buyers and sellers of labor meet and trade here.2 Also, while we strive to be rigorous, at a single price, as assumed in classical we emphasize issues rather models or equilibrium theory. Of course, economic methods per se. Hence the presentation models do not have to be realistic to be revolves around the ways in which the useful, and the supply-and-demand para- framework helps us think about substantive digm is obviously useful for studying many questions like those mentioned above. issues in labor economics. But it is equally The logical structure of the paper is as fol- clear that the simple supply-and-demand lows. We begin in section 2 with the problem approach is ill suited for discussing ques- of a single agent looking for a job, not only tions such as those raised in the previous because this is the way the literature started, paragraph. but because it is a building block for the We argue that even the earliest search equilibrium analysis to follow. Even this models, with their focus on a single worker, rudimentary model is consistent with two enhance our ability to organize observa- facts that do not come out of frictionless tions on employment histories. We then models: it takes time to find an acceptable examine more recent research that embeds job; and what one ends up with is at least par- the decision-theoretic model into an equi- tially a matter of luck, which means similar librium framework. Although there are agents may end up with different wages. In several important modeling decisions in section 3, we describe some generalizations equilibrium search theory, we argue that of this model designed to help understand two questions are paramount. First, how turnover and labor market transitions. This do agents meet? In particular, is search section also introduces tools and techniques random, so that unemployed workers are needed for equilibrium search theory. equally likely to locate any job opening, or We also spend some time here discussing directed, so that for example firms can the logical transition between decision theo- attract more applicants by offering higher ry and equilibrium theory. We first argue wages? Second, exactly how are wages that one can reinterpret the single-agent determined? Do matched workers and firms bargain, or are wages posted unilat- 1 Examples in monetary economics include Nobuhiro erally before they meet? We consider vari- Kiyotaki and Randall Wright (1993), Shouyong Shi (1995), ous alternative sets of assumptions and and Alberto Trejos and Wright (1995); examples in the marriage literature include Dale T. Mortensen (1988), indicate how the choice affects predictions. Kenneth Burdett and Melvyn G. Coles (1997, 1999), and We also emphasize how a common set of Robert Shimer and Lones Smith (2000); examples in IO methods and ideas are used in all of the include Steven C. Salop (1977), Boyan Jovanovic (1982), and Jovanovic and Glenn M. MacDonald (1994); examples different approaches. in finance include Darrell Duffie, Nicolae Garleanu, and Before proceeding, we mention that Lasse Pedersen (2002) and Pierre-Olivier Weill (2004). 2 search theory constitutes a very large field. Examples of theoretical work studying the question of whether frictionless competitive equilibrium is the limit of In addition to labor, it has been used in search equilibrium as the frictions get small include Ariel monetary theory, industrial organization, Rubinstein and Asher Wolinsky (1985, 1990), Douglas finance, the economics of the marriage Gale (1987), and Mortensen and Wright (2002). Theresa J. Devine and Nicholas M. Kiefer (1991), Kenneth I. Wolpin market, and other areas, all of which we (1995), and Zvi Eckstein and Gerard J. van den Berg must neglect lest this survey becomes (forthcoming) survey empirical work. de05_Article1 11/16/05 3:53 PM Page 961

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model as an equilibrium of a simple econo- wants to analyze policy, one would like to my. But in such a model several key vari- know whether the models rationalize a role ables, including the arrival rate and for intervention in a decentralized economy. distribution of wage offers, are essentially It is fine to say, for example, that a change in fixed exogenously. For some issues, one may some variable reduces unemployment, but want to know how these are determined in it is obviously relevant to know whether this equilibrium, and in particular how they are improves welfare. We derive conditions affected by labor market conditions or labor under which equilibrium with bargaining is market policy. There is no single way to pro- efficient. We also show that the combination ceed, but any approach requires us to con- of wage posting and directed search gener- front the two questions mentioned above: ates efficiency under quite general assump- how do workers and firms meet; and how do tions, providing a version of the first welfare they determine wages? theorem for economies with frictions. In section 4, we present a class of equilib- Finally, we finish in section 8 with a few rium models built on two main ingredients: concluding remarks.3 the matching function, which determines how workers and firms get together, and the 2. Basic Job Search bargaining solution, which determines We begin here with the familiar discrete- wages once they do. The matching function time formulation of the basic job search helps us in analyzing transitions from unem- model, and then derive its continuous-time ployment to employment as a function of, in analogue, which is used for the remainder of general, the behavior of all the workers and the paper. We then use the framework to dis- firms in the market. Bargaining is one of the cuss some issues relating to unemployment more popular approaches to wage determi- duration and wages. nation in the literature, and since it is a key ingredient in many models we take some 2.1 Discrete Time time to explain how the generalized Nash Consider an individual searching for a job solution works and how to interpret it in in discrete time, taking market conditions as terms of strategic bargaining theory. t given.4 He seeks to maximize E β x , In section 5, we consider an environment t=0 t where wages are posted ex ante, rather than 3 Earlier surveys of search theory as applied to labor bargained after agents meet, and in addition markets include Steven A. Lippman and John J. McCall where search is directed—i.e., workers do (1976a), Mortensen (1986), and Mortensen and not encounter firms completely at random Christopher A. Pissarides (1999a, 1999b). While there is naturally some overlap, there are important differences in but try to locate those posting attractive our approach. We build equilibrium models up from the terms of trade. Models with the combination decision problem, focusing on the role of random versus of wage posting and directed search, called directed search and the role of bargaining versus wage posting. In particular, previous surveys do not examine competitive search models, behave quite dif- directed search models, or explicitly discuss the efficiency ferently from those in section 4, although properties of different models, as we do. they can be used to analyze similar issues. In 4 While it is often said that the economics of search began with George Stigler (1961), his formulation was stat- section 6, we consider models where wages ic. The sequential job search model in this section was are posted but search is once again purely developed first by McCall (1970), Mortensen (1970), and random. This class of models has been wide- Reuben Gronau (1971), although others had analyzed related problems, including Herbert A. Simon (1955), who ly used in research on wage dispersion and discussed the housing market, and Samuel Karlin (1962), the distribution of individual employment/ who discussed asset markets. The presentation in this and unemployment spells. the next section is based on some lecture notes that con- tain many more details, examples and exercises, and can be In section 7, we discuss efficiency. This is found at http://www.ssc.upenn.edu/~rwright/courses/ important because, to the extent that one courses.html. de05_Article1 11/16/05 3:53 PM Page 962

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− where β ∈ (0,1) is the discount factor, x is w < wR and accept w ≥1wR (we adopt the t max{UWwdFw,. ( )} ( ) + U income at t, and E denotes the expectation. convention0 that he accepts1+ r when indiffer- Income is x = w if employed at wage w and ent). Substituting U = wR /(1 − β) and x = b if unemployed. Although we refer to w W(w) = w/(1 − β) into (2), we have as the wage, more generally it could capture = some measure of the desirability of the job, (3) wTwRR() depending on benefits, location, prestige, > ()1 −+ββbwwdFw∫ max{}(),. etc., and although we refer to b 0 as unem- 0 R ployment insurance (UI), it can also include the value of leisure or home production.5 The function T is easily shown to be a con- We begin with the case where an unem- traction, so there is a unique solution to = ( ) ployed individual samples one independent- wR T wR . This implies that if one fixes w0 = ( ) ly and identically distributed (i.i.d.) offer and recursively defines wN+1 T wN , the w → each period from a known distribution F(w). sequence converges to R as N . If the w = If an offer is rejected, the agent remains initial wage is 0 b, the worker’s reserva- unemployed that period. Assume previously tion wage in the final period of a finite hori- rejected offers cannot be recalled, although zon problem, wN has the interpretation of this is actually not restrictive because the being the reservation wage when N periods problem is stationary, so an offer that is not of search remain, after which the worker acceptable today will not be acceptable receives either b or the accepted wage w tomorrow. For now we assume that if a job is forever. accepted the worker keeps it forever. Hence, The optimal search strategy is completely we have the Bellman equations characterized by (3), but we also present some alternative representations that are (1) Ww()=+ wβ Ww () often seen in the literature. First, subtract- β ing wR from both sides of (3) and simplify- (2) Ub=+β ∫ max{} UWwdFw,, ( ) ( ) ing gives the standard reservation wage 0 equation β where W(w) is the payoff from accepting a (4) wb=+ ∫ ()() wwdFw− . R − β w R wage w (W stands for working) and U is the 1 R payoff from rejecting a wage offer, earning b, and sampling again next period (U stands for Using integration by parts, we can also write unemployed). this as − β β Since W(w) w/(1 ) is strictly increas- =+ − (5) wbR ∫ [()]1 Fwdw, ing, there is a unique wR, called the reserva- 1 − β wR tion wage, such that W(wR) U, with the property that the worker should reject which, as we shall see, is handy in some of the applications below.6 5 Our worker is interested in maximizing expected dis- counted income. This is the same as maximizing expected 6 Note that in the above analysis, as in most of what we utility if he is risk neutral, but also if he is risk averse and do here, it is assumed that the worker knows F. If he has consumption markets are complete, since then he can to learn about F while searching, the problem gets harder maximize utility by first maximizing income and then and a reservation strategy may not even be optimal. For = smoothing consumption. The case of a risk averse agent example suppose we know either: (a) w w0 with prob 1; = π = − π facing incomplete markets is more difficult. Early analyses or (b) w w1 with prob and w w2 with prob 1 . If > > π include John P. Danforth (1979) and John R. Hall, w2 w1 w0 and is small, it can be optimal to accept w0

Lippman, and McCall (1979); more recent studies include but not w1 since an offer of w1 signals that there is a good

Victor Valdivia (1996), James Costain (1997), Daron chance of getting w2. Michael Rothschild (1974) gives Acemoglu and Shimer (1999b, 2000), Martin Browning, conditions that guarantee a reservation strategy is optimal. Thomas F. Crossley, and Eric Smith (2003), and Rasmus See Burdett and Tara Vishwanath (1988a) for additional Lentz and Torben Tranaes (forthcoming). discussion and references. de05_Article1 11/16/05 3:53 PM Page 963

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2.2 Continuous Time that he gets an offer , times the expected We now derive continuous-time versions increase in value associated with the offer, of the above results. First, generalize the noting that the offer can be rejected. The reservation wage wR satisfies discrete-time model to allow the length of a = − __1__ W(w ) U, so equation (10) implies W(w) period to be ∆. Let β = and assume R 1 r U = (w − w )/r. Substituting this into (11) that the worker gets a wage offer with prob- R ability ∆ in each period.7 Then the payoffs gives the continuous time reservation wage to working and unemployment satisfy the equation following versions of (1) and (2) =+ − (12) wR b ∫ ()()wwdFwR . r wR 1 (6) Ww()=+ w Ww () Again one can integrate by parts to get 1 + r =+ − (13) wbR ∫ [()]1 Fw dw. r wR (7) Ub=+ ∫ 1+ r Although most of the models that we dis- 1 − cuss assume fixed search intensity, it can be ×∫ max{UWwdFw,. ( )} ( ) + U 0 1+ r endogenized. Suppose a worker can affect the arrival rate of offers , at cost g(), where Algebra implies g > 0 and g > 0. Unemployed workers choose to maximize rUw , where =+( ) R (8) rW() w1 r w =− + − (14) wbgR ()∫ (wwdFwR ) (). r wR =+( ) (9) rUrb1 The first order condition for an interior solution is +−∫ max {0,.Ww ( ) UdFw } ( ) 0 −= (15) ∫ ()()()wwdFwrgR . wR When ∆→0, we obtain the continuous time Worker behavior is characterized by a pair Bellman equations (wR, ) solving (14) and (15). It easy to show that an increase in b, e.g., raises w and rW() w= w R (10) reduces .

(11) rU=+ b∫ max {0,. W ( w ) − U } dF ( w ) 2.3 Discussion 0

Intuitively, while U is the value of being Traditional frictionless models assume that unemployed, rU is the flow (per period) a worker can costlessly and immediately value. This equals the sum of the instanta- choose to work for as many hours as he wants neous payoff b, plus the expected value of at the market wage. By relaxing these any changes in the value of the worker’s extreme assumptions, search models allow us state, which in this case is the probability to think about unemployment and wages in a different light. Consider unemployment duration. The probability that the worker has − 7 Alternatively, we may assume that the number of wage not found a job after a spell of length t is e Ht, offers per period is a Poisson random variable with mean where H = [1 − F(w )] is called the hazard . When the time period is short, the probability of R receiving multiple offers within a single period is negligible. rate and equals the product of the contact See Mortensen (1986) for details. rate and the probability of accepting de05_Article1 11/16/05 3:53 PM Page 964

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− 8 assumed above that when a worker accepts a 1 F(wR). The average duration of an unemployment spell is therefore job he keeps it forever. Yet according to Bruce Fallick and Charles A. Fleischman − 1 (16) DtHedt==∫ Ht . (2004), in the United States from 1994 to 0 H 2004, 6.6 percent of employment relation- Also, the observed distribution of wages paid = ⏐ ≥ ships ended in a given month (of these, forty is G(w F(w w wR). percent of workers switched employers Consider the impact of an increase in b, while the rest either became unemployed or say more generous UI, assuming for simplic- left the labor force). We now generalize the ity here that search intensity and hence are framework to capture such transitions. fixed. From (12), the immediate effect is to 3.1 Transitions to Unemployment increase wR, which has two secondary effects: the distribution of observed wages The simplest way of generating transitions G(w) is higher in the sense of first order sto- from employment into unemployment is to chastic dominance, since more low wage assume that jobs end for some exogenous offers are rejected; and the hazard rate H is reason; sometimes in the literature this is lower, which increases average unemploy- interpreted by saying that workers face lay- ment duration. Hence, even this elementary off risk. A tractable formulation is to assume model makes predictions about variables that this occurs according to a Poisson that would be difficult to generate using a λ 9 process with parameter , which for now is theory without frictions. an exogenous constant.10 Introducing exogenous separations does 3. Worker Turnover not affect the Bellman equation for U, Although the model in the previous sec- which is still given by (11), but now we have tion is interesting, there are important issues to generalize (10) to that it cannot address. For instance, we (17) rW() w=+ wλ [ U − W ()] w . 8 Because this simple model is stationary, H does not The reservation wage still satisfies W(w )U, change with the duration of unemployment. Several gen- R eralizations would overturn this, the simplest being to and the methods leading to (13) yield assume a finite horizon. More interestingly, Burdett (1979) and Mortensen (1977) allow UI to vary over time. (18) wb=+ ∫ [()]1 − Fw dw. R + λ w Lippman and McCall (1976b) and Lippman and John W. r R Mamer (1989) allow wage offers to vary over time. Salop λ (1973) studies systematic search where a worker first looks Notice that affects wR only by changing the for opportunities that are best according to some prior and effective discount rate to r + λ . However, a then, if unsuccessful, proceeds to other locations, typically worker now goes through repeated spells of lowering his reservation wage over time. The models dis- cussed earlier in which the worker learns about the wage employment and unemployment: when

distribution also predict wR and hence H vary over time. unemployed, he gets a job at rate Bruce D. Meyer (1990) and Wolpin (1987) are examples of = − H [1 F( wR)], and while an employed a large body of empirical work on hazard rates. λ 9 We can also study changes in F or . As pointed out he loses the job at rate . by Burdett (1981), for some experiments it is useful to A simple way to endogenize transitions to assume F is log-concave. Suppose we increase every w in unemployment is to allow w to change at a F either by a constant or proportionally. Perhaps surpris- ⏐ ≥ given job. Suppose that this happens accord- ingly, E[w w wR] may actually decrease, but it can be guaranteed to increase under log-concavity; see ing to a Poisson process with parameter λ, Mortensen (1986) or Wright and Janine Loberg (1987). and that in the event of a wage change a new Suppose we increase the arrival rate. One might expect

that this must raise the hazard, but since it increases wR, the net effect is ambiguous. One can show H/ > 0 under 10 An interesting extension of the basic model is to let log-concavity; see Christopher J. Flinn and James J. vary across jobs, which implies the reservation strategy Heckman (1983), Burdett and Jan Ondrich (1985), or van generally depends on the pair (w,); see Burdett and den Berg (1994). Mortensen (1980) or Wright (1987). de05_Article1 11/16/05 3:53 PM Page 965

Rogerson, Shimer, and Wright: Search-Theoretic Models of the Labor Market 965

⏐ w is drawn from F(w w). When the wage (22) rW() w=+ w ∫ max{ W ( w )− changes the worker can stay employed at w 1 0 or quit to unemployment. The exogenous − Ww(),0} dFw ( )+−λ [ U W (w)]. layoff model discussed above is a special case where w= 0 with probability 1, so that at λ The second term in (22) represents the rate the job effectively disappears. In this event that an employed worker gets an offer more general model, above his current wage. (19) rW() w=+ wλ ∫ max{ W ( w ) It is easy to see that W is increasing, 0 implying that unemployed workers use a = −−Ww(),|. U Ww ()} dFw ( w ) reservation wage satisfying W(wR) U, and employed workers switch jobs when- > = ⏐ ever w w. Evaluating (22) at w wR and A natural assumption is that F(w w2) first ⏐ combining it with (21), we get order stochastically dominates F(w w1) w > w W w whenever 2 1. This implies ( ) is (23) wb= increasing, and there is a reservation wage R wR such that unemployed workers accept if +− − ≥ ()[()()]()01∫ Ww WwR dFw . w wR and employed workers quit if their w ⏐ R wage falls to w wR. Hence separations are > > decreasing in w. In the simplest case where Observe that wR b if and only if 0 1. F(w|w) = F(w) (independence), we have Thus, if a worker gets offers more frequent- ly when employed than when unemployed, − λ (20) wb=+ ∫ ()()wwdFw− . he is willing to accept wages below b. R + λ w R r R To eliminate W from (23), use integration = + λ + − Notice that λ > implies w < b; in this case, by parts and insert W (w) {r 1 [1 R − workers accept a job paying less than unem- F(w)]} 1, which we get by differentiating ployment income and wait for the wage to (22), to yield change, rather than searching while unem- (24) wb= ployed. In any case the usual comparative R static results, such as ∂w /∂b > 0, are similar R ⎡ 1 − Fw( ) ⎤ to what we found earlier. +− ()01∫ ⎢ ⎥ dww. wR ⎣ rFw++λ []1 −( ) ⎦ 3.2 Job-to-Job Transitions 1 = To explain how workers change employers If 1 0, this reduces to the earlier reserva- without an intervening spell of unemploy- tion wage equation (13). Many results, like ∂ ∂ > ment, we need to consider on-the-job wR / b 0, are similar to what we found search.11 Suppose new offers arrive at rate above, but we also have some new predic- 0 while unemployed and 1 while tions. For instance, when wR is higher, work- employed. Each offer is an i.i.d. draw from ers are less likely to accept a low w, so they F. Assume that employed workers also lose are less likely to experience job-to-job transi- their job exogenously at rate λ. The Bellman tions. Thus, an increase in UI reduces equations are turnover.12 =+ − (21) rU b0 ∫ [() W w U ]() dF w wR 12 As in the basic model, we can endogenize search intensity here. Let g0( ) be the cost of achieving for an unemployed worker and g1( ) the cost for an employed 11 ≤ The on-the-job search model was introduced by worker. If g0( ) g1( ) for all , unemployed workers Burdett (1978). The presentation here follows Mortensen search harder than employed workers. In any case, search and George R. Neumann (1984). intensity decreases with w for employed workers. de05_Article1 11/16/05 3:53 PM Page 966

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= λ − − − 3.3 Discussion u˙ (1 u) 0[1 F(wR)]u. The model of the last subsection is a natu- Over time, this converges to the steady state ral framework in which to analyze individual λ transitions between employment and unem- ∗ = (25)u λ +− []( ) . ployment, and between employers. It makes 0 1 FwR predictions about the relationships between wages, tenure, and separation rates. For One can also calculate the cross-sectional example, workers typically move up the distribution of observed wages for employed wage distribution during an employment workers, denoted by G(w), given any offer spell, so the time since a worker was last ≥ distribution F(w). For all w wR, the flow of unemployed is positively correlated with his workers into employment at a wage no wage. Also, workers who earn higher wages − greater than w is u 0[F(w) F(wR)], equal to are less likely to get better opportunities, the number of unemployed workers times the generating a negative correlation between rate at which they find a job paying between wages and separation rates. And the fact that wR and w. The flow of workers out of this a worker has held a job for a long time typi- − λ + − state is (1 u)G(w){ 1[1 F(w)]}, equal cally means that he is unlikely to obtain a to the number of workers employed at w or better one, generating a negative relation- less, times the rate at which they leave either ship between job tenure and separation for exogenous reasons or because they get an rates. All of these features are consistent offer above w. In steady state, these flows are with the empirical evidence on turnover and equal. Using (25) and rearranging, we have wage dynamics summarized in, e.g., Henry 13 λ[]Fw()− Fw ( ) Farber (1999). ( R (26) Gw) = . With a slight reinterpretation, the frame- []11− Fw(){}λ +− [] Fw () work can also be used to discuss aggregate R 1 variables. Suppose there are many workers, We can now compute the steady state job-to- each solving a problem like the one dis- job transition rate, cussed above, with the various stochastic events (like offer arrivals) i.i.d. across work- − (27) 1 ∫ [(1 Fw)] dGw (). ers. Each unemployed worker becomes wR = − employed at rate H 0[1 F(wR)] and each λ This type of model has been used in a vari- employed worker loses his job at rate , so ety of applications. For example, Wright u the aggregate unemployment rate evolves (1986) uses a version with learning to discuss according to macro aggregates, showing how the combina- tion of search and learning generates consid- 13 Other extensions can also deliver similar prediction. erable persistence in the unemployment rate. One such class of models is the learning models in Jovanovic (1979a, 1979b) and Louis L. Wilde (1979). In Under the interpretation that the learning these models, workers have to learn about how good they comes from a signal-extraction problem, are at a job, and those with longer tenure have less to learn where workers see nominal wages and have to and hence are less likely to leave. Another class of models introduces human capital. Since we do not have space to learn the real wage, the model also generates do justice to this topic, we refer the reader to recent work a Phillips curve relation between inflation and by Gueorgui Kambourov and Iourii Manovskii (2004, unemployment. Unlike previous signal-extrac- 2005) that studies human capital within a search frame- work very similar to what is presented here. Other search- tion models, such as Robert E. Lucas (1972), based models with human capital include Acemoglu unemployment is persistent because search (1996), Lars Ljungqvist and Thomas J. Sargent (1998), provides a natural propagation mechanism. Adrian M. Masters (1999), Coles and Masters (2000), Burdett and Smith (2001), and Laing, Theodore Palivos, Jovanovic (1987) considers a variation that and Ping Wang (1995, 2003). allows workers who are not satisfied with de05_Article1 11/16/05 3:53 PM Page 967

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their current wage to either search for a bet- the idea.14 ter one, or to rest and return to work when In summary, we think it is silly to criticize w increases. This model can generate pro- the framework as being partial equilibrium cyclical quits and productivity, along with per se, since it is trivial to recast it as an equi- countercyclical unemployment, as in the librium model without changing the essence. data. Wright and Loberg (1987) use another The more pertinent question is, do we miss variation to analyze the impact of changes in anything of substance with these stories taxes on unemployment and wages of work- about lakes and islands? The models that we ers at different points in the skill distribu- present below introduce firms and equilibri- tion. Ljungqvist and Sargent (1998) add um considerations using more economics human capital accumulation during employ- and less geography. ment and deaccumulation during unemploy- 4. Random Matching and Bargaining ment and study the effects of policy. Kambourov and Manovskii (2005) use a ver- This section introduces a popular line of sion of the model to study life-cycle earnings research, emanating from Pissarides (1985, profiles. 2000), used to study the determinants of Although these applications all seem arrival rates, match formation, match disso- interesting, there is an old critique of the lution, and wages. We present a sequence of framework—which is that it is partial equi- models that emphasize different margins, librium because the distribution F is exoge- including entry, the decision to consummate nous (Rothschild 1973). From a logical point matches, and the decision to terminate of view, this critique is not compelling, as matches. Before we begin, there are two there are various ways to embed the model issues that need to be addressed: how do into an equilibrium context without chang- workers and firms meet, and how are wages ing the results. For instance, one can imag- determined. These models assume meetings ine workers looking for wages as fishermen are determined through a matching function looking for lakes. Then F is simply the distri- and wages through bargaining. bution of fish across lakes, which is some- 4.1 Matching thing we can logically take as fixed with Suppose that at some point in time there respect to most policy interventions. are v vacancies posted by firms looking for Another approach that involves only a workers and u unemployed workers looking small change in interpretation is to invoke for jobs. Building on ideas in Peter A. the island metaphor often used in search Diamond (1981, 1982a, 1982b), Mortensen theory. Imagine workers searching across (1982a, 1982b), Pissarides (1984, 1985), and islands, on each of which there are many elsewhere, assume the flow of contacts firms with a constant returns to scale tech- between firms and workers is given by a nology using only labor. The productivity of matching technology m = m(u,v). Assuming a randomly selected island is distributed all workers are the same and all firms are the according to F. If the labor market on each same, the arrival rates for unemployed work- island is competitive, a worker on an island ers and employers with vacancies are then with productivity w is paid w. This trivially given by makes the equilibrium wage distribution the same as the productivity distribution, F. In 14 We do not discuss the Lucas–Prescott model in detail, since it can be found in standard textbooks (Nancy fact, this is the Lucas and Edward C. Stokey, Lucas, and Prescott 1989, chapter 13; Ljungqvist Prescott (1974) equilibrium search model, and Sargent 2004, chapter 26). Applications include aside from some minor details—e.g., they Jeremy Greenwood, MacDonald, and Guang-Jia Zhang (1996), Joao Gomes, Greenwood, and Sergio Rebelo allow for decreasing returns to scale, which (2001), Fernando Alvarez and Marcelo Veracierto (1999), complicates the algebra but does not change and Kambourov and Manovskii (2004). de05_Article1 11/16/05 3:53 PM Page 968

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muv(),,muv() 4.2 Bargaining (28) ==and . weu v Consider the situation of a worker and a It is standard to assume the function m is firm who have met and have an opportunity continuous, nonnegative, increasing in to produce a flow of output y. Suppose that both arguments and concave, with if the worker gets a wage w, his expected m(u,0) = m(0,v) = 0 for all (u,v). It is also lifetime utility is W(w) while the firm earns convenient to assume m displays constant expected discounted profit J(π), where returns to scale, i.e., m(u,v) m(u,v). π = y − w. Again W stands for the value of While alternative assumptions are interest- working, and now J stands for the value to ing—e.g., Diamond emphasizes that the firm of a job that is filled. If they fail to increasing returns can generate multiple reach agreement, the worker’s payoff falls to equilibria—constant returns is consistent U and the firm’s to V. Again U stands for the with much empirical work.15 Also, constant value of unemployment, and now V stands returns generates a big gain in tractability, as for the value to the firm of a vacancy. We will it implies w and e depend only on the ratio soon determine U and V endogenously, but v/u, referred to as a measure of market tight- for now take them as given. We are of course > ness. Thus, w is an increasing and e a interested in situations where W(w) U and decreasing function of v/u, and there is a J(y − w) > V for some w, so that there is continuous, decreasing, 1 to 1 relationship something to bargain over. between w and e. A standard approach is to assume that w is The matching technology is meant to rep- determined by the generalized Nash bar- resent in a simple, if reduced-form, fashion gaining solution with threat points U and V, the notion that it takes time for workers and θ (29) wWwU∈−arg max [ ( ) ] firms to get together. Just as a production −θ function maps labor and capital into output, ×× [(Jy−− w ) V ]1 , m maps search by workers and firms into where ∈(0,1) is the worker’s bargaining matches. There are papers that model this power. The solution to the maximization more deeply, some of which we discuss problem satisfies below, but starting with an exogenous matching function allows us to be agnostic (30) θ[)]()Jy( −− w VWw about the actual mechanics of the process by =−θ − − which agents make contact. An advantage is ()[()]()1 Ww UJ y w, that this is a flexible way to incorporate fea- which can be solved for w. Since it is an tures that seem desirable—e.g., more search important building block in this class of by either side of the market yields more models, and since wage determination is one matches—and one can regard the exact of the main themes of this essay, we want to specification as an empirical issue. This discuss Nash bargaining carefully. might make matching a bit of a black box, John Nash (1950) did not actually analyze 16 but it is a common and useful approach. the bargaining process, but took as given four simple axioms and showed that his solu- 15 Early empirical work on matching goes back to tion is the unique outcome satisfying these Pissarides (1986) and Olivier J. Blanchard and Diamond axioms.17 The solution, while elegant and (1989); see Barbara Petrongolo and Pissarides (2001) for a recent survey. 17 Nash actually showed that the unique outcome con- 16 An interesting alternative is to assume the number of sistent with his axioms has = 1/2. Relaxing his symmetry matches depends on both the flows of newly unmatched axiom, (R-Nash) with any ∈(0,1) satisfies the other workers and firms and the stocks of existing unemploy- axioms, and this is what is called the generalized Nash ment and vacancies, as in Coles and Smith (1996, 1998). solution. See, e.g., Martin J. Osborne and Rubinstein See Ricardo Lagos (2000) for another approach. (1990). de05_Article1 11/16/05 3:53 PM Page 969

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practical, is again a black box. However, one This says that in terms of total lifetime can provide a game-theoretic description of expected utility, the worker receives his the bargaining process, along the lines of threat point U plus a share of the surplus, Rubinstein (1982), that has a unique sub- denoted S and defined by game perfect equilibrium with the following =−−+ − property: as the time between counteroffers (34) S Jy()wVWwU () in the game becomes small, the equilibrium −− = yrUrV outcome converges to the prediction of the + λ , Nash solution for particular choices of the r threat points and bargaining power that where the last equality is derived by using 18 depend on details of the underlying game; (31) and (32). see e.g., Osborne and Rubinstein (1990). From (31) and (32) we have W(w) − U = _w__w_R π − = ____R π For instance, suppose that each agent has a r and J( ) V r where wR and R given probability of proposing an offer (as are reservation wage and profit levels for the opposed to responding) in each round of worker and firm. Then (29) reduces to bargaining; everything else equal, this game θθ1− (35) wwwyw∈−−−arg max [ ] [π ] , generates the same outcome as the Nash RR solution in which the bargaining power which has solution equals that probability. θ =+−−θπ Of course this only pushes back one (36) w wyRRR() w. level—where does that probability come from? One position is to say that the nature Hence, in this model the Nash solution also splits the surplus in terms of the current of bargaining may well differ across indus- ≥ θ period utility. Notice that w wR if and only tries, countries, and so on, and varying is ≥ = π + π = − ≥ π if y yR R wR. Similarly, y w R if one way to try and capture this. Also, at least ≥ in simple models, as we vary θ between 0 and only if y yR. Hence, workers and firms agree to consummate relationships if and and 1 we trace out the set of bilaterally effi- ≥ cient and incentive compatible employment only if y yR. relationships, which would seem to cover the 4.3 Equilibrium cases of interest. Just as the matching func- tion is not the last word on how people meet, We now combine matching and bargain- Nash bargaining is not the last word on wage ing in a model where a firm’s decision to post determination, but it is a useful approach. a vacancy is endogenized using a free entry To proceed, suppose as usual that workers condition. There is a unit mass of homoge- and firms are risk neutral, infinitely lived, and neous workers, and unmatched workers discount future payoffs in continuous time at search costlessly while matched workers rate r, and that matches end exogenously at cannot search. We focus here on steady rate λ. Then we have states, and let u and v represent unemploy- ment and vacancies. The steady-state unem- = + λ − = λ / λ + (31) rW(w) w [U W(w)] ployment rate is u ( w), where = / w m(u,v) u and m is the matching tech- (32) rJ(π) = π + λ[V − J(π)]. nology. As we discussed above, assuming constant returns, once we know we know Ww()== J ()π 1 w This implies r+λ . Inserting since both are functions of u/v. these into (30) and rearranging gives e = (33) W(w) U 18 Notice that S is independent of the wage. Intuitively, S corresponds to the joint payoff available in the match, + θ[J(y − w) − V − W(w) − U]. while w simply divides this among the agents. de05_Article1 11/16/05 3:53 PM Page 970

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They value− ydFy)() of posting. a vacancy is outside of a match, given S. Here the only R such decision is whether to post a vacancy. rVk=− + [( Jπ ) − V ], (37) e Using (38) and the fact that bargaining π = − θ where k is a flow cost (e.g., recruiting costs). implies J( ) (1 )S, we have As free entry drives V to 0, we need not keep (42) kS=− ()1 θ . track of V, and we can rewrite (37) as e (38) Jk()π = . Equilibrium is completely characterized by e (41) and (42). Indeed, we can combine The value of unemployment satisfies them as r ++λθ yb− (39) rUb=+ [() WwU − ], w = w (43) −θ . ()1 e k while the equations for W and J are Under standard regularity conditions, a unchanged from (31) and (32). Formally, an unique solution for w exists. From this we equilibrium includes the value functions can recover the wage, (J,W,U), the wage w, and the unemployment and vacancy rates (u,v), satisfying the (44)w =−y ()()rS +λθ1 − . Bellman equations, the bargaining solution, free entry, and the steady-state condition.19 Finally, the steady state unemployment rate In terms of solving this model, one satisfies an equation analogous to (25), approach would be to try to find the equilib- accounting for the fact that all meetings rium wage. Start with some arbitrary w, result in matches: π solve (32) for J( ), and then use (38) to solve ∗ λ u = . for e and w. This determines W and U. λ + This w is an equilibrium if and only if the w implied values for J, W, and U are such that A number of results now follow easily. For the bargaining condition holds. While this example, an increase in b reduces the rate at works, here we bypass w by working directly which workers contact firms w, raises the with the surplus, which from (34) is rate at which firms contact workers e, reduces S, and raises w. The conclusion that (40) ()rSyr+=−λ U. unemployment duration and wages increase Now (33) allows us to rewrite (39) as with UI is similar to what we found earlier, = + rU b w S, and (40) gives but here the mechanism is different. In the single-agent model, an increase in b induced ++λθ =− (41) ()rSybw . the worker to raise his reservation wage, and to reduce search intensity if it is endoge- The next step generally in this method is nous. Now an increase in b raises the bar- to obtain expressions that characterize opti- gained wage, which discourages job mal choices for each of the decisions made creation, thereby increasing unemployment duration. 19 Rather than having entry, one can assume a fixed num- 4.4 Match-Specific Productivity ber of firms and then equilibrium determines V endoge- nously. Also, although we focus on steady states, dynamics In the above model, it takes time for work- here are straightforward. The key observation is that the free entry condition pins down e and therefore w. Hence, ers and firms to get together, but every con- given any initial unemployment rate, vacancies adjust so that tact leads to a match and w is the same in u/v jumps to the steady state level, which implies all other every match. This seems quite special when variables are constant along the path as u and v converge to their steady state levels. See Mortensen (1989, 1999), e.g., compared to what we did in sections 2–3, as for related models with more complicated dynamics. it corresponds to workers sampling from a de05_Article1 11/16/05 3:53 PM Page 971

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degenerate distribution. Moreover, in appli- +=λθ − − (50) ()rke (1 ).∫ ( yydFyR )() cations, changes in the probability that a yR contact leads to a match may be important. Here we extend the model so that not every We can now solve for yR and w from (49) contact results in a match and not every and (50). The first of these equations match has the same w. The easiest way to describes an increasing relationship between proceed is to assume that when a worker and w and yR (when it is easier for a worker to firm meet they draw match-specific produc- find a job, he is more willing to turn down a tivity y from a distribution F, where y is potential match with low productivity). The observed by both and constant for the dura- second gives a decreasing relationship tion of the match. From section 4.2, workers between yR and w (when yR is higher, ≥ and firms agree to match if and only if y yR, matches are less profitable for firms so they where yR is characterized below. post fewer vacancies). There exists a unique In equilibrium, workers in a match with equilibrium under standard conditions. One productivity y earn wage w(y) satisfying can again recover the wage function: since = = υ > the Nash bargaining solution. Let W (w) w(yR) yR and w (y) if y yR, we have y = + θ − be the value for an employed worker of a w(y) yR (y yR). match with productivity y earning w, Equilibrium also determines the − Jy(y w) the value for a firm with a filled observed distribution of productivity across job at productivity y earning profits y − w, existing relationships, or equivalently, given

and Sy the surplus in a job with productivity w(y), the observed wage distribution G(w). y. Generalizing (40) gives Since the distribution of match productivity +=−λ is F(y) truncated at yR, the equilibrium wage (45) ()rSyrUy . distribution G(w) is determined by yR, F(y), Only the Bellman equations for an unem- and w(y). ployed worker and the free entry condition It is easy to discuss turnover and wages. change appreciably, becoming For example, an increase in b shifts (49) but ∞ =+ − not (50), resulting in an increase in yR,a (46) rU bw ∫ { Wy [ w ()] y U } dF () y y reduction in w, and a reduction in R = − ∞ H w[1 F(yR)]. From a worker’s per- =+bSdF θ ∫ (y) spective, this closely resembles the single- w y y R agent problem, in the sense that he receives ∞ =− offers at rate w from a given distribution, (47) kJywydFye ∫ y[()]() yR and needs to decide which to accept, except ∞ =− θ now the arrival rate and distribution are e()1 ∫ SdFyy (). yR endogenous. To solve this model, combine (46) and (47) 4.5 Endogenous Separations α w k to get rU = b + α——− and substitute into (45): e(1 ) Mortensen and Pissarides (1994) endoge- θk (48) ()rSyb+=−−λ w . nize the separation rate by incorporating on- y −θ 20 e()1 the-job wage changes. The resulting = framework captures endogenously both the In particular, yR satisfies SyR 0, or

θk 20 This section mimics what we did in the single-agent (49) yb=+ w . R −θ problem in section 3. Other extensions discussed there can e()1 also be added, including on-the-job search (Pissarides 1984, 1994), and learning (Michael J. Pries 2004, Since Sy is linear in y, this implies = − + λ Giuseppe Moscarini 2005, and Pries and Richard Sy (y yR)/(r ), so (47) can be written Rogerson 2005). de05_Article1 11/16/05 3:53 PM Page 972

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flows into and out of unemployment. Given standard conditions. See Mortensen and that these flows vary a lot across countries Pissarides (1994, 1999b) for details of the and over time, it allows one to begin thinking argument. formally about factors that may account for these differences. To proceed, let y be cur- 4.6 Discussion rent productivity in a match, and assume There are many applications and exten- λ that at rate we get a new draw from sions of this framework. David Andolfatto F(y |y), where F(y |y2) first order stochasti- (1996), Monika Merz (1995, 1999), Harold > cally dominates F(y |y1) whenever y2 y1. It L. Cole and Rogerson (1999), Wouter J. den remains to specify the level of productivity in Haan, Gary Ramey, and Joel Watson (2000), new matches. One can assume they start at a Costain and Michael Reiter (2003), Shimer random y, but we assume all new matches (2005), and Robert E. Hall (2005) all study begin with the same y0. versions of the model quantitatively. There An equilibrium is defined as the natural is a literature that uses versions of the extension of the previous model, and we can model to study the behavior of worker and jump directly to the equation for the surplus job flows across countries as well as over the business cycle, including Stephen P. Millard ()rSyrU+=−λ (51) y and Mortensen (1997), Alain Delacroix (2003), Blanchard and Pedro Portugal + λ y ∫ SdFyy ()|. y (2001), and Pries and Rogerson (2005). yR There is also a literature that introduces = + heterogeneous workers and firms, including Since rU b Sy , this can be rewritten w 0 Acemoglu (1999, 2001), James Albrecht and (52) ()rSybS+=−−λθ Susan Vroman (2002), Mortensen and ywy0 Pissarides (1999c), Shimer (1999), and y Shimer and Smith (2000). Ricardo J. + λ ∫ SdFyy ()|. y Caballero and Mohamad L. Hammour yR (1994, 1996) and Gadi Barlevy (2002) study To close the model we again use free entry, whether recessions are cleansing or sullying in terms of the distribution of match quality (53) kS=−βθ ()1 . ey0 (do they lead to more good jobs or bad jobs). Moscarini (2001) also studies the Finding equilibrium amounts to solving (53) nature of match quality over the business and (52) for yR and w. cycle. We do not have space to do justice to The solution is more complicated here all the work, but this illustrates that it is an because we are now looking for a fixed point active and productive area. in a system of functional equations—(52) defines both yR and Sy as functions of w. 5. Directed Search and Posting Nevertheless, an increase in w reduces Sy for all y and hence raises the reservation We now move to models where some

wage yR. Thus, (52) describes an increasing agents can post wage offers, and other relationship between w and yR. At the same agents direct their search to the most time, (53) indicates that when w is higher attractive alternatives. Following Espen R.

Sy0 must be higher, so from (52) yR must be Moen (1997) and Shimer (1996), the com- lower, and this defines a decreasing rela- bination of posting and directed search is tionship between w and yR. The intersec- referred to as competitive search. Note that tion of these curves gives steady-state it is the combination of these features that equilibrium, which exists uniquely under is important; in section 6 we consider wage de05_Article1 11/16/05 3:53 PM Page 973

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posting with random search, which is quite must pay cost k, and we can either assume different. free entry (making v endogenous) or fix the The literature has proposed several equiv- number of vacancies. Any match within the alent approaches. One posits a group of period produces output y, which is divided agents called market makers who set up sub- between the worker and firm according to markets, with the property that any match the posted wage. At the end of the period, consummated in a submarket must be at the unmatched workers get b, while unmatched posted wage. Within each submarket there is vacancies get 0. Then the model ends. a constant returns matching function m(u,v), Consider a worker facing a menu of dif- so that the arrival rates w and e are deter- ferent wages. Let U denote the highest value mined by q = u/v, which is called the queue that he can get by applying for a job at some length (the inverse of market tightness). firm. Then a worker is willing to apply to a Each unemployed worker and each firm particular job offering a wage w ≥ b only if with a vacancy take as given w and q in every he believes the queue length q at that job— submarket, and go to the one offering the i.e., the number of workers who apply—is highest expected utility. In equilibrium, q in sufficiently small. In fact, he is willing to each submarket is consistent with agents’ apply only if w(q) is sufficiently large, in the expectations and no market maker can post a sense that different wage and attract both workers and (54) Uqwqb≤ ()+− [1 ()]. employers.21 ww Another approach supposes that employ- If this inequality is strict, all workers would ers themselves post wages, and unemployed want to apply to this firm, reducing the right workers direct their search to the most hand side. Therefore, in equilibrium, if any attractive firms. A high posted w attracts workers apply to a particular job, q adjusts to more applicants, which reduces workers’ satisfy (54) with equality. contact rate w and raises the employer’s To an employer, (54) describes how a contact rate e. In equilibrium, workers are change in his wage w affects his queue length indifferent about where to apply, at least q. Therefore he chooses w to maximize among posted wages that attract some work- =−+− ers. Firms choose wages to maximize expect- (55) Vkqywmaxe()( ) , ed profit. Still another approach assumes wq, that workers post wages, and firms direct taking (54) as a constraint. Note that each their search to them. As we said, these employer assumes he cannot affect U, approaches are equivalent, in the sense that although this is an endogenous variable to be they give rise to identical equilibrium condi- determined in equilibrium. Eliminating w tions. For brevity we consider only the case using (54) at equality, and also using = where firms post wages. e(q) q w(q), we get 5.1 A One-Shot Model (56) Vk=−max q We first discuss the basic mechanism in a +−−− ()(qy b ) qU ( b ). static setting. At the beginning of the period, e there are large numbers u and v of unem- The necessary and sufficient first order ∗ ployed workers and vacancies, and q = u/v is condition is the queue length. Each firm with a vacancy −=− (57) e ()(qy b ) U b. 21 The idea here is that market makers compete to In particular, (57) implies all employers attract workers and firms to their submarkets since they choose the same q, which in equilibrium can charge them an entrance fee, but in equilibrium this ∗ fee is 0 due to free entry into market making. must equal the economywide q . Hence (57) de05_Article1 11/16/05 3:53 PM Page 974

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pins down the equilibrium value of U. Then and worker 2 to firm 2; worker 1 applies to (54) at equality determines the market wage firm 2 and worker 2 to firm 1; and both workers use identical mixed strategies, ∗∗=+ε − (58) wb()( qyb ), applying to each firm with probability 1/2. ∗ ∗ ∗ ∗ q e (q ) One can argue that the coordination implied where ε(q ) ≡ _____∗_is the elasticity of (q ), e ( q ) e by the first two equilibria is implausible, at which is in (0,1) by our assumptions on the least in large labor markets, and so the matching function m. Comparing (58) with mixed-strategy equilibrium is the natural the results in section 4.2, notice that this outcome. This introduces a coordination wage rule operates as if the worker and firm friction, as more than one worker may apply bargained over the gains from trade, with for the same job. the workers’ share θ given by the elasticity ∗ Generalizing this reasoning, suppose (q ); this has important implications for the there are u unemployed workers and v efficiency of competitive search, as we dis- vacancies, for any u and v. If each worker cuss below. applies to each firm with equal probability, Substituting (58) into (55) pins down any firm gets a worker with probability − − _1 u (59) Vk=− 1 (1 v) . Taking the limit of this expres- sion as u and v go to infinity with q = u/v + ∗ − ∗ ∗ − [ e(q ) q e(q )](y b). fixed, in a large market a fraction e(q) If the number of vacancies v is fixed this = 1 − eq of firms get a worker. This is a stan- gives profit; or we can use free entry V = 0 to dard result in statistics. Suppose there are u ∗ endogenize v and hence q . In either case, balls independently placed with equal prob- the model is simple to use. Indeed, this ability into each of v urns. Then for large u model looks a lot like a one-shot version of and v, the number of balls per urn is a the random search and bargaining model. Poisson random variable with mean u/v, so a There is a key difference, however: here the fraction eu/v of the urns do not get any balls. surplus share is endogenously determined. For this reason, this process is often called an urn–ball matching function. 5.2 Directed Matching Because the urn–ball matching process Before generalizing to a dynamic setting, provides an explicit microeconomic story of we digress to discuss some related literature. both meetings (a ball is put in a particular James D. Montgomery (1991) studies a nas- urn) and matches (a ball is chosen from that cent version of the above model (see also urn), it is suitable for environments with het- Michael Peters 1984, 1991). He starts with erogeneous workers (not all balls are the

two unemployed workers and two firms. same). For instance, suppose there are u1 First firms post wages, then each worker type 1 workers and u2 inferior type 2 work- = + applies to one of them (possibly randomly). ers, with u u1 u2. Firms hire type 1 work- A firm receiving at least one application ers over type 2 workers when both apply; hires at the posted w; if more than one hence they hire a type 1 worker with proba- 22 applies the firm selects at random. bility 1 − e u1/v and a type 2 worker with prob- Suppose both firms offer the same w > 0. ability e u1/v(1 − e u2/v). They hire some Then there are three equilibria in the appli- worker with probability 1 − eu/v.23 cation subgame: worker 1 applies to firm 1 There are several generalizations of this matching process. Burdett, Shi, and Wright 22 By assumption here firms post wages rather than more general mechanisms. Coles and Jan Eeckhout (2003) 23 Therefore an increase in the number of undesirable relax this (e.g., they allow w to be contingent on the num- type 2 workers does not affect the matching rate for type 1 ber of applicants who turn up) and show it does not affect workers, but an increase in the number of desirable work- the main conclusions. ers adversely affects undesirable workers. de05_Article1 11/16/05 3:53 PM Page 975

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(2001) let some vacancies hire multiple Now suppose firms choose w and q to max- workers. Albrecht, Pieter Gautier, and imize rV, or equivalently by free entry, to max- − Vroman (2003), and Albrecht, Gautier, imize e(q)(y w). They take (62) as given. Serene Tan, and Vroman (2004) let workers Eliminating w using this constraint and again = make multiple applications. If workers can using e(q) q w(q), this reduces to simultaneously apply for every job, this yrU− effectively flips the urn–ball problem (66) max ()q −−qrU ( b ) . e + λ around, so a worker is employed if at least q r one firm offers him a job; see Benoit Julien, The necessary and sufficient first order John Kennes, and Ian King (2000). The condition, intermediate case in which workers can yrU− ()q =−rU b, apply for a subset of jobs delivers additional (67) e + λ possibilities. In general, these directed r search models provide a way to get inside has a unique solution, so all firms choose the the black box of the matching process.24 same q. Eliminating U and w from (62), (65), and (67) we get an implicit expression for q, 5.3 A Dynamic Model rq++λ () yb− To get something like the basic Pissarides e = (68) − . model with directed search, start with an ee()qqq () k unemployed worker. Suppose he anticipates an unemployment–vacancy ratio q and a This pins down the equilibrium q, or wage w. Then equivalently the arrival rates w and e. Under standard conditions the solution is rU=+ b ()[ q W ( w ) − U ] (60) w unique. We can again perform standard (61) rW() w=+ wλ [ U − W ()] w . exercises, such as changing b, with similar results: here this raises the u/v ratio, which It is convenient to combine these into reduces w, raises e, and increases w. But ()(qw− rU ) although the conclusions are similar to those (62) rU=+ b w . r + λ reached in the previous section, the mecha- Similarly, for firms nism is quite different. An increase in b in this model makes workers more willing to =− + − − (63) rV k e(q)[J(y w) V] accept an increase in the risk of unemploy- ment in return for an increase in w. Firms − = − + λ − − (64) rJ(y w) y w [V J(y w)]. respond by offering workers what they want—fewer jobs at higher wages. Free entry yields ()(qy− w ) 5.4 Discussion (65) k = e . r + λ Competitive search equilibrium theory provides arguably a more explicit explana- tion of the matching process and of wage 24 In these models, generally, one has to be somewhat careful about strategic interaction between firms. In the determination than the bargaining models in previous section, in maximizing (55) subject to (54), the section 4. Nash bargaining says w divides the firm takes as given that a worker’s market payoff is U. But, surplus into exogenous shares; here workers in general, if a firm changes its wage then U will change. Burdett, Shi, and Wright (2001) solve the model with finite face a trade-off between a higher wage and a numbers of agents, where each firm must compute the lower probability of getting a job, while firms effect of a change in its w on workers’ strategies and on the face a trade-off between profit and the prob- implied U. In the limit as u and v grow, they show that the problem is equivalent to one where each firm treats U ability of hiring. Competition among wage parametrically, as we assumed above. setters, whether these be firms, workers, or de05_Article1 11/16/05 3:53 PM Page 976

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market makers, leads to a point along the wage dispersion, which tries to understand indifference curves of agents trading off how workers with identical productivity can wages and arrival rates.25 be paid different wages. Note that the mod- However, a potential disadvantage of els in the previous two sections generate these models—indeed, of any model that wage dispersion only if workers are either ex assumes posting—is that it is a strong ante or ex post heterogeneous. A pure theo- assumption to say that agents commit to the ry of wage dispersion is of interest for a cou- posted terms of trade. If the markets is real- ple of reasons. First, the early literature ly decentralized, what prevents them from suggested that search is relevant only if the trying to bargain for a different w after they distribution from which you are sampling is meet? Again, this comes up in any posting nondegenerate, so theorists were naturally model. In the context of the wage posting led to study models of endogenous disper- model in the next section, Coles (2001) sion. Second, many people see dispersion as shows explicitly how to prevent firms from a fact of life, and for them the issue is reneging on their posted wage using reputa- empirical rather than theoretical. tional considerations, but there is more work As Mortensen (2003, p. 1) reports, to be done on the issue. “Although hundreds if not thousands of In terms of other extensions, Coles and empirical studies that estimate so-called Eeckhout (2000), Shi (2001, 2002), and human capital wage equations verify that Shimer (forthcoming) add heterogeneity, worker characteristics that one could view as which introduces wage dispersion among indicators of labor productivity are positively heterogeneous workers, and among identical related to wages earned, the theory is woe- workers at different firms. Acemoglu and fully incomplete in its explanatory power. Shimer (1999b) allow workers to be risk- Observable worker characteristics that are averse, which means it is no longer possible supposed to account for productivity differ- to solve the model explicitly. They show that ences typically explain no more that 30 per- an increase in risk aversion reduces wages, cent of the variation in compensation.” while an increase in b raises it. Building on Eckstein and van den Berg (forthcoming) this, Acemoglu and Shimer (2000) show that argue that “equilibrium search models pro- UI can enhance productivity. Much more vide a framework to empirically analyze the research is currently being done on directed sources of wage dispersion: (a) workers het- search models. Again, while we cannot do erogeneity (observed and unobserved); (b) justice to all of the work in the area, we want firm productivity heterogeneity (observed to say that it appears to have great potential. and unobserved); (c) market frictions. The equilibrium framework can . . . empirically 6. Random Matching and Posting measure the quantitative importance of each source.” 27 We now combine random matching, as in 26 Diamond (1971) is an early attempt to section 4, with posting, as in section 5. construct a model of dispersion, and This class of models has been used exten- although it did not work, it is useful to sively in the literature on the pure theory of understand why. Consider an economy where homogeneous workers each face a 25 In section 7, we show that competitive search equi- librium achieves the optimal trade-off between these factors. 27 As they report, van den Berg and Geert Ridder 26 The remaining combination—directed search and (1998) estimate that up to 25 percent of wage variability is bargaining—is not so interesting, at least if firms are homo- attributable to frictions, in the sense that this is what would geneous, since there is nothing for workers to direct their emerge from a posting model that ignored heterogeneity, search toward. Lawrence Uren (2004) studies directed while Fabien Postel-Vinay and Jean-Marc Robin (2002) search and bargaining with heterogeneous firms. estimate up to 50 percent. de05_Article1 11/16/05 3:53 PM Page 977

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standard search problem. We do not give all leisure—which leads to heterogeneity in the particulars, since the model is a special reservation wages and this makes the above case of what is described in detail below, but trade-off operational. the key is that the offer distribution F is gen- Consider two types of workers, some with = = > erated by wage-posting firms, each of which b b1 and others with b b2 b1. For any has a constant returns technology with labor wage distribution F there are two reserva- > as the only input, with productivity y. A firm tion wages, w1 and w2 w1. If Wi(w) is the hires any worker it contacts who is willing to value of a type i worker who is employed at

accept its posted w. Consider an individual wage w and Ui is the value of an unemploy- firm. Letting F be the distribution of wages ed type i worker, these wages satisfy = = posted by other firms, he wants to maximize W1(w1) U1 and W2(w2) U2. Generalizing expected profit taking F as given. An equi- our logic from the Diamond model, no firm

librium is defined as a distribution such that posts a wage other than w1 or w2. It is possi- every wage posted with positive probability ble that these two wages could yield equal earns the same profit, and no other wage profit, however, since low-wage firms can = earns greater profit. hire only workers with b b1 while high- Diamond provides a rather striking result: wage firms can hire everyone they contact. there is a unique equilibrium, and in it all In the Albrecht–Axell model, equal profits at employers set the same wage, w = b. The two different wages can occur in equilibrium proof is simple. Given any F, since workers for a large set of parameters.28 are homogeneous they all choose the same To see how this works, normalize the

reservation wage wR. Clearly, no firm posts measure of firms to 1, and let the measure of < = + w wR as this would mean it cannot hire, workers be L L1 L2 where Lj is the meas- > and no firm posts w wR as it can hire every ure with bj. Let be the endogenous frac- = worker it contacts at w wR. To see why it tion of firms posting w2. Any candidate turns out that w = b, consider an individual equilibrium wage distribution is completely > firm when all firms are posting w b. If it summarized by w1, w2, and . Observe first deviates and offers a wage slightly less than that the reservation wage of type 2 workers is = w, it still hires every worker it meets. Since w2 b2. It is clear that their reservation wage this is true for any w > b, we must have w = b cannot be any lower, since otherwise they in equilibrium. The model not only fails to would prefer to remain unemployed. On the rationalize wage dispersion, it fails to explain other hand, if their reservation wage exceeds

why workers are searching in the first place! b2, an argument like the one we used for the Diamond model ensures that a firm could

6.1 Heterogeneous Leisure reduce w2 and still attract these workers. To determine w1, note that type 1 workers = = Why might one expect to find pure wage accept both w w1 and w w2, and so given dispersion? One answer is that search fric- the arrival rate w their value functions satisfy tions produce a natural trade-off for a firm: while posting higher wages lowers your prof- (69) rU1 b1 w(1 )[W1(w1) U1] it per worker, it could allow you to hire work- w [W1(w2) U1] ers faster, and so, in the long run, you get λ more of them. In Diamond’s model, this (70) rW1(w1) w1 [U1 W1(w1)] trade-off is nonexistent, since when you 28 increase your wage above wR there is no Albrect and Axell do not actually have firms earning equal profit, but allow y to vary across firms and look for a increase in your hiring rate. Albrecht and Bo ∗ > ∗ = cutoff y such that firms with y y pay w w1 and those Axell (1984) allow for heterogeneity in work- > ∗ = with y y pay w w2 ; the economic implications are basi- ers—not in productivity but in their value of cally the same. de05_Article1 11/16/05 3:53 PM Page 978

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(71) rW (w )w λ[U W (w )]. λLb()− b 1 2 2 1 1 2 yb=+ 12 1and (77) 2 + λ Note that although type 1 workers accept ()w L2 w1, they get no capital gain from doing so _ rLbb ( − ) and suffer no capital loss when laid off, =+ w 12 1 = = y y ++λλ + . since W1(w1) U1. Then using w2 b2, we ()()rL ww2 can simplify and solve for w1 in terms of , When productivity is low all firms pay ()rb++λ b = (72) w = 12w . w1 b1, when it is high all firms pay 1 ++λ r w2 b2, and when it is intermediate there w ∈ Unemployment rates for the two types are is wage dispersion. When (0,1), we can = λ = λ solve T() = 0 for and use (72) to solve u1 α—w—+λ and u2 α—w—+λ. for w1 and the distribution of paid wages For firms, the expected value of contact- 30 ing a worker is the probability he accepts explicitly. Of course, all of this is for given arrival times the profit conditional on acceptance. __L_1_u1___ rates, and the value of that solves (76) The acceptance probability is for L1u1 L2u2 depends on w. Using the matching function, firms paying w1 and 1 for firms paying w2, __y__w_i_ we can endogenize while discounted profit is r . So expected mLu()+ Lu ,1 discounted profits from the two wages are = 11 22 (78) w , Lu yw− Lu+ Lu (73) = 11 1 11 22 1 e + + λ + Lu11 Lu 22 r where L1u1 L2u2 is the number of unem- ployed workers and we assume that all firms yb− are always freely posting a vacancy, so that (74) = 2 , = 2 e r + λ v 1, with the idea being that each one will hire as many workers as it can get. Since u2 29 where for now we are taking e as given. depends on , so does w. An equilibrium is 31 Inserting u1, u2, and w1, after some algebra, a pair ( w, ) satisfying (76) and (78). we see that − is proportional to 2 1 6.2 On-the-Job Search (75) Tr()=++ (λ )({ yb − )× w 2 In the previous section, firms may pay higher × [λλ++ ] −− λ} LLyb121()()w L1 wages to increase the inflow of workers. There also exist models where firms may pay higher − rLb()− b. w 12 1 wages to reduce the outflow of workers (e.g., Burdett, Lagos, and Wright 2003). In Burdett For an equilibrium, we need and Mortensen (1998), both margins are at work, and firms that pay higher wages both (76) = 0 and T(0) < 0; = 1 increase the inflow and reduce the outflow of and T(1) > 0; or ∈(0,1) and T() = 0. 30 Clearly, we get at most two wages here, but the argu- It is easy to show there exists a unique solu- ment can be generalized to many types of workers < < (Eckstein and Wolpin 1990). tion to_ (76), and 0 1 if and only if 31 See Damien Gaumont, Martin Schindler, and Wright _y < y < y where (forthcoming) for details, and for several other variations on the general theme of Albrecht–Axell models. They also discuss a problem with models based on ex ante hetero- 29 An alternative to having firms maximize expected geneity: Given type 2 workers get no surplus, if there is any discounted profit is to maximize the value of posting a search cost ε > 0, they drop out of the market, leaving only vacancy, which is more in line with sections 4 and 5; see type 1 workers. Then we are back to Diamond and the dis- Mortensen (2000). We adopt the criterion in the text tribution collapses (and of course the problem applies with because it facilitates comparison with the model in the any number of types). Gaumont, Schindler, and Wright next subsection. also discuss models that avoid this problem. de05_Article1 11/16/05 3:53 PM Page 979

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workers. We now present this model in detail, We now construct F explicitly. The key which is relatively easy here, because it is observation is that firms earn equal profits based on on-the-job search, and we can make from all posted wages, including the lowest substantial use of results derived in section 3. w = b: (w) = (b) for all w ∈ [b,⎯w]. Since = = λ / + λ + + λ The arrival rates are 0 and 1 while F(b) 0, (b) (yb) ( )(r ). unemployed and employed, and every offer Combining this and (80) gives an equation is a random draw from F(w). For ease of that can easily be solved for F(w). In the presentation, we begin with the case simplest case where r ≈ 0, the result is = = , which implies w = b by (24), and 0 1 R λ + ⎛ yw− ⎞ return to the general case later. Since all (81) F()w = 1 − . ⎝⎜ − ⎠⎟ unemployed workers use a common reserva- yb < tion wage, and clearly no firm posts w wR, We know the lower bound is b, and the the unemployed accept all offers and we upper bound⎯w can easily be found by solv- have u = λ /(λ + ) as a special case of (25). ing F⎯(w) = 1. This yields the unique distri- Also, the distribution of paid wages is bution consistent with equal profit for all λFw() wages posted. (79) Gw()= , In words, the outcome is as follows. All λ +−[]1 Fw() unemployed workers accept the first offer a special case of (26). they receive, and move up the wage ladder ≥ If a firm posts w wR, a worker he con- each time a better offer comes along, but tacts accepts if he is currently unemployed also return to unemployment periodically or currently employed but at a lower wage, due to exogenous layoffs. There is a nonde- which occurs with probability u + generate distribution of wages posted by (1 − u)G(w). The employment relationship firms F given by (81), and of wages earned then yields flow profit y − w until the work- by workers G, given by inserting F into (79). er leaves either due to an exogenous separa- The model is consistent with many observa- tion or a better offer, which occurs at rate tions concerning worker turnover, and also λ + [1 − F(w)]. Therefore, after simplifica- concerning firms, including, e.g., the fact tion, the expected profit from w is that high wage firms are bigger. See (80) ()w = Mortensen (2003) for details. There are many interesting extensions. λ()yw− . First, with ≠ the same methods lead to {}λλ+−[]11Fw( ) {} r++[] − Fw() 0 1 λ + ⎛ yw− ⎞ Fw()= 1 1 − , Again, equilibrium requires that any posted (82) ⎝⎜ − ⎠⎟ 1 ywR wage yields the same profit, which is at least = as large as profit from any other wage. where now wR is endogenous (with 0 1 < = > = Clearly no firm posts w wR b or w y. In we knew wR b). To determine wR, integrate (24) to get fact,_ one can show_ that the support of F is 2 [b,w] for some w y, and there are neither (λ + ) by+−( ) gaps nor mass points on the support.32 (83) w = 1 011. R (λ + )2 +−( ) 1 011 32 Suppose there were a mass point at some wc . Then a +ε firm posting wc would be able to hire away any worker it → contacts working from a wc firm, increasing its revenue dis- One can check that in the limit as 1 0, ε c _ = cretely with only an increase in cost, which means w does w wR, which means there is a single wage, not maximize profit. Suppose there were a gap in F, say w = w = b. This is the Diamond result as a between wc and wt . Then a firm postingwt could lower its wage R and reduce cost without reducing its inflow or increasing its special case when there is no on-the-job t → _ = outflow of workers, and again w could not maximize profit. search. Also, in the limit as 1 , w y and de05_Article1 11/16/05 3:53 PM Page 980

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G(w) = 0 for all w < y; hence all workers earn must consume w each period). They prove = → → w y. Moreover, as 1 , clearly u 0. that all equilibrium wage–tenure contracts are Hence, the competitive solution also emerges described by a common baseline salary scale, as a special case when 0 and 1 get large. which is an increasing, continuous relationship One can let firms be heterogeneous with between the wage and tenure. Firms offer dif- respect to y. In equilibrium there is a distri- ferent contracts in the sense that they start bution of wages paid by each type of firm, workers at different point on the scale. Thus, and all firms with productivity y2 pay more consumption smoothing reintroduces wage < than all firms with y1 y2. Thus, higher pro- dispersion, both in the sense that workers get ductivity firms are more likely to hire and different wages depending on their tenure, less likely to lose any worker. With heteroge- and in the more fundamental sense that firms neous firms, van den Berg (2003) shows offer different contracts. there may be multiple equilibria. Perhaps more significantly, firm heterogeneity is 6.3 Discussion important empirically, because with homo- The two models of wage dispersion we geneous firms the equilibrium wage distri- have presented are based on worker hetero- bution (81) has an increasing density, which geneity and on-the-job-search, respectively. is not in the data. With heterogeneity F can Of course, one can integrate them. This is have a decreasing density. Another very important empirically, because although on- important extension is to allow firms whose the-job-search models (with heterogeneous workers contact rivals to make counteroffers, firms) do a good job accounting for wages, as in Postel-Vinay and Robin (2002). they do less well accounting for individual Margaret Stevens (2004) allows firms to employment histories, especially the fact post wage–tenure contracts, rather than sim- that hazard rates tend to decrease with the ply a constant wage. She shows firms have an length of unemployment spells. Models with incentive to back load wages to reduce worker heterogeneity do better at account- turnover. If workers can make an up-front ing for this, but less well for wages. An inte- payment for a job, an optimal contract grated model can potentially account for extracts an initial fee and then pays w = y. If both; see Christian Bontemps, Robin, and firms are homogeneous this contract elimi- van den Berg (1999). Many other extensions nates all voluntary quits. In equilibrium all and applications are possible, and this is a firms demand the same initial fee, and this productive area for both theoretical and leaves unemployed workers indifferent empirical research.33 about accepting the position. Stevens also shows that if initial payments are impossible, 7. Efficiency say because of liquidity constraints, there is an equilibrium where all firms offer a con- We now move on to efficiency. Of course, tract that pays the worker 0 for a fixed peri- in an economy with many agents there are od and then pays w = y. If firms are many Pareto optimal allocations. Here we homogeneous, they extract all of the surplus, focus on those that maximize the sum of there are no job-to-job transitions, and all agents’ utility, or equivalently, that maximize contracts are identical. 33 We mention some related work. Masters (1999) stud- However Burdett and Coles (2003) show ies a wage-posting model and uses it to analyze changes in the minimum wage. Delacroix and Shi (forthcoming) con- that Stevens’s results can be overturned if sider directed search in an environment similar to workers desire smooth consumption. They Burdett–Mortensen and show it also gives rise to wage dis- allow firms to commit to wage–tenure con- persion, although for different reasons. Finally, some peo- ple consider as an alternative to random search models tracts, but assume workers are risk-averse and where workers are more likely to meet large firms, as in do not have access to financial markets (they Burdett and Vishnawath (1988b). de05_Article1 11/16/05 3:53 PM Page 981

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the present discounted value of output net + (1 − θ∗)b. If is too high, for example, of the disutility of work and search costs. then w > w∗ and v is too low. 7.1 A One-Shot Model Now consider the competitive search model from section 5. From (58), in com- Initially, u workers are unmatched. Firms petitive search equilibrium w = w∗, and decide how many vacancies v to post, each at equilibrium is necessarily efficient. That is, cost k, and then m(u,v) matches form. Let the Hosios condition holds endogenously— q = u/v and assume constant returns, so although agents do not bargain, the surplus = = m(u,v) vm(q,1) v e(q). Each match pro- is still being split, and the equilibrium split duces y at an opportunity cost b to each work- implies efficiency. To understand why, it is er. Assume provisionally that wages are useful to think in terms of competition determined by bargaining, w = θy + (1 − θ)b. between market makers, as discussed above. Then the economy ends. Firms post vacancies With free entry, market makers effectively until the free entry condition maximize workers’ expected utility, − (84) ()(qybk1 −−=θ )( ) w(q)(w b), recognizing that firms must e − = break even to participate, e(q)(y w) k. holds. This is a static version of the basic Using the constraint to eliminate w, this is Pissarides model. identical to the planner’s problem. Now consider a planner who posts vacan- Now consider extending the model to cies to maximize output, net of workers’ endogenize workers’ search _intensity. The_ opportunity cost and the cost of posting total number of matches is m(su,v), where s = vacancies. Equivalently, he chooses q u/v, is average intensity. Assuming constant knowing that each worker will contact a returns, a firm hires with probability _ _ vacancy with probability w(q), to maximize = e(sq) m(sq,1), while a worker with search −− intensity s gets hired with probability (85) uqybvkw ()( ) _ _ _ s (sq) = s (sq)/sq. An unemployed work- =−−u[] w e e ()(qy b ) k. er chooses s to maximize q ssqyb ()(θ − ) The necessary and sufficient condition for a (88) bgs−+() e . solution is sq − − = In equilibrium all workers choose the same (86) [ e(q) q e(q)](y b) k. _ ∗ s = s, where Denote the planner’s solution by q . The following is immediate from (84) ()(sqθ y− b ) (89) g()s = e , and (86): the planner’s solution and the sq decentralized solution coincide if and only if and free entry by firms implies ∗∗ (87) θε= ()q, (90) k =−− ()(sq1 θ )( y b ). ∗ ∗α ∗ e ε = q e(q ) where (q ) —α —(q∗—) was defined in section 5 e The planner solves as the elasticity of e(q). This is the Hosios condition (Arthur J. Hosios 1990), determin- ()(sq y−− b ) k (91) maxbgs−+() e u , ing the share of the surplus that must go to qs, q workers for bargaining to be efficient.34 which has necessary and sufficient conditions Alternatively, in terms of wages,∗ ∗ equilibrium is efficient if w = w = θ y =− α ′′α (92) gs() ( sqy )( b ) qqe ()+=qqw() e 34 = α α 1 Because e(q) q w(q), we have e ()q w()q ; hence the condition can equivalently be stated as firms’ − – – – bargaining power 1 must equal the elasticity of w(q). (93) k [ e(sq) sq e(sq)](y b). de05_Article1 11/16/05 3:53 PM Page 982

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Notice something interesting: (89) and (92) where a0 can be interpreted as the value of θ = ε _ coincide if the Hosios condition (sq) an unemployed worker and a1 the surplus holds, while (90) and (93) coincide under the from a match.35 same condition. That is, bargaining equilibri- The first order condition from (94) simpli- um achieves efficient search intensity and fies to entry under the Hosios condition. This =−[] (95) ka1 ee() q qq (). seems genuinely surprising,_ as there are two variables to be determined, s and v, and only Using the fact that Y(e) = a + a e and the one parameter, υ. 0 1 envelope theorem, we can differentiate both Since we have already shown that in com- sides of (94) to get petitive search equilibrium we get the ⎡ ⎤ Hosios condition endogenously, competitive k e ()q (96) ra=−+ y b −a ⎢ +λ⎥ . search equilibrium achieves efficient search 11q ⎣ q ⎦ intensity and entry. As suggested above,

market makers effectively choose the terms Combining (95) and (96) to eliminate a1, we of trade to ensure that both workers and arrive at firms behave optimally. There is nothing rq++λ () yb− special about endogenous entry decisions or (97) e = . − k search intensity. We could consider various ee()qqq () other extensions (e.g., match-specific pro- This completely characterizes the optimal ductivity with the reservation match value yR policy q = q(e); notice that in fact q does not determined endogenously), and we would depend on the state e, only on exogenous find the same result: bargaining equilibrium parameters. is efficient if and only if the Hosios condition How does this compare with equilibrium? holds, and competitive search equilibrium Recall that equilibrium in section 4.3 satis- generates this condition endogenously. fies (43). It is easy to see that the solutions Rather than go through these exercises we are the same, and hence bargaining equilib- now move to dynamics. rium coincides with the planner’s solution, if = ε 7.2 A Dynamic Model and only if (q). Now looking at (68) from the competitive search version of the Here we formulate the planner’s problem model, we see that competitive search equi- for the benchmark Pissarides model recur- librium is necessarily efficient. Hence the sively. The state variable is the measure of results from the static model generalize matched workers, or the employment rate, e. directly, and again carry over to more gen- The current payoff is y for each of the e eral models with endogenous search intensity, − employed workers, b for each of the 1 e match-specific productivity, and so on. unemployed workers, and − k for each of the v = (1 = e)/q vacancies. The employment rate 7.3 Discussion α = __e _(q_) − − λ follows a law of motion e˙ q (1 e) e. We have demonstrated in several contexts Putting this together, the planner’s problem is that competitive search is efficient, while (94) rY() e=+−max ye b (1 e ) bargaining is efficient if and only if the q Hosios condition holds. Although these results are in some sense general, it would be ke()1 − ⎡ ()(qe1 − ) ⎤ − + Ye()⎢ e − λe⎥ . q ⎣ q ⎦ 35 The mapping defined by (94) is a contraction and takes affine functions into affine functions. Since the set of such functions is closed, the result follows immediately. One can show that Y(e) is affine; i.e., This method also works and is especially useful when there = + Y(e) a0 a1e for some constants a0 and a1, is a distribution of productivity across matches. de05_Article1 11/16/05 3:53 PM Page 983

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misleading to suggest that we can get effi- there is no reason to necessarily expect effi- cient equilibria in all search models. A gen- ciency (Mortensen 2000). In general, there is eral understanding of the nature of much more work to be done on this topic. efficiency in these models is still being devel- oped. Mortensen and Wright (2002) provide 8. Conclusion some results, but only for constant returns matching functions; without this, equilibria In contrast to standard supply-and- are unlikely to be efficient. Shimer and demand models, search theory emphasizes Smith (2001) show that in some models with frictions inherent in the exchange process. heterogeneity, even with constant returns, Although there is no one canonical search the efficient outcome may not even be com- model, and versions differ in terms of wage patible with a steady state, but exhibits determination, the matching process, and cycles. other assumptions, we have tried to show The efficiency of search equilibria with ex that there is a common framework underly- ante investments is an interesting topic. For ing all of the specifications. We have used example, in Acemoglu (1996) and Masters the different models to discuss several (1998), agents decide on how much capital issues, mainly related to worker turnover to acquire prior to searching. Generically and wages. We have also used them to dis- there is no θ that makes the equilibrium out- cuss some simple policy experiments, such come under bargaining efficient—the value an increase in UI. Different models empha- of θ that provides the right incentive for size different margins along which such poli- investment in human capital by workers is cies work, including the choice of different from the value of that provides the reservation wage, search intensity, entry, and right incentive for firms (whether firms so on. choose the number of vacancies, the types of The approach, as we have seen, is consis- jobs to create, or physical capital). However, tent with many interesting observations. For Acemoglu and Shimer (1999a) show com- a start it predicts the unemployment rate is petitive search equilibrium with ex ante not zero, which is perhaps a low hurdle but investments is efficient. Another case where not one that can be met by many alternative there is no θ such that bargaining yields the theories. The reason the model predicts this efficient outcome is discussed by Smith is obvious, but also obviously correct—it (1999), who assumes firms have concave takes time for workers to find jobs. It also production functions and hire a large num- takes time for firms to fill vacancies, and a ber of workers, but bargain with each one nice property of these models is that there individually. can be coexistence of unemployed workers One can also ask about the efficiency of the and unfilled vacancies. The framework can wage-posting models in section 6. In general, be used to organize thinking about individual if firms commit to pay the same w no matter transitions between unemployment and the circumstances, it is unlikely to yield effi- employment, as well as job-to-job transitions. cient outcomes. In Albrecht and Axell (1984), Different search-based models can be e.g., a planner would want all meetings to used to help explain the relationships result in matches, which does not happen. In between tenure, wages, and turnover, the simplest Burdett and Mortensen (1998) including versions that incorporate on-the- = model, with 0 1, efficiency does result job search, learning, or human capital. because all meetings involving unemployed Several models can be used to discuss the workers result in matches and other meet- distribution of wages, and some of the mod- ings are irrelevant from the planner’s per- els are consistent with observations such as spective. In extended versions, however, the fact that high wage firms tend to have de05_Article1 11/16/05 3:53 PM Page 984

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more workers, or high productivity firms Vroman. 2003. “Matching with Multiple Applications.” Economics Letters, 78(1): 67–70. tend to pay more. We also discussed the wel- Albrecht, James W., Pieter A. Gautier, Serene Tan, and fare properties of the models. Bargaining Susan B. Vroman. 2004. “Matching with Multiple models achieve efficiency if and only if the Applications Revisited.” Economics Letters, 84(3): 311–14. bargaining weights satisfy a condition related Alvarez, Fernando, and Marcelo Veracierto. 2000. to the matching technology, while in com- “Labor-Market Policies in an Equilibrium Search petitive search models, this condition Model,” in NBER Macroeconomics Annual 1999, Volume 14. Ben S. Bernanke and Julio J. Rotemberg, emerges endogenously. eds. Cambridge and London: MIT Press, 265–304. We have only scratched the surface, but Andolfatto, David. 1996. “Business Cycles and Labor- hopefully we have conveyed the main ideas. Market Search.” American Economic Review, 86(1): 112–32. The goal was never to claim that search the- Barlevy, Gadi. 2002. “The Sullying Effect of orists fully understand all of the issues, and Recessions.” Review of Economic Studies, 69(1): indeed there are many interesting directions 65–96. van den Berg, Gerard J. 1994. “The Effects of Changes for future research. One involves further of the Job Offer Arrival Rate on the Duration of quantifying the models, using both micro Unemployment.” Journal of Labor Economics, 12(3): and aggregate data. On a more conceptual 478–98. van den Berg, Gerard J. 2003. “Multiple Equilibria and front, the models we have presented do not Minimum Wages in Labor Markets with offer a cogent theory of the firm, and do not Informational Frictions and Heterogeneous distinguish between worker and job flows. Production Technologies.” International Economic Review, 44(4): 1337–57. The welfare results are incomplete. 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