Edgar Preugschat‡

March 31, 2014

Abstract We show that the directed search equilibrium is not constrained efficient in a dynamic setting with symmetric incomplete information about workers’ productivity and where firms discriminate among applicants after testing them. There is a twofold intertemporal inefficiency linked to the effects of the marginal vacancy on vacancy creation in the following period. In particular, firms can imperfectly detect unskilled workers, while workers are still uncertain about their type. Firms benefit from this asymmetry of information and offer inefficiently low wages. Our simulation work suggests that the welfare loss reaches its maximum when testing is perfectly informative. Constrained efficiency can be attained by properly taxing the entry of firms.

Keywords: Constrained Efficiency, Directed Search, Informational Frictions, Selection JEL Codes: J64, J68

1

Introduction

In the standard competitive (or directed) search model of the labor market (Moen (1997)), the equilibrium allocation is constrained efficient as wages price employment uncertainty and job offers must promise the workers’ market value to attract them. Firms are price-takers in Fern´andez-Blanco gratefully acknowledges financial support from the Spanish Ministry of Science and Technology under Grant No. ECO 2010-19357. † Department of Economics, University Carlos III of Madrid, c/ Madrid, 126, 28903 Getafe, Spain (email: [email protected]) ‡ Department of Economics, University of Konstanz, Box 145, 78457 Konstanz, Germany (email: [email protected]). ∗

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the sense that they take the market value as given in equilibrium. This efficiency result has proved to be robust to a number of extensions. In particular, Shi (2002) and Shimer (2005) show that constrained efficiency is also attained in a static economy with heterogeneous workers and where firms rank applicants by productivity. A few recent articles have shown that the efficiency result does not carry over to a competitive search model with informational frictions. This is the case in the presence of adverse selection, e.g. Michelacci and Suarez (2006) and Guerrieri, Shimer, and Wright (2010), or moral hazard, e.g. Guerrieri (2008). 1 We contribute to this literature by analyzing the welfare properties of the equilibrium allocation when firms get better informed than applicants about their market productivity. By using a simplified version of Fern´andez-Blanco and Preugschat (2013), we analyze an economy with symmetric incomplete information about workers’ productivity, multilateral meetings, and where firms observe an imperfect signal about the type of the applicant before hiring. We show that it is the combination of informational frictions and selection what leads to the inefficiency outcome. To be more concrete, we study a two-period frictional economy where agents are uncertain about the market productivity of workers. Recruiting firms have access to an imperfect screening technology. Upon meeting an applicant, they observe a private signal and decide whether or not to shortlist her. Meetings are multilateral. That is, firms may receive several applications, imperfectly test the candidates and select the most profitable one, if any. In this directed search framework, firms commit to contractual offers to attract applicants and coordination frictions arise. Wages can be set contingent on expected productivity. Firms find it optimal to rank candidates by expected productivity in equilibrium. Therefore, the expected productivity of applicants is endogenous in the second period and all agents learn about a worker’s productivity over time. The combination of incomplete information and firms testing and selecting applicants leads to a twofold intertemporal inefficiency. First, the intensive margin: an additional vacancy in the first period reduces the average productivity of the matches created in the second period. Second, the extensive margin: the marginal vacancy affects the expected number of effective candidates queueing for jobs in the second period. The first externality (or composition effect) has been previously identified by Albrecht, Navarro, and Vroman (2010) and Ch´eron, Hairault, and Langot (2011). These two models are built in a random search framework with wage-bargaining. They find that the so-called Hosios condition does not suffice to attain constrained efficiency as wages are negotiated after learning the charFaig and Jerez (2005) study a directed search model of a retail market in which buyers are privately informed about their preferences. They find that constrained efficiency is not attained unless sellers can charge different prices. 1

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acteristics of the worker at hand, while firms’ entry decision hinges on the workers’ average productivity. Instead, in a directed search setting, wages are posted to attract candidates, and are allowed to be type-contingent. 2 To see that it is the composite effect of informational frictions and selection what leads to an inefficiency outcome, we can shut down one element at a time. First, under complete information, constrained efficiency is attained in equilibrium as has been established by the aforementioned competitive search literature. Second, if the screening technology is not informative and, hence, selection does not take place, then our setting turns out to be equivalent to the complete information economy. The limit case with a perfect screening technology is of particular interest as we show that the equilibrium remains constrained inefficient. The result is fairly intuitive. If candidates’ types are (arbitrarily close to be) perfectly learned through screening, but workers remain ignorant, recruiting firms benefit from the information asymmetry: the market value of workers is lower than optimal, firms post inefficiently low wages, and firms’ entry becomes inefficiently large. Instead, the social planner only cares about the mass of effective workers, and pays no attention to the information asymmetry. Our simulation work suggests that welfare losses increase monotonically with the accurateness of the screening technology, reaching its maximum when this technology is perfect. Contrary to the informational cascade literature, firms do use their private signals and complement them with the publicly available information. Thus, the inefficiency result found in that literature based on the herding behavior of the agents who decide not to use their own information is not present in our setting. See e.g. Bikhchandani, Hirshleifer, and Welch (1992) and Banerjee (1992). Finally, we argue in Section 5 about the empirical grounds of the key assumptions of the model. In particular, Barron, Bishop, and Dunkelberg (1985) and van Ours and Ridder (1992) report that firms interview several applicants for a given vacancy. Furthermore, Macan (2009), among others, provides evidence on the imperfectness of the screening technology of firms. The paper is organized as follows. We set up the model and characterize the equilibrium in Section 2. Constrained efficiency is analyzed in Section 4, where we also discuss the main results and the related literature. Finally, Section 6 concludes. 2

See Section 5 for a further discussion of these papers.

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2

Model

The economy lasts for two periods, t ∈ {1, 2}. A unit mass of unemployed workers are born every period. There is free entry of firms in each period. All agents are risk-neutral and discount future payoffs at factor β ∈ (0, 1). Unemployed workers are identified by a pair (τ, i), with τ denoting their elapsed duration of unemployment. We assume that newborn workers enter the labor market with one period of unemployment, τ ∈ {1, 2}. We refer to workers with unemployment duration τ = 2 in the second period as the long-term unemployed. The index i ∈ {`, h} stands for their market skills. We normalize home productivity to 0. The net market productivity of a type i worker amounts to yi , with yh > y` = 0. A worker is born skilled with probability μ ∈ (0, 1). Let uit (τ ) denote the measure of unemployed workers of type (τ, i) at the beginning of period t. As a result of the Law of Large Numbers, a mass uht (1) = μ of newborn workers are skilled.3 The two-dimensional unemployment distribution ut ≡ (uit (τ ))i,τ is the aggregate state variable in this economy in period t. At the beginning of every period, workers can be either employed or unemployed. The latter seek job opportunities. The employed produce according to their type, and consume their wage. Employment is an absorbing state for simplicity. 4 Each firm is identified with one job. By posting a vacancy, recruiting firms incur a fixed cost k. A vacant firm may receive multiple applications. While unemployment duration of any given candidate is public information, the worker’s productivity is unobservable to both the worker herself and potential employers. Seeking firms imperfectly screen job candidates, however. The screening expenses are included in the cost k. An unskilled applicant passes the test with probability λ` = λ ∈ (0, 1), whereas skilled applicants always succeed at the test, i.e. λh = 1.5 A worker who has been unemployed for τ periods expects to succeed at the test in period t with probability pt (τ ) =

uht (τ ) + λu`t (τ ) . uht (τ ) + u`t (τ )

(1)

We refer to a worker successful at the test as an effective applicant. Neither workers nor other firms have access to the test results and only learn the hiring decision. Recruiters use For notational consistency, ui1 (2) = 0 for i ∈ {`, h}. The model can be extended to allow for firm-initiated separations with no significant difference for our goal. 5 The key assumption is that skilled workers are not only more productive at the market, but also perform better at the screening stage. For simplicity, we assume that the skilled always pass the test. 3 4

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the screening technology to discriminate among candidates. Since unskilled applicants are not profitable, only workers who pass the test are shortlisted. The expected productivity of the effective type τ candidates in period t is y t (τ ) =

yh uht (τ ) + y` λu`t (τ ) , uht (τ ) + λu`t (τ )

(2)

whereas the actual productivity of the selected applicant is instantaneously revealed upon hiring.

2.1

First Period Search and Value Functions.

For expositional reasons, we omit the reference to the duration of unemployment, τ , in the first period. Firms post and fully commit to job offers. A job offer consists of a single wage w1 . There is perfect information about firms’ offers. After observing the distribution of wage offers, workers direct their search. By directed search we mean that agents rationally anticipate that higher wages, and more generally larger employment values, are associated with higher job-filling rates and lower job-finding rates. Formally, we consider the urn-ball matching protocol. Agents may find no partner because of coordination frictions. If workers could coordinate among themselves, there would be no unemployment. It is natural to assume lack of coordination when dealing with a continuum of agents. As a result, some firms receive a number of applications, while others receive none. Each firm posting a job w1 expects q1 (w1 ) effective applicants. To simplify notation we will omit the dependence of q on the offer w1 , unless needed for clarity. The probability of filling the job is η1 (q) = 1 − e−q . Given that the mass of newly employed workers must coincide with the mass of newly filled firms, the job-finding probability conditional on passing the test becomes ν1 (q) = η1q(q) . The actual job-finding probability, hence, is the composite event of passing the test and being selected for the job. That is, h1 (q) = p1 (1)ν1 (q). Unemployed workers apply to job w1 if the expected value derived from w equals their market value U1 , which is defined as the value they can obtain elsewhere. Otherwise, q(w) = 0. That is, U1 and

≥ q1 (w1 ) ≥ 0,

h1 (q1 (w1 )) (w1 (1 + β) − βU2 (2)) + βU2 (2)

(3)

with complementary slackness.

This equilibrium condition, analogous to subgame perfection, is required to pin down rational 5

expectations on queue lengths out of the equilibrium. The expected gains of a firm posting vacancy w1 amount to V1 (w1 ) = −k + η1 (q1 ) (y 1 − w1 ) (1 + β)

2.2

(4)

Second Period Search and Value Functions.

Analogously to the first period, search is directed and an urn-ball matching protocol is assumed. The key difference with respect to period 1 is twofold. First, as meetings are multilateral and applicants are heterogeneous in their unemployment duration, firms can compare and discriminate among them. It can be shown that it is optimal for firms to rank applicants by expected productivity in equilibrium. Since expected productivity falls with unemployment duration because of the mechanic sorting effects, firms discriminate against long-term unemployed workers. We will then assume hereafter that the more productive workers are also the more profitable. 6 Second, firms can commit to a contractual job offer, w2 = {w2 (τ )}τ ≤2 , which may stipulate wages contingent on expected productivity. We do not allow wages to be contingent on ex-ante unobservables such as e.g. the ex-post revealed actual productivity or the number of applications the firm received of each type. We think of those events as unverifiable by a third party and therefore non-enforceable. Each firm posting a job w2 expects q2 (τ, w2 ) effective applicants of duration τ . Let q2 (w2 ) ≡ (q2 (1, w2 ), q2 (2, w2 )). Again, we will omit the reference to w2 unless necessary. The probability of filling a job with a type τ worker is η2 (τ, q2 ) = e−

P

τ 0 <τ

q2 (τ 0 )

1 − e−q2 (τ ) .

The first factor of this expression stands for the probability that no worker with a shorter unemployment spell than a candidate of type τ either applies to the firm or, if applied, performs well in the test. Whereas the second term captures the coordination frictions as in period 1. Notice that this expression captures both the firm’s ranking strategy and the fact that unsuccessful workers at the test are never hired. The actual matching probability for a worker of duration τ is defined as h2 (τ, q2 ) = p2 (τ )ν2 (τ, q2 ), where the conditional probability 2) . Therefore, the exit rate from unemployment has two components. The ν2 (τ, q2 ) = η2q(τ,q 2 (τ ) first factor captures the sorting effects, while the second one introduces the firms’ ranking by duration. 6

See Fern´andez-Blanco and Preugschat (2013) for a formal proof in a more general setting.

6

The counterpart of the equilibrium condition (3) is, for each duration τ , U2 (τ ) ≥ h2 (τ, q2 (w2 ))w2 and q2 (τ, w2 ) ≥ 0, with complementary slackness,

(5)

The value function of the vacant firms posting offer w2 is V2 (w2 ) = −k +

X τ

η2 (τ, q2 (τ, w2 )) (y 2 (τ ) − w2 (τ ))

There are two resource constraints in this second period. First, the law of motion of available labor resources is ui2 (2) = ui1 (1) (1 − λi ν1 (q1 )) , ∀i ∈ {`, h}.

(6)

We are interested in a symmetric allocation, where identical agents make identical decisions. This translates into a single labor market open in this second period. Thus, the mass of vacancies does not depend on the unemployment duration of applicants. It follows that q2 (1) q2 (2) = ` μ + λ(1 − μ) + λu2 (2)

uh2 (2)

3

(7)

Equilibrium.

We now define the equilibrium concept. Definition 1 A directed search symmetric equilibrium consists of the unemployment values U1 , U2 (τ ) ∈ R+ , distributions of unemployed workers ut ∈ [0, 1]2×2 , wages w1 ∈ [0, yh ] and w2 ∈ [0, yh ]2 , and expected queue length functions Q1 (∙) : [0, yh ] → R+ and Q2 (τ, ∙) : [0, yh ]2 → R+ , such that: i) Unemployment values U1 and U2 (τ ) satisfy the functional equations (3) and (5). ii) Firms maximize expected profits at contracts w1 and w2 , and profits are 0 because of free entry. iii) Workers search optimally. That is, conditions (3) and (5) hold. iv) Resource constraints (6) and (7) hold. Conditions (6) and (7) help to characterize the equilibrium allocation by determining the expected queue lengths off-the-equilibrium path by making workers indifferent between the 7

alternative job offer and the equilibrium one. Therefore, firms rationally anticipate the expected number of workers queueing for any given job offer, and then design the contract that maximizes profits. That is, firms maximize profits given workers’ optimal search behavior. Now, we characterize the equilibrium allocation. The optimal decisions of firms are solved out by maximizing profits subject to the optimal search behavior of job-seekers. Given the market value of workers, the firm’s problem in period t becomes (P1 ) maxq1 ,w1 η1 (q1 ) (y 1 − w1 ) (1 + β) s. to q1 U1 ≤ q1 h1 (q1 ) w1 (1 + β) − βU2 (2) + q1 βU2 (2) (P2 ) maxq2 ,w2

X τ

s. to

η2 (τ, q2 (τ )) (y 2 (τ ) − w2 (τ ))

q2 (τ )U2 (τ ) ≤ q2 (τ )h2 (τ, q2 )w2 (τ )

The next proposition establishes the existence of equilibrium as well as a unique solution for the two firms’ problems. The proof is in the Appendix. Proposition 3.1 Given U1 , U2 (∙), problems (P1 ) and (P2 ) have a unique solution. Furthermore, second period firms find it optimal to employ workers of all unemployment durations. There exists a symmetric equilibrium.The equilibrium wages are q1 e−q1 (y (1 + β) − βU2 (2)) + βU2 (2) 1 − e−q1 1 q2 (1)e−q2 (1) −q2 (2) w2 (1) = y (1) − (1 − e )y (2) 2 2 1 − e−q2 (1) q2 (2)e−q2 (2) w2 (2) = y (2) 1 − e−q2 (2) 2

w1 (1 + β) =

(8) (9) (10)

As is usual in search models, first period wages reward a share of the worker’s productivity net of the continuation value of the worker if unemployed. So do second period wages for long-term unemployed. In contrast, the equilibrium second period wage w2 (1) pays the marginal value of the newly unemployed worker as the firm can potentially fill the vacancy with a long-term unemployed worker. In all cases, wages weigh worker’s productivity by the probability of receiving only one application of unemployment duration τ conditioned on receiving at least one from type τ applicants. For the purpose of our welfare analysis, it suffices to derive the private net returns of a vacancy in the first period. To obtain the profits that firms make in equilibrium, we 8

substitute out the equilibrium wages into the value function (4), using expression (5) for the continuation value of unemployment U2 (2). They are 1 (q1 ) 2 ) 2 ∂ν1 (q1 ) y 1 (1 + β) + p2 (2) ∂η∂q2 (2,q q1 ∂q1 βy 2 (2) V1∗ = η1 (q1 )y 1 (1 + β) − k − q1 ∂η∂q 1 2 (2)

(11)

The interpretation of expression (11) is straightforward. The first two terms amount to the expected discounted output net of the vacancy cost. The remaining two terms are the wage bill the firm incurs. The first part of these wage costs stands for the externality the marginal firm creates on the other vacancies in period 1, whereas the second one captures the intertemporal effects on period 2 firms. Free entry implies V1∗ = 0 in equilibrium.

4

Constrained Efficiency

We argue that a benevolent social planner can improve upon the decentralized equilibrium. It can be shown that the second period equilibrium allocation is constrained efficient, conditional on an efficient entry of firms in the first period. This is not surprising because the economy starting in the second period does not differ from Shi (2002) and Shimer (2005) conditional on the first period decisions. Since the inefficiency outcome results from the dynamic externalities it suffices to show that the private and social gains of period 1 firms do not coincide with each other. Having determined the profits of period 1 firms in the previous section, we now turn to the social planner problem. As is standard, given risk neutrality of the workers’ preferences, the goal of the planner is to maximize total output net of recruitment costs. The planner sets the mass of vacancies and the hiring strategies given the heterogeneity in productivity for each period. The planner is subject to the same constraints specified above for the decentralized economy. First, as the planner cannot assign workers to jobs, coordination frictions arise. Second, the planner faces the same incomplete information problem and has access to the same screening technology. Aiming to maximize output, the planner also discriminates against candidates with longer unemployment spells. The planner’s problem is max v1 (η1 (q1 )y 1 (1 + β) − k) + βv2 s. to

vt =

μ+λ(1−μ) , qt (1)

P2

τ =1

η2 (2, q2 )y 2 (τ ) − k

and constraints (6) and (7).

The following result follows. The proof is in the Appendix.

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(12)

Lemma 4.1 The social returns of the marginal vacancy in period 1 are 1 (q1 ) y 1 (1 + β) Vˆ1 = η1 (q1 )y 1 (1 + β) − k − q1 ∂η∂q 1 q2 β ∂η2 (2,q2 ) − q21(1) η2 (2, q2 ) ∂y∂q2 (2) + y (2) 2 ∂q1 1

(13)

The first two terms stand for the expected output net of vacancy costs. The third term is the standard negative externality on other firms in period 1 as they are less likely to fill their vacancies. When comparing to the expression (11) for the private returns, we see that this effect is internalized in equilibrium as it is usually the case in directed search models. The last term of expression (13) captures the reduction in the expected returns of period 2 firms. This intertemporal effect occurs through two channels that correspond to the two addends within this last term. First, the intensive margin: posting one more vacancy affects negatively the composition of the pool of unemployed and reduces the expected returns in period 2. Second, the extensive margin: the marginal firm reduces the mass of effective job-seekers in period 2, making it more difficult to fill jobs in that period. After some simplifications, the difference between the private and social returns of a marginal vacancy becomes V1∗

− Vˆ1 = =

βq12 q2 (1)

η2 (2, q2 ) ∂y∂q2 (2) 1

−βq12 dν1 (q1 ) q2 (1) dq1

+

∂η2 (2,q2 ) y 2 (2) ∂q1

+

2 ) dν1 (q1 ) q2 (1)y 2 (2) ∂η∂q2 (2,q p2 (2) dq1 2 (2)

2) h η2 (2, q2 ) μ(1−μ)(1−λ)λy + q2 (1)y 2 (2) ∂η∂q2 (2,q (uh (2)+λu` (2))2 2 (2) 2

2

μ+λ2 (1−μ) μ+λ(1−μ)

− p2 (2) (14)

The next proposition is based on the fact that this difference is positive as the last term between brackets is positive. Thus, there is excessive entry of firms in equilibrium. Furthermore, notice that the efficient allocation can be implemented through a tax on the entry cost, or equivalently on firms’ profits, equal to the amount in (14). Proposition 4.2 Constrained efficiency is not attained in the market economy. There are too many vacancies in equilibrium. By implementing a tax on firms’ profits or a fee on posting vacancies, the equilibrium allocation becomes constrained efficient.

5

Discussion

To understand the inefficiency result, it is instructive to look at the two limit cases in which unemployment duration becomes worthless information. If λ is either arbitrarily small or close to 1, the efficiency loss associated with the intensive margin is negligible, ∂y∂q2 (2) ∼ 0, 1 and the first term of expression (14) becomes 0. That is, there are no composition effects 10

as either firms almost perfectly detect the skilled applicants, if λ ∼ 0, or the test is barely informative, if λ ∼ 1. In the latter case, i.e. when firms cannot detect unskilled workers, the loss in period 2 output due to the marginal increase of the mass of period 1 vacancies is neutralized by the increase in total output in period 1. As a result, the equilibrium is constrained efficient. Notice that, in this case, the setup does not differ from the standard directed search model. In other words, symmetric incomplete information by itself does not generate inefficiencies. However, if the screening technology is almost perfect, λ ∼ 0, the equilibrium is still not constrained efficient even though there are no efficiency losses on the intensive margin. The inefficiency outcome results from the fact that the intertemporal effects on the extensive margin are not totally captured by the equilibrium wages. When posting vacancies, the planner looks at the expected number of effective units of labor, whereas firms are subject to the market value constraint. The difference between these two economies is captured by the last term of expression (14). If λ is arbitrarily close to 0, the first probability within brackets of this last term is 1, whereas the second probability becomes strictly lower than 1. Put differently, in the directed search framework, competition comes from firms taking the market value of workers as given, which amounts to the continuation value they can obtain elsewhere. If firms perfectly learn the applicant’s type and, hence, unskilled workers are not employable, the unskilled do not alter the applicants’ expected productivity, but do affect the worker’s market value. In contrast, the planner decision is not affected by the unskilled as vacancy creation is determined by the number of effective working units. Thus, firms in the market economy benefit from this asymmetry of information leading to inefficiently low wages and an excessively large entry. Notice that the equivalent setting to the planner’s problem would be a market economy in which there would be only skilled workers. In this alternative world, both the planner and firms in the market economy would behave identically, and the last term of expression (14) would vanish. For intermediate values of λ, the inefficiency result is due to the intertemporal externalities on both the intensive and extensive margins not captured by the equilibrium wages. It is worth investigating the dependence of the magnitude of the inefficiency on the accurateness of the screening technology, λ. As remarked above, there are no efficiency losses through the intensive margin if λ ∼ 0 or λ ∼ 1, whereas the loss is positive for values of λ in between. In contrast, the extensive margin externality and, hence, the asymmetric information inefficiency becomes monotonically stronger the more accurate the test is. As a result, there will be welfare losses for relative high levels of λ because of the inefficiencies resulting from both margins. Theoretically, the welfare effects are ambiguous for medium levels of accurateness

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Percentage difference in welfare

0,3

0,25

0,2

0,15

0,1

0,05

0 0

0.1

0.2

0.3

0.4

0.5

λ

0.6

0.7

0.8

0.9

1

Figure 1: Welfare difference between the planner’s and the market economy as percentage of welfare in the latter for different values of the accurateness of the screening technology.

instead as the two intertemporal effects move in opposite directions. To obtain further insights about the efficiency losses for intermediate values, we simulate data for our model and compute the welfare losses for different levels of accurateness of the screening technology. Figure 1 depicts the expected discounted output difference between the planner’s solution and the equilibrium allocation as a percentage of the latter for different values of the parameter λ.7 We obtain that welfare losses monotonically increase with the accurateness of the screening technology. That is, the inefficiency due to asymmetric information dominates the effects through the intensive margin for medium levels of λ. Moreover, the maximum welfare loss takes place when screening is almost perfect.

5.1

Related literature.

The constrained efficiency result has proved to be robust within the directed search framework. A number of extensions departing from Moen (1997) have obtained efficiency, even in settings where the so-called Hosios condition does not suffice to make the equilibrium constrained efficient in a Diamond-Mortensen-Pissarides (DMP) world. This is the case, for example, of the economy in which firms must invest in capital prior to seeking an employee, as shown by Acemoglu and Shimer (1999). The intuition is that by posting prices, entering firms internalize the externalities on the other agents in the economy. However, a few We take β = 0.95, μ = 0.4, yh = 1, and k = 0.2. A montonically decreasing shape seems robust for other parameter combinations. 7

12

papers have shown that the constrained efficiency results no longer hold in the presence of informational frictions. See for instance Michelacci and Suarez (2006), Guerrieri (2008) and Guerrieri, Shimer, and Wright (2010). This paper shows that inefficiencies result from the combination of informational frictions and selective hiring. A similar inefficiency has been found in the DMP literature. Albrecht, Navarro, and Vroman (2010) allow for endogenous labor force participation, which together with differences in market productivity across workers have effects on the expected productivity. Ch´eron, Hairault, and Langot (2011) analyze a life-cycle model and find an intergenerational externality. The Hosios condition does not suffice to attain constrained efficiency in either case. The intuition is that wages are ex post bargained over upon observing the characteristics of the worker at hand, whereas the firms’ entry decision is based on the expected value of the vacancy. Our understanding is that to the extent that these two inefficiencies are related to the effects on expected productivity of applicants, they are similar to our intensive margin channel. Notice also that Ch´eron, Hairault, and Langot (2011) states that the equilibrium in an age-segmented market is constrained efficient under the Hosios condition. Instead, we show that, in a directed-search setting, the labor market is not segmented and firms prefer to target all workers’ types as they can discriminate among them. Empirical evidence related to assumptions and results. There are two key assumptions in our model: namely, symmetric incomplete information and firms meeting and testing several applicants. Obviously, the latter implies a lack of information regarding the candidates for the job. It may be argued that applicants have better information about their skills than firms prior to matching. We abstract away from this potential adverse selection problem, and assume that firms and workers are equally informed about the market abilities of the latter prior to searching. Arguably, there are several features of real world markets that support this assumption. For example, economic conditions and job requirements vary over time, worker’s value from a firm’s perspective may differ from the worker’s one, and that match-specific components are empirically relevant. 8 A number of empirical studies show that firms interview several applicants for a given vacancy. For example, using the 1980 Employer Opportunity Pilot Project survey, Barron, Bishop, and Dunkelberg (1985) estimate an average of six applicants interviewed per job. van Ours and Ridder (1992) find this number to be above 12 at the first interview using establishment data from the Netherlands. As a result of meeting heterogeneous applicants, firms find it optimal to discriminate against workers with long unemployment spells in pe8

Nagyp´al (2007) concludes that differences in match-specific productivity are significant.

13

riod 2 in equilibrium. This is the case because unemployment duration is informative about the expected productivity of the candidate. Kroft, Lange, and Notowidigdo (2013) report that unemployment duration could be estimated for about 75% of the actual resumes they collected from job boards. Likewise, by conducting field experiments, Oberholzer-Gee (2008) and Kroft, Lange, and Notowidigdo (2013) provide suggestive evidence that firms discriminate against applicants with longer unemployment spells. Finally, imperfectness of applicant screening is widely supported by the human resource management literature. See e.g. Macan (2009), and Posthuma, Morgeson, and Campion (2002) for a summary.

6

Conclusions

This paper shows that the directed search equilibrium is not constrained efficient in a dynamic economy with informational frictions and firms testing and selecting among applicants. The combination of informational frictions and selective hiring creates a twofold intertemporal inefficiency leading to an excessive entry of firms. Efficiency can be attained by properly taxing the entry of firms.

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Appendix

Proof of Proposition 3.1. The proof follows closely Fern´andez-Blanco and Preugschat (2013). We start by looking at problem (P1 ). We use the constraint to replace wages in the objective function. We make (P1 ) a maximization problem with q1 as the only control variable: (P1 ) maxq1

q1 η1 (q1 ) y 1 (1 + β) − βU2 (2) − U1 − βU2 (2) p1

Notice that the new objective function is a continuos, strictly concave function of q1 . Therefore, given market values U1 and U2 (2), there exists a unique solution. Wages are then determined by the constraint. We then obtain expression (8) for wages. Consider now problem (P2 ). Again, we can substitute out wages by using the constraints, and obtain the equivalent maximization problem (P2 ) maxq2 η2 (1, q2 (1))y 2 (1) −

q2 (1) q2 (2) U2 (1) + η2 (2, q2 )y 2 (2) − U2 (2) p2 (1) p2 (2)

Notice that if second period firms prefer not to employ workers of duration τ , the wage offer would be low enough to make the queue length q2 (τ ) = 0. This last expression of problem (P2 ) is consistent with this case and implies that firms would obtain zero profits from applicants of this duration. The Hessian of the objective function is −e−q2 (1) (y (1) − y (2)) + x x 2 2 |H| = , x x where x = −e−q2 (1)−q2 (2) y 2 (2). It is easy to see that the Hessian is definite negative. Thus, there exists a unique solution of problem (P2 ). By using the two constraints, we can derive wages and obtain expressions (9) and (10). It only remains to show that there exists an equilibrium. Let Iτ ≡ [0, ∞]τ . Let us define correspondence T1,τ on the domain Iτ as T1,τ (x) = (qτ , wτ ), where (qτ , wτ ) are the solution of problem (Pτ ), for a given U2 (2) if τ = 1. 16

Likewise, we define the correspondence T2,τ : [0, ∞]2τ ⇒ Iτ as T2,1 ((q1 , w1 )) = h1 (q1 ) w1 (1 + β) − βU2 (2) + βU2 (2), given U2 (2) T2,2 ((qτ 0 , wτ 0 )) = (h2 (1, q2 )w2 (1), h2 (2, q2 )w2 (2))

We proved above that correspondence T1,τ is indeed a function. It can be shown that it is a continuous one, indeed. Likewise, T2,1 and T2,2 clearly are continuous functions. Therefore, the composite correspondence Tτ = T2,τ ◦ T1,τ turns out to be a continuous function defined on a compact set. Brouwer’s Fixed Point Theorem applies and ensures the existence of equilibrium.k Proof of Lemma 4.1. We can write the Lagrangian of the planner’s problem (12) expressing q1 as a function of v1 by using the resource constraint. Notice that the derivative of the Lagrangian with respect to the mass of period 1 vacancies, v1 , corresponds to the social returns of a vacancy. It is easy to see that such returns amount to the value in expression (13).k

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