Wage-Vacancy Contracts and Multiplicity of Equilibria in a Directed Search Model of the Labor Market∗ Nicolas L. Jacquet†
John Kennes‡
Singapore Management University
Aarhus University
Serene Tan§ National University of Singapore August 2016
Abstract This paper studies a directed search model of the labor market which is standard in all aspects except two. First, we allow firms to post wage-vacancy contracts advertising the number of workers they would pay as well as the payment all will receive. Second, we consider two cases: one where workers are risk neutral, and one where workers are risk averse, both in finite and large economies. Our paper shows that when firms post wage-vacancy contracts whether workers are modeled as risk neutral or risk averse matters: the types of symmetric equilibria and the nature of multiplicity of equilibria are different. Somewhat surprisingly, when there are finite numbers of risk neutral workers and firms, we obtain a finite number of symmetric equilibria, but when workers are risk averse we obtain a continuum of equilibria. Furthermore, our paper sounds a cautionary note on using large economies as an approximation of finite economies: when workers are risk neutral the nature of equilibrium is preserved going from a finite to a large economy, but the nature of equilibrium is different when workers are risk averse. JEL Classification: C78; D40; J41 Keywords: Directed Search; Wage-Vacancy Contracts; Multiplicity of Equilibria ∗
This paper was previously circulated with the title “Does Risk Aversion Matter in Directed Search
Models.” We would like to thank seminar participants at Monash University, Australian National University, and the University of Tokyo, and workshop participants at the 2014 Singapore Search and Matching Conference, and especially Ken Burdett, Pedro Gomis-Porqueras, Jacob Wong, and Randy Wright for their helpful comments. † Email:
[email protected]. School of Economics, Singapore Mangement University, 90 Stamford Road, S178903, Singapore. Tel: +65-68280293. ‡ Email:
[email protected]. Department of Economics and Business, Aarhus University, Fuglesangs Allé 4, 2632, 133, 8210 Aarhus V, Denmark. Tel: +45-87165564. § Corresponding author. Email:
[email protected]. Department of Economics, National University of Singapore, AS2 #06-02, 1 Arts Link, S117570, Singapore. Tel: +65-65163964.
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1
Introduction
In random search models where “searchers are randomly colliding as particles in space” (Lagos, 2000) trying to form matches with each other, frictions are a feature of the environment that is taken as given. In contrast, directed search models remove this informational problem: it assumes that searchers on one side of the market know exactly where all other searchers on the other side of the market are, and not only that, allows searchers on one side to signal its “attractiveness” to the other side in order to entice them to come to it. In pioneering work by Burdett et al. (2001) and Peters (1991) sellers have one good to sell, and they post a price they commit to. All buyers want to buy one good, and observe all prices offered by all sellers, and then they decide which seller to go to. Frictions in directed search models are coordination frictions. Imagine an environment with equal numbers of buyers and sellers; if buyers could coordinate among themselves, then they could come up with an arrangement where each buyer goes to one seller so that each buyer obtains a good. But when buyers cannot coordinate their search, which is a reasonable assumption especially in large markets, some sellers will receive no buyers while others will receive more than one buyer. Hence, in equilibrium some buyers will typically be unsuccessful in obtaining a good.1 Early directed search models assume that each seller posts a fixed price to be paid by the successful buyer - we call these price contracts, and it turns out there is a unique symmetric equilibrium. However, there can be multiplicity of symmetric equilibria once more general postings are allowed, as in Coles and Eeckhout (2003), where the price paid by the buyer is contingent on the number of buyers who show up at the seller - firms post a contingent price contract.2 Postings where the price paid can be contingent on realized demand are certainly interesting, not least because they are relevant for some markets, for, as was first shown by Coles and Eeckhout, they admit auctions as one of the equilibria. However, these types of postings are not widely observed in labor markets: instead, what is commonly observed is that an employer typically advertises the number of positions available for a certain kind of job, and the wage associated with that position. In the civil service, for instance, salaries are often determined by a pay scale, and all applicants hired for a given type of job are often hired at the same rank. Also, in the academic job market for junior hires in disciplines 1
The literature typically focuses on symmetric equilibria to capture agents’ inability to coordinate. Since symmetric equilibria are such that all buyers go to each seller with the same probability, typically some buyers will be rationed. 2 Geromichalos (2012) considers more general mechanisms than Coles and Eeckhout (2003) and also obtains multiplicity.
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like Economics, institutions typically advertise the number of positions to be filled and the identical wage paid to each candidate.3 Moreover, apart from papers interested in differences in firms size, e.g., Lester (2010), Hawkins (2013), and Tan (2012), the directed search literature has so far ignored the possibility of firms competing on the probability of receiving payment and not just the payment if successful. In this paper, we study a directed search model of the labor market where firms are restricted to post contracts with a fixed wage, like in Burdett et al. (2001), but can compete on the probability for workers to receive a payment.4 In particular, firms post contracts of the form ( ), where firms can advertise the number of positions that they have and the (fixed) wage all hired workers will receive. We call these contracts wage-vacancy contracts. One way to think about the contracts firms post is that there is one job to be filled, workers have no disutility from work, and if firms post 1 and workers are indeed hired that one worker is asked to do the job while all other ( − 1) workers receive
side payments.5 Another way to think about this type of contract is that a job is a task
and is perfectly divisible, and firms can choose how to divide a task up, and the wage each of these workers will receive.67 By allowing firms to post 1 firms are effectively able to compete on the probability that workers can receive a payment, even though they do not directly choose this probability. We consider two cases, the standard one where workers are risk neutral and another where workers are risk averse, in both a finite economy and in a large economy, which is the limit of a finite economy. In fact, firms compete not only on the wage they post but also on the probability of employment through their choice of the number of vacancies to post, and risk averse and risk neutral agents trade off the expected value and variability of payoffs differently. Hence, it cannot be assumed that firms will choose the same types of contracts in the two cases. And it turns out that equilibria can look very different in the two cases. In the finite economy, the equilibria look very different when workers are risk neutral 3
In practice, junior hires can be paid differently even though they were hired in the same year by the same employer, but the difference in pay typically reflects differences in outside options or perceived quality of the job candidates. Such differences do not arise in our model since workers are identical and restricted to sending one application. 4 We can allow firms to post wages contingent on the position of the worker in the queue among those who are hired, but that will not materially affect the main results, so we abstract from this. 5 One interpretation of this is that the ( − 1) workers are hoarded. 6 Some may argue that firms that choose the latter option look like the civil service in some countries, with low-paying, but few-hours type of jobs. 7 We abstract from firms having larger capacities than unity because we want to keep as close as possible to the standard directed search framework.
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than when workers are risk averse. In particular when workers are risk neutral we get a finite number of symmetric equilibria, while there is a continuum of equilibria when workers are risk averse. When workers are risk neutral and supposing there are workers, there will be equilibria, with a unique wage for each of these equilibria. One equilibrium has all firms promising payment to only one applicant, another equilibrium has all firms promising payment to two applicants, and so on. The multiplicity here arises because there are too many degrees of freedom. When workers are risk neutral a lower wage can be exactly compensated by an increased probability of payment, implying there are as different ways to offer a given level of expected utility to workers: given the posting of all other firms, a firm is indifferent between posting (1 = 1), (2 = 2),,( = ) for appropriately chosen 1 . And for each there is a unique such that if all other firms post with , then the last firm is not strictly better off by posting something else. These equilibria are not payoff equivalent and can be ranked by each type of agents. Workers prefer equilibria where firms post larger ’s because they obtain a larger expected utility due to a competition effect: there are more vacancies chasing the same number of workers. Conversely, firms prefer equilibria with smaller ’s due to this competition effect. If we allow for free entry and consider efficiency, the unique efficient equilibrium is the one where all firms post vacancies. This corresponds to the “perfectly competitive” outcome where the last firm does not have an impact on the expected utility of workers in applying to other firms, since all other firms already guarantee employment to workers. Hence this equilibrium is efficient. In all other equilibria where firms post strictly less than vacancies, the last firm, by changing its wage, can affect workers’ expected utility of applying to other firms, and hence this last firm can be thought of as having some monopoly power. When workers are risk averse, in a finite economy all symmetric equilibria are such that firms promise payment to all applicants, and there is a continuum of symmetric equilibria. That is, there exists a continuum of wages such that all firms promise payment to all applicants at the same wage. Firms promise payment to all workers in all symmetric equilibria because, for a given level of expected utility promised to workers, a firm can lower its expected wage bill by increasing the number of applicants it promises to pay. In fact, a firm can charge an insurance premium to workers for lowering the risk they face of not receiving payment, and the higher the the higher the premium. The existence of a continuum of equilibria comes from a trade-off firms face between the insurance premium a firm can capture and the control it has over the probability of application of workers. If a firm posts vacancies like other firms, then it has no control over the probability with which it is applied
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to, for all firms offer the same utility to workers. If the firm instead posts a contract which promises payment to − 1 workers, it can charge a lower insurance permium, which ceteris
paribus reduces its profit, but it can control the probability it is applied to by its choice
of wage to be paid, which ceteris paribus increases its profit. It turns out that the loss in profit coming from the lower insurance premium that the firm can charge by posting − 1
vacancies exceeds the gain coming from the ability to control the probability of application as long as the wage charged by all firms falls within a range. As in the case of risk neutral workers equilibria can also be ordered by each type of
agents. In this case it is clear that workers prefer equilibria with higher wages, and the opposite holds true for firms. If we allow for free entry of firms and consider efficiency, only one of the equilibria is efficient. This therefore means that there is a continuum of inefficient equilibria, some with too much and others with too little entry. This is in contrast with the risk neutral case where all inefficient equilibria are characterized by too much entry. In a large economy, when workers are risk neutral we still have countably many equilibria with a unique wage associated with each equilibrium: one equilibrium has all firms posting one vacancy, another has all firms posting two vacancies and so on, including an equilibrium where all firms hire all who show up. Interestingly, all equilibria are payoff equivalent and efficient when free entry of firms is allowed. However, when workers are risk averse, all firms post as many vacancies as there are workers and the continuum of wages obtained in the finite economy collapses to one wage - and this wage is exactly the wage posted by firms when workers are risk neutral and when they post as many vacancies as workers. In other words there is a unique equilibrium in the limit when workers are risk averse, which is efficient, and this outcome corresponds to one of the equilibrium outcomes in the case where workers are risk neutral. Hence, although all symmetric equilibria are efficient in large economies, there is a great difference in outcomes between the risk neutral and risk averse case. Note that for these differences to arise one must consider general contracts and not restrict firms to posting wage contracts. Another way to think about these results is in terms of the change of nature of equilibrium when we go from a finite economy to a large economy. When workers are risk neutral, the nature of equilibrium does not change when we go from a finite to a large economy, since we go from a finite number of symmetric equilibria to a countable number of symmetric equilibria with the same features. However, when workers are risk averse, the nature of equilibrium changes dramatically when we go from a finite economy to a large economy: the continuum of equilibria in a finite economy collapses to a unique equilibrium in the large
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economy. Hence, using large economies as an approximation of finite economies with many agents, which can be convenient because a market utility property holds in large economies, is not problematic with risk neutral agents, but it can be with risk averse agents. So our paper sounds a cautionary note on using large economies as an approximation of finite economies. Our paper is particularly related to the literature that studies directed search models where firms are allowed to post contracts more general than wage or price contracts like in the Burdett et al. (2001) model. On one hand we focus on a more restrictive choice of contract offerings than what McAfee (1991) and Coles and Eeckhout (2003) allow. But on the other hand, our paper is more general than these papers because we allow “unsuccessful” workers to be compensated; in these papers only one buyer can be served and firms cannot compensate unsuccessful buyers. Geromichalos (2012) also allows for general contracts, and in addition allows firms to choose their capacity. But like the two previous papers, firms do not compensate unsuccessful buyers. Although related to Jacquet and Tan (2012), this paper differs from that paper in several respects. First, this paper considers differences in equilibrium outcomes when workers are risk averse compared to when they are risk neutral, whereas Jacquet and Tan (2012) consider only the case of risk averse workers. Moreover, they allow for very general contracts where payments made by firms can vary with the position of a worker in the queue and with realized demand whereas in this paper the types of contracts we allow firms to post are limited to contracts that we think are more empirically relevant. The paper is structured as follows. Section 2 lays out the model setup. Section 3 solves the Social Planner’s problem. Section 4 characterizes equilibrium when workers are risk neutral. Section 5 characterizes equilibrium when workers are risk averse. Section 6 compares the results in sections 4 and 5. Section 7 discusses in what ways the differences in results matter. All proofs not in the main text can be found in the Appendix.
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Setup
Agents and Payoffs - There is a large number of identical, profit maximizing, and risk neutral firms8 and identical workers. Firms can be active by paying a cost ∈ (0 1). We
denote by the number of active firms, which are indexed by ∈ {1 }. Each active firm has one productive job, and if the job is filled, the product to be split is 1. If the job is
unfilled, the firm is idle and nothing is produced. Each firm can choose to make payments 8
Firms are commonly thought of as less risk averse than workers. Hence, modeling firms as risk neutral can be thought of as a normalization.
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to = {1 } workers;9 and we assume that firms must make the same payment to all
workers who receive a payment. We restrict ourselves to non-negative payments, and since in equilibrium a firm will never post a payment greater than the product of a match, we know that payments are bounded above by 1. We denote by = ( ) the contract posted by
firm , which states that they will make payment of to workers.10 We assume that firms have deep pockets and therefore there are no restrictions on how negative the realized profit of a firm can be. When firm posts and workers have applied, its profit is ( 1 − if ; and ( ) = 1 − if ≥ Workers are indexed by ∈ {1 }. Each worker can make only one application,11
possibly with mixed strategies. For a worker who has applied to firm , the utility of a worker is (), where = 0 if he is not paid, and = if he receives payment from . We assume that (0) = 0 and 0 () 0 for all ≥ 0.
Timing of events - There are two stages. In the first stage, the entry stage, firms decide
whether or not to be active. Given a number of active firms, the second stage, which is a game we call the directed search game, unfolds: this is itself a two-stage game. In the first stage, the contract posting game, each firm decides on the contract to post, and all firms simultaneously post their announcements, which are publicly observable. As is standard in directed search, we assume that firms can fully commit to their announcements. In the second stage, the application subgame, once workers have observed the postings of all firms, they choose their application strategy, and we denote by ∈ [0 1] the probability with
which a worker applies to firm . If a firm receives less applicants than the it had posted,
it will hire all of them, and then randomly pick one worker to produce. If a firm receives strictly more applicants than , it will randomly choose workers to make payments to, and again, one of these workers will produce. In other words, we assume that from the firms’ point of view, all workers are anonymous to it, so it treats all of them the same way. We also assume that firms are anonymous to workers, so that if two firms offer the same contract, workers treat them the same way and will apply to them with the same probability. 9
Since there are only workers, there is no gain for a firm to promise payments to more than workers. Another way to think about ( ) is that a firm is posting vacancies at wage . We use these two interpretations interchangeably. 11 Allowing workers to make multiple applications complicates the analysis considerably, especially in the finite case (see Albrecht et al. (2004)). To deliver our point as succintly as possible we abstract from these complications. 10
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Definition of Equilibrium - Restricting ourselves to the case where firms use pure strategies and workers use symmetric strategies, we define an equilibrium as follows. Definition 1 Given a number of active firms , a Subgame Perfect Nash Equilibrium of the directed search game is a strategy profile (σ θ ), where σ ≡ ( ) =1 and θ ≡ ( )=1 ,
such that:
(i) given σ , θ is the Nash equilibrium in the application subgame; and (ii) given θ in the subgame, σ is the Nash equilibrium in the contract posting game. This is the definition of a directed search game, taking as given the numbers of workers and firms, which is standard. In the first stage, there is free entry, and the number of active firms is such that all active firms make zero profit net of the cost of entry. In other words, is such that = , but we abstract from whether the number of active firms is an integer.12 We solve this model by first taking as given the numbers of workers and firms and solving for the equilibrium in the directed search game, and then we solve for the equilibrium number of active firms with free entry. Note that given a number of active firms, the directed search game that ensues is just like the Burdett et al. (2001) framework, except that firms can make payments to more than one worker. Discussion of Assumptions - There are four assumptions we would like to briefly discuss. First, we assume that firms have deep pockets. We think it is not an unreasonable assumption for the following three reasons. This is not unlike what is implicitly assumed in the Diamond-Mortensen-Pissarides model of the labor market where a firm must pay a cost to post a vacancy as long as its job is not filled, and there is no restriction on how long that may be. Moreover, one could think of each firm in our paper as being a division or establishment of a larger firm, and that there is some pooling of risk across divisions. Or, one could open a market for contingent securities so that firms can insure each other against the possibility of having a wage bill in excess of the product of the match, which is feasible since each firm makes non-negative profits in expected terms. Second, we maintain the assumption that firms are able to perfectly commit to their postings. Some might say that this is a very strong assumption given that firms can make losses when they hire many workers. However, we want to stay as close as possible to the standard directed search models assuming wage contracts. Moreover, even with wage 12
It is possible to more precisely define the number of active firms to be integer-valued, but this unnecessarily complicates the analysis without yielding any insight.
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contracts, firms have an incentive to renege on their posting. In fact, if two or more workers apply to a firm it can auction the job and offer it to the worker asking for the lowest wage, and when only one worker applies, either the firm or the worker will have an incentive to ask for a change in the wage (depending on the wage determination mechanism that would be used if the wage can be renegotiated ex post). In other words, dropping the assumption of commitment, although interesting, brings other issues and delivers equilibria different from what one obtains with commitment even with wage contracts. Hence, we do not see the assumption of commitment as being stronger in our setup as it is with a large part of the directed search literature.13 Third, we consider a static model and thereby ignore the role that self-insurance through savings can play for workers who face the risk of being unemployed. This choice stresses riskaversion over the willingness to intertemporally substitute consumption, very much like in implicit contract models. A by-product of this choice is that we avoid issues with a potential lack of analytical tractability which often arise in models with savings and risk.14 Finally, there are two ways to think of our assumptions regarding production and the disutility of effort. The usual way to think of production is to have one worker produce one unit of good, so if a firm makes payments to 3 workers, only one worker is asked to produce while the other two are not. Another way, though, is to allow labor input to be perfectly divisible, so that if the firm were to make payments to 3 workers, then all 3 workers are asked to work 1/3 of the time in order to produce the one unit of good. For ease of exposition we use the former interpretation in this paper.
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The Social Planner’s Problem
3.1
Risk Neutral Workers
We assume that the social planner wants to maximize the sum of expected utilities of all workers, and that he faces the same constraints and restrictions as agents. In particular, just as workers cannot coordinate their applications in equilibrium, the planner cannot simply direct workers to go to specific firms. When workers are risk neutral, the social planner’s problem is equivalent to maximizing the total expected output net of the cost of setting up firms. In order to do so in the finite 13 See Camera and Selcuk (2009) for a directed search model of the labor market where ex post renegotiations are allowed. 14 Endowing agents with savings would not alter the analysis in our static setup, for an agent would still consume different amounts whether unemployed or employed.
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economy, the planner will ask workers to apply to each firm with probability 1 so as to minimize the expected number of firms receiving more than one application. Given this, the planner will choose to solve ∙ µ ¶ ¸ 1 max 1 − 1 − − .
(1)
Since the probability that no worker applies to a firm is (1 − 1) , the terms in the square
brackets 1 − (1 − 1) is the probability that at least one worker shows up at a firm, which is also the expected output of a firm, since the firm only needs to hire one worker to produce. The solution to the planner’s problem is thus given by , which uniquely solves µ ¶ µ ¶−1 1 1 − = 1− 1− 1−
(2)
and has a nice interpretation. When the planner is considering whether to have an extra firm in the economy this has two effects. First, if the firm has at least one worker applying to it output increases by one, and this event happens with probability 1 − (1 − 1) . The
second effect is that the entry of this firm creates an externality on other firms. An extra
firm reduces the probability with which other firms receive applications, and as the expected queue length at each firm is the last firm also expects to receive workers. Since a worker’s expected marginal product is (1 − 1)−1 , the total externality imposed on other
firms through reduced output is () (1 − 1) .
The reason why (1 − 1)−1 has the interpretation of a worker’s expected marginal
product is because when a worker has applied to a firm, the probability that at least one £ ¤ other worker has applied to this firm is 1 − (1 − 1)−1 in which case this worker’s
marginal product is 0, and with probability (1 − 1)−1 no other worker has applied in
which case this worker’s marginal product is 1.
As the entry of an additional firm has two effects, the planner chooses the right number of firms to be active in the market such that the net gain from entry of this extra firm exactly offsets the cost of entry . In the large economy, which is taken to be the limit of the finite economy as the number of agents goes to infinity at a given rate, i.e., , → ∞ at rate = , and taking the
limit of the net output per worker, the planner’s problem is to choose to maximize max
1 − − −
and his unique choice solves 1 − − − − = 10
(3)
We summarize the results of the planner’s problem when workers are risk neutral below. Proposition 1 When workers are risk neutral the planner will: () ask workers to apply to all firms with the same probability; and () in the finite economy choose the number of active firms to be , the unique solution to equation (2), in the large economy choose the measure of workers relative to active firms to be , the unique solution to equation (3).
3.2
Risk Averse Workers
When workers are risk averse the planner’s problem is more complicated. As the number of productive matches is a random variable, we allow the social planner to insure workers against this randomness. This is achieved by assuming that the social planner in turn is insured by firms with deep pockets, just like firms can insure workers in equilibrium by offering payments in excess of the output produced. As this was done in Jacquet and Tan (2012), we do not repeat their analysis here,15 but will just state the results. Proposition 2 When workers are risk averse the social planner will do parts () and () of proposition 1, and in addition, () allocate to each worker for all possible realizations in the number of productive matches his expected marginal product, which is (1 − 1)−1 in the finite economy, or − in the large economy.
Parts () and () are the same for the planner whether workers are risk neutral or risk averse. The only difference is that if workers were risk averse the planner would want each worker to, in addition, get the same share of output, which is their expected marginal product, no matter the realization of the number of productive matches, which is part (). To see that workers do indeed get their expected marginal product no matter what, equation (2) can be rewritten as µ ¸ ¶−1 ∙ µ ¶ 1 1 1− = − 1− 1−
(4)
The right hand side of equation (4) is the total expected output net of cost per worker, and the left hand side is the expected marginal product of a worker. What the planner wants, since workers are risk averse, is for workers to get the same no matter how many productive matches there are, which is intuitive. Hence, the planner ensures that any variability to workers does not occur. What workers end up getting is an equal share of the expected 15
Please see the proof of Lemma 1 in Jacquet and Tan (2012).
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total output, and from equation (4), it is clear that the expected total output when divided equally across all workers corresponds to each worker’s expected marginal product. For the large economy, we have − =
1 − − −
which is the limit counterpart of equation (4), where the expected marginal product of a ¡ ¢ worker, − , equals the total output net of cost per worker, 1 − − −
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The Equilibrium with Risk Neutral Workers
In this section we assume that () = .
4.1
The Directed Search Game
Suppose the number of active firms is . Denoting () as the probability of workers turning up at a firm when workers apply to that firm with probability , µ ¶ () = (1 − )− the profit of firm which posts ( ) is:
( ; ) =
P −1
=1
() (1 − ) +
³P
=
´ () (1 − )
where the first set of terms is the profit the firm makes if up to − 1 workers shows up,
in which case the firm produces 1 and pays out a wage to all applicants; the second set
of terms are the profit the firm makes if at least workers show up, in which case the firm produces 1 and pays to randomly chosen applicants so that the total wage bill is . It turns out that ( ; ) can be rewritten as ( ; ) = [1 − (1 − ) ] − Ω ()
(5)
where the terms in the square brackets equal the probability of at least one person applying to firm , and Ω () is the probability that a worker who applies to firm is hired when the firm promises payment to applicants, where ¤ £ P ( − ) () − =0 Ω () = 12
Equation (5) is very intuitive, as it says that a firm’s expected profit is simply the expected output minus the expected wage bill. The probability that the firm hires at least one worker is [1 − (1 − ) ], and since the output in this event is 1 [1 − (1 − ) ] is also the
expected output. The expected wage bill to a firm is the expected number of hires, Ω (), multiplied by the wage paid per worker. Furthermore, it is useful to note that (5) can be re-expressed as ( ; ) = [1 − (1 − ) ] − ( ; ) ,
(6)
where ( ; ) = Ω () is the expected utility of applying to firm given its posting and that workers apply to it with probability (given the postings of other firms). Finally, let ( − ) |− =1 ≡ −
− | =1 ∈ [−1 0]. − −
is the elasticity, evaluated at = 1, of the expected utility of applying to any firm with vacancies with respect to the probability − with which workers apply to any other firm. We then have the following proposition, which characterizes the set of symmetric equilibria of the directed search game. Proposition 3 When workers are risk neutral there are symmetric equilibria where firms post the same number of vacancies ∈ {1 } at a wage ∗ () =
(1 − 1)−1 £ ¤ Ω (1) 1 − ( − ) |− =1
(7)
and workers apply to each firm with probability = 1 in each of these equilibria. In an equilibrium with a given , each firm’s profit is ¶ ¸ ∙ µ 1 ∗ ∗ − () × Γ () |=1 , () = 1 − 1 −
(8)
where Γ () ≡ Ω (). To understand this result, it is useful to first assume that firms do not internalize the impact of their posting on the expected utility of workers, i.e., firms take in expression (6) as given.16 The profit function expressed this way is strictly concave in , and there therefore 16
This way of solving a directed search game assumes what is called a market utility property, a technique first used by Montgomery (1991), and which holds in large economies but not in finite economies as Burdett et al. (2001) showed.
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exists a unique ∗ () that maximizes the firm’s profit for a given . This implies that firm chooses its posting such that workers apply to it with probability ∗ (). But there are different ways to do so: firm can offer (1 = 1) such that Ω(∗ (); 1)1 = , or (2 = 2) such that Ω(∗ (); 2)2 = , or any ( ), ∈ {1 }, such that Ω(∗ (); ) = .
In fact, for a given , the higher the employment guaranteed by firm to a worker when it offers a higher is exactly offset by the lower wage it posts to give workers the same . And since all these contracts induce the same probability of application from workers and implies the same level of expected utility of application, the expected wage bill is the same and therefore firm makes the same profit for these different postings. Importantly, this is because workers are risk neutral. Since in a symmetric equilibrium all firms post the same contract and each worker applies to each firm with probability 1, a posting ( ) is an equilibrium posting only if, given that all other firms post this contract, the last firm chooses the same posting, which is equivalent to choosing to be applied to with probability 1. However, for a given number of vacancies posted by other firms − , there is a unique wage (− ) posted by all other firms such that firm chooses to be applied to with probability 1. There are contracts
that deliver this, but the potential candidate for a symmetric equilibrium is the posting where firm posts as many vacancies as other firms, and the associated wage is equal to the wage posted by other firms. Given the posting profile we can compute the expected utility of workers ∗ (). There are two complications when characterizing properly the set of symmetric equilibria. First, in finite economies firms do not take as given the expected utility they have to offer workers, i.e., the market utility property does not hold. Although the best-response of firm is still a -point correspondence, with each element having a different number of vacancies posted, the failure of the market utility property implies that for a given posting of all other firms the optimal choice of for firm varies with . The fact that firms take into account the impact of their posting on the expected utility of applying to other firms through the ∗ probability of application is captured in the equilibrium wages () by the elasticity
in the denominator. This is explained in more detail below when we discuss the properties of symmetric equilibria in Section 4.1.1. Second, when all other firms’ posting is such that − = , then firm is no longer able to fully control the probability it is applied to if it also chooses to post = . In fact, if it posts a wage lower than all other firms, no worker will apply; if instead it posts a wage above what other firms post, then all workers apply with probability 1; and finally, if it posts
14
the same wage as other firms then workers apply with probability 1. This highlights a feature of the model which is crucial when we consider risk averse workers: there is a cost to a firm in posting = vacancies, and this cost is in the restriction that this choice of vacancies imposes on the probability a firm is applied to. Equation (7) is a closed form expression for the wage posted by firms in all the symmetric equilibria. When all firms post = 1 the equation corresponds to the expression in Burdett et al. (2001) (put in the labor market context). In particular, when all firms post = , which means that they will pay everyone who shows up, the wage posted by firms is simply ¶−1 µ 1 ∗ () = 1 − , (9) that is, each worker is paid his expected marginal product. The reason why there is multiplicity of equilibria as shown in Proposition 3 is because firms have too many degrees of freedom, like in Coles and Eeckhout (2003). In Coles and Eeckhout (2003) the extra degree of freedom relative to Burdett et al. (2001) comes from the fact that firms can post contingent contracts, whereas in this paper the extra degree of freedom comes from the number of workers who can be compensated. The reason why there is a finite number of equilibria here, as opposed to a continuum of equilibria like in Coles and Eeckhout (2003), is because firms are choosing from a finite number of possible ’s.17 4.1.1
Properties of Symmetric Equilibria
∗ Comparison of ()
Suppose all other firms posted = 1 at some wage , and the last firm is trying to choose the to post. It is true that on one hand the higher the the last firm posts, the lower the wage it has to offer because of an “insurance” effect: taking as given what other firms post, the last firm can offer the same expected utility to workers even though it is lowering the wage because the increase in raises the probability that workers receive a payment. One may then think that a symmetric equilibrium offering more ’s necessarily has a lower associated wage. However, there also exists a competition effect: as increases, more “vacancies” are chasing the same number of workers, and the greater competition for workers drives the wage up. 17
Whereas in Coles and Eeckhout (2003), in a goods market context, firms, when deciding which prices to post, can choose prices along an interval. So given what all other agents are doing, instead of having an -point correspondence like in our paper, their best-response correspondence is an interval of prices.
15
∗ Recall the expression for () in equation (7) is ∗ () =
(1 − 1)−1 £ ¤. Ω (1) 1 − ( − ) |− =1
The insurance effect is captured through Ω (1), which increases with , while the competition effect is captured by ( − ). The absolute value of this elasticity | | is a measure of a firm’s “monopoly power”: a
firm is said to have monopoly power if by changing the price it posts it can affect a worker’s probability of being employed at other firms. To see this, let us suppose we are in the Burdett et al. (2001) environment where all firms are restricted to posting = 1, and now imagine
that a firm wants to try to reduce the wage it posts given what all other firms are doing. By doing so, workers will find it relatively more unattractive applying to this firm, and will thus apply with higher probability to other firms, but when all workers do that, the probability of being employed at other firms falls. Hence, this firm is said to have some monopoly power because the fall in labor supply to the firm when it decreases its wage posted is limited by the fact that it becomes harder for a worker to be employed elsewhere. Another way to think of this is that firms do not face a perfectly elastic labor supply curve. This monopoly power disappears, i.e., = 0, in two cases: the first is when each firm is too small to matter, which, as we will show shortly, is true in the large economy; and the second case is when all firms post = , because then this last firm cannot affect the probability with which workers are employed at other firms. Intuitively, one would expect | | to be decreasing in , for as increases competition
increases and therefore a firm’s monopoly power should decrease. Although we have not been able to establish this result analytically, numerical results suggest it is indeed the case. The following lemma establishes that as long as there are not too many vacancies relative to the number of firms, then the wage is decreasing in the number of workers that firms promise a payment to. ∗ ∗ Lemma 1 For given , ( + 1) ≤ () for all = 1 − 1, with strict
inequality for − 1.
Since firms do not offer greater than the number of workers , we also have the following corollary. ∗ ∗ ∗ Corollary 1 For , ( + 1) () for all , and if = then ( + 1) ≤
∗ () for all , with strict inequality for all − 1.
16
Note that when firms go from posting = − 1 to = the wage does not change
when = , thereby implying that in this case the competition effect exactly compensates the insurance effect. Furthermore, numerical results also suggest that when is very low ∗ ∗ relative to , then there is a critical such that ( + 1) () for ≤ , and
∗ ∗ ( + 1) () for other , so the competition effect for vacancies dominates ∗ for . In the Appendix, we provide an example, Example 1, where () changes
non-monotonically as changes. Comparison of ∗ ∗ ∗ The expected wage a worker receives in equilibrium, () ≡ Ω (1) × (), is
simply:
∗ () =
(1 − 1)−1 . 1 − |=1
(10)
∗ When workers are risk neutral, the expected wage is also their expected utility, i.e., () = ∗ ().
Comparing equations (9) and (10), and recalling that a worker’s expected marginal product is (1 − 1)−1 , there is no “wedge” between expected wage and the expected product
if and only if the elasticity is zero, which in the finite economy, is true in the equilibrium where all firms post = . In other words, in all other equilibria where , because a firm has some monopoly power, there is a wedge between a worker’s expected wage and his expected marginal product. ∗ Proposition 4 () is (strictly) increasing in if and only if | | is (strictly) decreasing
in .
Proof. The result follows directly from expression (10). As mentioned above, we have not been able to establish that the elasticity | | is decreas-
ing in , but numerical results show that it is the case, and thus numerical results show that
∗ () is strictly increasing in , which in turn implies that ∗ () is strictly decreasing
in . 4.1.2
Large Economy
∗ Let and go to infinity at rate ≡ , define ∞ (), ∗∞ (), and Ω∗∞ () as the limit
∗ ∗ ∗ of (), ∗ () and Ω∗ () respectively, and ∞ () = Ω∗∞ () ∞ ().
17
Lemma 2 In the large economy, the wage posted by firms in the symmetric equilibrium where payments are promised to workers is ³ P−1 =1 − =0
∗ () = P ∞
∗ () is decreasing in . and ∞
!
´
(11)
Even though is not always monotonically falling in in the finite economy, it is true in the limit. From the above equation, it is clear that the more firms post, the more positive terms are added to the denominator, and hence the lower the equilibrium wage. The reason behind the difference between the finite and large economies lies in the competition effect: in the large economy the competition effect is at its maximum for all possible number of vacancies posted: = 0 for all because a firm is too small to have any impact on the expected utility of applying to other firms. It is easy to show that in the limit, Ω∗∞ (), the probability of being hired at a firm posting
, conditional on applying there, is Ω∗∞ and hence
µ ∙ X−1 ¶¸ − X − () = =1 =0 ! ∗ ∞ () = −
which is independent of . As for firms’ profit, ∗∞ () = ∞ = 1 − − − − ,
(12)
which is also independent of . ∗ () and Ω∗∞ () move in opposite directions when changes in such a Mathematically, ∞
∗ way that they cancel out exactly, so ∞ is independent of , and hence ∗∞ is also independent
of .
∗ The intuition why ∞ is independent of is that in the large economy a firm is too small
to matter in affecting a worker’s payoff in applying elsewhere, so the elasticity is zero for all ’s, and hence, the expected wage of a worker, which is just the expected utility of a worker, is exactly equal to a worker’s expected marginal product, and is independent of .18 18
We can see this from equation (10) when we take the expression at the limit with = 0.
18
4.2
The Entry Stage and Efficiency
In the finite economy, the number of firms that enter ∗ () is such that ∗ () = ,
(13)
where ∗ () is defined in equation (8). When = , it is easy to show that ∗ () is strictly decreasing in for all , and that there thus exists a unique ∗ () solving the above equation. However, we have not been able to show analytically that ∗ () is strictly decreasing in for all , though numerical results suggest that for given a , ∗ () is strictly decreasing in , and hence there is a unique ∗ () that solves ∗ () = . Given the entry stage, how does this affect the symmetric equilibria laid out in Section 4.1? Numerical results show that in the finite economy, the equilibrium wage with free-entry ∗ ∗ () is decreasing monotonically in the number of vacancies posted, and that workers’
expected utility is non-monotonic in the number of vacancies posted - In the Appendix we provide an example, Example 2, where the expected utility is non monotonically changing ∗ in even though ∗ is monotonically decreasing in .
In the large economy, the ratio of firms to workers, , must satisfy ∗∞ () = ,
(14)
where ∗∞ () is defined in (12), and it is easy to show that ∗∞ () is decreasing in . Since ∗∞ () is independent of , there is a unique ∗ that solves (14) for all values of .
Comparing the symmetric equilibria to the allocation chosen by the social planner, we
have the following proposition. Proposition 5 When workers are risk neutral: (i) in the finite economy there is a unique efficient symmetric equilibrium, and it is where ∗ ∗ −1 firms post = at wage , where ∗ solves (13) for = ; and ∗ = (1 − 1 )
(ii) in the large economy, all equilibria are payoff equivalent, and all equilibria are efficient.
19
5
The Equilibrium with Risk Averse Workers
Assume that 00 0.
5.1
The Directed Search Game
We will first establish that a firm never wants to post less than vacancies, and then we will show that there exists a continuum of wages for which a symmetric equilibrium exists. Lemma 3 Any posting where firms post cannot be a symmetric equilibrium posting. The idea is that if firms other than firm post some − , then firm can fully control the probability it is applied to. Moreover, it has an incentive to deviate to post − : by lowering the variability in payoffs through a higher probability of a payout, the firm can charge risk averse workers an “insurance premium” by lowering the expected wage and earn more profit because of the lower expected wage bill without suffering from any change in the expected number of applicants. Since all firms posting cannot be part of an equilibrium, the only possible symmetric equilibria are such that all firms post = . The rest of this section analyzes the existence and multiplicity of these equilibria. Let us suppose that all firms other than post the contract (− ) and firm posts ( ). Then we have that the profits of firm ⎧ ⎪ ⎪ ⎨ 0 = e ⎪ ⎪ ⎩ b
are given by if − ; if = − ; if − ,
e = [1 − (1 − 1) ]− . In fact, if firm posts − where b = 1− and
then no worker will apply to firm since this worker can go to any other firm and be hired for sure, so firm has zero profit; if − then all workers will go to firm so its profit
is b = 1 − ; and if firm posts = − , workers will go to each firm with equal
probability 1 and the firm will enjoy a profit e = [1 − (1 − 1) ] − . We then
have the following result.
Lemma 4 Assume all firms other than post (− ) and firm posts vacancies. Then firm posts the same wage as all other firms if and only if £ ¤ − ∈ (1) (1 − 1)−1 () [1 − (1 − 1) ] . 20
Proof. We have that b≥ e if and only if − ≤ (1) (1 − 1)−1 , while b ≥ 0 if and
only if − ≤ 1 (which is strictly greater than (1) (1 − 1)−1 ) and e ≥ 0 if and only if ≤ () [1 − (1 − 1) ]. The result follows.
What we have left to check is to rule out all possible deviations by a firm from = , given that all other firms are posting − = at − . In general, we have to check that the deviating firm does not want to deviate to post = 1 2 − 1. However, the following
lemma establishes that we only need to check that firm does not wish to deviate to post = − 1.
Lemma 5 Assume all firms other than post vacancies. If firm does not want to deviate to post − 1 vacancies, then it will not want to deviate to post any − 1 vacancies. The reason we only have to check that a firm does not wish to deviate to post = − 1
vacancies is because its profit as a function of the probability with which workers apply, (; ), is increasing in , as illustrated in figure 1 below. In fact, the expected wage bill of a firm decreases with for a given because of the lower variability in payoffs offered to workers.
( ; v n 1)
( ; v n 2)
( ; v 1)
Figure 1 To understand why posting − 1 rather than vacancies may be attractive for a firm
given that all other firms are posting = , recall that the last firm cannot control fully the probability it is applied to if it chooses = , whereas it does if it chooses = − 1.
In figure 2 below, the profit of a firm if it were to post with = like all other firms
implies a profit of e and = 1; but if it deviated to = − 1 the profit function is
now continuous in , and in fact the firm would pick = and since e the firm would prefer to deviate.
21
Proposition 6 Assume workers are risk averse. Then there exists ∈ [(1)(1 −
∗ ∗ ∗ 1)−1 ()) and ∈ ( () ()[1 − (1 − 1) ]], where () is given £ ¤ in (9), such that all firms posting ( ) for ∈ are equilibrium postings, i.e.,
there exists a continuum of symmetric equilibria.
Note that the symmetric equilibrium wage when workers are risk neutral with = , ∗ (),
is still an equilibrium wage when workers are risk averse.
d ~n ,m ( ; v n)
d ( ; v n 1) 1/ m
d
Figure 2 It maybe appear somewhat surprising that there is a continuum of symmetric equilibria when workers are risk averse, especially because when workers are risk neutral there are as many equilibria as there are vacancies. Take the case where there are two risk neutral workers and two firms: when workers are risk neutral then there will be exactly two symmetric equilibria. The intuition one might then have about how the equilibrium looks like when workers are risk averse is that we would obtain a unique equilibrium, since with curvature in the utility function workers are no longer indifferent about the many options that are payoff-equivalent in the risk neutral case. This intuition is indeed correct when considering the ’s that firms will pick - they would offer, as shown in lemma 3, workers certainty of employment through offering as many ’s as there are workers over any other . But in terms of the wage that is posted by firms, this intuition fails. To understand why a continuum of equilibria exists in the case where workers are risk averse, it is easier to start by going back to the case where workers are risk neutral. When workers are risk neutral, given that all other firms are posting vacancies at wage , the last firm is indifferent between posting ( − 1) vacancies or posting vacancies at wage if 22
and only if = (1 − 1)−1 which is a worker’s expected marginal product, and its profit
is e . At all 6= (1 − 1)−1 the last firm prefers to deviate to post ( − 1) vacancies.
The intuition as to why this is true only at = (1 − 1)−1 is because when all firms
post vacancies, no firm has any “monopoly power”, and by posting = a firm no longer controls fully the probability with which it is applied to since it can either be applied to with probability 0, 1, or 1. If, instead, a firm deviates and posts = −1 vacancies then it can still fully control the probability it is applied to. Hence, if the firm’s preferred application probability is different from 1, which is true when all other firms post 6= (1 − 1)−1 ,
then the firm strictly prefers to post = − 1 vacancies. And if = (1 − 1)−1 the firm is indifferent between posting vacancies at wage = (1 − 1)−1 and be applied to with
probability 1 or posting − 1 vacancies. Hence, there is “continuity in a deviant firm’s
profit” when it goes from posting to ( − 1) vacancies. See Figure 3. Risk neutral workers
~n,m
d ( ; v n 1) 1/ m
Figure 3 Now take the case where workers are risk averse, and all other firms post vacancies at the same = (1 − 1)−1 . The last firm, when deciding whether to deviate by posting ( − 1) vacancies or posting the same and vacancies, is facing a trade-off: posting =
vacancies implies a lower expected wage bill for = 1, but it also means the firm is not able to fully control the probability with which workers apply to it as it can if it posts = −1. When = (1 − 1)−1 the last firm wants to be applied to with probability 1,
and so at this wage the firm strictly prefers to post = rather than = − 1 since there
is no cost in terms of probability of application and there is a net gain in terms of a lower expected wage bill. By a continuity argument, for wages close enough to = (1 − 1)−1
the gain in terms of expected wage bill with = dominates the loss due to the restriction 23
to having workers apply with probability 1 because the desired is close enough to 1. In other words, there is no continuity in profit when going from to ( − 1) vacancies. See
Figure 4. Which is why there is a continuum of symmetric equilibria when workers are risk averse as stated in proposition 6.
Risk averse workers
~n,m
d ( ; v n 1) 1/ m
Figure 4 The above deals with existence of a continuum of equilibria. We have not been able to obtain a general analytical characterization of the set of equilibria, even with a simple functional form for utility () = , ∈ (0 1). In fact, we have not been able to prove that
the interval of equilibrium wages indicated in the above proposition is the only set of wages
for which a symmetric equilibrium exists. However, numerical results suggest that with this CRRA utility function it is indeed the case.19 Moreover, numerical results also show that the bounds () and () are strictly increasing and decreasing in respectively,
and they converge to (1 − 1)−1 as converges to 1.
Figure 5 below plots the two bounds for the case where there are 5 firms and 4 workers
as functions of the risk-aversion parameter . The top line indicates the upper bound on wages, and the bottom line indicates the lower bound on wages. 19
Please see Example 3 in the Appendix for an example with 2 workers and 2 firms where the bounds are solved for explicitly.
24
Figure 5
5.2
Large Economy
It turns out that in the limit we can say more about what these lower and upper bounds on wages are. More precisely, it turns out that these upper and lower bounds on wages posted by firms when posting = converge to one wage. Proposition 7 In a large economy with risk averse workers there is a unique symmetric equilibrium where all firms post as many ’s as there are workers, at wage ∗ ∞ = − ,
(15)
the expected queue length at each firm is , and each firm makes a profit ∗∞ = 1 − − − − .
(16)
When there are finite numbers of workers and firms, say 2 workers and 2 firms, when one firm, say firm 1, is considering whether to post = 2 or to deviate and post = 1, given that the other firm is posting = 2, it has to calculate the respective profits and then decide what to do. If it posts = 1, then it has to trade it off by posting a higher wage than the other firm, but what it gains by posting = 1 is that he can control, or choose, , the probability with which workers apply to him. When the economy is small, the fact that a firm can post one less and choose can be important.
25
However, when the economy becomes larger, then whether a deviating firm can post = − 1 matters less: if it were to post one less , then its wage will be a bit larger than
what all other firms are posting, but it can still choose . But in the limit, this “gain” the deviating firm can have vanishes: there will be only one wage where a deviating firm is indifferent between posting as many ’s as there are workers or one less, and this wage is that defined in equation (15). This value is the same for all values of . This is because as the economy becomes larger the wage premium a firm has to offer by posting = − 1
instead of = decreases, since the risk to a worker of not receiving a payment decreases, which increases the set of wages for which a firm prefers to deviate and post = − 1 rather
than = (because posting = − 1 enables the firm to indirectly control the probability of application). Or in other words, the set of wages for which a firm does not wish to deviate
becomes smaller and eventually becomes a singleton as the economy becomes arbitrarily large. But what is this wage that the bounds converge to? From equation (9), when = and we take the limit we get that ∗ lim ( = ) = −
→∞
In other words, the bounds converge to the wage posted by firms if workers were risk neutral, which is just a worker’s expected marginal product.
5.3
The Entry Stage and Efficiency
In the finite economy with risk averse workers, in the directed search game all firms will post = at the same wage that lies within an interval and earn a profit as defined earlier, e = [1 − (1 − 1) ] −
.
Given , e is strictly decreasing in , and so there is a unique ∗ solving e = .
In the limit, there is a unique ∗ satisfying
∗∞ = since ∗∞ = 1 − − − − is decreasing in .
Comparing these results above when workers are risk averse to what the social planner
would do, the following proposition is immediate. 26
Proposition 8 Assume workers are risk averse. () In the finite economy, there is a unique efficient symmetric equilibrium, and it is where firms post = at wage (1 − 1)−1 ; and
() in the large economy the unique symmetric equilibrium is efficient. We thus obtain that in the finite economy there can be too little or too much entry, depending on whether the wage posted by firms in the directed game is above or below the expected product of a worker. And the unique efficient equilibrium is such that firms post as wage the expected marginal product of a worker. The next section discusses in detail this result.
6
Comparison of Results
6.1
Finite Economy
In the finite economy, when workers are risk neutral, there is a finite number of symmetric equilibria which are not payoff equivalent. There is an equilibrium where all firms post = 1, another equilibrium where all firms post = 2, and so on, including an equilibrium where all firms post = . Only one of these equilibria, the one where all firms post = is efficient, and in this case, workers are paid their expected marginal product, (1 − 1)−1 .
When workers are risk averse the equilibrium is such that firms post as many ’s as there
are workers, and there is an interval of wages posted by firms which support all firms posting = or that there is a continuum of equilibria. Like in the risk neutral case, only one of these equilibria is efficient, and it is the one where all firms post = and workers are paid their expected marginal product, (1 − 1)−1 .
Even though there is a multiplicity of inefficient equilibria in both cases, the nature
of inefficiency is very different. When workers are risk neutral, in all inefficient equilibria there is too much entry because the wage posted is “too low” - this arises because in finite economies when firms post firms have monopoly power, whereas when firms post = this monopoly power vanishes. When workers are risk averse, all equilibria have firms posting = , but in some of the inefficient equilibria the wages posted exceed (1 − 1)−1 , and in other equilibria the wages posted are lower than (1 − 1)−1 . The inefficient equilibria here are not due to
“monopoly power” of the firms since they do not have any when = . In fact, in this case firms take as given the expected utility to offer to workers since they face a perfectly elastic 27
labor supply. Yet equilibria can be inefficient. On the one hand it is costly to deviate and post a contract with = − 1 rather than because a firm then faces a larger expected
wage bill for a given probability of application. And on the other hand, a firm can control fully the probability it is applied to when it chooses = − 1 whereas it cannot when = .
As long as the wage posted by all firms is close enough to the expected marginal product
of workers, implying that firms want to be applied to with probability close enough to 1, then firms are strictly better off not posting = − 1 because the reduction in profits due to the higher expected wage bill exceeds the gain in profit from being able to control the probability of application. Note that in both the risk neutral and risk averse cases the multiplicity can be considered to be a coordination failure, because if firms could coordinate they would all agree on the contract posting yielding the highest level of expected profit.
6.2
Large Economy
When workers are risk neutral there are as many equilibria as there are workers; all of them are payoff equivalent; and all are constrained efficient. There is one equilibrium where all firms post = 1, another equilibrium where all firms post = 2, and so on. The intuition why all equilibria are efficient is because firms no longer have any monopoly power since they are each too small to matter. However, note that even though all of these efficient equilibria are payoff equivalent, all of them have different wages posted: there are some equilibria where all firms post high wages, with a low , and where the expected unemployment is high; and there are other equilibria where all firms post low wages, with a high , and where the expected unemployment is low. When workers are risk averse, there is a unique equilibrium where all firms post as many ’s as there are workers, and all workers are paid their expected marginal product, so this equilibrium is also efficient. Which means that in the risk averse case, the prediction of the model is much sharper since there is a unique equilibrium, whereas there is multiplicity of equilibria with risk neutral workers.
7
Concluding Remarks
It is clear that when firms can post contracts more general than wage contracts, assuming that workers are risk neutral or risk averse matters. In both cases there exist multiple symmetric equilibria in finite economies, but the nature of the multiplicity is different in 28
the two cases. For large economies, even though all symmetric equilibria with risk neutral workers are payoff equivalent, since workers receive the same expected wage and the number of firms entering is the same as the expected profits are the same, and these payoffs are the same as with risk averse workers, there are significant differences in outcomes. In fact, the number of “unemployed” workers can be very different in the different symmetric equilibria when workers are risk neutral if one think of payments made by firms as representing wages paid to workers hired. Looking at the results from a different angle, this paper shows that there is a countable number of symmetric equilibria when workers are risk neutral in both finite and large economies. But when one considers economies with risk averse workers the multiplicity of equilibria obtained in finite economies disappear in large economies. Hence, using large economies as an approximation of finite economies with many agents, which can be convenient since the market utility property holds in large economies, is not problematic with risk neutral agents, but it can be with risk averse agents. Moreover, it is important to note that allowing for more general contracts than the wage contracts that most of the literature focuses on matters for whether risk-aversion matters for the nature of equilibria. If one restricts firms to posting wage contracts, then whether workers are risk averse or risk neutral does not matter for the nature of equilibria: there is a unique symmetric equilibrium, although the wage paid is different. Naturally, the assumption of risk averse workers also matters for the efficiency of the equilibrium, since the equilibrium is efficient with risk neutral workers whereas it is not if instead workers are assumed to be risk averse (see Acemoglu and Shimer, 1999; Jacquet and Tan, 2012). If instead, one allows firms to post more general contracts than wage contracts, just like we do in this paper, then the assumption made regarding the risk aversion of workers is important. Hence, the combination of assumptions regarding the types of contracts allowed and the degree of risk aversion is what matters and highlights the fact that assuming that workers are risk neutral is not problematic if one restricts firms to posting wage contracts, but that it is not an innocuous assumption once firms are allowed to post more general contracts.
29
References [1] Albrecht, James, Pieter Gautier, Susan Vroman, and Serene Tan (2004), “Matching with Multiple Applications Revisited,” Economics Letters, 84(3), 311-314. [2] Acemoglu, Daron, and Robert Shimer (1999), “Efficient Unemployment Insurance,” Journal of Political Economy, 107(5), 893-928. [3] Burdett, Kenneth, Shouyong Shi, and Randall Wright (2001), “Pricing and Matching with Frictions,” Journal of Political Economy, 109(5), 1060-1085. [4] Camera, Gabriele, and Jaehong Kim (2014), "Uniqueness of Equilibrium in Directed Search Models," Journal of Economic Theory, 151, 248-267. [5] Camera, Gabriele, and Cemil Selcuk (2009), "Price Dispersion with Directed Search," Journal of the European Economic Association, 7(6), 1193-1224.. [6] Coles, Melvyn, and Jan Eeckhout (2003), “Indeterminacy and Directed Search,” Journal of Economic Theory, 111(2), 265-276. [7] Geromichalos, Athanasios (2012), “Directed Search and Optimal Production,” Journal of Economic Theory, 147(6), 2303-2331. [8] Hawkins, William (2013), “Competitive Search, Efficiency, and Multi-worker Firms,” International Economic Review, 54 (1), 219-251. [9] Jacquet, Nicolas, and Serene Tan (2012), “Wage-Vacancy Contracts and Coordination Frictions,” Journal of Economic Theory, 147(3), 1064-1104. [10] Lagos, Ricardo (2000), “An Alternative Approach to Search Frictions,” Journal of Political Economy, 108(5), 851-87. [11] Lester, Ben (2010), “Directed Search with Multi-Vacancy Firms," Journal of Economic Theory, 145 (6), 2108-2132. [12] McAfee, R. Preston (1991), “Mechanism Design by Competing Sellers,” Econometrica, 61(6), 1281-1312. [13] Montgomery, James D. (1991), “Equilibrium Wage Dispersion and Interindustry Wage Differentials,” Quarterly Journal of Economics, 106(1), 163-179.
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[14] Peters, Michael, “Ex Ante Price Offers in Matching Games: Non-Steady States,” Econometrica, 59, 1425-1454. [15] Tan, Serene (2012), “Directed Search and Firm Size,” International Economic Review, Vol. 53(1), 95-113.
31
Appendix Proof of Proposition 3 Step 1. We first look for symmetric equilibria where . Suppose that all other firms post e vacancies at wage e and workers apply to those firms with probability e . The last
firm, firm , is considering to post ≤ vacancies also at some wage , and workers apply to this deviant firm with probability . Firm ’s profit is
( ; ) = 1 − (1 − ) − Γ () , or
( ; ) = 1 − (1 − ) − Ω () .
(17)
If this deviant firm is to applied to with some probability ∈ (0 1],20 it must be that this
deviant firm’s posting satisfies
Ω () = Ω e (e ),
where e = (1 − )( − 1).Hence, (17) can be rewritten as a function of (and ) e only as follows:
(; ) e = 1 − (1 − ) − Ω e [e ()].
(18)
When all other firms post e , then the last firm is indifferent between posting = 1 2
for a given probability of being applied to . In fact, for a given there exists different ways to offer workers the same expected utility of applying : the firm can post = 1 at wage 1 or any other at wage such that Ω () = . It can be proven (see Lemma A.1 in the Supplementary Appendix) that the profit function is strictly concave in , thereby implying that the first-order condition is necessary and sufficient for a maximum. Differentiating (18) with respect to and dividing by throughout, we obtain that the optimal choice of is such that Ω(e ) e (1 − )−1 − Ω = 0 e (e ) − e e
(19)
Rewriting the above, and noting that in a symmetric equilibrium e = , e = = 1, and e = −1( − 1), we have that for the only wage for which firms do not wish to deviate, because it is the unique wage for which a deviant firm would like to be applied to with probability 1, is:
20
(1 − 1)−1 ∗ io h () = n . () 1 Ω () 1 − Ω() Ω | =1 −1
We are looking for symmetric equilibria, so firms must be making non-negative expected profits so a deviant firm cannot want not to be applied to.
32
If we define by ( − ) the elasticity evaluated at = − = 1 of the expected utility of applying to any firm with respect to the probability − with which workers apply to any other firm, i.e., ( − )|=1 ≡ −
− 1 Ω ( ) |=1 ∈ [−1 0] , |=1 = − − 1 Ω( )
∗ we obtain that the expression for () can be rewritten as
∗
(1 − 1)−1 () = . Ω (1) [1 − ] |=1
(20)
Step 2. Let us now consider the case when all but one firm post e = at wage . e In this case,
the deviant firm no longer fully controls the probability it is applied to if it chooses to post
vacancies as well: it will either be applied to with probabilities 0, 1, or 1. In fact, when all other firms post = , the expected utility offered by other firms is independent of what the last firm posts since Ω () = 1 for all . If this deviant firm is to applied to with some probability ∈ (0 1], we again have that the deviant firm’s profit can be expressed as
a function of (and ): e
(; ) e = 1 − (1 − ) − . e
Then, if firm could “freely” choose , it must be such that = (1 − )−1 − e = 0,
which yields that the deviant firm would like to be applied to with probability () e =1− e1(−1) .
(21)
If () e 6= 1, since the deviant firm cannot be applied to with the desired probability by posting vacancies, it is strictly better off by posting less than vacancies at a wage such that the expected utility of applying to it with probability () e is . e And if () e =
1, then the deviant firm is indifferent between the different ways it can reach this probability of application, one for each possible number of vacancies. Expression (21) yields that () e = 1 if and only if e = (1 − 1)−1 . Hence, no firm wants to post a contract
different from other firms when they all post = vacancies if and only if they do so at wage e = (1 − 1)−1 . And since ( − ) = 0 for = and Ω () = 1 for all , we
get that expression (20) holds also for = .¥
33
Proof of Lemma 1 Let ∗ ∗ ∗ ∆ () ≡ () − ( + 1) .
Since ¢ 1 −1 io Ω () 1 |=1 Ω () −1
¡ 1−
∗ h () = n Ω () 1 −
∗
h ( + 1) = n Ω+1 () 1 −
¡ 1−
¢
¡ 1−
=h Ω () −
1 −1 io Ω+1 () 1 |=1 Ω+1 () −1
∗ we want to ask when ∆ () ≥ 0 for = 1 − 1.
¢
1 −1 i () 1 |=1 Ω −1
and
¢ 1 −1 i () 1 |=1 Ω+1 −1
¡ 1−
=h Ω+1 () −
∗ ∆ () ≥ 0 if and only if ¸ ¸ ∙ ∙ Ω+1 () 1 Ω () 1 |=1 − Ω () − |=1 ≥ 0, Ω+1 () − −1 − 1
which, since Ω () = Γ () (), is equivalent to ¸ ∙ Γ () Γ +1 () [Γ+1 () − Γ ()] − − ≥ 0.
(22)
Moreover, we have that P Γ+1 () − Γ () = 1 − =0 () P Γ () = =0 () , and that P+1 P Γ+1 () Γ () +1 − = () =0 () − =0 () = ( + 1)
(23)
(See Lemma A.2 in the Supplementary Appendix for the derivation of equation (23)). Therefore equation (22) reduces to ¤ £ ¤ £ P 1 − =0 () − ( + 1) +1 () = +1 () + + () − ( + 1) +1 () ≥ 0.
The inequality above is strict for ( + 1) and weak for ≥ ( + 1). Since takes values from {1 − 1}, the largest value can take is = − 1, which means that equation (22) is strictly positive for . The result follows.¥
34
Example 1: non-monotonicity in wages (in section 4.1.1) Suppose = 3, = 10 Hence, firms can post = 1 10
∗ ()
1
006060785416
2
003361454879 &
002675690152 &
3
002515846887 &
4
002531802054 %
5
002573782605 %
6
002594894848 %
7
002600410377 %
8
002601183233 %
9
10 002601229487 % Proof of Lemma 2 We can rewrite equation (7) as ∗ ()
First remember that:
=h Γ () −
¡ 1− ³
1
¢ 1 ´i Γ () | =1
(24)
µ
¶ ³X ´ Γ () |=1 = () |=1 =0 ¡ ¢ Let → ∞ at ratio = . Since () = (1 − )− and = , finite, −
then as → ∞, () → ! this implies that as → ∞ ¶¸ ∙ µ X − 1 Γ () = 0, |=1 → 0 × | =0{z ! }
and
h X Γ () = −
=0
∙ i X ( − ) () |=1 → −
Hence, using equation (24) in the limit we obtain that ∗ ∗ () ≡ lim () = ∞ →∞
−
¸ − ( − ) . =0 !
− =P − =0 ( − ) !
P
35
³ P−1 =1 − =0
!
´
∗ Lastly, it is obvious that ∞ () decreases with since we are adding terms to the
denominator as increases.¥
Example 2: comparative statics with free entry (in section 4.2) Suppose = 3 = 15 and = 01. This means that ∈ {0 15}.
∗
1
2820457181
07608324183
05981566893
2
2786455394 &
06177003495 &
05981955772 %
2775123973 &
05983569606 &
05981982729 %
3 4 5 6 7 8 9
2776767657 &
2774941489 &
2774927179 &
2774926441 & 2774926441 −
2774926441 −
∗ ∗ ()
06002103917 &
05982078714 &
05981987166 &
05981982915 &
05981982794 &
05981982647 &
05981982748 %
05981982789 %
05981982790 −
05981982791 −
15 2774926441 −
05981982659 &
05981982791 −
12 2774926441 −
14 2774926441 −
05981981946 %
05981982790 %
05981982791 −
13 2774926441 −
05981982791 &
10 2774926441 −
11 2774926441 −
∗ ∗ ()
05981982791 −
05981982791 −
05981982791 −
05981982790 −
05981982790 −
05981982790 −
05981982790 −
05981982790 −
Proof of Proposition 5 We are left with considering entry. () For all equilibria but the one with = , the number of active firms such that = does not correspond to the entry implied by the Planner’s problem in equation (2). In fact, ∗ ∗ we have that () = (1 − 1)−1 () for all , and a firm’s expected profit
is
∗ ()
¶ µ ¶−1 µ 1 1 1− =1− 1− − ∗ () for all .
Clearly, we have that ∗ () = is just equation (2), and thus the number of active firms ∗ = when = , and ∗ for all . () In the limit, since all equilibria are payoff equivalent and a firm’s profit is given by equation (12), which, when equated to , yields the same equation as the planner’s entry condition, equation (3), the result follows immediately.¥ 36
Proof of Lemma 3 Suppose there exists an equilibrium where all firms post e at some wage . e We will
show that a firm has an incentive to deviate and post e vacancies. Suppose all but one firm post e 6= . Their profit is
(; e) = 1 − (1 − ) − Ω e ()
If a firm deviates and posts e at some wage so that workers still apply with probability , its profit is
(; ) = 1 − (1 − ) − Ω ()
Since the deviant firm is applied to with the same probability if and only if it offers an expected utility ) Ω () , = () Ω () = (e
it means that
µ
¶ (e ) = Ω () µ ¶ −1 () = Ω () ³ ´ ³ ´ For a convex function such that (0) = 0 we have that 1 1 2 2 for 1 2 0. −1
Hence, if e Ω () Ω () (since by offering more vacancies you offer workers a higher
probability of being employed given that they have applied), so µ µ ¶ ¶ −1 −1 Ω () Ω () Ω () Ω ()
which implies that
−1
(; e ) = 1−(1 − ) −Ω ()
µ
µ ¶ ¶ −1 (; ) = 1−(1 − ) −Ω () Ω () Ω ()
In other words, a firm is always better off deviating by posting more vacancies than what everyone else is posting, for all 0.¥ Proof of Lemma 5
Suppose all but one firm post vacancies at wage . e This means that the last firm, to
be applied to with some positive probability, has to offer an expected utility to workers of 0
= (). e Using the techniques from the proof of lemma 3, it is clear that for µ µ ¶ ¶ () e () e −1 0 −1 Ω () Ω ( ) Ω () Ω (0 ) 37
which for all 0 is equivalent to 0 () (). This means that if the last firm wants to deviate, the larger the number of vacancies posted the higher the profit for all 0, and therefore a deviant firm would never choose − 1 (since in a candidate equilibrium
active firms must be making positive profit, implying a deviant firm would not aim to receive zero applications).¥ Proof of Proposition 6
Preliminaries - Let us denote by ( |0 0 ) the expected profits of a firm posting the contracts ( ) when all other firms post the contracts (0 0 ). First, notice that the profit function of a firm that posts the contract ( ) like all other firms is ( | ) = 1 − (1 − 1) − () × , which implies that ( | ) =− . (25) If instead a firm deviates and posts = − 1 vacancies at a wage (| b ) such that it is
applied to with probability ∈ (0 1] is
−1
((| b ) − 1| ) = 1 − (1 − ) − ( − )
∙
¸ () . − −1
(26)
In fact, if the deviant firm posts = − 1 vacancies and chooses a wage such that it is
applied to with some probability ∈ (0 1], it must be that workers are indifferent between applying to either type of firms, so that ∙ ¸ ¡ ¢ −1 − 1 −1 + 1− ((| b )) = (),
or
∙
¸ () , (| b ) = − −1 and the expected number of workers paid is − so the expected wage bill is ( − −1
)(| b ).
Taking the derivative of (26) with respect to yields 0 () ((| b ) − 1| ) h i´ ≤ 0. = − ³ 0 −1 −1 () −
If we define
Γ ≡ min −
∗ 0 ( ()) ³ h i´ , ∗ ()) 0 −1 −−1 (
38
then for all we have that ((| b ) − 1| ) ∈ [Γ 0]. Hence, if we define by
b∗ ( ) the optimal wage to post when posting − 1 vacancies when all other firms post ( ), we have that for any ,21
( ( b∗ ( ) − 1| ) − ( b∗ ( + ) − 1| + )) ∈ [0 −Γ].
(27)
Moroever, we know from Step 2 of Proposition 3 that when all other firms post vacancies, then the last firm, if it could control the probability it is applied to, would choose to be ∗ applied to with probability 1 if and only if the wage posted is () = (1 − 1)−1 . ∗ Moreover, we know that that when all other firms post vacancies at wage () =
(1 − 1)−1 , then a firm that deviates and post = − 1 vacancies and chooses the opti-
mal wage would be making strictly less profit than by posting the same contract as all other firms. Let us denote the profit differential by ∆. That is, ∗ ∗ ∗ ∗ ( () | () ) − ( b∗ ( () ) − 1| () ) = ∆
(28)
∗ ∗ () - Assume all firms but one post ( ) with = () + Wages greater than
for 0. From (27) we have that for the last firm, ∗ ∗ ( b∗ ( ) − 1| ) ≤ ( b∗ ( () ) − 1| () ),
and from (25) that
∗ ∗ ( | ) = ( () | () ) −
.
Hence, these two expressions together with (28) imply that for all ∈ [0 ∆] and =
∗ () + , we have that
( | ) ≥ ( b∗ ( ) − 1| ),
with strict inequality for all ∈ [0 ∆). That is, if all firms but one post a contract
∗ ( ) with = () + for ∈ [0 ∆], then the last firm has no incentive to deviate,
and the preference not to deviate is strict for ∈ [0 ∆).
∗ ∗ Wages smaller than () - Assume all firms but one post ( ) with = () −
∗ for ∈ (0 ()). From (27) we have that for the last firm,
21
∗ ∗ ( b∗ ( ) − 1| ) ≤ ( b∗ ( () ) − 1| () ) − Γ.
If 0, then the interval becomes [−Γ 0].
39
Furthermore, we know that ∗ ∗ ( | ) = ( () | () ) +
.
Using (28) we therefore obtain that b∗ ( ) − 1| ) ≤ ( | ) − [∆ + (Γ + (
)].
∗ Hence, if Γ ≥ −, then ∆ + (Γ + ) 0 for all ∈ (0 ()), and we therefore
∗ have that ( b∗ ( ) − 1| ) ( | ) for all ∈ (0 ()), implying that
∗ ∗ if all other firms post ( ) with = () − for ∈ (0 ()), then the last firm
is strictly better off posting the same contract. If, however, Γ −, then we still have
that ( b∗ ( ) − 1| ) ≤ ( | ) for all ∈ (0 −∆(Γ + )], with strict
inequality for ∈ (0 −∆(Γ + )). That is, in this case if all other firms post ( ) with
∗ ∗ = () − for ∈ (0 ()), then the last firm is better off posting the same contract
if ∈ (0 −∆(Γ + )], and strictly better off for ∈ (0 −∆(Γ + )).¥ Example 3: derivation of 2 worker - 2 firm case (in section 5.1)
If firm just mimicked firm and posted = ∈ [14 34], then ’s profit is
∗ = 34 − .
But if firm deviated to post one vacancy, for a worker to be indifferent between applying to firms and , it has to be that his expected utilities from applying to both firms are equalized, or that
= ⇔ ( ) = 05 [1 + ] ( ) ⇔ 2 ( ) − ( ) = ( ) Firm ’s profit maximization problem is ¡ ¢ max 1 − 2 (1 − ) if ∈ [05 ]
( ) : ( ) [ ( ) − ( )] − ( − 1) 0 ( ) [2 ( ) − ( )] = 0 which implicitly defines ( ) If we let () = , 1, the above F.O.C. becomes 0 = [2 − (2 − 1)] + [ ( − 1) − ]
40
which can be manipulated to yield = 1 + (1 − ) where ≡ 1. + 2 (1 − ) When = 05, the above is just =
µ
1 + 2
¶2
(29)
Equation (29) is actually firm ’s best response in posting , given firm ’s posting of , or ( ). Theoretically we can then substitute ( ) into firm ’s profit maximization problem to compute ( ) and then compare this to ∗ to see which values of imply that the firm prefers deviating or not. However, in equation (29), is defined implicitly. So we will solve for deviation profits in terms of ( ) and then check what are the -values such that firm prefers to deviate, and then use (29) to compute the implied -values for which firm prefers to deviate. Since ( ) = (1 + ) ( − 1)2 firm prefers to deviate and post 1 vacancy if and only if ( ) ∗ ⇔ 2
(1 + ) ( − 1)
∙ ¸2 (1 + ) 3 3 − = − (from (29)) ⇔ 4 4 2
¤ 1£ 3 2 5 − 2 − 3 + 1 0 ⇔ 4 0320 or 0832 And from (29), this is equivalent to
0139 or 0698 then ( ) ∗ Hence, if ∈ [ 14 0698], then firm prefers not to deviate.
Therefore, = = 2 is an equilibrium if wage posted ∈ [ 14 0698]¥
41
Proof of proposition 7 Let us suppose that all other firms are posting = at wage , e and the last firm is
deciding whether to post = at wage , e or to deviate to post ( − 1). It is useful to reason
in terms of queue length rather than probability of application. Let us define ∆ to be the difference between posting − 1 vacancies and choosing the queue length and posting
vacancies like the other firms and therefore face the marketwide queue length = , given that all other firms are posting . e
¶ ³ µ ´ ∆ = −1 () − () = 1 − − 1− Ã ! ³ ³ ´ ´ 1 −1 e , + − − ¡ ¢−1 () 1−
and in the limit, we get that
∆ ∞ ≡ lim ∆ = − − − − ( e − ). →∞
A firm will not want to deviate from posting vacancies at the same wage of other firms, and therefore face queue length = , and instead post − 1 and face the queue length of its choice if and only if ∆ ≤ 0, which in the limit as → ∞ is equivalent to − −
− − ( e − ) ≤ 0.
It is easy to verify that ∆ ∞ is strictly concave in , and that the maximum of ∆ ∞ is
obtained when e = − . Hence, the maximum value for ∆ ∞ is
e−1 ) = − − e + ( e − ln e−1 ) ≡ (). e ∆ ∞ ( = ln
It turns out that is such that
≥ 0 if and only if e ≥ − . e
e − , ∆ ∞ 0. Hence, if e = − , or that = , max∆ ∞ = 0, and if However,
≤ 0 if and only if e ≤ − , e − and if e = , or that = , max∆ ∞ = 0, and if e − , ∆ ∞ 0.
In other words, for all wages e 6= ∗ = − , given that all other firms post vacancies at
wage e a firm is better deviating and posting − 1 vacancies so as to be free to induce the
queue length of its choice, while for ∗ = − the last firm is indifferent between deviating or not. Hence, the unique symmetric equilibrium in the large economy is such that all firms post vacancies at wage ∗ = − ¥ 42
