Job Auctions and Hold-Ups

Transcription

1 From the SelectedWorks of Marcel Jansen 2009 Job Auctions and Hold-Ups Marcel Jansen, Universidad Carlos III de Madrid Available at:

2 Job Auctions and Hold-Ups Marcel Jansen Universidad Carlos III de Madrid January 20, 2009 Abstract We consider a labor market with search frictions in which firms need to invest in capital before they can post a vacancy. This assumption creates a natural scope for hold-up problems, but the innovation of our study is that we allow for competition among the applicants who apply for the same job. In our economy all applicants are paid their actual marginal product. Nonetheless, with random search there exists a hold-up problem, leading to underinvestment in capital. On the contrary, if workers can direct their search towards firms with different capital levels, the equilibrium is efficient. This result contrasts sharply with the predictions of models with ex-post bargaining that never yield an efficient allocation. Moreover, our results extend the efficiency of auction mechanisms to an environment with non-contractible investments. Keywords: hold-up, search, efficiency, auctions. JEL Classification: C78, D44, D83. This paper is based on the first chapter of my PhD dissertation at the EUI in Florence. I thank Fabien Postel-Vinay (the editor) and two anonimous referees for their comments and I would also like to thank Ramon Marimon (my advisor), Giuseppe Bertola, Klaus Kultti, Claudio Michelacci, Dale Mortensen and seminar participants at the SED (Alghero), EUI, University of Helsinki, Carlos III and IUI in Stockholm for valuable comments and suggestions. Financial support from the Spanish Ministry of Education (project SEJ ) is gratefully acknowledged. All remaining errors are mine. Correspondence address: Department of Economics, Universidad Carlos III de Madrid. C\ Madrid 126, Getafe (Madrid), Spain. jansen@eco.uc3m.es 1

3 1 Introduction How do frictions in the labor market affect the investment decisions of firms? To answer this question, we construct an equilibrium model of the labor market in which firms need to invest in capital before they can search for a worker. This assumption creates a natural scope for hold-up problems. The future employees of a firm will refuse to share in the cost of capital, while they may be able to capture a share of the returns to capital because the frictions generate rents. 1 In a conventional matching model the above setup would inevitably lead to distortions (Davis, 2001). The combination of random matching and bilateral bargaining generates a conflict between the conditions for efficient entry and efficient investments in capital: Either the workers have some bargaining power and wages increase with capital, creating a hold-up problem, or all bargaining power is vested in firms, leading to very low wages and excessive entry. Moreover, similar results have been derived for the case of investments in education (Laing et al., 1995), in general training (Acemoglu, 1997) and for complementary investments in human and physical capital (Acemoglu, 1996; Masters, 1998). We depart from the above studies by allowing for the possibility that a firm attracts more than one applicant. Our aim is to show that the competition between these rival applicants may offer the firms protection against the risk of rent appropriation. Second, we explicitly model the meeting process. 1 Hold-up problems are due to a lack of complete contingent contracts. With complete contracts all those who benefit from an investment could be forced to pay their share of the cost (Groult, 1984). Moreover, the proposed solutions to the hold-up problem typically require that the parties can write an enforceable contract before the actual investments are made (e.g. MacLeod and Malcomson, 1993). Here these solutions are ruled out because the parties do not know the identity of their future employees when they make their investment. 2

4 Unemployed workers are allowed to apply for one job per period, and to capture the coordination problems in a large market, we assume that they use identical mixed strategies. Hence, even if the workers have perfect information about the available jobs, there will always be some jobs that attract more than one applicant while the rest of the jobs attract zero or one applicant. The potential competition between rival applicants is important because it reduces the bargaining power of the individual workers. To capture this feature, we model the hiring process as an auction and we assume that firms award a job to the applicant who demands the lowest wage. Our main result contrasts sharply with the predictions of Davis (2001). We show that the equilibrium is (constrained) efficient if the unemployed workers can condition their application decisions on the investment levels of the firms. The frictions lower the efficient investment level, but they need not distort the investment decisions of firms. Our analysis builds on a strand of the search literature that explores the role of auction mechanisms in markets with frictions. Most of these studies analyze retail markets, but there also some interesting applications to labor markets. 2 Here we follow Shimer (1996) by assuming that wages are set in sealed bid second price auctions, and initially we maintain his assumption that firms cannot commit to a reservation profit above the level of their endogenous outside option. These assumptions lead to a simple and attractive wage rule: either the firm has two or more applicants who obtain their reservation value (irrespective of whether they are hired or not), or the firm has a single applicant who appropriates the entire surplus of the job. We demonstrate that these payments reflect the actual productivity of 2 The main debate is centered around the question who is able to organize the auctions. We follow Shimer (1996) by assuming that the firms act as sellers. By contrast, Julien, Kennes and King (2000) assume that workers publicly announce their reservation wages and that firms decide which worker they want to contact. 3

5 applicants. For any given level of capital, the firms will therefore create the efficient number of jobs. Nonetheless, the match surplus may accrue to a single applicant and this creates a scope for hold-ups. We firstanalyzethecaseinwhichtheworkerscannotdiscriminatebetween jobs and choose to apply at a randomly selected firm. In this case, the equilibrium is never efficient. For each choice of capital the firms face the same likelihood of rent-extraction by a single applicant, while the surplus of a match is increasing in capital. As a result, the expected wage costs of a firm are increasing in its choice of capital and this creates a hold-up problem. In a second step we show that the hold-up problem disappears if the workers can assign different application probabilities to jobs with different capital stocks. A sufficient condition for directed search is that the unemployed workershaveperfectinformationaboutthecapitalstocksoftheavailable jobs. This assumption creates a competitive environment in which each job opening needs to offer the unemployed workers the same expected income and firms make efficient investments. Under directed search, the income of a single applicant is still an increasing function of the firm s investment level, but this is now exactly compensated by a rise in the firm s expected number of applicants and hence a fall in the probability that the firm will face a single applicant. The expected wage costs of a firm are therefore independent of its investment level. Finally, in a last step we give firms the option to announce a reservation profit that limits the surplus share of single applicants. We show that this option does not change the results. As long as both the capital stocks and the reservation profits are freely observable, we find that the equilibrium is efficient and that each firm announces its true outside option. Our efficiency result is therefore not driven by a convenient choice of the auction rule. When granted a choice, the firms select the efficient mechanism. In the literature on competing auctions, the same result is obtained as 4

6 a limit result for large markets with a continuum of buyers and sellers (e.g. Peters 1997, 2000). One could try to analyze hold-up problems in this setting. However, with a finite number of agents on both sides of the market, the ex ante investments in capital would lead to very complicated strategic interactions. Any change in the investment level of one firm would affect the equilibrium choices of all the other agents in the economy. To avoid these complications, we construct a competitive search model with a continuum of firms and workers. Under perfect information, each individual firm takes the market utility of unemployed workers as given and it chooses the capital stock that maximizes its profits. Moreover, to make sure that the firms make the right decisions, we assume that they have rational expectations about their expected number of applicants for each possible value of the capital stock. More precisely, for any of the investment levels that are not offered in equilibrium, the firms anticipate a ratio of applicants to jobs so that the workers who would apply for these jobs obtain the equilibrium level of utility. This assumption has become standard since the seminal work of Moen (1997). Moreover, from the contributions of Peters (2000) and Burdett et al. (2001) we know that the resulting allocation typically coincides with the limit equilibrium of a finite economy in which the firms compete directly with each other. The rest of the article is organized as follows. Section 2 introduces the environment and formalizes our urn ball meeting process. Section 3 characterizes the efficient resource allocation. This is followed in Section 4 by a discussion of the equilibrium with random search. The equilibrium with directed search is derived in Section 5 and in Section 6 we discuss the outcome with endogenous reservation profits. Finally, in Section 7 we briefly discuss the relationship between our study and the contribution of Acemoglu and Shimer (1999). They show that the hold-up problem is resolved if firms can post wages. Their mechanism also relies on directed search but the assump- 5

7 tion of wage posting eliminates the room for wage bargaining. Nonetheless, in the equilibrium with perfect information the two pricing mechanisms are equivalent. We briefly explain this equivalence result and at the end of section 7 we describe some situations in which our auction mechanism dominates wage posting. Section 8 concludes. 2 The model 2.1 Agents, preferences and technology We consider an economy with a continuum of identical workers with measure normalized to one and a larger continuum of identical firms. All agents are risk neutral and seek to maximize their expected lifetime income. Time is discrete and the common discount factor of firmsandworkersisequalto β (0, 1). Workers may be in one of two states, employed and producing or unemployed and searching. Firms, on the contrary, are inactive until they buy some capital k>0at a constant marginal cost p. After the purchase of capital a firmmayattempttohireaworkerbypostingavacancy. The production technology is the same for all firms. A filled job with k units of capital is able to produce f(k) units of the final good per period. The price of the final good is normalized to one and f is assumed to be strictly increasing and concave function that satisfies the usual Inada conditions with f(0) = The hiring process The assumption of irrevocable ex ante investments in capital creates a natural scope for hold-ups. The reason is that workers cannot be induced to pay for capital while they may be able to capture part of the returns to capital because matches generate rents. The firms in our economy try to avoid this 6

8 rent-sharing by exploiting the competition among rival applicants. To allow for this competition we follow Shimer (1996) and we model the hiring process as a second price auction with an endogenous number of participants. 3 The exact timing of events is as follows. At the start of any period t all inactive firms have the option to invest in capital and to post a vacancy together with all the other firms who became active in previous periods but who have not yet been able to hire a worker. The total mass of vacant jobs is denoted by v t while the fraction of vacant jobs with a capital stock k k 0 is denoted by G t (k 0 ). In the second stage of the hiring process, each unemployed worker can contact one firm. In an equilibrium with perfect coordination each worker would contact a different firm. However, there is a consensus among search theorists that this is an implausible outcome in large market economies. Throughout the analysis we shall therefore restrict attention to equilibria in which all job seekers use the same mixed strategy. As a result, there will always be some jobs that attract more than one applicant, while other identical ones attract none. In the next stage, the firms solicit a (sealed) bid from each of their applicants. The bid needs to specify what share of the match proceeds the applicant is willing to offer to the firm in case she is hired. All firms commit to hire the worker who submits the highest acceptable bid and to pay this worker according to the second-highest bid. In case of a tie, the winner is chosen at random among the candidates who submitted the winning bid. The chosen worker begins her job in the next period and to rule out renegotiation, we assume that contracts are perfectly enforceable. Hence, the wages of workers are determined once and for all at the start of a relationship. 4 3 The restriction to second price auctions is inessential. The same results could be derived with a firstbestauction. 4 The case in which contracts can be renegotiated by mutual consent is analyzed in 7

9 Finally, turnover is exogenous. At the end of each period the capital stock of every filled and unfilledjobbreaksdownwithprobabilitys. Theworkers who are affected by these shocks loose their jobs and become unemployed with zero income. Further details of the hiring process are discussed in section 4. In the remainder of this section we derive the expression for the matching rate of firms when workers randomize over the available jobs. This strategy is optimal when G t is degenerate or when the job seekers cannot discriminate betweenjobswithdifferent capital stocks. The case in which job seekers are able to direct their search to firms with different capital stocks is explained in Section in Matching probabilities Suppose that the ratio between the measure of unemployed workers, u t,and the measure of vacant jobs, v t,isequaltoq t u t /v t [0, ). If workers randomize over jobs, q t is also the (market) queue length or the expected number of applicants of each vacancy, while the actual number of applicants of a firm, denoted by n, follows a Poisson distribution with parameter q t : 5 Pr{n = x q = q t } = qx t e qt. (1) x! We define the meeting rate of a firm by η(q t )=1 e qt. This is the probability that a firm attracts at least one applicant. Similarly, let µ(q t ) denote the probability that a typical worker is hired if firms randomize over their Jansen (2003). This assumption puts restrictions on the timing of payments, but it does not alter any of our results. Here we therefore ignore this possibility. 5 This is a standard result in the matching literature. For a formal derivation of the (limiting) distribution of n see Peters (2000). 8

10 applicants. Since a match involves one worker and one job, it must be the case that v t η(q t )=µ(q t )u t, and so µ(q t )=η(q t )/q t =(1 e qt )/q t. The properties of the (urn ball) meeting technology imply that the mass of firms with at least one applicant, m(v t,u t )=v t η(q t ), is linearly homogeneous in u t and v t while the meeting probability of an individual firm (worker) is monotonically increasing (decreasing) in q t with lim qt 0η(q t )= lim qt µ(q t )=0and lim qt η(q t )=lim qt 0 µ(q t )=1. Finally, for future referenceitisworthwhiletomentionthattheelasticityofm(v t,u t ) with respect to u t is equal to q t e qt /(1 e qt ). This ratio of probabilities measures the share of potential matches m(v t,u t ) in which the firm is contacted by one applicant. We now have all the necessary ingredients to determine the efficient outcomes. 3 Efficient allocations An allocation is (constrained) efficient if it maximizes the discounted value of aggregate output minus the cost of vacancy creation. 6 The set of efficient allocations is derived using the construct of a social planner who faces the same informational restrictions and coordination problems as the private agents. Hence, the planner may be able to instruct workers to apply for jobs with a particular level of capital, but she cannot send each worker to a different firm. Formally, an efficient allocation consists of a time path for the capital stock of jobs, k t, the market queue length, q t, and the unemployment rate, u t. However, in the Appendix we demonstrate that the optimal values of k t and q t are constant over time (and independent of u t ). Moreover, due to the concavity of f(k) it is never optimal to assign more than one type of job tothesamemarket. Belowwethereforesuppressthetimeindicesandwe 6 The analysis in this section draws on Section 3 in Acemoglu and Shimer (1999). 9

11 restrict attention to markets with homogeneous jobs. Let λ be the shadow value of an unemployed job seeker. In an efficient steady state we show that λ e = λ/β satisfies the following recursion (1 β(1 s)) λ e = h max µ(q)(1 s)β Π(k) pk e i λ 1 (1 β(1 s))pk, k,q q (2) where Π(k) = P τ=0 [β(1 s)]τ f(k) =f(k)/(1 β(1 s)). Thetermµ(q)(1 s) measures the probability that an unemployed worker will be employed at the start of next period. The current value of the resulting output stream is βπ(k). From this we need to subtract the cost of capital, βpk, andthe shadow value of the worker, λ = βλ, e to obtain the social gain from successful match creation. Finally, in the efficient allocation the planner opens 1/q vacancies per unemployed worker. Each of these vacant jobs reduces net output in every period by (1 β(1 s))pk. 7 Solving (2) for λ e leads to a convenient characterization of the efficient allocations: Proposition 1 (Acemoglu and Shimer 1999) An efficient steady state solution exists. It is characterized by a pair (q S,k S ) (0, ) 2 that solves: max {k,q} Proof. See Appendix. β(1 s)η(q)(π(k) pk) (1 β(1 s))pk. (3) (1 β(1 s))q + β(1 s)η(q) The above maximand is not jointly concave in k and q, andsothefirstorder conditions are not sufficient to define a maximum. Nonetheless, since 7 The implicit assumption of a fictitious rental price r =(1 β(1 s))p per unit of capital is without loss of generality since P τ=0 [β(1 s)]τ r = p. Thus, the planner faces the same expected cost of capital as the private firms in our economy. For details, see Appendix. 10

12 the planner s problem has an interior solution, they are useful to recognize inefficient allocations. Inserting Π(k) =f(k)/(1 β(1 s)) and η(q) =1 e q into (3), this yields the following two conditions: Corollary 2 Any efficient interior allocation (q S,k S ) satisfies: β(1 s)(1 e qs ) 1 β(1 s)+β(1 s)(1 e qs ) f 0 (k S ) 1 β(1 s) = p (4) β(1 s) ³1 qs e qs q S e f(k 1 β(1 s)+β(1 s) S 1 q S e qs )/k S = p. (5) 1 β(1 s) Proof. See Appendix. Condition (4) characterizes the efficient investment level as a strictly increasing function of the market queue length q. Thefirst fraction on the left hand side is the expected current value of a unit of output at the start of production, while the second fraction is the discounted marginal product of capital overtheexpectedlifetimeofthejob. Thelefthandsideof(4)therefore defines the marginal revenue of an additional unit of capital. In an efficient allocation this must be equal to the marginal cost of capital p. Similarly, in an efficient allocation the social benefit of an additional vacancy needs to be equal to the cost pk of creating this vacancy. This second requirement gives rise to condition (5) which defines the efficient queue length as a strictly increasing function of k. Notice that the social proceeds from an additional vacancy are proportional to m(u, v)/ v =1 e q qe q. The total increase in the number of filled jobs is smaller than η(q) because each additional job congests the market for the existing vacancies. The efficiency conditions are illustrated in Figure 1. Thecurvethatde- scribes the efficient investment levels is labeled K S, while Q S depicts the 11

13 efficient queue lengths. Both curves start in the origin and are strictly increasing on the entire domain (q, k) (0, ) 2. In our example there is a unique efficient allocation, but in general there may be more than one efficient outcome. Nonetheless, in welfare terms this issue is irrelevant because all the efficient allocations generate the same value for aggregate net output. k k S Q S K S q S q Figure 1: The constrained efficient allocation 4 Random search Inthenextsectionswecomparetheefficient allocations to the equilibrium allocations with job auctions. For the moment we fix the auction rule and we assume that all firms use the auction rule of Shimer (1996). Hence, the firms cannot protect their returns to capital by announcing a reservation profit above the level of their endogenous outside option. This assumption will be relaxed in Section 6. The purpose of this first section is to characterize the outcome when job seekers observe the capital stock of a potential employer after they have contacted the firm. At the time when the job seekers make their application 12

14 decisions all jobs therefore look alike. Given this lack of information about the investment levels of the firms, the equilibrium strategy of the job seekers is to randomize over the available jobs. Each firm with a vacant job will therefore expect to attract q = v/u applicants, and the firmswilltakethis queue length as given when they choose k. Below we show that this feature gives rise to a hold-up problem even though the applicants are paid their true marginal product. Formally, let K denote the set of equilibrium investment levels and let J V (k) denote the asset value of a vacant job with k units of capital. A steady state equilibrium with random search must then satisfy the following conditions: (1 ) All agents have rational expectations about q and G. (2 ) The firmsthatenterthelabourmarketmakeaprofit-maximizing investment in capital, and so any k K maximizes J V (k 0 ) pk 0.(3) New entrants make zero profits, and so J V (k) pk =0if k K. (4 ) The wage payments of workers are the outcome of a sealed bid second prize auction. The model is solved backwards starting with wages. For the moment we assume that all jobs generate a surplus if they are filled by a worker so that match formation is always mutually beneficial Wages Consider a firm that attracts n 1 rival applicants who submit a bid after they observe k. We denote the equilibrium continuation value of the firm and the chosen worker by J F (k, n) and J E (k, n), respectively. The match payoffs are discounted to the start of production and we rule out any transfers before this date. Finally, the total proceeds of a match are denoted by Γ(k) = J F (k, n) +J E (k, n), and we let J U denote the asset value of an 8 When we characterize the equilibrium investment decisions, we will show that this is indeed a feature of the equilibrium. Firms never find it optimal to create a job that is subsequently refused by all workers. 13

15 unemployed worker who behaves optimally in all periods. Theunique(weakly)dominantstrategyofeachapplicantistooffer the firm Γ(k) J U at the start of production. 9 If n 2, this is also the equilibrium continuation value of the firm while the randomly chosen worker obtains her reservation value J U. On the contrary, when the firm attracts a single applicant, its second-best option is to leave the job vacant. In this case the firm is forced to pay the worker a payoff of J E (k, 1) = Γ(k) J V (k) because we ruled out reservation profits above the level of J V (k). Accordingly, the outcome of the auction can be summarized as follows: Γ(k) J V (k) if n =1 J E (k, n) = J U if n 2, J V (k) if n =1 J F (k, n) = Γ(k) J U if n 2. The above payoff functions illustrate the difference with the case of bilateral bargaining. Depending on the realization of n the entire surplus of a match, Γ(k) J V (k) J U, either goes to the firm or to the worker. Moreover, the wage payments reflect the actual productivity of applicants. The first worker who contacts a firm creates a surplus of size Γ(k) J V (k) J U.Anyfirm with a single applicant is therefore willing to pay this worker up to Γ(k) J V (k) to form a match. On the contrary, if a firm has more than one applicant, 9 At the optimal bid the workers are indifferent between employment and unemployment because Γ(k) J U is the maximum amount that the workers would be willing to pay to become the owner of the the job. This strategy is strictly dominant if the applicants learn the realization of n after they have submitted their bid. In contrast, a worker who knows that she is the only applicant can submit any acceptable bid in the range between J V (k) and Γ(k) J U. (6) (7) 14

16 the marginal productivity of each worker is zero because the firm can hire at most one of the identical workers. 4.2 Asset values We are now in a position to derive the asset value equations for J V (k), J U and Γ(k). Let us start with the case of a vacant job. If the job survives until next period, the continuation value of the job changes to Γ(k) J U if the firm attracts more than one applicant, while it stays at J V (k) if n 1. Hence, since Pr{n 2} =1 e q qe q, we can write the asset value equation for J V (k) as: J V (k) =β(1 s) (e q + qe q )J V (k)+(1 e q qe q )(Γ(k) J U ), (8) where all payoffs are discounted to account for time preference and the risk of a breakdown. The asset value equation for unemployed workers is slightly more complicated than (8) because we need to take into account that vacant jobs may have different capital stocks. With random search, a worker s asset value thus depends on the expected value of k andonthenumberofrivalapplicants who contact the same firm. It is easy to show that this variable follows thesamedistributionasn. 10 In particular, the probability that a worker will face zero rival applicants is e q. The (conditional) probability that a worker manages to extract the surplus of a match is therefore equal to e q times the survival probability 1 s. In all the other events the worker obtains her 10 This result follows from the fact that the decisions of a single worker do not affect the value of q. The conditional probability that the chosen firm attracts m = n 1 additional applicants is therefore equal to the unconditional probability that the firm attracts a total of m applicants. 15

17 reservation value at the start of the next period. Accordingly, the asset value of an unemployed job seeker satisfies Z J U = β (1 e q )J U + e q [(1 s)(γ(k) J V (k)) + sj U ]dg(k), (9) where G(k) denotes the distribution of capital over the vacant jobs. The properties of this distribution need to be determined as part of the equilibrium. Finally, the joint proceeds of a match with capital-intensity k satisfy the following standard Bellman equation: Γ(k) =f(k)+β[(1 s)γ(k)+sj U ]. (10) Inserting this expression into (8) delivers an intermediate solution for J V (k) in terms of k, q and J U : J V β(1 s)(1 e q qe q ) f(k) (1 β)j U (k) = 1 β(1 s)+β(1 s)(1 e q qe q ). 1 β(1 s) (11) The above equation defines J V (k) as the value that a firm can guarantee for itself if it would refuse to match with a single applicant. 4.3 Equilibrium investments and hold-up The derivation of the equilibrium investments is now straightforward. Each firm that enters the labor market maximizes its expected profits J V (k 0 ) pk 0 taking q and the investment decisions of all other firms as given. Since the decisions of a single firm do not affect the outside option of workers, this leads to the following first order condition: Notice that the second order conditions are always satisfied since f(k) is strictly concave. 16

18 β(1 s)(1 e q qe q ) 1 β(1 s)+β(1 s)(1 e q qe q ) f 0 (k) 1 β(1 s) = p. (12) A comparison with (4) immediately reveals the hold-up problem. The profit-maximizing investment level is a strictly increasing function of the likelihood that a firm attracts at least two applicants. By contrast, the efficient investment level depends on the probability that a job is filled, which occurs with probability η(q) =1 e q. Thus, in equilibrium the firms ignore the part of the marginal returns that may accrue to a single applicant and due to the concavity of f this leads to under-investment in capital. Formally, Lemma 3 For any given q (0, ), firms under-invest in capital as compared totheefficient allocation. Proof. See Appendix. Finally, given that all firms choose the same level of investment, it is straightforward to show that the expression for J U simplifies to J U = β(1 s)e q 1 β(1 s)+β(1 s)e q f(k) (1 β(1 s))j V (k) 1 β, (13) while the zero profit condition of firms, J V (k, q) pk =0, can be written as µ β(1 s)(1 e q qe q ) f(k)/k 1 β(1 s)+β(1 s)(1 qe q ) = p, (14) 1 β(1 s) which coincides with (5). Condition (14) shows that the entry margin is efficient. Hence, for any given value of k, firms create the efficient number of jobs. However, from Lemma 3, we know that the reverse is not true. Conditional on a efficient value for q, firms are not willing to invest the efficient amount k S because the 17

19 workers appropriate part of the returns on their investments. The equilibrium allocation is therefore never efficient. This point is illustrated in Figure 2. k Q S K S K R k R q R q Figure 2: The equilibrium with random search ThecurvewiththelabelK R depicts the equilibrium investment decisions of firms. Due to the hold-up problem, this curve lies everywhere below its efficient counterpart K S. On the contrary, the equilibrium entry locus coincides with the efficient entry locus Q S. In the case of a unique efficient allocation, we therefore find that k R <k S and q R <q S. The individual firms invest too little in capital and at the aggregate level this gives rise to excessive job creation. Our results are summarized in the following Proposition: 12 Proposition 4 (Hold-ups) (i) A steady state equilibrium with random search is characterized by a pair (k R,q R ) (0, ) 2 that solves (12) and (14). (ii) The equilibrium allocation never coincides with an efficient allocation (k S,q S ). (iii) If the efficient allocation is unique, then k R <k S and q R <q S. 12 See Jansen (2003) for a detailed discussion of the conditions for existence and uniqueness of an interior equilibrium. 18

20 Proposition 4 provides a useful benchmark. It demonstrates that job auctions cannotatthesametimeguaranteeefficient entry and efficient investments in capital if workers are randomly allocated to jobs. The problem with random search is that all firms attract the same queue of applicants. The likelihood that a firm attracts a single applicant is therefore unrelated to its investment in capital, while the implicit cost of search (the output foregone during search) increases with k. As a result, both the match surplus and the expected wage bill of a firm are increasing functions of k, creatingahold-up problem Directed search Inthissectionwestudythesameenvironmentasbefore,butnowweassume that workers have perfect information about the capital stock of the available jobs. Thanks to this change in the information structure, the job seekers are able to adopt an optimal application strategy. This strategy specifies the probability that a worker applies for a job at each measurable set of firms withthesamestockofcapital. A first implication of directed search is that the labor market may break up into several homogenous sub-markets for jobs with different capital stocks. Second, if the firms happen to offer jobs with different capital levels, then the queue length in each sub-market will adjust so that the workers obtain the same value J U in each of these markets. Below we show that these features create a competitive environment in which the expected wage costs of a firm are independent of its choice of capital. The remuneration of a single applicant is still an increasing function of the firm s choice of k, but 13 This is similar to the hold-up problem in Davis (2001), except that he examines a model with pairwise matching and ex post bargaining. workers obtain the same exogenous share of the surplus. As a result, in his model all 19

21 this is now exactly compensated by an increase in the value of q. In the rest of the analysis, we denote the queue length for jobs with k units of capital by q(k). Anyfirm that plans to enter the labor market can observe the realized values of q(k) for all k K. On the contrary, for levels of k that are not offered in equilibrium, the firms need to form beliefs about the associated queue length. To ensure that firmsrecognizeallprofitable deviations, we assume that these out-of-equilibrium beliefs are consistent with rational expectations. That is, for any k 0 / K the firms anticipate a queue length q(k 0 ) so that the workers who apply for this job obtain the equilibrium value J U. This approach is standard in models of competitive search (e.g. Moen 1997) and from the work of Peters (2001) we know that this outcome corresponds to the formal limit of a finite economy in which the firms offer competing auction mechanisms. Accordingly, the equilibrium allocation with directed search needs to satisfy the following five conditions (1 ) The application strategy of workers maximizes the value of J U given G. (2 )Eachfirm that enters the labor market makes a profit-maximizing investment in capital. (3 ) New entrants make zero profits. (4 ) Firms beliefs about q(k) are consistent with rational expectations starting at any decision node. (5 ) The wage payments are the outcome of a sealed bid second price auction. 5.1 Analysis Since the wage rule is the same as before, we start the analysis by defining the Bellman equations. The asset value of a vacant job with capital-intensity k satisfies Once again, we assume that a job is filled if it attracts at least one applicant. This assumption is natural in the context of a directed search model since the workers will only apply for those jobs that make them (weakly) better off. 20

22 J V (k, q(k)) = β(1 s) (e q(k) + q(k)e q(k) )J V (k, q(k)) +(1 e q(k) q(k)e q(k) )(Γ(k) J U ). (15) The above expression is identical to (8) except that it allows the queue lengths to depend on k, so that it may be the case that q(k) 6= q. Likewise, the asset value of a job seeker who applies for a job with k units of capital satisfies J U (k, q(k)) = β((1 e q(k) )J U +e q(k) (1 s) Γ(k) J V (k, q(k)) + sj U ). (16) This Bellman equation has a similar structure as (9), but the value of J U is now defined by the highest value that a worker can obtain during unemployment: J U =supj U (k, q(k)). (17) k K The optimal application strategy of workers needs to satisfy two conditions. (i) All sub-markets in the support of the application strategy must offer the workers the same value J U. (ii)nosub-marketcanoffer workers a higher value than J U and fail to attract to any applicants. Taking into account that the job distribution is endogenous, this leads to the equilibrium condition that J U (k, q(k)) = J U k K. Inserting this condition into (16), we obtain the following expressions for J V (k, q(k)) and J U : J V (k, q(k)) = β(1 s) 1 e q(k) q(k)e q(k) 1 β(1 s)+β(1 s)(1 q(k)e q(k) ) f(k) 1 β(1 s) (18) J U = β(1 s)e q(k) 1 β(1 s)+β(1 s)(1 q(k)e q(k) ) f(k) 1 β. (19) 21

23 The payoff function for J V (k, q(k)) gives rise to the same zero profitcondition as in Section 4 (see below). The difference with respect to the previous section concerns the choice of capital. For any k K, condition (19) defines a strictly increasing relationship between k and q. Moreover, the same indifference relation also defines the queue length that results if some firm would deviate from the equilibrium by offering a job with a k/ K. Formally, let k denote the level of capital that solves (19) for q(k) =0. Our restriction on the out-of-equilibrium beliefs of firms can then be formalized as follows: q(k) = 0 if k<k J U (k, q(k)) = J U if k k. (20) The firms anticipate that their jobs will attract zero applicants if they invest less than k because J U (k, 0) < J U for all k < k. On the contrary, for any k above this threshold level, the firms anticipate a queue length that is consistent with the optimal application strategy so that workers obtain J U on all the available jobs. Inspection of (19) shows that the resulting queue length function q(k) is continuous, strictly increasing in k and strictly decreasing in J U on (k, ) (0, ). An equilibrium can now be defined succinctly as a pair of value functions for J V (k, q(k)) and J U that satisfy (18) and (19), a queue length function q(k) that satisfies (20) and a non-empty set K arg max k [J V (k, q(k)) pk] so that all firms maximize their expected profits taking J U and q(k) as given and new entrants earn zero profits A complete description of the equilibrium allocations also includes G. Thisisrelevant when K is not a singleton. But even then we do not need G to calculate the equilibrium values of k, q and J U as demonstrated in Lemma 5. 22

24 k J U Q S k D k q D q Figure 3: The equilibrium with directed search The following Lemma shows that the equilibrium allocation can be characterized as the solution to a simple constrained optimization problem. Lemma 5 (K,q(k)) with k D K and q D = q(k D ) is a steady-state equilibrium with directed search if and only if (k D,q D ) solves subject to J U =max {k,q} β(1 s)e q 1 β(1 s)+β(1 s)(1 qe q ) f(k) 1 β (21) β(1 s)(1 e q qe q ) 1 β(1 s)+β(1 s)(1 qe q ) f(k)/k p 0. (22) 1 β(1 s) Proof. See Appendix. Notice that the constraint coincides with (5). Hence, Lemma 5 states that any pair (k, q) that is observed in equilibrium must maximize the value of the representative job seeker subject to the efficient zero-profit condition of firms. 23

25 k J U J U (k E ) Q S k D k E E q E q q Figure 4: Profitable deviations This feature is illustrated in Figure 3. Once again the zero-profit condition is represented by the efficient locus Q S.Totheright(left)ofthislocus,new entrants make profits (losses), because they attract more (fewer) applicants than is needed to break even. Workers, on their part, value high investment levels and short queues. Their indifference curves have a positive slope, and indifference curves that are located further away from the origin correspond to higher levels of utility. Thus, the unique equilibrium is located at a point of tangency between the zero-profit condition and the highest attainable indifference curve of the workers. The (k, q) combinations on this indifference curve form the choice set of an inactive firm that wants to enter the labor market. Hence, in equilibrium each entrant will choose to invest k D because any other investment level would generate strictly negative profits. It is also easy to show that the other (k, q) combination on Q S cannot be supported as an equilibrium. Suppose that the economy is located in point E in figure 4. In this situation a deviant firm can attract a queue of q 0 applicants if it invests the efficient amount k D, and since the pair (k D,q 0 ) 24

26 lies to the right of Q S the firm will obtain a strictly positive profit. 16 The above arguments are formalized in the Appendix. Here we want to stress that the equilibrium choices of the firms maximize the shadow value of the job seekers. This leads to our main efficiency result. Proposition 6 (Efficiency) Any equilibrium allocation (K,q(k)) coincides with an efficient allocation. Conversely, for any efficient allocation (q S,k S ) there exists an equilibrium with directed search such that k S K and q S = q(k S ). Proof. See Appendix. Proposition 6 demonstrates that the introduction of directed search resolves the hold-up problem. The actual wage payments of a single applicant still depend positively on the capital-intensity of the firm, but these higher wages are now exactly compensated by a longer queue of applicants and hence a lower probability that the firm will face a single applicant. At the margin the expected wage costs of a firm are therefore independent of its choice of capital. 6 Competing auction rules The previous sections served to identify the two necessary conditions for efficiency. Job seekers must be able to direct their search towards firms with differentcapitalstocksandtheauctionrulemustbesuchthatapplicants 16 The positive profits of the deviant are due to two different forces. First, since q 0 >q D the deviant firm has a better chance to appropriate the surplus than in the equilibrium allocation. Second, since J U (k E ) <J U the actual surplus is bigger than in the efficient allocation 25

27 are paid their marginal product. So far, the latter condition was satisfied by assumption because the firms could not commit to a reservation profit above the value of their endogenous outside option. The aim of this section is to show that our efficiency result still holds if we relax this assumption. To be more precise, suppose that firms can credibly announce a reservation profit π J V (k). If a firm decides to use this option, the payoff of a single applicant drops to J E (k, 1) = Γ(k) π. However, the next proposition shows that this extension does not affectourresultsaslongasworkers can freely observe the reservation bids before they make their application decisions. 17 Proposition 7 Suppose firms can credibly announce a reservation bid and unemployed workers can freely observe the reservation bids and the capital stock of all the firms before they make their application decisions. Then, the equilibrium allocation is efficient and each firm announces its true outside option value, i.e. a firm with capital-intensity k announces a reservation profit π(k) =J V (k) =pk. Proof. See Appendix. Proposition 7 is our most general result. It shows that our predictions do not depend on a convenient choice of the auction rule. If we endogenize the reservation profit, firms end up choosing the auction rule that decentralizes the efficient allocation. As mentioned in the Introduction, the prediction that firms announce their true outside option value is a common result in the literature on competing auction mechanisms. The value-added of Proposition 7 is that we 17 Proposition 7 is derived under the condition that firms have rational expectations about the queue length for all possible combinations of k and q. Appendix. For details, see the 26

28 extend this result to an environment with non-contractible investments. In equilibrium the firms discard the option to protect their returns on capital by announcing a high reservation profit. Such a strategy would limit the surplus extraction by single applicants, but the overall effect on profits would be negative since the firm would attract fewer applicants. This completes our analysis of job auctions. In the last section we briefly explain the connections with the earlier work of Acemoglu and Shimer (1999) who analyzed a model of wage posting. 7 Job auctions vs wage posting It is well-known that auctions and posted prices are revenue-equivalent mechanisms in markets with a continuum of identical and risk neutral buyers and sellers (Kultti, 1999). Below we show that a similar prediction can be derived for our economy with ex ante investments. Next, we identify some situations in which this equivalence breaks down and auctions dominate wage posting. The study of Acemoglu and Shimer (1999) demonstrates two important results. Their main result shows that the hold-up problem is resolved if firms can post wages and if workers can direct their search towards their preferred wage. Since the workers only care about the wage, the firms can freely adjust capital. In equilibrium, this feature guarantees that the firms obtain the marginal returns on their investment, while the assumption of directed search obliges the firms to post a wage that maximizes the value of the job seekers subject to the zero-profit condition of the firms. Next, in an extension they show that the same allocation is obtained if the firms can post surplus shares and workers observe both the investments and the surplus shares before they make their application decisions. In the latter case the firms make efficient investments and they post a surplus share that satisfies the Hosios (1990) condition. 27

29 This second result resembles our Proposition 7. Under the efficient auction rule the workers appropriate the entire surplus in a share θ = qe q /(1 e q ) of the matches. It is easy to show that the risk neutral workers and firms are indifferent between this lottery and the case in which firms commit to pay workers a fixed proportion θ of the surplus. 18 The only difference is that the applicants would be paid their expected rather than their actual marginal product. The fixed surplus share satisfies Hosios condition because θ is equal to the elasticity of aggregate matching function m(v, u) (see Section 2.3). Moreover, in both cases the workers also need to observe the investment levels. Finally, as suggested by Acemoglu and Shimer (1999), the firms could go one step further and commit to pay workers their expected marginal product in the efficient allocation. In this case it would be unnecessary for the workers to observe capital. This brief discussion indicates that the equivalence between auctions and posted prices survives if the applicants can observe capital. This result is certainly useful, but we want to stress that there are many situations in which the equivalence breaks down. Here we want to stress the role of private information. 19 Wage posting clearly dominates auctions when the investment levels of firms are not freely observable. In this case, auctions lead to hold-ups while wage posting is still efficient. On the contrary, job auctions are more efficient than wage posting when workers have private information about their skills 18 If all firms post a surplus share θ, we obtain the following equilibrium asset value equations (1 β)j U = µ(q)θ Γ(k) J V (k) J U (1 β(1 s))j V (k) = η(q)(1 θ) Γ(k) J V (k) J U. Substituting θ = qe q /(1 e q ) into these equations yields (15) and (16). 19 Theexistingliteraturefocusedontheroleofriskattitudesandthesizeofthemarket. 28

30 when they apply for jobs. In a second price auction the applicants choose to reveal this information. In an economy with homogenous jobs, this leads to an efficient outcome. The firms can always hire the best candidate and pay this worker her shadow value (e.g. Shimer, 1999). By contrast, in the case of wage posting this link between wages and shadow values is often lost because the firms are constrained to make unconditional wage offers (e.g. Inderst, 2005; Michelacci and Suarez, 2007; Peters, 2008). The firmsmaystillbeabletoseparatetheworkersintohomogenous sub-markets with different wages. However, this ex ante screening comes at a cost. The best workers only apply for some high-wage jobs while the efficient outcome is achieved if they randomize over all the jobs and if the firms screen their applicants ex post. 20 Furthermore, the high-wage firms must have a credible threat to reject or fire any undesired worker once her productivity becomes observable (Inderst, 2005). As a result, they may be forced to pay the best workers a premium on top of their shadow value and this may lead to a further reduction in the job finding rate of these workers. The introduction of ex ante investments in capital is likely to aggravate these problems. The most interesting case is when physical capital and the skills of workers are complements in production. In this case the efficient allocation typically features some mismatch (e.g. Shi, 2002; Shimer 2005) This argument is formalized in Eeckhout and Kircher (2008) for the case of product markets. The same study also shows that price posting typically survives as an efficient pricing mechanism when all meetings are bilateral. In this case ex ante screening is optimal because it avoids that buyers with a low private valuation can crowd-out buyers with a higher valuation. 21 In these studies the frictional assignment does not feature perfect assortative matching because the matching rates of the best jobs can be raised by matching them to some lowproductivity workers who do not congest the market for the best workers. Shi (2001) takes adifferent route by assuming that firms need to commit their machines to one type of worker. Under this condition he demonstrates that the efficiency results of Acemoglu and Shimer (1999) continue to hold in an environment with two-sided heterogeneity. 29

31 This efficient matching pattern can be decentralized if firms can post typecontingent wages, or if firms use auctions and applicants can direct their search to firms with different capital stocks. In contrast, with non-contingent wages, the relatively high wage offers of capital-intensive jobs may attract too many applications from low-productivity workers. This may discourage the investment in capital because some of the returns to capital may accrue to low-productivity workers, or it may force the firms with capital-intensive jobs to pay a wage premium. In both cases we should expect a suboptimally low supply of capital-intensive jobs. Finally, it is important to notice that job auctions may also prevent underinvestment in human capital. Workers often make a large investment in human capital before they enter the labor market. Now suppose, as is often claimed, that the return to schooling is positively correlated with innate ability. If this ability is private information, then firms cannot tie their wage offers to productivity unless the firms manage to create separate markets for each level of innate ability. However, in an economy with homogenous jobs this ex ante screening reduces the returns to education because the most talented workers only match with a subset of jobs. By contrast, in the case of job auctions, firms can always hire the most productive candidate, and since wages reflect the actual marginal productivity of applicants this should lead to efficient investments in human capital. In sum, while wage posting allows firms to signal their investments in capital, auctions are useful to elicit private information about workers skills and their returns to investments in human capital. This may explain why job announcements for white collar workers often leave some room for wage negotiation. 30