Strategic Wage Setting and Coordination Frictions with Multiple Job Applications

Transcription

1 Strategic Wage Setting and Coordination Frictions with Multiple Job Applications Pieter A. Gautier Vrije Universiteit Amsterdam and Tinbergen Institute José L. Moraga-González University of Groningen and Erasmus University Rotterdam First version: October 2003 Revised version: December Abstract We study a labor market where workers choose the number of jobs to apply for and firms simultaneously post a wage. There exist two types of equilibria in the market. If the cost of applying for jobs is high, workers apply for just one job and firms offer the minimum wage; this number of job applications is adequate or insufficient from a welfare viewpoint. If the cost of applying for jobs is low, workers apply for two or for more jobs, there is wage dispersion and the number of applications workers send out is excessive from a welfare perspective. A minimum wage can be welfare improving because it compresses the wage distribution and reduces congestion effects caused by excessive applications. Since wage dispersion leads to rent-seeking behavior, an appropriately set mandatory wage can eliminate the congestion externality completely. However, if vacancy supply is determined by free entry there exists no wage policy that restores efficiency. We thank Jim Albrecht, Ken Burdett, Pierre Cahuc, Melvyn Coles, Andrea Galeotti, Eric Maskin, Rob van der Noll, Misja Nuyens, Vladimir Karamychev, Zsolt Sandor, Rob Shimer, Otto Swank and Susan Vroman for their comments. The paper has also benefited from presentations at the CESifo 2005 Conference on Employment, ESEM 2004 (Madrid), ASSET 2004 (Barcelona), IZA workshop on search and matching models, the IZA-SOLE transatlantic meeting and the 2004 Zeuthen workshop in Copenhagen. Department of Economics, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, <pgautier@feweb.vu.nl>. Gautier acknowledges financial support from NWO through a VIDI grant Department of Economics, University of Groningen, Postbox 800, 9700 AV Groningen, The Netherlands, <j.l.moraga.gonzalez@rug.nl>. 1

2 1 Introduction Both the academic job market and the Internet job search platforms (e.g. monster.com) have the following two features: (i) posted vacancies give job descriptions but no wage information and (ii) workers observe all posted vacancies for a certain occupation within a given region and apply for various jobs at a time. This paper presents a labor market model with these characteristics. Specifically, we consider a large number of firms and workers. Each firm has a single vacancy and sets a wage to maximize profits. Workers decide whether to apply for just one job or for several jobs simultaneously. Once wages and applications are decided upon, firms with candidates pick one worker at random, offer the job to her and the market closes. 1 Workers who receive multiple offers take the one with the highest wage. We are interested in the efficiency properties of the Nash equilibria of this game. This model has a number of novel features. The number of applications a worker sends out when looking for a job and the wages that the firms set are endogenously and simultaneously determined in equilibrium. 2 Job search causes congestion effects which are not internalized by individual workers and therefore the equilibrium tends to be inefficient. 3 The existence of wage dispersion increases the individual incentives to apply for jobs even more. This makes policies that compress the wage distribution like a minimum or a mandatory wage socially desirable in our model. We first study how job applications influence welfare. Our model has the same two coordination frictions as in Albrecht et al. (2003). First, since a vacancy can only be filled by one worker, the applications of a worker impose a negative externality on the other workers. Second, when workers send two or more applications, multiple firms can select the same candidate while only one of the firms will hire the worker. The efficient number of job applications balances these forces by maximizing the number of matches minus total application costs. This optimal number is first increasing in the vacancy-to-unemployment ratio and then decreasing. When there are many workers per vacancy, coordination frictions among workers are severe relatively to those 1 We use the same recruiting technology as the one explored in Albrecht et al. (2003). Gautier, Moraga-González and Nuyens (2005) explore technologies where firms can approach several candidates. 2 The model is therefore not subject to the partial-partial critique of Rothschild (1973). 3 Usually, search intensity is treated as a technology parameter (e.g. Pissarides, 2000) that increases the match probability of a worker. For the aggregate number of matches it does not matter whether the number of workers or the search intensity is doubled. In our model, more search leads to more coordination problems between firms, as in Albrecht et al. (2004), and may thus lead to fewer matches; this may help explain a recent finding in Kuhn and Skuterud (2004) that Internet search does not reduce unemployment duration. 2

3 among firms. An increase in the vacancy-to-unemployment ratio then reduces the first coordination friction while it increases the second one only moderately. As a result, marginal social gains from applications increase. When there are relatively few workers per vacancy, the coordination problem among firms is large relatively to the first coordination problem and the opposite result follows. Depending on the cost involved in completing and submitting a job application and on the minimum wage, there may be equilibria where workers either apply for just one job or apply for several jobs. When workers apply for just one job we get the standard Diamond (1971) result that the wage distribution is degenerate at the minimum wage. A single wage equilibrium can only be sustained when application costs are neither too high, which prevents worker participation, nor too low, which gives a worker incentives to deviate by applying for multiple jobs. The incentive to apply for various jobs arises from the fear of a worker to be rationed and remain unemployed. This contrasts with Burdett and Judd (1983) and Acemoglu and Shimer (2000) where a monopsony equilibrium always exists. For some parameters, this single wage equilibrium is efficient but sometimes one job application is insufficient to maximize social welfare. When application costs are low, workers send multiple applications and the equilibrium is characterized by wage dispersion. 4 Besides the fear from being rationed and remain unemployed, wage dispersion is an additional motivation to apply for multiple vacancies, namely, to get a better paid job. It turns out that the number of job applications workers send is excessive from a social welfare perspective. This result contrasts with Acemoglu and Shimer (2000), where there is too little search in equilibrium because some workers free-ride on the search effort of others. The difference in results arises from the different motives workers have to send out applications in our setting: intensive search increases not only the expected wage but also the hiring probability. The way the minimum wage affects workers incentives to send out applications hinges upon labor market tightness: when there are few (many) firms per worker, an increase in the minimum wage leads to more (less) applications. The intuition for this result is the following. When the wage distribution is skewed towards the competitive wage, an increase in the minimum wage reduces workers incentives to apply for additional jobs by compressing the wage distribution even more. Therefore, an increase in the minimum wage is desirable if the vacancy-to-unemployment ratio is 4 The nature of the wage dispersion is similar to Burdett and Judd (1983), Lang (1991) and Acemoglu and Shimer (2000) where some workers observe more than one wage while others sample just one. In our model, some workers receive one offer and others receive two or more. In Burdett and Mortensen (1998), by contrast, wage dispersion arises because a firm s wage offer may arrive either at an employed worker (who are choosier) or at an unemployed worker, while in Albrecht and Axell (1984) it arises because workers have different reservation wages. 3

4 relatively large because in that case the equilibrium number of applications is reduced and welfare rises. This mechanism is new to our knowledge. When there is already a lot of competition for workers, extra applications are not desirable. Introducing a minimum wage reduces the number of applications and this can increase employment by reducing coordination frictions. By contrast, if the number of vacancies is small relative to the number of workers, then a minimum wage increase gives workers incentives to send more applications in equilibrium, which worsens the situation. Interestingly, efficiency can be achieved in our setting by imposing a particular mandatory wage that eliminates wage dispersion altogether. However, if we allow for free entry of vacancies then there is no single wage instrument that can restore efficiency. Our model is related to the directed search models of Montgomery (1991), Burdett, Shi, Wright (2001), Albrecht et al. (2003) and Galenianos and Kircher (2005) in the sense that workers know the location of firms. In directed search models, workers also observe the wages offered in the market before deciding where to send their applications. By contrast, here workers only learn the wage of a firm after the contact has been made. 5 This generates atomless wage distributions in equilibrium. Whether search is random or directed is ultimately an empirical question. However, casual evidence shows that some job advertisements do post a wage and some don t (see Finally, our work is related to Shapiro (2004). He treats search intensity as exogenous and assumes that workers always apply for two jobs. The focus of our paper is different from all of the papers mentioned above. We are mainly interested in the interaction between endogenous search intensity and the wage distribution and in the possible inefficiencies that are driven by this interaction. We conclude for example that a minimum wage cannot restore efficiency while in Shapiro s model it can. The difference in results is caused by the endogeneity of the number of applications in our model. Our analysis is further related to the consumer search models of Burdett and Judd (1983) and Janssen and Moraga-González (2004). The fundamental difference between those models and our labor market model is that here vacancies can only be occupied by a single worker whereas in consumer markets, rationing is usually less important. As opposed to those models where consumers tend to search less than what is collectively desirable, here workers anticipate the labor market frictions that arise from rationing and send an excessive amount of applications. 5 For example, even though assistant professors may have some idea about the starting salaries at different universities, the amount of effort they have to put in the job depends on the teaching load, types of courses that must be taught, management duties, etc., which typically are revealed after the contact has taken place. 4

5 The paper is organized as follows. Section 2 presents the labor market model. In section 3 we discuss the social optimum. Section 4 presents the decentralized labor market equilibria as well as their efficiency properties. Section 5 discusses the implications of some wage policies, and includes an analysis of free-entry in our model. Finally, section 6 concludes. 2 The labor market We study a single period labor market with a large number, u, of unemployed workers and a large number, v, of firms. Each firm has exactly one vacancy. Let θ be the number of vacancies per worker; we refer to θ as the vacancy-to-unemployment ratio, or as labor-market tightness. Firms, indexed by i, post wages. A strategy for a firm i is a wage offer, denoted F i (w). Posted wages must be above a minimum wage w. Varying w allows for the study of the effects of minimum wage policies. 6 We normalize the value of output of a matched firm-worker pair to one. Workers, indexed by j, who get a job receive gross utility equal to the wage w they are paid. To be able to receive wage offers, workers must send applications to the vacancies. Applying for jobs is costly. Let k > 0 be the cost a worker must incur to complete and submit an application. A strategy for a worker j is the number of vacancies to apply for: a j = {1, 2,..., v}. Workers randomly send their applications to the vacancies. Thus, if worker j sends a j applications, the probability that a randomly selected vacancy gets an application from the worker in question is a j /v. 7 If a worker receives two or more wage offers, she chooses the vacancy that offers the highest wage. Workers and firms play a simultaneous moves game. A firm chooses a wage offer taking as given the wages offered by the rest of the firms as well as the number of applications sent by each worker. A worker decides how many applications to send taking as given the wages offered by the firms as well as the number of applications sent by the other workers. We focus on symmetric Nash equilibria. Modelling the interaction between firms and workers as a simultaneous moves game captures a situation where a typical worker knows the locations of the firms with vacancies but only learns the wage of a firm after she has applied there and her application results in a job offer. Once applications have been received, each firm randomly picks a candidate and offers the job to her. If the candidate has no better offers, she accepts it while if she has better offer(s) she rejects 6 Alternatively, w can be interpreted as an unemployment benefit. In such case, wages must be quoted above w to attract workers, and changes in w allow us to examine the implications of distinct levels of unemployment benefits. 7 Equilibria where, say, worker 1 applies for firm A s vacancy, worker 2 for firm B s job, worker 3 for firm C s vacancy, and so on and so forth, are not so interesting because they require a lot of coordination between the workers, which is difficult to achieve in large labor markets. 5

6 it. After this, the market closes. This recruitment technology describes a situation where frictions are large so that a firm that fails to hire its first candidate closes its vacancy (Albrecht et al., 2003, 2004). An alternative assumption would be that screening workers is costless so that a firm that fails to hire its first candidate can always go back to the next candidate and so on till the last. In Gautier, Moraga-González and Nuyens (2005) these two recruitment technologies are compared; it turns out that, qualitatively, the results of this paper carry over to that framework. The labor market described above is similar to the nonsequential consumer search model of Burdett and Judd (1983) but with the important modification that each firm has just one vacancy to offer in the market. This assumption, which is quite common in labor markets, brings two important novelties: first, the incentive of a firm to overbid other firms wages is weakened; second, when determining her optimal search strategy, a worker must take into account that she may be rationed at some or at all vacancies she applied for. These two implications have a bearing on the nature of equilibria and their efficiency properties. 3 Efficiency We start by discussing the efficient number of applications. The planner s objective is to maximize aggregate output net of total application cost. Total output is simply equal to the number of matches times the value of a match. Since all workers are identical we can focus on welfare per worker. Suppose every worker sends a job applications. Let z(a, θ) denote the probability that one of the a applications of a worker results in a job offer. A worker gets no job offer with probability (1 z(a, θ)) a so the gross welfare per worker is: W (a, θ) = 1 (1 z (a, θ)) a (1) We now calculate z (a, θ). Let ρ(a, θ) be the ( urn ball ) probability that a firm gets at least one applicant when each worker submits a applications: ρ(a, θ) = ( ( lim 1 1 a ) u ) = 1 e a u,v, v θ u =θ v Conditional on having sent applications to firms i and l, in a large labor market (u, v, v u = θ), getting an offer from firm i is independent of getting an offer from firm l (see Albrecht et al., 2004). Then z(a, θ) is simply equal to the number of firms with applications divided by the total number 6

7 of applications in the market: z(a, θ) = lim u,v, v u =θ ( 1 ( 1 a v ) u ) v au = ρ(a, θ)θ a For future reference, we note that z(a, θ) is monotonically decreasing in a, with lim a 0 z(a, θ) = 1 and lim a z(a, θ) = 0; obviously, the opposite properties hold with respect to θ. Let S(a, θ) = (1 z(a, θ)) a (1 z(a + 1, θ)) a+1 denote the gross social gains obtained from an extra job application per worker when every worker sends a applications. For a given a, these social gains depend only on market tightness θ so we can plot the social gains of extra job applications for different number of applications. To illustrate, Figure 1a shows S(a, θ) for a = 0, 1, 2. The shapes of these curves can be explained in terms of the two types of coordination frictions that exist in this market. First, since workers send job applications randomly, the applications of a worker exert a negative externality on the other workers. These congestion effects between the workers weaken as the firm-worker ratio goes up. Second, when each worker sends two or more job applications a firm may choose the same candidate as some other firm(s), in which case only one of the firms hires the worker and the others remain idle. These coordination problems between the firms strengthen as the firm-worker ratio goes up. The curve S(0, θ) denotes the social gains from labor market participation. When a = 0 only the congestion effects between the workers are at work and thus S(0, θ) is monotonically increasing in the market tightness parameter. By contrast, when a 1, there are also coordination frictions between firms and as a result, the S(a, θ) curves first increase and then decrease in θ. When the firm-worker ratio is initially low each firm receives many applications and therefore the first coordination friction is large relatively to the second one because the probability that multiple firms consider the same candidate is low while the probability that multiple workers apply to the same vacancy is high. In this case, an increase in θ reduces the congestion effects between workers more than when θ is already large. In the latter case, applications are spread across many vacancies and therefore the congestion effects are practically absent but the firms coordination problem is severe. An increase in θ then aggravates the firm coordination problem further and this reduces the marginal social gains from applications. Since the marginal social cost of sending an extra application is k, we can easily obtain the set of points (θ, k) for which sending a applications is socially optimal. The regions of parameters for which 0, 1 and 2 applications are optimal are represented in Figure 1a with the labels â = 0, 7

8 â = 1 and â = 2, respectively. The socially optimal number of job applications balances the two coordination problems mentioned above to maximize the number of matches net of job applications costs. (a) Parameter regions for which different number of applications are efficient (b) Efficient number of applications and θ (k = 0.001) Figure 1: The social optimum Proposition 1 (a) Let θ, k satisfy S(0, θ) k < 0. Then it is optimal for the social planner to close down the labor market. (b) Otherwise, for any k and θ there exists a unique socially optimal number of job applications â(θ, k). This number is independent of the minimum wage and (weakly) decreasing in k. (c) Let k max θ S(1, θ). Then â(θ, k) is non-monotonic in the vacancyto-unemployment ratio θ; first it is increasing and then it is decreasing in θ. For a formal proof see the Appendix. The non-monotonicity of the efficient number of job applications with respect to market tightness can be seen in Figure 1a. Suppose the cost of sending applications is k 2. For a low θ, say 0.5, the efficient number of applications is 1. If the vacancy-tounemployment ratio increases, say to 3, the efficient number of applications goes up to 2. A further increase in θ, say to 6, reduces the efficient number of applications back to 1. This non-monotonicity of the efficient number of applications is related to the two types of coordination frictions that exist in this market. Assuming the number of job applications workers send is a continuous variable, Figure 1b shows the socially optimal job search intensity of a worker. 4 Labor market equilibrium We now turn to the private incentives in the labor market. We start by describing some general features of wage equilibria. 8

9 Lemma 1 (a) If an equilibrium exists where workers send just one job application, the equilibrium wage distribution must be degenerate at the minimum wage w; (b) if an equilibrium exists where 2 a v, there must be wage dispersion and the wage distribution must be atomless. The proof of this result, which shows the influence of workers search activity on wage setting and vice versa, is in the appendix and follows Burdett and Judd (1983) and Lang (1991). Workers are more likely to send multiple job applications when there is wage dispersion. Moreover, wages can only be dispersed if the worker a firm is trying to hire can also be hired by some other firm, which requires that the applicant sends multiple applications. Thus wage dispersion and multiple job applications arise together in our setting. We now move to the existence and characterization of equilibria. We start with a consideration of degenerate wage equilibria. 4.1 Single wage equilibrium Suppose that workers send just one job application. Since firms offer the minimum wage, a worker s expected payoff is equal to: where z(1, θ) is the probability that a worker gets a job offer. Eu(a = 1) = z(1, θ)w k, (2) For an equilibrium with a single job application to exist, the private gains from participation in the labor market must exceed the cost of an application. In addition, it must be the case that a potential deviant worker, say j, who sends an extra application (or more) obtains gains that fall short of application cost. This leads to: Proposition 2 Let z(1, θ)w k ( 1 (1 z(1, θ)) 2) w. Then there exists a unique symmetric equilibrium in pure strategies where all firms post the minimum wage w and workers apply for only one of the vacancies. In equilibrium, a worker is unemployed with probability 1 z(1, θ), a vacancy remains unfilled with probability 1 ρ(1, θ) and a firm s expected profit is π i = ρ(1, θ)(1 w). The proof is in the Appendix. A single wage equilibrium can only be sustained when, for a given level of the minimum wage, application costs are neither too high, which prevents worker participation, nor too low, which gives a worker incentives to deviate by applying for multiple jobs. This incentive to deviate arises from the fact that a job can only be occupied by one worker, which contrasts earlier work where there is no rationing and as a consequence a monopsony equilibrium exists always (Burdett and Judd, 1983). In the absence of a minimum wage, there is no a = 1 9

10 equilibrium for similar reasons as in Diamond (1971). The region of parameters for which applying for one job is part of an equilibrium is represented in Figure 2a. (a) Existence region of an equilibrium with a single job application (w=0.8) (b) Efficiency of a single application equilibrium (w=0.8) Figure 2: Single wage equilibrium and welfare We now ask whether the single wage equilibrium described in Proposition 2 is socially efficient. A single job application is socially optimal if and only if z(1, θ) k 1 z(1, θ) (1 z(1, θ)) 2. These conditions differ from those in Proposition 2 in two regards. First, the private gains from participation in the labor market depend on the minimum wage while this is not the case for the social benefits because the social planner doesn t care about the distribution of surplus between the worker and the firm. As a result the private value of participation is lower than the social benefits. This is depicted in Figure 2b where social and private incentives to send a job application are compared. In region A the social benefits from participation exceed the private gains because the minimum wage is too low. In region C, one job application is a Nash equilibrium while it would be desirable that each worker sends 2 applications instead. The second problem is that an individual worker does not internalize the externalities his/her application imposes on other workers while the social planner does. This implies that, for some parameters, it is efficient to apply for just one job but this is not viable in equilibrium because an individual worker would prefer to deviate by sending 2 applications. This occurs in region E of Figure 2b. Proposition 3 Assume that application cost and the minimum wage are such that workers apply for one job only in equilibrium. From the point of view of social welfare maximization, this number of applications is adequate or insufficient, but never excessive. 10

11 4.2 Dispersed wage equilibrium We now turn to the existence and characterization of wage-dispersed equilibria with multiple job applications. We start by examining wage setting behavior by firms, given workers strategies. A typical firm may receive one or more applications. If a firm receives more than one application, the firm picks one candidate at random. The (randomly chosen) applicant has sent a total of a applications, so she can potentially receive 1, 2, 3,..., up to a total of a 1 other wage offers. The firm does not compete for the worker in question with probability (1 z(a, θ)) a 1, competes with just one other firm with probability (a 1)z(a, θ)(1 z(a, θ)) a 2, and so on. Thus the expected payoff to a firm i offering a wage w given that the other firms offer a wage from F ( ) is: a 1 ( ) π i (w; F ( )) = ρ(a, θ) a 1 z(a, θ) j (1 z(a, θ)) a 1 j F (w) j (1 w) (3) j j=0 We can use the binomial theorem to simplify (3) to: Eπ i (w; F ( )) = ρ(a, θ) [1 z(a, θ) + z(a, θ)f (w)] a 1 (1 w) Since in a mixed strategy equilibrium, all firms must be indifferent between strategies, profits of any firm must be equal to the profits of a firm that offers the lower bound of the support of F ( ); this lower bound must be equal to the minimum wage because otherwise a firm offering such wage would gain by decreasing it: Eπ i (w; F ( )) = ρ(a, θ) (1 z(a, θ)) a 1 (1 w) (4) Solving Eπ i (w; F ( )) = Eπ i (w; F ( )) yields the equilibrium wage distribution: F (w; a, θ, w) = 1 z(a, θ) z(a, θ) [ [1 ] 1 ] w a w (5) The upper bound of the support of the wage distribution can be found by setting F (w; a, θ, w) = 1 and solving for w, which yields: w = 1 (1 w)(1 z(a, θ)) a 1 (6) The maximum wage offered in the market is increasing in w and decreasing in a and θ. Equations (5) and (6) characterize wage setting behavior by firms given worker strategies. A close look at the equilibrium wage distribution reveals that an increase in θ results in a rightward 11

12 shift of the distribution. Thus, wages become more competitive as market tightness increases. An increase in the minimum wage w has a similar influence on the wage distribution. This follows from three observations: (i) the upper bound of the distribution of wages is less than 1, increasing in a and in θ; (ii) F ( ) is monotonically decreasing in z(a, θ) and in w; and (iii) z(a, θ) is monotonically increasing in θ. These results are summarized next: Lemma 2 For any value of θ and a 2, there exists a unique equilibrium wage distribution given by (5) with support [ w, 1 (1 w)(1 z(a, θ)) a 1]. For any > 0, the equilibrium distribution of wages F (w; a, θ, w + ) dominates in a first-order stochastic sense the distribution of wages F (w; a, θ, w); further, F (w; a, θ +, w) first-order stochastically dominates F (w; a, θ, w). One of the reasons why workers find it attractive to apply for several jobs is that this increases the chance to get an offer. The other motive is that applying for various jobs increases the probability of getting several offers, i.e., a better paid job. Therefore, it is important to clarify how wage dispersion depends on the parameters of the model. A result that will be useful later is that wage dispersion is non-monotonic in market tightness. When θ 0, the probability that two or more firms compete for the same worker is arbitrarily low so firms have no incentives to offer wages above the minimum wage. As θ increases, firms expect to face competition for a worker with positive probability and this leads to significant wage dispersion. When θ every worker receives more than a wage offer almost surely so firms have no incentives to offer wages below the workers productivity. To characterize an equilibrium with wage dispersion and multiple applications fully, we need to study workers optimal strategies. A worker maximizes utility taking firm wage setting and other workers strategies as given. The (gross) expected utility of a worker sending a l applications when the rest of the workers sends a applications is: Eu(a l ; a) = w w a l ( ) al z(a, θ) j (1 z(a, θ)) a l j j F (w) j 1 wf(w)dw (7) j j=1 This expression tells us that a worker who sends a l applications receives a single offer with probability a l z(a, θ)(1 z(a, θ)) a l 1 and in this case the wage the worker expects to get is the mean wage; the worker gets two offers with probability ( a l 2 ) z(a, θ) 2 (1 z(a, θ)) a l 2 and then the wage the worker expects to get is the maximum of a random sample of size 2; and so on and so forth. 8 8 In the empirical job search literature an important distinction is made between the offer distribution, F and 12

13 Using the binomial theorem, equation (7) can be rewritten as Eu(a l ; a) = a l z(a, θ) and integration by parts yields: w Eu(a l ; a) = w w(1 z(a, θ)) a l w [1 z(a, θ)(1 F (w))] a l 1 wf(w)dw w w [1 z(a, θ)(1 F (w))] a l dw Changing the variable of integration appropriately, we can rewrite the worker s utility as: 1 Eu(a l ; a) = 1 (1 w)(1 z(a, θ))a l 1 dy (1 z(a,θ)) a l a l 1 a y l Integration of this expression yields: Eu(a l ; a) = 1 (1 z(a, θ)) a l a l(1 w)(1 z(a, θ)) a 1 ( 1 (1 z(a, θ)) a l+1 a ) a l + 1 a (8) Let (a) = Eu(a; a) Eu(a 1; a) denote the marginal gains to a worker from applying for a 1 jobs rather than for a jobs, given the rest of the workers apply for a jobs. Similarly, let + (a) = Eu(a + 1; a) Eu(a; a) denote the marginal gains from sending one application more than the rest of the workers. Applying for a 2 jobs is a worker equilibrium if the following two inequalities hold: 9 (a) > k > + (a) (9) For a given minimum wage and a given number of applications, (a) and + (a) can be represented as a function of θ. Figure 3a shows the region of parameters for which an equilibrium where workers send 2 applications exists. There are three issues we would like to put forward here. The first is the existence of multiple equilibria. This can be seen in Figure 3b where the existence region for the equilibrium with 1 job application overlaps the existence region of the equilibrium with 2 applications. A similar picture arises if one compares the existence regions of equilibria with 2 and 3 applications (see Figure 4a). The second remark pertains to the non-existence of equilibrium in pure worker-strategies. In ( Figure 3b there is a region of parameters above + (2) and below 1 (1 z(1, θ)) 2) w for which the distribution of accepted wage offers which is in our case very similar to the bracketed term in (7): G (a, θ, w) = z(θ, a) j (1 z(θ, a)) a 1 j F (w) j. This is basically a combination of the wage offer distribution F (w) and Σ a 1 j=0 a 1 j the probability that the worker accepts the offer. 9 To be sure, if deviating by sending one job application more (less) than the rest of the workers is not profitable, neither is deviating by sending b applications more (less) than the rest of the workers. 13

14 (a) Existence region of an equilibrium with 2 job applications 1 and 2 job (b) Existence regions for the equilibria with applications Figure 3: Equilibria with multiple job applications (w = 0.8) no equilibrium in pure worker-strategies exists. The same problem appears in Figure 4a. The source of these two problems, multiplicity of equilibria and non-existence of equilibrium, is that workers can only apply for a discrete number of vacancies. To be able to proceed further we treat the number of job applications as being a continuous variable so we can use differentiation. (a) Existence regions for the equilibria with 2 and 3 job applications (b) Equilibria with 1, 2 and 3 applications Figure 4: Equilibria with multiple job applications (w = 0.8) (cont.) Then the marginal gains to a worker sending a l applications are given by deu(a l ; a) da l = (1 z(a, θ)) a l (1 w)(1 z(a, θ))a 1 log[1 z(a, θ)] + (a l + 1 a) [ 2 (a 1)(1 (1 z(a, θ)) a l +1 a ) + a l (a l + 1 a) log[1 z(a, θ)](1 z(a, θ)) a l+1 a ] In symmetric equilibrium, a worker should continue to send applications till the marginal gains from applying equal the marginal cost k. Invoking symmetry in (10) yields the equilibrium condition: (10) Ψ(a; θ, w) = k (11) 14

15 where Ψ(a; θ, w) = (1 z(a, θ)) a log[1 z(a, θ)] +(1 w)(1 z(a, θ))[(a 1)z(a, θ) + a(1 z(a, θ)) log[1 z(a, θ)] (12) It is easy to see that (i) Ψ(a; θ, w) is continuous in a and strictly positive everywhere 10 and (ii) marginal gains converge to zero as a approaches infinity. These two conditions guarantee existence of equilibrium. 11 As a result, we can state: Proposition 4 Assume the number of job applications a is a continuous variable; let 0 < k Ψ(2, θ, w). Then an equilibrium exists where workers send a ( 2) applications and firms randomly choose wages from the set [ w, 1 (1 w)(1 z(a, θ)) a 1 ] according to the wage distribution F (w; a, θ, w) = 1 z(a, θ) z(a, θ) [ [1 ] 1 ] w a 1 1 ; 1 w the number a solves Ψ(a ; θ, w) k = 0. The equilibrium unemployment rate is (1 z (a, θ)) a. This result characterizes labor market equilibria with multiple applications and wage dispersion. We note that the number of job applications workers send in equilibrium increases as k falls and becomes arbitrarily large as the cost of applications converges to zero. This contrasts with the work of Burdett and Judd (1983) and Acemoglu and Shimer (2000) where workers sample a maximum of two wages in equilibrium. We now come to the efficiency of wage dispersed equilibria. Assuming a is continuous, the efficient number of applications maximizes W (a, θ) ak, where W (a, θ) is given by (1). The first order condition is (1 z (a, θ)) a log[1 z (a, θ)] + (1 z (a, θ)) a 1 ( 1 z(a, θ) ) az(a, θ) = k (13) θ A comparison of the workers equilibrium condition (11) with (13) yields the following result: 10 The latter remark follows from noting that Ψ(a, θ, w) > (1 w)(1 z) a log[1 z] + (1 w)(1 z)[(a 1)z + a(1 z) log[1 z] = (1 w)(1 z) a 1 (a 1)[z + (1 z) log(1 z)] > 0 where the last inequality follows from z(a, θ) (0, 1). 11 For uniqueness of equilibrium with multiple job applications we need to check that, for any θ and w, dψ(a)/da < 0. Analytically this is difficult but numerical analysis of this derivative suggests that this is the case. 15

16 Proposition 5 Suppose k, θ and w are such that workers send multiple applications in equilibrium. Suppose also that a is a continuous variable. Then the equilibrium number of applications is excessive from the point of view of social welfare maximization. The proof is in the Appendix. This result, illustrated in Figure 5, shows that workers send too many applications in equilibrium. The graph plots the equilibrium number of applications a (θ) and the efficient amount â(θ). As mentioned above the social planner cares about the number of matches and thus internalizes the congestions caused by applications; by contrast, workers only care about their individual utility, which results in excessive job search. Figure 5: Private and social incentives to send job applications (k = 0.008, w = 0) 5 Wage policies Given the inefficiencies in the decentralized market economy, we now ask whether there is a role for the minimum wage to improve or restore efficiency? For this purpose we distinguish between the single application equilibrium and the equilibrium with wage dispersion. In subsection 5.1 we continue to take labor market tightness as given and show that for high θ, a positive minimum wage is desirable but it can never restore efficiency because as long as there is wage dispersion there is inefficient rent-seeking behavior. We then show that a unique and appropriate mandatory wage, ŵ can restore efficiency. However, in subsection 5.2, we show that, if there is free entry of vacancies, ŵ doesn t attract the right number of vacancies. In other words, there is no wage policy that gives constraint efficiency when vacancy creation is also endogenous. 16

17 5.1 Fixed vacancy-to-unemployment ratio Inspection of Figure 2b reveals that if the cost of sending job applications is sufficiently large and the equilibrium involves one job application only, then an increase in the minimum wage may induce worker participation in what otherwise is an inactive market. In that case, a minimum wage is desirable. Consider now the more realistic case of low application costs. Focusing on the wage-dispersion equilibrium, we examine how the number of applications a changes with the minimum wage. Proposition 6 Assume a is a continuous variable and that k, θ and w are such that workers send multiple job applications in equilibrium. (a) There exists θ such that a is increasing in w for all θ < θ. (b) There exists θ such that a is decreasing in w for all θ > θ. As a result, an increase in w reduces welfare for θ < θ and increases welfare for θ > θ. For fixed a, an increase in the minimum wage does not affect the probability that a worker is hired but it does affect the wage a worker expects to get. When θ is low, the wage distribution is close to the degenerate distribution at the minimum wage; an increase in the minimum wage shifts the wage distribution upwards, which results in larger marginal gains from sending applications. In this case, a raise in the minimum wage is welfare reducing. By contrast, when θ is large the wage distribution is close to the degenerate distribution at the competitive wage and, thus, an increase in the minimum wage reduces the marginal gains from applications by making the wage distribution even more compressed. In this case, a raise in the minimum wage is socially desirable. 12 Proposition 6 is illustrated in Figure 6 where we have computed the equilibrium number of applications as a function of market tightness for different values of the minimum wage. Since minimum wage policies cannot restore efficiency in our labor market, we close this Section by discussing whether mandatory wage policies can. A mandatory wage eliminates wage dispersion altogether and as such eliminates one source of inefficiency here. Suppose the wage is exogenously fixed to ŵ. Using (7) we can write the expected (gross) utility of a worker sending a l applications 12 This mechanism is quite different from others currently emphasized in the literature: a minimum wage increase raises the likelihood a job is accepted (e.g. Burdett and Mortensen, 1998), increases the search intensity or drives out less productive firms (e.g. Van den Berg, 2003). For evidence on minimum wage increases compressing the wage distribution see Teulings (2003). 17

18 Figure 6: Equilibrium number of applications, market tightness and minimum wage. given the rest of the workers apply for a jobs. Eu(a l ; a) = (1 (1 z(a, θ)) a l )ŵ. (14) This expression is very similar to the expression for welfare per worker in (13). If the government sets a mandatory wage very close to ŵ = 1 it is obvious that the number of job applications would be excessive because workers do not internalize the congestion effects caused by their applications. As ŵ decreases, workers incentives to send applications fall and the congestion effects are mitigated. The other extreme is when the mandatory wage ŵ is very small so that the private incentives fall short of social gains. This suggests the that a finely-tuned mandatory wage can align the incentives of individual workers with the collective interest. Proposition 7 For any given θ, k, let a be the solution to (13). Then there exists a mandatory 1 z(a,θ) az(a,θ) θ wage ŵ = 1 (1 z(a,θ))log[1 z(a,θ)] for which the market equilibrium is efficient. The proof is in the Appendix. The reason we can only get efficiency when we eliminate wage dispersion completely is that search is partly a rent-seeking activity. For a given mean wage, a higher variance of the wage distribution increases the incentives to apply for jobs because the worker s pay-off depends not on the average wage offer but on the expected highest offer. 18

19 5.2 Free entry of vacancies In this Section, we examine the free-entry long-run equilibrium. We start with a consideration of firm entry for an exogenously given number of job applications a > 1, i.e., we focus on the dispersed wage equilibrium. Let c be the cost of opening a vacancy. For a given number of workers u, and a given number of applications a, the free-entry equilibrium is given by the vacancy-to-unemployment ratio θ that solves Eπ i (w; F ( )) c = 0 (along with the equilibrium wage setting behavior given in Proposition 2). Or, using (4), a θ z(a, θ) (1 z(a, θ))a 1 (1 w) = c (15) The planner s problem consists of maximizing W (a, θ) cθ, where W (a, θ) is defined in equation (1). The optimal vacancy-to-unemployment ratio is then given by the following first order condition: dw (a, θ) dθ = a (1 z (θ, a)) a 1 dz (θ, a) c (16) [ dθ ] = a (1 z (θ, a)) a 1 z(a, θ) e a θ c = 0 θ θ Comparing (15) with (16) reveals that firms incentives to enter the labor market are not socially efficient in the absence of a minimum wage since a θ z(a, θ) (1 z(a, θ))a 1 a (1 z (θ, a)) a 1 = a ( ) θ (1 z(a, θ))a 1 1 e a θ > 0, [ ] z(a, θ) e a θ θ θ Call the socially optimal level of labor market tightness θ (solution to (16)). Since the minimum wage reduces firms incentives to enter the market, there must be a minimum wage w = w at which θ = θ. From (15) and (16) we get that: w = 1 z(a, θ) e a θ z(a, θ) = a θ e a θ 1 e. (17) a θ Proposition 8 For any given number of applications a, firm entry is socially excessive in the absence of a minimum wage; setting a minimum wage equal to w in (17) firm entry is optimal. Interestingly, this minimum wage is exactly equal to the minimum wage that is required for efficiency in the directed search framework with ex-post competition of Albrecht et al. (2003). Since in our setting all firms obtain the same expected pay-off we can just consider a firm that offers the minimum wage. In Albrecht et al. (2003), all firms are in the same position as a firm that offers 19

20 the minimum wage in our framework because when multiple firms compete for the same worker, Bertrand competition eliminates firm profit. Bertrand competition also drives down the ex ante posted wage to zero in their model so that a minimum wage is always binding. So surprisingly, the socially optimal minimum wage under multiple applications does not depend on whether search is directed or random. The socially optimal minimum wage is equal to the probability that a vacancy receives exactly one applicant ( a θ e a θ ) conditional on receiving one or more applicants (1 e a θ ). I.e., a worker only contributes to the matching process if she applies to a firm without other applications. Moreover, she contributes more the lower is a θ and therefore the minimum wage is decreasing in a θ. Albrecht et al. (2003) show that when workers apply for multiple jobs, efficiency requires both full ex-ante competition for candidates and full ex-post competition. We have neither of these. Since search is random here, high-wage firms do not attract more candidates than low-wage ones and since wages and the number of applications are determined simultaneously, there is no full ex-post (Bertrand) competition for candidates with multiple offers. A natural question is whether there exists a centralized wage policy that restores efficiency under free entry. From the previous section we know that there is only one mandatory wage, ŵ (given in Proposition 7) that sets search intensity optimally. We now study whether this mandatory wage also sets entry optimally. Under a mandatory wage, each vacancy faces the same probability to hire a worker. This probability is equal to the total number of matches per worker as given by (1) divided by θ. Therefore, under free entry and a mandatory wage, ŵ, θ is implicitly determined by: [1 (1 z (a, θ)) a ] θ Comparing the LHS of (18) with the LHS of (16) determines ŵ ŵ = 1 Finally, comparing ŵ with w yields: (1 ŵ ) = c (18) [ ] a (1 z (a, θ)) a 1 az(a,θ) θ (1 z(a, θ)) 1 (1 z (a, θ)) a. Proposition 9 Assume there is free entry of vacancies. Then, ŵ > ŵ, a, θ, so a mandatory wage alone can never generate the constraint efficient outcome. The proof is in the Appendix. As an additional instrument consider the case of a corporate tax. It is easy to see that this tax must equal t = (ŵ (a, θ ) ŵ(a, θ )) where (a, θ ) is the joint solution to (13) and (16). In this case, the pay-off of a firm, conditional on hiring a worker, would 20

21 be equal to 1 ŵ t = 1 ŵ. Other instruments that could restore efficiency are income taxes or entry fees. In Pissarides (2000, section 8.3) there does exist a unique mandatory wage that restores efficiency. The difference is due to the fact that his matching function is exogenous and therefore invariant to changes in the minimum wage while ours is endogenous. 6 Final remarks We have presented a simple labor market where workers decide upon the number of jobs they apply for and firms simultaneously post a wage. We have made explicit the fact that firms are typically capacity constrained and can hire one worker at most. The fear from being rationed and remain unemployed gives workers incentives to send multiple job applications. But multiple applications lead to wage dispersion, which in turn increases the desire to apply for multiple vacancies and stimulates rent-seeking behavior. socially excessive. As a result, the number of job applications in the market is An increase in the minimum wage can improve welfare when the vacancy-to-unemployment ratio is large since in this case it reduces the number of job applications workers send in equilibrium. An appropriately chosen mandatory wage eliminates wage dispersion altogether and can therefore align the private and the social incentives for a given level of labor market tightness. However, under free entry no mandatory wage instrument can restore efficiency. References [1] Acemoglu, D. and R. Shimer (2000), Wage and Technology Dispersion, Review of Economic Studies 67, [2] Albrecht, J.W. and B. Axell (1984), An equilibrium model of search unemployment, Journal of Political Economy, 92, [3] Albrecht, J.W., P.A. Gautier, and S.B. Vroman (2003), Equilibrium Directed Search with Multiple Applications, Tinbergen Institute discussion paper 4/3. [4] Albrecht, J.W, F. Tan, P.A Gautier, and S.B. Vroman (2004), Matching with Multiple Applications Revisited, Economics Letters 84, [5] Berg van den, G.J. (2003), Multiple equilibria and minimum wages in labor markets with informational frictions and heterogeneous production technologies, International Economic Review, 44-4, [6] Burdett, K. and K. L. Judd (1983), Equilibrium Price Dispersion, Econometrica 51, 4,

22 [7] Burdett, K. and D. Mortensen (1998), Wage Differentials, Employer Size and Unemployment, International Economic Review, 39, [8] Butters (1977), Equilibrium Distributions of Sales and Advertising Prices, Review of Economic Studies, 44-3, [9] Burdett, K., S. Shi, and R. Wright (2001), Pricing and Matching with Frictions, Journal of Political Economy, 109, [10] Diamond, P. (1971), A Model of Price Adjustment, Journal of Economic Theory, 3, [11] Galenianos, M. and P. Kircher (2005), Directed Search with Multiple Job Applications, mimeo, University of Pensylvania. [12] Gautier, P.A. and J.L. Moraga-González (2004), Strategic Wage Setting and Coordination Frictions with Multiple Applications, Tinbergen Institute Discussion Paper # /1, The Netherlands. [13] Hosios, A. (1990), On the Efficiency of Matching and Related Models of Search and Unemployment, Review of Economic Studies, 57, [14] Janssen M. and J.L. Moraga-González (2004), Strategic Pricing, Consumer Search and the Number of Firms, Review of Economic Studies 71, [15] Shapiro, J. (2004), Income taxation in a frictional labor market, Journal of Public Economics 88, [16] Kuhn, P. and M. Skuterud (2004), Internet Job Search and Unemployment Durations, American Economic Review, 94-1, [17] Lang, K. (1991), Persistent wage dispersion and involuntary unemployment, The Quarterly Journal of Economics, 106, [18] Montgomery, J. (1991), Equilibrium wage dispersion and interindustry wage differentials, Quarterly Journal of Economics, 106, [19] Pissarides, C. (2000), Equilibrium Unemployment Theory, 2 nd ed., MIT Press. [20] Teulings, C.N. (2003), Contributions of minimum wages to increasing wage inequality, Economic Journal, 113, Appendix Proof of Proposition 1: To prove (a) and (b) we first note that, for a fixed θ, the gross marginal social gains from sending job applications decrease in a. Rewrite welfare as W (a, θ, k) = 1 ( ) xθ 1 1 e x x, where x = a/θ. Next, treat a as being continuous and take the derivative of W ( ) with respect to a. This yields dw (a, θ)/da = g(x) θ, where g(x) = log [1 1 ] e x + 1 e x xe ( x ) (1 1 ) x e x x x 1 1 e x x x 22