Interviews and the Assignment of Workers to Jobs

Transcription

1 Interviews and the Assignment of Workers to Jobs Benjamin Lester Federal Reserve Bank of Philadelphia Ronald Woltho University of Toronto February 8, 2016 Preliminary version Abstract This paper studies the eect of screening costs on the equilibrium allocation of workers with dierent productivities to rms with dierent technologies. In the model, a worker's type is private information, but can be learned by the rm during a costly screening or interviewing process. We characterize the planner's problem in this environment and determine its solution. A rm may receive applications from workers with dierent productivities, but should in general not interview them all. Once a suf- ciently good applicant has been found, the rm should instead make a hiring decision immediately. We show that the planner's solution can be decentralized if workers direct their search to contracts posted by rms. These contracts must include the wage that the rm promises to pay to a worker of a particular type, as well as a hiring policy which indicates which types of workers will be hired immediately, and which types will lead the rm to keep interviewing additional applicants. The views epressed here are those of the authors and do not necessarily reect the views of the Federal Reserve Bank of Philadelphia or the Federal Reserve System. 1

2 1 Introduction A central challenge to labor economists is to understand how labor markets allocate workers of dierent abilities to rms with dierent technologies. This challenge was rst addressed within the contet of a frictionless environment by Becker (1973), and has recently been re-eamined within the contet of environments that incorporate various frictions within the matching process (see Shi, 2001, 2002; Shimer, 2005; Peters, 2010 and Eeckhout & Kircher, 2010). However, all of these papers abstract from an important, realistic feature of modern labor markets: the screening or interviewing of applicants. In this paper, we construct a model in which rms with dierent technologies post vacancies, and workers with dierent abilities observe the contracts being oered at these rms and apply for a single position. Following the literature on directed search, we assume that workers cannot coordinate their application strategies, so that multiple workers may apply to the same rm, and these workers may be of dierent abilities. Unlike the eisting literature, however, we assume that a rm cannot distinguish one applicant from another prima facie. Instead, in order to discover a worker's ability, we assume that a rm must interview the applicant, which is a costly process. Given this cost, a rm may not choose to interview each applicant, but rather it may implement a hiring policy that indicates which types of workers will be hired immediately, and which types will lead the rm to keep interviewing additional applicants. In this environment, we characterize the optimal wages and hiring policies of each rm, the optimal application behavior of workers, and the equilibrium allocation of workers to rms. We show that the equilibrium allocation is constrained ecient, i.e. a social planner would use the same hiring policy and would choose the same allocation of workers to rms. The model provides testable implications with respect to various labor market outcomes and can be used to analyze to what etent these implications depend on the magnitude of the screening costs. For eample, the model predicts how unemployment rates and wage dispersion vary with worker skill. Further, the model is informative of how an inow of low-skilled workers into the pool of job seekers aects the equilibrium allocation and wages. 2

3 This paper proceeds as follows. Section 2 introduces the environment and eplains how we model recruitment. We solve the planner's problem in section 3. In section 4, we derive the market equilibrium and show that it decentralizes the planner's solution. Section 5 concludes. 2 The Environment Players. We consider a static economy populated by a measure 1 of workers and a measure v of rms. Each worker is endowed with a single, indivisible unit of labor, and each rm requires one such unit of labor to produce. Workers and rms are risk-neutral and heterogeneous with respect to productivity. In particular, each worker can be represented by a characteristic which is distributed according to the cumulative density function F (), which has support on a closed and connected interval X [, ] R +. A worker's type is private information. Like workers, rms dier in a characteristic y distributed according to G(y) over the interval Y [ y, y ] R +. However, unlike workers, a rm's type is publicly observable. Both F () and G(y) are assumed to dierentiable, with f() = F () and g(y) = G (y). Matching. Workers can visit a single rm in attempt to match, but the matching process is frictional. In particular, workers cannot coordinate with one another when choosing a rm to visit, and hence the number of buyers to arrive at each seller, n, will be stochastic. As is customary in the literature on directed (or competitive) search, we formalize this by restricting attention to symmetric strategies: all workers of type apply to (one of the) type y rms with the same probability (and each type y rm with equal probability). As a result, the number of type workers to apply to a particular type y rm is a random variable distributed according to the Poisson distribution. 1 We denote by λ(, y) the epected number, or queue length, of type workers to arrive at a rm of type y, so that the probability that a rm of 1 This can be derived eplicitly by analyzing the game with a nite number of workers and rms, and taking the limit as the number of agents tends to innity; see, for eample, Burdett et al. (2001) and Peters (2010). 3

4 type y receives n applications from workers of type is given by the usual [ λ(, y)e λ(,y)] /n!. It will be convenient to dene L(, y) = λ(, y)d and Λ(, y) = λ(, y)d as the queue of applicants who are more or less productive, respectively, than a worker of type. Given this notation, Λ(, y) = L(, y) represents the total queue length of workers applying to a type y rm, while Λ(, y) = L(, y) = 0. Interview Costs. When workers apply across dierent types of rms with positive probability, a rm will typically receive applications from workers of dierent productivity levels. However, since a worker's type is private information, the rm will not be able to distinguish these dierent types of workers prima facie. In order to overcome this informational asymmetry, we assume that a rm can learn a worker's type through a job interview. Job interviews are costly; rms incur a cost k for each interview. 2 Output and Payos. Once the interviewing process is over and matches are formed, production occurs. A match between a worker of type and a rm of type y yields z(, y) units of output. We assume that z(, y) 0 for all (, y) X Y, z(, y)/ > 0, and z(, y)/ y > 0. Firms that do not match produce zero units of output, and workers that are unmatched receive payo zero. In the market equilibrium, the rm and the worker divide the output according to the contract that they have written. Their payos are linear in the amount of output that they consume. 3 The Planner's Problem In this section, we will characterize the decision rule of a (constrained) benevolent planner whose objective is to maimize net social surplus, subject to the constraints of the physical environment. These constraints include the frictions inherent in the matching process, as well as the requirement that worker types are costly to learn. The planner's problem can be broken down into two components. First, the planner has 2 To keep the analysis as simple as possible, we will assume that interviewing is a necessary condition for hiring. That is, a rm cannot avoid the interview cost by hiring a worker at random. 4

5 to assign queue lengths of workers to each rm, subject to the constraint that the sum of these queue lengths across all rms cannot eceed the total measure of workers available of each type. Second, the planner has to specify selection rules for rms to follow after the number of workers that arrive at each rm is realized. Working backward, we begin by characterizing these rules at an arbitrary rm, taking the queue lengths as given, and return later to characterize the optimal assignment of queue lengths across rms. Optimal Selection Rule Consider a rm of type y with queue lengths {λ(, y)}. Let the total number of applicants be n. The rst result that we present in this section is that it is optimal for the planner to learn the workers' productivities sequentially (i.e., one at a time), preserving the option to stop after each interview i n and to hire either one of the i workers who has been interviewed. Intuitively, this sequential search strategy is optimal because it allows the decision of whether an additional worker should be interviewed to be contingent on the realization of previous workers' productivities; the planner can economize on k if the epected gain from interviewing one or more additional workers is small. The proof of this result is straightforward and follows very closely to that in Morgan & Manning (1985). Lemma 1. The optimal strategy for the planner is a sequential search strategy. We now characterize the planner's optimal selection rule. In general, after a rm of type y has interviewed i workers with productivities ( 1,..., i ), when n i total workers have arrived, the planner's decision rule is a function that dictates whether a rm should (a) keep interviewing additional workers (if there are any) or (b) hire worker i {1,..., i}. However, several features of the environment simplify this function immediately. First, notice that the only relevant information in the vector ( 1,..., i ) is the maimum productivity; clearly, if the planner instructs the rm to stop interviewing workers, it must hire the worker with the highest productivity. Given this, it will be convenient to denote by i ma{ 1,..., i }. Second, while the cost of an additional interview is a constant (k), we conjecture and later conrm that the continuation value of interviewing the (i + 1) th worker is increasing in i, 5

6 so that the planner's optimal selection rule is a cuto strategy. Let us denote the planner's cuto by µ p i,n (y) for n N\{1} and i {1, 2,..., n 1}, so that the rm will stop interviewing workers and hire the worker with productivity i if, and only if, i µ p i,n (y). Otherwise, if i < µ p i,n (y), the rm will continue to interview the net worker. Below we establish that, in fact, the optimal stopping rule is stationary; in particular, it does not depend on the number of workers who have already been interviewed, i, or the number of workers still available for interviewing, n i. 3 Lemma 2. Suppose n N workers arrive at a rm. Letting µ (y) satisfy k = µ (y) λ (, y) L(, y) [z (, y) d z (µ (y), y)], (1) the planner maimizes the social surplus implementing the following rule: (i) If n > 1 and i {1, 2,..., n 1}, the rm of type y should stop interviewing workers and hire the worker with productivity i if, and only if, i µ p i,n (y) = µ (y). Otherwise, the rm should interview the net worker. (ii) If n=1 or i = n, the rm hire the worker with productivity n. The proof of Lemma 2 utilizes an induction argument; we sketch the rst step here, for intuition, and present the formal proof in the appendi. Suppose an arbitrary number n 2 workers arrive at a rm of type y, and let H n j,n (, y) denote the net epected surplus from continuing to interview workers, given that the maimum productivity of the n j workers interviewed so far is, for some j {1, 2,..., n 1}. For notational convenience, we will drop the second subscript n, so that H n j,n (, y) H n j (, y), µ p n j,n (y) µp n j (y), and so on; it should be understood that the analysis is for an arbitrary value of n. Figure 1 plots H n 1 (, y), H n 2 (, y), and. Notice three facts: (i) H n 1 (, y) > z (, y) if and only if < µ (y); (ii) H n 2 (, y) > H n 1 (, y) if and only if < µ (y); and (iii) H n 2 (, y) = H n 1 (, y) for µ (y). The rst of these facts is by construction: µ (y) 3 See Lippman & McCall (1976) for a similar result. It is worth noting that the stationarity of p i,n implies that whether or not the planner can observe i and/or n is ultimately immaterial, as he would choose the same cuto rule in either environment. 6

7 is dened as the value of that makes the planner indierent between interviewing one additional worker and stopping. Hence, it is should be obvious that µ p n 1 (y) = µ (y) is optimal. INSERT FIGURE 1 HERE Now consider the second and third facts. Intuitively, H n 2 (, y) H n 1 (, y) 0 captures the option value of being able to interview two more workers instead of only one. However, if n 2 µ (y) = µ p n 1 (y), then this option is never eercised after n 1 productivities have been discovered, and thus H n 2 (, y) = H n 1 (, y); since the n th worker is never interviewed, both H n 2 (, y) and H n 1 (, y) denote the epected surplus from interviewing just one more worker. Alternatively, if n 2 < µ (y) = µ p n 1 (y), then the option to interview two more workers instead of one can be valuable, but this additional value is realized only in the event that n 1 ( n 2, µ (y)): if n 1 < n 2 then n 2 = n 1, and if n 1 > µ (y) then the n th worker is never interviewed. Hence, as n 2 µ (y), this option value converges to zero and H n 2 ( ) H n 1 ( ). As a result, clearly H n 2 (, y) > z (, y) if and only if < µ (y); that is, µ p n 2 (y) = µ (y) is optimal. This completes the sketch of the rst induction step; the remainder proceeds in much the same way. Interviewing Costs We now derive the epected interviewing costs incurred by a rm of type y with queue lengths {λ(, y)} and cuto µ p (y). In what follows, we will suppress the argument of µ p (y) whenever possible for notational simplicity, though it should be understood that the hiring policy will, in general, depend on the rm's type. Given a hiring policy µ p, the workers that apply to a rm can be divided into two groups: those with type < µ p and those with type µ p. For lack of a better label, we refer to the rst type of worker as a bad applicant and the latter as a good applicant. The queue lengths of the respective groups are Λ(µ p, y) and L(µ p, y). To derive the epected interviewing costs incurred by the rm, consider the event that n total applications are received. Then the number of good applicants, n G, conditional 7

8 ( on n total applicants, is distributed according to the binomial distribution B n, L(µp,y) L(,y) Similarly, conditional on n, the number of bad applicants, n B, is distributed according to ( ) the binomial distribution B. Therefore, given n, the epected interviewing costs can be written { n i=1 n, Λ(µp,y) L(,y) ( ) Λ(µ p i 1 ( ) }, y) L(µ p, y) Λ(µ p n L(, y) L(, y) i + n, y) k. (2) L(, y) The rst term in brackets in equation (2) captures the possibility that the rst good applicant is discovered in interview i {1,..., n}, in which case the rm will incur cost ik (since the rst interview is costless). The second term in brackets captures the event that all n applicants are bad, in which case the rm incurs cost nk. Since n is distributed according to the Poisson distribution with parameter L(, y), one can take epectations across all possible realizations of n. Doing so reveals that the total epected costs of interviewing for a rm of type y with queue lengths {λ(, y)} who has chosen µ p is ). 4 K(µ p, y) = [ 1 e ] L(µp,y) L(, y) k. (3) L(µ p, y) Match Output and Social Surplus In order to derive the total social surplus, we begin by constructing the epected output at a rm of type y with queue lengths {λ(, y)} and a hiring policy µ p (y). Again, we suppress the argument of µ p whenever possible for notational convenience. There are three relevant events that can occur for any rm after choosing a hiring policy µ p : (1) at least one good applicant arrives; (2) no good applicants arrive but at least one bad applicant arrives; and (3) no applicants arrive. Since e L(µp,y) is the probability that the rm receives no applicants with productivity µ p, it follows that the rst event occurs with probability 1 e L(µp,y). In this event, recall that the rm continues to interview workers until it nds the rst good applicant. In particular, if there are multiple good applicants, each of them is equally likely to be hired 4 The proof of this result is standard and therefore omitted. 8

9 since the applicants are interviewed in a random order. Hence, the productivity of the worker that is hired is a random draw from the distribution of good workers. Given the properties of the Poisson distribution, this conditional distribution has density φ(, y) = λ(, y) λ(, y) = λ( µ, y)d L(µ p, y) p dened over the range [µ p, ]. As a result, the output of a rm of type y, conditional on receiving at least one good applicant, can be written µ p φ(, y)z(, y)d = µ (4) λ(, y) L(µ p, y) z(, y)d. (5) Since the rst event occurs with probability 1 e L(µp,y) and the third event occurs with probability e L(,y), the second event occurs with probability 1 [ 1 e L(µp,y) + e L(,y)] = e L(µp,y) e L(,y). In this event, the rm interviews all applicants and hires the worker with the highest productivity. Thus, the productivity of the selected worker is drawn from the density function 5 ϕ(, y) = λ(, y)e L(,y), (6) e L(µp,y) e L(,y) which is dened over the range [, µ p ]. Therefore, conditional on receiving no good applicants and at least one bad applicant, the output of a rm of type y is equal to ˆ µp ϕ(, y)z(, y)d = ˆ µp λ(, y)e L(,y) e L(µp,y) e L(,y) z(, y)d. (7) By assumption, no output is produced when a rm receives zero applicants. Therefore, given the results above, it is straightforward to construct the total epected output at a rm with productivity y and hiring policy µ: Z(µ p, y) = [ 1 e L(µp,y) ] λ(, y) µ L(µ p p, y) z(, y)d + ˆ µp λ(, y)e L(,y) z(, y)d. (8) Hence, the total social surplus, integrating across all rm types, can be written S = ˆ y y g(y) [Z (µ p (y), y) K (µ p (y), y)] dy, (9) where we've written µ eplicitly as a function of y to remind the reader that hiring policies will, in general, depend on a rm's type. 5 To understand this density function, let ma denote the maimum value of amongst the applicants. The probability that ma <, conditioning on the occurrence of the second event, is equal to e L(,y) e L(,y) e L(µp,y) e L(,y). Dierentiating with respect to yields the density function in equation (6). 9

10 Optimal Queue Lengths The social planner maimizes S by choosing the queue lengths of each type of worker at each type of rm, λ(, y), subject to the constraint that, for all X, f() = ˆ y y λ(, y)g(y)dy. (10) Equation (10) is a typical resource constraint: it requires that the measure of workers of type allocated across all rms must be equal to the measure of workers of type available in the pool of unemployed workers. Given (9) and (10), the Lagrangian of the optimization problem can be written ˆ y [ L = g(y) Z (µ p (y), y) K (µ p (y), y) y ] ξ()λ(, y)d dy + where ξ() denotes the Lagrange multiplier for type workers. ξ()f()d, (11) For each (, y) X Y, the rst order condition with respect to λ(, y) reduces to [ ] ξ() e L(,y) z(, y) λ(, y)e L(,y) z(, y)d k 1,y) e L(µp (12) L(µ p, y) if < µ p (y), and ˆ µp ξ () e L(µp,y) z (, y) λ (, y) e L(,y) z (, y) d k 1,y) e L(µp (13) L (µ p, y) + 1 [,y) e L(µp [1 + L (µ p ˆ, y)] ] λ (, y) z(, y) L (µ p, y) µ L (µ p p, y) z (, y) d L(, y) + L(µ p, y) k (14) if µ p (y). In both cases, we require that λ(, y) 0, with complementary slackness. The rst order conditions on λ(, y) are fairly intuitive. For eample, for the case of < µ p (y) and λ(, y) > 0, equation (12) implies that the shadow value of adding an additional worker of type to the applicant pool at a rm of type y is equal to the epected marginal gain in productivity (should the worker be hired) less the epected increase in interviewing costs. 6 To see this more clearly, note that (12) can be re-written in the following 6 Note that, at the optimum, the shadow value of adding an additional worker of type must be equated across all rms with type y such that λ(, y) > 0. 10

11 form (assuming λ(, y) > 0), which decomposes the benets and costs into four possible cases: ξ() = e L(,y) [z(, y) k] + [ e L(,y) e L(,y)] [ z(, y) + [ e L(µp,y) e L(,y)] [ k] + [ 1 e L(µp,y) ] { 1 e L(µp,y) [1 + L(µ p, y)] [1 e L(µp,y) ] L(µ p, y) ] λ(, y)e L(,y) e L(,y) e L(,y) z(, y)d k } [ k]. First, it could be the case that no other worker applies to this rm, in which the introduction of a worker of type yields a marginal gain of z(, y) and a cost k. Second, it could be the case that at least one other worker applies to this rm, but that the worker of type is the most productive. In this case, he will be interviewed with certainty, so that the interviewing cost k will be incurred with probability one. He will also be hired with certainty, and the epected net gain in output is z(, y) minus the epected output had this type worker not applied; remember that ϕ(, y) = [λ(, y)e L(,y) ]/ [ e L(,y) e L(,y)] is the density of the maimum value of. The third possibility is that the worker is not the best worker to apply, but that there are no good applicants. In this case, the worker is interviewed with certainty, so that the cost k is incurred, but there is no marginal gain in output. Finally, it may be the case that a good worker applies to this rm. In this case, a worker with type < µ p (y) will never be hired, but he may or may not be interviewed. Indeed, the probability that a worker with < µ p (y) is interviewed, conditional on at least one good worker applying, is equal to 1 e L(µp,y) [1 + L(µ p, y)]. [1 e L(µp,y) ] L(µ p, y) Similar reasoning can be used to understand the rst order condition with respect to λ(, y) when µ p (y). The rst line of equation (13) reects the epected marginal increase in output, less the increase in epected interviewing costs, from adding a worker of type µ p (y) to a rm of type y when no other good workers apply. The second line of equation (13) reects the epected benet of this worker being hired when at least one other good worker also applies. 7 In this event, the epected change in output may be positive 7 Note that there is neither a cost nor a benet of adding this worker to the pool of applicants if (i) at 11

12 or negative; since the rm stops interviewing after meeting with the worker of type, the productivity of the other good worker that would have been hired could be greater or less than. However, in this event, the addition of the worker of type to the applicant pool unambiguously decreases the epected costs of interviewing, which eplains the nal term in (13). 4 Decentralized Equilibrium In this section, we show that the planner's solution can be decentralized. We start by discussing the contracts that rms need to post, before deriving the epected payos and characterizing the equilibrium. Contracts A rm of type y posts a contract which consists of two elements. The rst element of the contract is a wage schedule w(, y), indicating the wage that the rm will pay if it hires a worker of type. The second element is a hiring policy µ(y) X which species how the rm will select a worker among the applicants. In particular, the rm commits to use a sequential search strategy and to hire the rst applicant whose type eceeds µ(y). If none of the applicants are of type µ(y) or higher, the rm will interview all applicants and hire the most productive. 8 We focus our attention on equilibria in which rms play symmetric, pure strategies. Therefore, the contract of a type y rm can be summarized by a pair {µ(y), w(, y)}, while the number of workers that the rm attracts will be distributed according to the Poisson distribution with mean (or epected queue length) λ(, y). least one other good worker applies, and (ii) the worker of type µ p (y) is not hired; in this case, both the output and the costs of interviewing are unchanged. 8 Notice that this contracting space allows the rm considerable eibility in shaping the distribution of workers that apply and the type of the worker that is ultimately hired. For eample, a rm of type y that wants to discourage workers of type from applying (so as to avoid congestion at the interviewing stage) can specify a wage w(, y) = 0. In addition, the choice of µ(y) allows the rm a variety of policies ranging from hiring an applicant at random (µ(y) = ) to hiring the best applicant (µ(y) = ). 12

13 Optimal Application Behavior and Equilibrium Queue Lengths The rst step is to characterize how λ(, y) is determined. To do so, it is rst convenient to dene the epected payo to a worker of type from applying to a rm of type y, taking as given both the contract posted by the rm and the application behavior of the other workers. In particular, if < µ(y), the worker is only hired if no better worker applies, which occurs with probability e L(,y). Alternatively, if µ(y), the worker is hired if he is selected from the pool of good applicants, which occurs with probability [ 1 e L(µ,y)] /L(µ, y). Thus, for a type worker, the epected payo from applying to a type y rm can be written V (, y) = { e L(,y) w (, y) if < µ 1 e L(µ,y) L(µ,y) w (, y) if µ. (15) Let us denote by V () = sup y Y V (, y) the maimal payo, or market utility, of a type worker. In a competitive search equilibrium, the application behavior of workersand thus queue lengthsadjust so that workers are eactly indierent between all rms to which they apply with strictly positive probability. That is, given some market utility V (), the equilibrium queue length of type workers at a type y rm must satisfy λ(, y) 0 (16) V () V (, y) (17) λ(, y) [ V () V (, y) ] = 0. (18) Optimal Posted Contracts and Equilibrium Prots Consider a rm of type y that has posted a contract {µ(y), w(, y)} and epects queue lengths {λ(, y)}. Recall that the rm obtains a payo π (, y) = z (, y) w (, y) when it matches with a worker of type, and incurs epected interviewing costs given by K (µ, y). Hence, the rm's epected prots can be derived using the same logic used in the derivation 13

14 of social surplus, yielding Π = ˆ µ λ (, y) e L(,y) π (, y) d + [ 1 e L(µ,y)] = Z(µ, y) K (µ, y) ˆ µ λ (, y) e L(,y) w (, y) d [ 1 e L(µ,y)] µ λ (, y) π (, y) d K (µ, y) L (µ, y) µ λ (, y) w (, y) d. (19) L (µ, y) Firms are innitesimally small; they take as given the behavior of other rms, and thus the market utility of workers. Hence, given V (), equations (16)(18) dene an implicit map between the contracts that rms post and the epected number of applicants of type that apply to their rm. In particular, a rm of type y with hiring policy µ (y) that wishes to attract a queue λ (, y) > 0 of workers of type must oer a wage w (, y) that yields eactly the epected utility V (). If it oers less, it will attract no applicants, λ (, y) = 0. Taking this relationship as given (i.e., as a constraint), we can use (15) and (18) to substitute the wages out of (19) and re-write the rm's objective function as Π = Z(µ, y) K (µ, y) λ(, y)v ()d, (20) where now the rm maimizes with respect to µ (y) and λ (, y) for all. Notice that (20) resembles a standard prot maimization problem in which the rm purchases queues of type workers at the price V (). Decentralized Equilibrium and the Planner's Solution Denition 1. A (symmetric strategy) directed search equilibrium is characterized by contracts {µ(y), w(, y)} for each y Y, queue lengths {λ(, y)} for each (, y) X Y, and market utilities {V ()} for each X such that: 1. Firms behave optimally: for each y Y, given {V ()} and (16)(18), the contract {µ(y), w(, y)} maimizes epected prots Π. 2. Workers behave optimally: for each (, y) X Y, given {V ()} and {µ(y), w(, y)}, queue lengths {λ(, y)} satisfy (16)(18). 3. Markets clear: for each X, {V ()} is such that f() = y y 14 λ(, y)g(y)dy.

15 We will now establish that a decentralized equilibrium also constitutes a solution to the planner's problem, so that competitive search equilibria are constrained ecient. To see this, let us denote by {λ (, y)} and {µ (y)} the queue lengths and hiring policies, respectively, in a competitive search equilibrium. Also, let {ˆλ(, y)} and {ˆµ(y)} denote an alternative feasible solution to the planner's problem; that is, let {ˆλ(, y)} be such that f() = ˆ y y ˆλ(, y)g(y)dy for each X. Since each type y rm maimizes (20) in a competitive search equilibrium, it must be that Z (λ (, y), µ (y), y) K (λ (, y), µ (y), y) λ (, y)v ()d ) ) Z (ˆλ(, y), ˆµ(y), y K (ˆλ(, y), ˆµ(y), y ˆλ(, y)v ()d. Integrating across all rms and simplifying yields ) ) Z (λ (, y), µ (y), y) K (λ (, y), µ (y), y) Z (ˆλ(, y), ˆµ(y), y K (ˆλ(, y), ˆµ(y), y since ˆ y V () λ (, y)g(y)dyd = y ˆ y V () ˆλ(, y)g(y)dyd = y V ()f()d. Hence, a competitive search equilibrium is a solution the planner's problem, with ξ() = V () for each. We summarize this in the following proposition. Proposition 1. A directed search equilibrium eists and it is constrained ecient. (21) 5 Conclusion This paper studies the eect of screening costs on the equilibrium allocation of workers with dierent productivities to rms with dierent technologies. The novel element of the paper is that a worker's type is private information, but can be learned by the rm during a costly screening or interviewing process. We characterize the planner's problem in this 15

16 environment and determine its solution. We nd that in general workers with dierent productivities may apply to the same rm. The rm should not interview all its applicants: once a suciently good applicant has been found, the epected gain in match surplus from interviewing additional applicants is smaller than the interviewing cost. We show that the planner's solution can be decentralized if workers direct their search to contracts posted by rms. These contracts must include two components: (i) the wage that the rm promises to pay to a worker of a particular type and (ii) the rm's hiring policy which indicates which types of workers will be hired immediately, and which types will lead the rm to keep interviewing additional applicants. 16

17 Appendi Proof of Lemma 1. In general, if n workers arrive at a rm of type y, the planner might want to rst learn the productivities of i 1 n workers simultaneously at cost i 1 k. Then, depending on the realization ( 1,..., i1 ), the planner could choose to either hire one of these i 1 workers, or alternatively the planner could choose to continue and sample i 2 n i 1 workers, in which case this iterative process would continue. This is precisely a special case of the problem that Morgan & Manning (1985) study, so we sketch the intuition for our result below and refer the reader to their paper for a more rigorous treatment. To see that it can never be optimal to learn the productivity of more than one worker at a time, suppose n workers arrive at a rm, and consider the planner's decision of whether to learn the productivities of the rst two workers sequentially or simultaneously. Let H i ( i, y) denote the net epected surplus from continuing to learn workers' productivities (under the optimal policy) given that the maimum productivity of the i workers interviewed so far is i ma { 1,..., i }. We will show that, for any realizations of the rst two workers' productivities ( 1, 2 ), the net social surplus is weakly larger from learning these two productivities sequentially than it is from learning them simultaneously. To see this, realize that the net social surplus from learning these productivities simultaneously is 2k + ma {z ( 2, y), H 2 ( 2, y)}, (22) whereas the surplus from learning these productivities sequentially is k + ma {z ( 1, y), k + ma {z ( 2, y), H 2 ( 2, y)}} = 2k + ma {z ( 1, y) + k, z ( 2, y), H 2 ( 2, y)} (23) Clearly, the epression in (23) is weakly larger than (22) for any 1 and 2, and strictly larger for some 1 and 2. Taking the epectation over all possible realizations therefore implies that a planner would want to learn these productivities sequentially e ante. It is straightforward to etend this argument to learning the productivities of m > 2 workers 17

18 simultaneously after any number i n m workers have already inspected the good. Proof of Lemma 2. The proof of Lemma 2 utilizes an induction argument. We begin with the rst step. If the rm has met with all n workers, the decision is trivial: the rm should match with the worker with productivity n and output is h n,n ( n, y) = z ( n, y). As in the tet, to economize on notation we will drop the second subscript n below, so that h i,n ( i ) h i ( i ). Further, dene q(, y) = λ(,y) L(,y) and Q(, y) = Λ(,y) L(,y). Working backward, consider the planner's problem when the rm has interviewed with only n 1 of the n workers. If the planner instructs the rm to stop interviewing workers, again the worker with productivity n 1 is hired, yielding output z ( n 1, y). Alternatively, if the rm meets with the net worker, the epected output is H n 1 ( n 1, y), where so that H n 1 (, y) = k + H n 1 (, y) = k + z (, y) Q (, y) + ma {h n (, y), h n (, y)} q (, y) d, Notice immediately that H n 1 (µ, y) = z (µ, y) and 0 < H n 1(,y) z (, y) q (, y) d. (24) µ p n 1 = µ is the optimal cuto after interviewing with n 1 workers, and { z (, y) for µ h n 1 (, y) = H n 1 (, y) for < µ. < 1, so that clearly Now consider the planner's problem after the rm has met with n 2 workers. establish three important properties of H n 2 (, y): (1) H n 2 (, y) = H n 1 (, y) for all µ ; (2) H n 2 (, y) > z (, y) for all < µ ; and (3) lim µ H n 2 (, y) = H n 1 (µ, y) = z (µ, y). Given these three properties, along with the fact that H n 2( ) We = H n 1( ) (0, 1) for µ, it follows immediately that z (, y) H n 2 (, y) if, an only if, µ, and hence p n 2 = µ is optimal. After meeting with n 2 workers, the epected surplus from another meeting when n 2 = is H n 2 (, y) = k + h n 1 (, y) Q (, y) + 18 h n 1 (, y) q (, y) d. (25)

19 If µ, then h n 1 (, y) = z (, y) and thus H n 2 (, y) = H n 1 (, y). Alternatively, if < µ, then h n 1 (, y) = H n 1 (, y) > z (, y) and H n 2 (, y) = k + H n 1 (, y) Q (, y) + > k + Q (, y) + ˆ µ = H n 1 (, y) > z (, y). ˆ µ z (, y) q (, y) d + H n 1 (, y) q (, y) d + Finally, note that lim µ n 2 (, y) = k + H n 1 (µ, y) Q (µ, y) + = k + µ Q (µ, y) + = H n 1 (µ, y) = µ. µ z (, y) q (, y) d µ z(, y)q (, y) d µ z(, y)q (, y) d µ z (, y) q (, y) d Therefore, the optimal cuto after meeting with n 2 workers is n 2 = µ. We have established that the following is true for j = 2: 1. H n j (, y) = H n j +1(, y) for all µ. 2. H n j (, y) > for all < µ. 3. lim µ H n j (, y) = H n j +1(µ, y) = µ, so that { z(, y) for µ h n j (, y) = H n j (, y) y for < µ. Now, suppose this is true for all j {2, 3,..., j}. We will establish that it is also true for j + 1. After meeting with n j 1 workers, the epected surplus from another meeting when n j 1 = is H n j 1 (, y) = k + h n j (, y) Q (, y) + h n j () q (, y) d. (26) If µ, then h n j (, y) = z(, y) and thus H n j 1 (, y) = H n 1 (, y). Moreover, given the rst assumption in the induction step, H n j (, y) = H n 1 (, y), so that H n j 1 (, y) = H n j (, y). 19

20 Alternatively, if < µ, then H n j 1 (, y) = k + H n j (, y) Q (, y) + Finally, note that > k + Q (, y) + ˆ µ H n j (, y) q (, y) d + q (, y) d = H n 1 (, y) > z(, y) lim µ n j 1 (, y) = k + H n j (µ, y) Q (µ, y) + z(, y)q (, y) d µ = H n 1 (µ, y) = z(µ, y). µ z(, y)q (, y) d Therefore, we have that the optimal cuto after meeting with n j 1 workers is µ p n j 1 = µ. 20

21 References Becker, G. S. (1973). A theory of marriage: Part I. Journal of Political Economy, 81(4), Burdett, K., Shi, S., & Wright, R. (2001). Pricing and matching with frictions. Journal of Political Economy, 109, Eeckhout, J. & Kircher, P. (2010). Sorting and decentralized price competition. Econometrica, 78, Lippman, S. A. & McCall, J. J. (1976). The economics of job search: A survey. Economic Inquiry, XIV, Morgan, P. & Manning, R. (1985). Optimal search. Econometrica, 53, Peters, M. (2010). Unobservable heterogeneity in directed search. mimeo. Shi, S. (2001). Frictional assignment I: Eciency. Journal of Economic Theory, 98, Shi, S. (2002). A directed search model of inequality with heterogeneous skills and skill-biased technology. Review of Economic Studies, 69(2), Shimer, R. (2005). The assignment of workers to jobs in an economy with coordination frictions. Journal of Political Economy, 113(5),

22 Figure 1: Intuition for stopping rule 22