Competitive Search, Efficiency, and Multi-worker Firms William B. Hawkins∗ January 24, 2012
Abstract I study competitive search equilibrium in an environment where firms operate a decreasing-returns production technology and hire multiple workers simultaneously. Firms post wages, possibly several of them. The equilibrium can feature wage dispersion even though all firms and workers are ex ante identical. Unlike the benchmark where firms hire a single worker, hiring is constrained inefficient. Efficiency requires that firms commit to the number of hires, pay all applicants, or pay wages that depend on the number of applicants. Under wage-posting, the inefficiency is highest at intermediate levels of labor market tightness. JEL: E24, J41, J64 Keywords: competitive search, multi-worker firms, efficiency, wage posting
∗ University of Rochester, RC Box 270156, Rochester, NY 14627-0156,
[email protected]. This paper is a substantial revision of part of my dissertation written at the Massachusetts Institute of Technology; I am grateful to my advisors, Daron Acemoglu and Iv´ an Werning, for helpful and timely suggestions and encouragement. I thank the editor, Jan Eeckhout, and three anonymous referees, for a very insightful set of comments and suggestions. I also thank (in alphabetical order) Mark Aguiar, Herman Bennett, Olivier Blanchard, Claire Bowern, Glenn Ellison, Veronica Guerrieri, Nobuhiro Kiyotaki, Marek Pycia, Casey Rothschild, Robert Shimer, and participants at many seminars for comments.
1
1
Introduction
The study of directed search models of frictional labor markets, following Montgomery (1991), Peters (1991), Moen (1997), and Shimer (1996), has focused on the case where matching between firms and workers is oneto-one: firms can productively employ only a single worker.1 However, in reality most workers are employed by large firms,2 and accordingly there has been increasing recent interest in understanding the role of labor market frictions in accounting for patterns of firm size, growth, and wage payments both in the cross section and over time. In part for reasons of tractability, directed search models have been prominent in this recent literature,3 and it is therefore desirable to understand the microfoundations and assumptions of these models in an environment with production by multi-worker firms. Providing a better understanding of the structure and properties of competitive search equilibrium, especially in the important case where firms post wages, is the aim of this paper. I consider a search-theoretic model of a labor market in which firms operate a decreasing returns to scale production technology. Search is directed: to attract workers, firms post wages—possibly several of them—and commit to pay any worker hired the wage to which the worker applied. At the end of the application process, each firm has some number of successful applicants and must decide how many of them to hire, subject to the commitments it made ex ante through its posted wages. Unlike the more recent applied literature on directed search with multi-worker firm mentioned above, I do not make the unrealistic simplifying assumption that firms’ employment is a continuous variable: instead, employment takes on integer values. There are three main contributions of the paper. First, I establish the existence of a competitive search equilibrium for an arbitrary decreasing returns production function, and I characterize some general properties of the types of wage-posting strategies used by profit-maximizing firms. In the special case in which firms wish to hire at most two workers, I give a complete characterization of equilibrium. A notable feature of the equilibrium I characterize is that, depending on parameter values, a single firm may post multiple different wages each of which receives applications from workers: thus, I show that there can be equilibrium wage dispersion in my model even though all firms and all workers are ex ante identical. I then ask whether the equilibrium is constrained efficient, and I show in general that it is not. This is in direct contrast with the results of Moen (1997) and Shimer (1996), who showed that, when a firm wishes to hire a single worker, wage posting decentralizes the constrained efficient allocation. The reason why is that 1 In studying one-to-one matching, the directed search tradition was consistent with the earlier random search literature (Diamond, 1982; Mortensen, 1982; Pissarides, 1984). 2 For example, the 2002 Economic Census enumerated 5.5 million U.S. firms with a total of 110.8 million paid employees, an average of around 20 employees per firm. 3 Two recent examples are Kaas and Kircher (2011) and Schaal (2010); further references are given in Section 7.
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wages play a dual role. Because search is directed, wages determine where workers apply for jobs; because of decreasing returns to labor in production, wages also determine how many applicants it is profitable for the firm to hire. The same wage contract cannot, in general, guarantee the efficiency of both decisions. In the benchmark single-worker environment, the determination of how many workers to hire was trivial: firms never in equilibrium post a wage greater than the output of the single worker. In the environment of this paper, the posted wage puts a ‘ceiling’ on the number of applicants that the firm will be willing to hire ex post at this wage, and because this ceiling binds with positive probability in equilibrium, the equilibrium is not constrained efficient. The natural next step is to ask whether, if firms could post more general contracts than just wages, the equilibrium would then be constrained efficient. I show that this is indeed the case: if firms can commit to the number of workers they hire as well as to the wage, or pay applicants as well as hired workers, or even simply pay workers according to a wage schedule, efficiency is restored. However, I argue that contracts like these are not observed in practice, perhaps due to considerations of moral hazard. This suggests that the inefficiency I study in this paper may be prevalent in real-world labor markets. The remainder of the paper is structured as follows. Section 2 introduces the model, while Section 3 gives a very simple characterization of constrained efficiency. The heart of the paper is in Section 4, in which I define competitive search equilibrium, establish existence, and prove the characterization results referred to above. The efficiency of equilibrium is the subject of Section 5. In Section 6 I give an extensive discussion of the main assumptions of the model and their role, including the assumptions that firms post wages rather than contracts, that firms post multiple wages, and that production exhibits decreasing returns; and I show how to endogenize the number of firms. Section 7 discusses additional related literature, and Section 8 concludes.
2
Model
There is a continuum of measure µ of risk-neutral, identical workers and a continuum of measure ν of riskneutral, identical firms. Each worker has a distinct ‘name’ i ∈ [0, µ], and similarly each firm has a distinct name j ∈ [0, ν]. There is no necessary relationship between µ and ν. The timing of events within the single period of the model is as follows. 1. Each firm j posts a finite number of wages, wj ≡ (w1,j , w2,j , . . . , wNj ,j ), with w1,j < w2,j < . . . < wNj ,j . 2. Each worker observes all posted wages (and the identity of the firm posting each wage) and submits a single application to some wage offered by some firm.
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3. Each firm receives applicants to each of its wages and chooses how many to hire. 4. Production takes place and payments to workers are made. Unemployed workers earn unemployment income b. The production technology is common to all firms. I assume that y(0) = 0, that y(h) is increasing and strictly concave in h, and that the marginal product of the hth worker is less than the unemployment income for large enough h:
¯ ≡ min {h | y(h + 1) − y(h) < b} < ∞. h
(1)
¯ is also less (Of course, because of the strict concavity of y, the marginal product of the hth worker for h > h than b.) Assumption (1) ensures that there are nontrivial decreasing returns to scale, in the sense that a firm that receives sufficiently many applicants will reduce aggregate output if it hires them all. Note that h takes on integer values: I do not make the simplifying assumption that firm-level employment is a continuous variable.4 The labor market is frictional. I model this by assuming that a firm may only post wage contracts, as already described.5 Specifically, contracts that make wage offers contingent on worker identity are ruled out. Other types of payment structures are also ruled out even if they can be implemented anonymously: for example, paying workers directly for applying is assumed not possible. (I consider generalizations of this type in Section 6.1.) I also assume that hiring decisions may not depend on the identity i of the worker, but only on the wage for which that worker has applied, together with the (realized) number of applications made to each of the wages posted by the firm. In addition, I assume that all workers use the same application strategy.6 As a result, the expected number of applications which arrive at a particular wage wk,j posted by 4 Note also that all workers are assumed perfect substitutes for each other in production: the firm does not care which workers it hires. Thus, it is better to think of a secretarial firm hiring several secretaries or an economics department hiring several search theorists, rather than cases where workers are less substitutable, as when an economics department wishes to hire a secretary and a search theorist, or even a search theorist and an econometrician. 5 What I call a wage contract need not literally be a wage. It is simply any contract whose expected value is the same for firm and worker ex ante and such that any uncertainty regarding this value at the time workers apply for the job is not resolved before the firm makes its hiring decision. For example, in a richer version of the model in which workers need to exert costly effort while employed, what I call a wage contract could incorporate essentially arbitrary provisions on how payments will be structured ex post in order to induce effort. 6 As noted by Shimer (2005), the assumption that firms post only wage contracts and that hiring decisions may not depend on the identity of the worker is a restriction on the strategy space. I make this assumption because I wish to study the properties of equilibria in which firms post wage contracts, not where they post personalized contracts for specific workers. The assumption that workers use the same strategy is an equilibrium refinement. It is standard to justify this assumption in the wage posting literature by observing that other equilibria require much greater coordination among workers, who each need to know their ‘role’ in the game in order to support any asymmetric equilibria. This seems implausible in a large labor market: thus Montgomery (1991, p. 167) argues that this kind of coordination is ‘nearly impossible,’ while Burdett et al. (2001, p. 1066) call it ‘hard to imagine.’ See further Peters (1991) and the discussion in Section VII of Shimer (2005). All these arguments apply with full force in the environment I study in this paper: the fact that firms wish to employ multiple workers does not affect the idea that asymmetric equilibria can be supported only by a degree of pre-play coordination that is unreasonable in a large market.
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firm j will depend only on the set of wages wj , and not on the identity of the firm. Once workers have submitted their applications, those firms who have received at least one applicant to at least one of the wages they posted may choose to hire any or all of these applicants. Any applicant who is hired must be paid the wage to which he applied. However, he need not be hired: this is at the discretion of the firm, which chooses which workers to hire, subject to the wage commitment associated with hiring each. At the conclusion of the hiring process, production takes place, wages are paid, and workers who were not hired receive unemployment income b. I postpone an extensive discussion of the assumptions of the model to Section 6.
3
Constrained efficient allocations
Before defining and characterizing equilibrium, I first study briefly the problem of a social planner who wishes to maximize output in this economy. (Because firms and workers are risk-neutral, this is the same as maximizing a utilitarian social-welfare function with equal weights on all agents.) The planner can instruct workers on where to apply, and firms on which workers to hire. If the planner were unconstrained and able to condition her instructions on the identities of firms and workers, then she could implement the frictionless allocation.7 However, I assume that the planner faces the constraint that she must treat identical workers and identical firms identically in the application process. Because in fact all firms and all workers are identical, this means that the planner can only direct workers to apply at random to some firm, with equal probability across all firms. By a standard abuse of the law of large numbers, each firm therefore receives on average q =
µ ν
applications. Because the realization of
which firm each worker applies to is independent, the probability with which each firm receives m applicants (where m ∈ N ≡ {0, 1, 2 . . .} is
1 m −q . m! q e
Again abusing the law of large numbers, I assume that this is also
the fraction of firms receiving precisely m applications.8 After applications have arrived at firms, the planner directs firms to hire workers until the unemployment income b exceeds the marginal product of labor, y(h) − y(h − 1). That is, she directs a firm that received m ¯ workers, where h ¯ is defined by (1). Output is therefore given, where q = µ/ν, applicants to hire min{m, h} 7 In this environment, one way of implementing the frictionless allocation would be for a measure x = b µ c + ν − µ of firms ν to receive b µ c applications and ν − x to receive b µ c + 1 applications. Each firm then hires the smaller of all its applicants and ν ν ¯ of h defined by (1). 8 Since I assumed for simplicity that there are continua of workers and firms, the argument given in this paragraph is formally not correct: for example, the probability that any individual worker i applies to a particular firm j is zero. A more rigorous way to proceed would be to follow Burdett et al. (2001) and start with the assumption that there are finitely many firms and workers. If µ and ν are positive integers, then each worker applies to each firm with probability ν1 . Therefore the probability qν µ! µ 1 m ν−1 µ−m 1 m that a firm receives precisely m applicants, for m ∈ {0, 1, . . . , µ}, is m = (µ−m)!µ 1 − ν1 where m m! q ν ν 1 m −q q= µ . This converges to m! q e as µ, ν → ∞ with q = µ constant. Similar arguments starting with the finite economy and ν ν passing to the limit apply to the study of competitive search equilibria, and are again omitted for the sake of brevity.
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by ¯ h ∞ m −q X X q m e−q q e q m e−q ¯ + (m − h)b ¯ . =ν + y(h) ν y(m) max [y(h) + (m − h)b] 0≤h≤m m! m! m! ¯ m=0 m=0 ∞ X
(2)
m=h+1
Using the identity that µ/ν = q = q
P∞
1 m−1 −q e m=1 (m−1)! q
=
P∞
m=0
1 m −q m m! q e , this can also be written as
∞ m −q X q m e−q q e ¯ − hb ¯ . µb + ν [y(m) − mb] + y(h) m! m! ¯ m=0
(3)
¯ h X
m=h+1
4
Competitive search equilibrium
In this section I define competitive search equilibrium, characterize equilibrium, and establish that an equilibrium exists.
4.1
Definition
In a competitive search equilibrium, each firm j posts a finite sequence of wages wj = w1,j < w2,j < . . . < wNj ,j ).9 It anticipates that in equilibrium, it will receive an expected number of applicants qk,j to each wage wk,j that depends on wj and on aggregate variables. The most important question to answer in defining competitive search equilibrium is how the queue lengths qj = (q1,j , . . . , qNj ,j ) are determined. To answer this question, I first need to consider the optimal hiring behavior of the firm once applications have been received. Therefore, suppose that firm j receives mk,j ∈ N applicants to each wage wk,j it posts. Write mj = (m1,j , . . . , mNj ,j ). The firm chooses which applicants to hire in order to maximize profits. It is clearly profit-maximizing to hire the applicants who applied to the lowest wages first. The number of workers hired by the firm, denoted h(mj , wj ), is given by the largest value of h such that the hth-lowest wage commitment among the applicants the firm received is less than or equal to the marginal product of the hth worker, y(h) − y(h − 1). It may be helpful to think informally in terms of the intersection of a labor supply curve defined by X
ls (mj , w) =
mk,j
wk,j ≤w
and a labor demand curve derived from the production function as
ld (w) = max{h | y(h) − y(h − 1) ≥ w}. 9 The
notation and form of the definition in this section closely follow Shimer (2005).
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h(mj , wj ) is then given by the value of ls (mj , w) at the largest value of w for which ls (mj , w) ≤ ld (w).10 The resulting profit earned by the firm, denoted π ˜ (mj , wj ), is given by y(h(mj , wj )) less the sum of the lowest h(mj , wj ) wage commitments associated with applicants. Now consider an applicant to a particular wage wk,j . Let h(wk,j ) be the unique value of h such that wk,j ∈ (y(h + 1) − y(h), y(h) − y(h − 1)], so that the firm is willing to hire h(wk,j ) applicants at wages at most wk,j . (It will be convenient to introduce the notation Ih = (max{b, y(h + 1) − y(h)}, y(h) − y(h − 1)].) Let Pk−1 Mk−1,j = n=1 mn,j be the number of applicants to wages less than wk,j . Then if Mk−1,j ≥ h(wk,j ), the firm does not hire any workers who submitted applications to wk,j , since it received enough applicants who demanded lower wages—and who are therefore hired preferentially to those workers who applied to wk,j — that hiring workers who applied to wage wk,j is not profitable. On the other hand, if Mk−1,j +mk,j ≤ h(wk,j ), the firm did not receive many workers who applied to lower wages than wk,j . In this case it is willing to hire all the applicants to wk,j (in addition to all applicants to lower wages). Finally, in the intermediate case where Mk−1,j < h(wk,j ) < Mk−1,j + mk,j , the firm is willing to hire h(wk,j ) − Mk−1,j workers from among the mk,j applicants to wage wk,j . The assumption that the firm’s hiring strategy cannot be conditioned on workers’ identities implies that the firm selects each applicant to wk,j to be hired with equal probability 1 mk,j [h(wk,j )
− Mk−1,j ].
Because the lowest-wage applicants are hired preferentially, it is convenient to define the expected queue of applicants who applied to lower wages than wk,j by
Qk−1,j =
k−1 X
qn,j ,
n=1
with Q1,j = 0. Denote the probability that an applicant to wage wk,j is hired, given that (in expectation) Qk−1,j applicants applied to lower wages and qk,j applicants applied to this wage, by P (wk,j , Qk−1,j , qk,j ). Then the expected value of applying to wage wk,j is given by
(4)
U (wk,j , Qk−1,j , qk,j ) = b + (wk,j − b)P (wk,j , Qk−1,j , qk,j ),
10 I assume that hiring occurs when the firm is indifferent; this is convenient for establishing the existence of equilibrium, and intuitively optimal for the firm, which attracts longer queues to a given wage if it is more likely that applicants to that wage will be hired. This assumption also guarantees that there is a largest w for which ls (mj , w) ≤ ld (w), since it ensures that labor demand, like labor supply, is upper hemicontinuous.
7
where P (wk,j , Qk−1,j , qk,j ) is given by11
P (wk,j , Qk−1,j , qk,j ) h(wk,j )−1
∞ m −qk,j Qnk−1,j e−Qk−1,j X qk,j e = min{m, h(wk,j ) − n} qk,j n=0 n! m! m=1 h(wk,j )−n h(wk,j )−1 m −qk,j X X Qnk−1,j e−Qk−1,j qk,j e 1 h(wk,j ) − n − . [h(wk,j ) − n − m] = qk,j n=0 n! m! m=0
1
(5)
(6)
X
Note that P (wk,j , Qk−1,j , qk,j ) is strictly decreasing in qk,j ∈ [0, ∞) and satisfies P (wk,j , Qk−1,j , 0) = Ph(wk,j )−1 1 n −Qk−1,j and limqk,j →∞ P (wk,j , Qk−1,j , qk,j ) = 0.12 n=0 n! Qk−1,j e It is now straightforward to characterize the queue lengths qk,j that arise in competitive search equilibrium. Following Montgomery (1991) and Acemoglu and Shimer (1999a,b), denote by U the market utility, that is, the expected income a worker would receive from applying to some other firm than firm j. Since firm j is infinitesimal, it chooses its wage offer wj under the belief that its choice has no effect on U . The queue lengths qk,j are determined so that each applicant receives expected income U , that is U (wk,j , Qk−1,j , qk,j ) = U for any wage receiving a positive measure of applicants. More concretely, because applicants to lower wages are hired preferentially by the firm, first determine 11 To understand equations (5) and (6), consider a worker and the firm to which he has applied. The probability this firm 1 −Qk−1,j received precisely n applicants to wages below wk,j is given by n! Qn . The worker can only be hired if n is less k−1,j e 1 m −qk,j than h(wk,j ), in which case the firm is willing to hire up to h(wk,j ) − n applicants to wage wk,j . With probability m! qk,j e (independently of the realization of n) the firm received precisely m additional applicants to wage wk,j (in addition to the worker under consideration). It then hires the lesser of m + 1 and h(wk,j ) − n of the applicants to wk,j , choosing the workers 1 to hire at random from the applicant pool. The worker is therefore hired with probability m+1 min{m + 1, h(wk,j ) − n}. Thus the probability the worker is hired conditional on applying to firm j is h(wk,j )−1
X n=0
∞ X 1 m −qk,j 1 n 1 Qk−1,j e−Qk−1,j min{m + 1, h(wk,j ) − n} qk,j e . n! m + 1 m! m=0
Equation (5) now follows by rewriting ∞ X m=0
∞ 1 m −qk,j 1 X 1 1 min{m + 1, h(wk,j ) − n} qk,j e = min{m + 1, h(wk,j ) − n} q m+1 e−qk,j , m+1 m! qk,j m=0 (m + 1)! k,j
then relabeling the index of summation on the right side. Equation (6) rewrites min{m, h(wk,j ) − n} = h(wk,j ) − n − P 1 m −q max{h(wk,j ) − n − m, 0} and then uses the identity 1 = ∞ to generate an expression in which the sum on the m=0 m! q e right side of (5) contain only finitely many terms. Pk 1 m −q 12 Define φ (q) = k − . To establish the claims about P (·) in the text, it suffices to show that for each k m=0 [k − m] m! q e 1 k ≥ 1, q φk (·) is strictly decreasing in q ∈ [0, ∞) with limq→0 1q φk (q) = 1 and limq→∞ 1q φk (q) = 0. The last of these is immediate P 1 m −q since φk (q) ≤ k. To establish the other two, first observe that φk (q) = ∞ is strictly positive for q > 0 m=1 min{m, k} m! q e P 1 m −q and satisfies φk (0) = 0. Differentiating the definition of φk (q) shows after extensive cancelation that φ0k (q) = k−1 m=0 m! q e 1 1 00 k−1 −q and φk (q) = − (k−1)! q e . To see that q 7→ q φk (·) is strictly decreasing, observe that the derivative of this expression is 1 [qφ0k (q) − φk (q)]. q2
This is strictly negative for q > 0 if qφ0k (q) − φk (q) < 0 for q > 0. At q = 0, this expression evaluates to be zero, and its derivative is qφ00 (q), which is strictly negative for q > 0; this establishes the claim. Finally, apply L’Hˆ opital’s rule to calculate that limq→0 1q φk (q) = limq→0 φ0k (q) = 1.
8
q1,j so that
U = b + (w1,j − b)
(7)
∞ 1 X
q1,j
min{m, h(w1,j )}
m=1
1 m −q1,j ; q e m! 1,j
(The right side of this equation is the specialization of (4) and (5) to the case where k = 1 and accordingly Q1,j = 0.) Alternatively, if U ≥ b + w1,j − b (or more simply, if w1,j ≤ U ), workers do not apply to wage w1,j so that q1,j = 0. This is because applying elsewhere gives higher expected income even if the worker gets the job for sure. Next determine qk,j for k > 1 inductively: knowing Qk−1,j , determine qk,j so that
U = b + (wk,j − b)
(8)
1
h(wk,j )−1
qk,j
∞ X 1 m −qk,j 1 n −Qk−1,j min{m, h(wk,j ) − n} qk,j Q e e . n! k−1,j m! m=1
X n=0
1 Alternatively, if U ≥ b + (wk,j − b) qk,j
Ph(wk,j )−1 n=0
1 n −Qk−1,j , n! Qk−1,j e
so that the expected income from
applying to wk,j is less than U even if no other workers apply to this wage at this firm, then set qk,j = 0. Note that in either case, qk,j is the unique13 nonnegative solution to h(wk,j )−1
qk,j (U − b) = (wk,j − b)
(9)
X n=0
∞ X 1 n 1 m −qk,j Qk−1,j e−Qk−1,j min{m, h(wk,j ) − n} qk,j e . n! m! m=1
Finally, to proceed to the next step, calculate Qk,j = Qk−1,j + qk,j . Equation (9) can be interpreted as saying that in order to attract a queue of length qk,j to wage wk,j , firm j must compensate each hired worker for their opportunity cost of employment, b, and in addition, must pay an expected total additional wage bill to the applicants to this wage of qk,j (U − b). That is, the ‘price’ of a unit queue length is U − b. The fact that this ‘price’ is independent of the identity of the firm and of the posted wage vector is the sense in which search is competitive. A more compact notation for equation (9) is given by using (5) to write
(10)
U − b ≥ (wk,j − b)P (wk,j , Qk−1,j , qk,j )
and
qk,j ≥ 0
with complementary slackness. Denote by q(w, U ) the sequence of queue lengths q determined according to (7), (8), and (9) for a given wage tuple w and a given market utility U . Conditional on the queue lengths, the expected profit of firm j, denoted π(wj , U ), is given by summing 13 Uniqueness
follows because q 7→ P (wk,j , Qk−1,j , q) is a strictly decreasing function, as established in footnote 12.
9
the profits made when the realized vector of applicants is mj , weighting by the probability of the realization P QNj mk,j −qk,j 1 π ˜ (mj , wj ). To write this more compactly, recall mj , that is, π(wj , U ) = k=1 mk,j ! qk,j e mj ∈NNj that a firm hires an hth worker if and only if it receives at least h applicants to wages no higher than ˜h = P y(h) − y(h − 1). Accordingly, if for each h I define Q k | wk,j ≤y(h)−y(h−1) qk,j , then the probability the Ph−1 1 ˜ m −Q˜ h firm hires an hth worker is 1 − m=0 m! Qh e , so that total expected output of the firm is ¯ h X
(11)
h=1
"
# 1 ˜ m −Q˜ h 1− Q e (y(h) − y(h − 1)). m! h m=0 h−1 X
Next use (10) to write the expected payment made by firm j to an applicant to wage wk,j as
wk,j P (wk,j , Qk−1,j , qk,j ) = U − b + bP (wk,j , Qk−1,j , qk,j ).
Multiplying by qk,j gives the total wage bill for applicants to wk,j , since total expected hires among such applicants is qk,j P (wk,j , Qk−1,j , qk,j ). Summing over all its posted wages then gives the firm’s total expected PNj wage bill. The resulting expression can be simplified using the observation that k=1 qk,j P (wk,j , Qk−1,j , qk,j ) Ph¯ h Ph−1 1 ˜ m −Q˜ h i . It follows is just the firm’s expected employment level, which also equals h=1 1 − m=0 m! Qh e that the firm’s total expected wage bill can be written
(U − b)
(12)
Nj X k=1
qk,j + b
¯ h X
"
h=1
# 1 ˜ m −Q˜ h 1− Q e . m! h m=0 h−1 X
Combining (11) and (12) establishes that the expected profit of the firm is
(13)
π(wj , U ) =
¯ h X
"
h=1
# Nj X 1 ˜ m −Q˜ h 1− qk,j . Q e (y(h) − y(h − 1) − b) − (U − b) m! h m=0 h−1 X
k=1
A competitive search equilibrium is a set W of increasing tuples of wages w = (w1 < w2 < . . . < wN ) (possibly of different lengths), a measure φ on W, and a market utility U such that (i) conditional on U , if w ∈ W, then w maximizes π(w, U ); and (ii) the measure φ and the queue lengths satisfy the resource constraints Z
X
q dφ(w) = µ
and
φ(W) = ν.
W q∈q(w,U )
4.2
Existence
In this section I prove the existence of a competitive search equilibrium. I first establish a preliminary lemma that allows me to reduce the dimensionality of the wage vectors that
10
I need to consider. Up to now, I have allowed for firms to post wage vectors w of arbitrary finite length. However, it turns out that a firm cannot gain from posting a wage vector that contains more than one wage in an interval of the form Ih = (max{b, y(h + 1) − y(h)}, y(h) − y(h − 1)]. This is the subject of the following Lemma.14 Lemma 1. Suppose that w and q satisfy (9), and that there are k and h such that
y(h + 1) − y(h) < wk < wk+1 ≤ y(h) − y(h − 1). Then there exists w0 ∈ [wk , wk+1 ] such that w0 = (w1 , . . . , wk−1 , w0 , wk+2 , . . . , wN ) and q0 = (q1 , . . . , qk−1 , qk + qk+1 , qk+2 , . . . , qN ) also satisfy (9), and such that the firm’s profit is unchanged. The proof of this result is straightforward (the formal proof, as all subsequent formal proofs, is in the Appendix). If a firm posts multiple wages in some Ih but one of them attracts no applicants, then the wage that attracts no applicants can be dropped. If two wages both attract positive queues, then the presence of the lower wage wk (whose applicants are hired preferentially to those applicants who applied to wk+1 ) makes the higher wage wk+1 less attractive and reduces the queue length there. If the firm instead posted a wage vector that dropped wk+1 , then wk would still attract the same number of applicants as before, qk ; on the other hand, if the firm posted a wage vector that dropped wk , then the number of applicants to wk+1 can be shown to exceed qk + qk+1 . By continuity I can find a wage w0 ∈ (wk , wk+1 ) that generates queue length qk + qk+1 when posted in place of both wk and wk+1 . Because (9) says that the expected wage bill paid by a firm depends only on the total of all its queue lengths, this costs the same in wage payments as before; because w0 , like wk and wk+1 , lies in the interval Ih , expected hiring and expected profits are unchanged also. The construction in Lemma 1 establishes not only that profits are unchanged when the firm posts w0 in place of w, but also that the total amount of applicants attracted by the firm is unchanged when it makes this change. Therefore, if there is an equilibrium where a positive measure of firms post w, then there is also an equilibrium where the same measure of firms post w0 (with no change to the posted wage tuples of other firms). Repeated application of this argument implies that if an equilibrium exists, then there is an equilibrium in which no posted wage vector w contains no more than one wage in each interval Ih . Moreover, allowing for multiple wages to be posted in such an interval does not affect either firm or worker payoffs. In addition, if a firm posts a wage tuple w which contains a wage w which receives no applicants, then 14 It
is worth noting that (a simplified version of) the proof of Lemma 1 also applies in the case where firms wish to hire only a single worker, as in Moen (1997) and Shimer (1996). Thus, in the standard competitive search model, firms cannot gain from posting more than one wage, even in the absence of any commitment to hire. This observation is new to the literature to my knowledge, although it is not surprising in view of the constrained efficiency of equilibrium in the single-worker case.
11
its profits and queue lengths are unchanged from those that it would receive if it posted the tuple obtained by deleting w from w. Therefore it is essentially without loss of generality to restrict attention to the class of equilibria in which each firm posts no more than one wage in each interval Ih , and in which each posted wage attracts a positive queue length. I will therefore study only equilibria with these two properties, which ¯ (1) guarantees that y(h) − y(h − 1) < b. Because I call simple equilibria, from now on. Note that for h > h, workers will not apply to wages below b, Lemma 1 ensures that in a simple equilibrium firms post at most ¯ wages, one in each interval of the form Ih for h = 1, 2, . . . , h. ¯ h I can now establish the existence of a (simple) equilibrium. Proposition 1. A competitive search equilibrium exists. The proof consists of two main steps. I first show that for any U , there exists a (possibly non-unique) maximizer of π(w, U ). This is non-trivial because the intervals Ih are not closed. However, it is possible to show that it is not profit-maximizing for a firm to post a wage just above the left limit of such an interval (intuitively because decreasing the wage slightly would then allow the firm to hire an additional worker cheaply when it gets enough applicants to do so). The remainder of the proof then follows from an application of the Theorem of the Maximum. One result needed in the proof of Proposition 1 is interesting enough to record it as the following Lemma. ¯
Lemma 2. Define a function Y : [0, ∞)h by
Y (q1 , . . . , qh¯ ) =
¯ h X h=1
where for each h, Qh =
Ph−1
n=1 qn .
"
# −Qh Qm h e 1− (y(h) − y(h − 1) − b) m! m=0 h−1 X
Suppose that (q1 , . . . , qh¯ ) and (q10 , . . . , qh¯0 ) are such that Qh ≤ Q0h for each
h, with Qh¯ = Q0h¯ . Then Y (q1 , . . . , qh¯ ) ≤ Y (q10 , . . . , qh¯0 ). If in addition there is h such that Qh < Q0h , then Y (q1 , . . . , qh¯ ) < Y (q10 , . . . , qh¯0 ). The value Y (q1 , . . . , qh¯ ) defined in Lemma 2 is the expected value of the surplus generated by a firm ¯ − 1; that has attracted applicant queues of length qi to some wage in the interval Ih−i for each i = 0, . . . , h ¯ Ph¯ according to (13), it differs from the profit by (U − b) k=1 qk . The Lemma claims that this expected value is higher if for each k there are longer queues at wages in the intervals Ik ∪ · · · ∪ Ih¯ . Intuitively this is clear since applicants who queue at lower wages are more valuable to the firm, because more of them can be hired.
4.3
Characterization
I now turn to a partial characterization of competitive search equilibrium. A complete characterization in general is not available; the structure of the optimal wage-posting strategy of a firm can be quite complex. 12
However, I am able to prove one general result, and to characterize equilibrium completely in the case where firms wish to hire only up to two workers. Lemma 2 indicates that, conditional on the total measure of applicants attracted to all its posted wages, a firm would like to attract as many of these applicants as possible to wages that are as low as possible. A natural conjecture for the optimal wage-posting strategy of a firm, then, might be that if U is sufficiently low, the firm only posts wages in the interval Ih¯ ; by Lemma 1, it is enough for the firm to post a single ¯ − y(h ¯ − 1), the firm might not be able to attract wage in this interval. If U is higher, but still below y(h) enough applicants to any wage in this interval. One might then conjecture that the firm would post a pair ¯ − y(h ¯ − 1) together with some w ∈ I¯ . This would intuitively allow the firm to attract as of wages, y(h) h−1 many applicants as possible with a wage that allows for the most efficient possible ex post hiring, and then some additional applicants with the higher wage w. Then, if U was higher still, the firm might post a triple ¯ − y(h ¯ − 1), y(h ¯ − 1) − y(h ¯ − 2), w), with w ∈ I¯ , and so on. (Of course, wages weakly less than U (y(h) h−2 attract no applicants, so the firm could drop any wages in that range.) This intuition is tempting but wrong. The reason why the intuition mentioned above is wrong is that it ignores the effect one wage the firm posts has on the queue lengths attracted to other wages the firm posts. If the firm changes a low wage that it posts, this affects the queue length at this low wage, and therefore the queue length at higher wages that the firm posts (because applicants to higher wages are only hired if there are not sufficient applicants to lower wages). This interaction is in general quite complicated, and means that I cannot give a general characterization of a firm’s optimal wage posting strategy. However, the intuition of Lemma 2 does allow me to prove that two successive wages in an optimal wage tuple cannot both lie in the interior of intervals of the form Ih . Proposition 2. Let w = (w1 , . . . , wN ) be a wage tuple. Suppose that for some k < N and some h, h0 , wk ∈ Ih \{y(h)−y(h−1)} and wk+1 ∈ Ih0 \{y(h0 )−y(h0 −1)}. Suppose that the queue lengths qk and qk+1 are both strictly positive. Then w is not consistent with firm profit maximization (that is, w ∈ / arg max π(·, U )). The proof of Proposition 2 is simple and in the spirit of the preceding heuristic argument. If the firm raises wk , it follows from (9) that qk rises. However, qk+1 falls, as the additional applicants to wk make it less likely that an applicant to wk+1 will be employed. If the firm additionally raises wk+1 so as to keep the total queue attracted by the two wages (that is, qk + qk+1 ) constant, then it follows immediately from Lemma 2 that profits strictly increase. Proposition 2 establishes that any wage tuple offered in equilibrium (which must maximize π(w, U )) does not contain any successive wages both of which are in the interior of intervals of the form Ih . In the case ¯ ≤ 2, so that the firm wishes to hire only at most two workers because y(3) − y(2) < b, this allows a when h
13
sharp characterization of all wage posting strategies than can arise in a simple equilibrium. Proposition 3. Suppose that y(3) − y(2) < b. Denote the market utility by U . There are U1∗ ≤ U2∗ < y(2) − y(1) such that in any simple equilibrium 1. if U < U1∗ , then each firm posts a single wage w ∈ (U, y(2) − y(1)); 2. if U ∈ [U1∗ , U2∗ ] then each firm posts the single wage w = y(2) − y(1); 3. if U ∈ (U2∗ , y(2) − y(1)) then each firm posts a wage pair of the form (y(2) − y(1), w) with w ∈ (y(2) − y(1), y(1)); and 4. if U ≥ y(2) − y(1), then each firm posts a single wage w ∈ (U, y(1)). All simple equilibria are symmetric, in the sense that each firm posts the same wage tuple. In the special case where firms wish to hire at most two workers, the result of Proposition 3 is strictly stronger than that of Proposition 2, and gives a complete typology of all simple equilibria. To prove the Proposition, I first use Proposition 2 to restrict the possible wage tuples consistent with profit maximization to three possible cases: either the firm posts a single wage, a pair of wages of the form (y(2) − y(1), w) for w ∈ (y(2) − y(1), y(1)), or a pair of wages of the form (w, y(1)) with w ∈ (b, y(2) − y(1)]. The last case can be disregarded since posting a wage of y(1) does not earn the firm any profits. In the remaining cases, I can explicitly write the problem of the firm conditional on which form of wage tuple it posts. For example, if the firm posts a wage tuple of the form w = (y(2) − y(1), w) with w ∈ (y(2) − y(1), y(1)], its profit is given by
(14)
π(w, U ) = (1 − e−q1 )[y(1) − (y(2) − y(1))] + e−q1 (1 − e−q2 )[y(1) − w],
where q1 and q2 are defined implicitly by the specializations of (9) to this environment:
(15)
q1 (U − b) = (y(2) − y(1) − b)[2 − 2e−q1 − q1 e−q1 ]
and
q2 (U − b) = (w − b)e−q1 [1 − e−q2 ].
Equation (14) is just the specialization of (13) to this case. A firm makes profits only if it receives at least one applicant to wage y(2) − y(1), which occurs with probability 1 − e−q1 , or if it receives no applicants to that wage but does receive at least one applicant to the higher wage w, which occurs with probability e−q1 (1 − e−q2 ). (A second applicant to wage y(2) − y(1) will be hired if present but generates no additional profit.) Substituting from the constraints to eliminate the wages and solving for the optimal queue lengths allows the optimal profit from posting this type of wage tuple to be calculated. Repeating this argument
14
Wage tuple
b+1 b+0.5
Total queue length
b
6 3 0
Profit
1.5 1 0.5 0
b
b+0.25
y(2)−y(1) Market utility, U
b+0.75
b+1
Figure 1: Equilibrium in a specific example for different values of the market utility U in the simpler case where the firm posts a singleton wage and comparing the answers allows the firm’s profit-maximizing wage-posting strategy to be determined. It may be helpful in understanding the structure of equilibrium to refer to Figure 1. Holding fixed a particular example production function and the value of unemployment income b, the figure traces out how wages, queue lengths, and profits vary with labor market tightness (proxied by the market utility). The case shown arises if, for some b > 0, y(1) = b + 1, y(2) = 2b + 1.5, and y(h) − y(h − 1) < b for h ≥ 3. The top panel shows the equilibrium wage tuple as a function of the market utility; the middle panel shows the queue length at each posted wage; and the last panel shows firm profit at this allocation. The three vertical red lines in each panel, from left to right, are U = U1∗ , U = U2∗ , and U = y(2)−y(1) = b+0.5. The horizontal red line in the top panel is w = y(2) − y(1). The four cases described in Proposition 3 can be seen, in particular the case when U ∈ (U2∗ , y(2) − y(1)) for which the firm optimally posts a pair of wages rather than just one, as when U takes any other value. As already noted, the variable on the horizontal axis in the Figure is U , the market utility, rather than the more primitive µ/ν, the ratio of firms to workers. However, using the market clearing condition allows the equilibrium for a particular (µ, ν) pair to be determined. It can be found graphically by using the second panel to locate the value of U that leads to a total queue length per firm of µ/ν; the rest of the equilibrium can then be read off the top and bottom panels. In this case the equilibrium is unique since total queue length is strictly decreasing in U .
15
A noteworthy feature of equilibrium is that even though all firms and all workers are identical, there can be wage dispersion. In particular, it is clear from Proposition 3 and from Figure 1 that even when firms wish to hire only two workers, there can already be two wages posted. The reason for wage dispersion arises from the tradeoff the firm faces between the two roles of the wage: a high wage attracts many workers conditional on the firm’s hiring behavior, but simultaneously restricts the firm’s hiring behavior. If a low wage does not attract enough workers, the firm may post additional wages in order to attract additional applicants, even though hiring these additional applicants is more expensive. This type of wage dispersion should be distinguished from existing models of wage dispersion, which rely on ex ante heterogeneity (Albrecht and Axell, 1984), on-the-job search (Burdett and Mortensen, 1998), or the fact that some workers receive multiple offers (Burdett and Judd, 1983). The source of heterogeneity is somewhat reminiscent of that in Gaumont et al. (2006), in the sense that there is ex post heterogeneity (a worker is ‘more productive’ if there are few other applicants to the firm to which she applied), but the details of the two models are quite different. Beyond the case where firms wish to hire at most two workers, closed-form characterization of equilibrium becomes intractable, because there are many possible cases to check and Proposition 2 does not reduce this set sufficiently to enable general results to be proved. (For any specific production function, however, it is straightforward but time-consuming to check all possible cases, the same way as is done in the proof of Proposition 3.) The one thing worth observing is that optimal wage-posting strategies can be quite complex. For example, even if y(4) − y(3) < b, so that firms wish to hire at most three workers, it is possible for the optimal wage-posting strategy to require posting a wage triple (w1 , w2 , w3 ), and even for two elements of this triple to be ‘interior’ to the intervals Ih containing each. (That is, w1 ∈ (b, y(3) − y(2)), w2 = y(2) − y(1), and w3 ∈ (y(2) − y(1), y(1)).)15 Many other possible configurations are also possible. Therefore, rather than proceed further with characterizing equilibria in general, I now turn to studying the efficiency of equilibrium.
5
Efficiency
This section asks whether competitive search equilibria are constrained efficient. As foreshadowed in the Introduction, in general, they are not. This is in stark contrast to the case when, as in Moen (1997) and Shimer (1996), the firm wishes to employ only one worker. In addition to this general inefficiency result, I prove two additional results. First, the equilibrium is constrained efficient in a weaker sense, that if the planner is constrained to have firms post wage tuples and then make hiring decisions that are privately 15 An example of parameters for which the equilibrium wage tuple is of this form is given by taking y(1) = 2, y(2) = 3.0045, y(3) = 4.00553, y(m) < m + 1 for m ≥ 4, µ = 6.90776, ν = 1, and b = 1. In this case numerical optimization shows that U = 1.001 and the optimal wage tuple is (w1 , w2 , w3 ) = (1.00102, 1.0045, 1.56809), attracting queues (q1 , q2 , q3 ) = (0.99988, 4.86649, 1.04139). Details of the calculation are available on request. Note that the fact that three wages can be posted in equilibrium underlines that the mechanism that generates wage dispersion in this paper is separate from that in Gaumont et al. (2006), where a ‘law of two wages’ holds.
16
optimal to the firm once applicants have arrived, then she would not choose to have firms post different wage tuples than they do in the decentralized equilibrium. Second, I show that the inefficiency associated with wage-posting is hump-shaped: in markets that are very imbalanced, in the sense that either µ ν or ν µ, the welfare in a competitive search equilibrium is close to that in the constrained efficient allocation. The main result of the Section is Proposition 4, which establishes that the equilibrium is constrained inefficient for a large set of parameter values. Proposition 4. A simple equilibrium is constrained inefficient if and only if a positive measure of firms ¯ − y(h ¯ − 1)]. post wage tuples wj that do not take the form of a single wage w ∈ [b, y(h) ¯ − y(h ¯ − 1), then The proof of the Proposition is straightforward. If a firm posts a wage greater than y(h) ¯ − 1 applicants to this wage. With positive probability, the firm receives it will be willing to hire at most h more applicants than this to this particular wage (and no applicants to any of its other posted wages, if any). ¯ workers, but this does not occur because w In this situation the planner would prefer that the firm hire h ¯ − y(h ¯ − 1). Conversely, if all firms post a single wage no greater than y(h) ¯ − y(h ¯ − 1), then the exceeds y(h) unique equilibrium is symmetric and exhibits efficient hiring, which are sufficient conditions for efficiency.16 Proposition 4 gives a necessary and sufficient condition for the equilibrium to be constrained efficient, but the condition is expressed in terms of an endogenous variable, the posted wage tuple. Therefore, it is useful to give a sufficient condition on the primitives of the model to ensure that inefficiency arises. Intuitively, if workers are scarce relative to firms, then the market utility will be high, which ensures that wages offered in equilibrium (which must be no less than the market utility) will also be high. This guarantees that the ¯ > 1, or equivalently, that y(2) − y(1) > b. I record this conditions of Proposition 4 obtain, provided that h result as the following Proposition. Proposition 5. Suppose that y(2) − y(1) > b. Then the equilibrium is constrained inefficient if µ/ν is sufficiently small. It may be surprising that efficiency does not always obtain, in view of the common intuition that competitive search equilibrium usually generates constrained efficient allocations. The usual intuition for efficiency is that under directed search workers’ applications are in effect bought by firms in a competitive market. This competitive pricing mechanism also operates in my model: firms understand that an additional worker can be attracted to queue only by posting wages high enough to guarantee her an expected utility U . However, competitive pricing is not sufficient to decentralize the efficient allocation: firms must also make the ‘correct’ hiring decisions. In the standard model where firms wish to hire only one worker each, this condition is 16 If the model is augmented by allowing for free entry by firms, the observation that all firms post a single wage no greater ¯ − y(h ¯ − 1) is no longer sufficient for efficiency. See Section 6.3 below. than y(h)
17
trivially satisfied whenever the wage is less than the value of the product of a matched firm-worker pair, which is always true if it is efficient to have any active firms. When firms must also choose the number of workers to hire from their applicant queues, an additional margin is introduced. If firms can commit only to wages paid conditional on hiring, then once applications have been made, firms prefer not to hire those workers whose marginal product exceeds the wage to which they applied. If any posted wage exceeding ¯ − y(h ¯ − 1) attracts applicants, then the equilibrium will not be constrained efficient. y(h) The inefficiency studied here arises because the number of workers arriving at a firm is stochastic. Models of multi-worker firms in the directed search tradition frequently assume that firm-level employment is a continuous variable.17 The results of this section show that this assumption is not innocuous. Another way to think about the inefficiency result is that the firm’s ability to commit only to a wage tuple w represents a constraint on its technology for transforming applications into output. I have shown that in general, this constraint binds and does not allow for efficient hiring ex post. At any wage greater ¯ − y(h ¯ − 1) which is posted as part of an equilibrium wage tuple and which attracts a positive thatn y(h) queue length, there are some realized applicant numbers m for which, if m workers arrive at a firm, then the firm with posted wage w will choose ex post to hire some number of them different from the number which maximizes output, y(h) + (m − h)b. This intuition suggests that if firms can commit to contracts that do not take the form simply of commitment to a posted wage, efficiency will be restored. I investigate this further in Section 6.1. However, I first prove some additional results about the inefficiency associated with the equilibrium with wage posting. Propositions 4 and 5 show that for a large set of parameters, no competitive search equilibrium is constrained efficient. However, they are not informative about the circumstances under which the inefficiency is large. It turns out that the inefficiency is hump-shaped as a function of the market tightness, that is, the ratio of firms to workers (or equivalently, in terms of the ratio of workers to firms). When the market is very imbalanced, in the sense that there are many more workers than firms or vice versa, then the difference in social welfare (that is, total output) between the constrained efficient allocation and the corresponding quantity in any equilibrium becomes small. To make this statement more formally, denote by Y ∗ (µ, ν) the value of output in the constrained efficient allocation, as in (2). Denote by Y e (µ, ν) the value of total output in equilibrium (or, more formally, since the equilibrium is in general not unique, the infimum of the value of 17 Firm-level
employment is a continuous variable in Kaas and Kircher (2011) and in Schaal (2010), for example.
18
output over all equilibria). Next, define the welfare cost of the equilibrium inefficiency by18
(16)
∆(µ, ν) =
[Y ∗ (µ, ν) − µb] − [Y e (µ, ν) − µb] Y ∗ (µ, ν) − Y e (µ, ν) = . ∗ Y (µ, ν) − µb Y ∗ (µ, ν) − µb
With this notation, I can state the following Proposition. Proposition 6. Fix µ > 0. (a) ∆(µ, ν) → 0 as ν → ∞. (b) There is ν¯ > 0 such that if ν ≤ ν¯, then ∆(µ, ν) = 0. Note that an immediate corollary of Propositions 5 and 6 is that the welfare cost of inefficiency is maximized at some interior value of µ/ν. The intuition for Proposition 6 is twofold. First, in part (a) of the Proposition, there are many more firms than applicants; this implies that the market utility must be large as well. For ν sufficiently large, it rises above y(2) − y(1). In this case, the inefficiency associated with the equilibrium is easy to understand: all firms which receive any applicants hire only one of them. However, a property of urn-ball matching is that the fraction of firms who receive more than 1 applicant, among those that receive any, becomes small as the number of applicants per firm falls. In this case the welfare cost of inefficiency vanishes. In part (b) of the Proposition there are many more applicants than firms. In this case the market utility falls to its lower ¯ − y(h ¯ − 1), and bound, b. The profit-maximizing strategy of firms is to post a single wage w less than y(h) in this case the inefficiency studied in this section does not arise: the equilibrium is constrained efficient. Proposition 6 implies that the inefficiency associated with wage posting is likely to be small in dynamic versions of the model. There are two reasons for this. First, if hiring proceeds sequentially, so that firms only ever attempt to hire a single worker at a time, then wage posting is constrained efficient for the same reasons as in the benchmark model (Moen, 1997; Shimer, 1996). Second, if firms hire multiple workers at the same time, then the market may tend to a situation where no agents on the short side of the market are unmatched. This is exactly the situation described by Proposition 6 of an very unbalanced market. This intuition can be made precise: the interested reader is referred to Hawkins (2006) for details. Finally, note that the equilibrium is constrained efficient in the more constrained problem in which the planner must post wage tuples rather than choosing hiring directly. More precisely, suppose the only 18 Note that ∆(µ, ν) is homogeneous of degree 0 in (µ, ν) and so depends only on the ratio µ/ν. This is because any equilibrium of an economy where the measure of workers is µ and the measure of firms is ν corresponds in the obvious way to an equilibrium of an economy which is identical except that these measures are replaced with λµ and λν for some λ > 0. There are other possible definitions of the welfare cost, given by replacing the denominator of (16) with either Y ∗ (µ, ν) or with the measure of agents µ + ν, the measure of workers µ, or the measure of firms ν. Analogous results to Proposition 6 hold for these definitions of the welfare cost also, but the result stated in Proposition 6 is strictly stronger.
19
instrument available to the planner is to alter the wage tuples posted by firms, taking as given that each firm will determine how many workers to hire privately optimally according to its posted wages and the realizations of the number of applicants to each wage. It is straightforward to verify that the planner’s solution is a competitive search equilibrium (the market utility is given, as usual, by the shadow value of an additional worker to the planner).19
6
Discussion
In this section, I discuss the role played by some of the key assumptions of the model. I first focus on the assumption that firms can post only multiple wages; I consider allowing firms to post richer contracts, or to post only a single wage. Next, I consider in turn the roles played by the assumptions that the number of firms and workers are fixed and that there are strictly decreasing returns to labor in production. Finally, I compare my results to other work which allows for a different concept of vacancies.
6.1
Posting Contracts rather than Wages
In Section 5 I showed that competitive search equilibrium under wage posting is generally inefficient. It is natural to ask whether in alternative, richer contracting environments an efficient competitive search equilibrium might exist. In this section I show this is the case. Intuitively, the inefficiency under wage posting arises precisely because a firm’s posted wage is a single instrument that plays the dual role both of attracting applicants and of determining hiring from among those applicants. If the firm can commit to a richer contract that allows for these aspects not to be in conflict, then the equilibrium can be efficient. I will restrict myself for concreteness to the case of contracts that can be written in the form of a pair (H, W ). H : N → 2N denotes the hiring commitment of the firm: for each m, H(m) ⊆ {0, 1, . . . , m} indicates a commitment by the firm that if it receives m applicants, it will hire some number h ∈ H(m) of these. W : N2 → [0, ∞) denotes the payment commitment of the firm: if it receives m applicants and hires h of these, then it will make total payment W (m, h) to all applicants. Provided that each hired worker receives at least a payment of b, the distribution of W (m, h) across workers does not matter since all agents are risk neutral. In this notation, posting a wage of w as in the benchmark model corresponds to setting H(m) = {0, 1, . . . , m} for each m and W (m, h) = wh. Some examples of contracts of this form include the following. 19 This result is reminiscent of Moen and Ros´ en (2004). In their paper workers and firms cannot overcome a coordination problem, but the competitive search equilibrium is still constrained efficient provided the planner takes this coordination problem as technological. Note also that because I have not shown that the competitive search equilibrium is unique, I cannot make the stronger statement that any competitive search equilibrium is constrained efficient.
20
(A) Firms commit to a single wage w and a number M , and commit to hire h = min{m, M } workers each at wage w; (B) Firms commit to pay each applicant a fee a and each hired worker a wage w, but do not commit to anything else; (C) Firms commit to a wage vector (w1 , w2 , . . . , wN ) such that each hired worker is paid a lottery with h equally likely prizes {w1 , . . . , wh } if h ≤ N workers are hired. These can be written in the general notation introduced above respectively as H(m) = min{m, M } and W (m, h) = wh; H(m) = {0, . . . , m} and W (m, h) = am + wh; and H(m) = {0, . . . , min{m, N }} and Ph W (m, h) = j=1 wh . The first environment directly gives the firm the possibility of committing ex ante to its ex post hiring behavior. The second environment adds an additional instrument for firms to make payments to workers. The last environment may seem unintuitive; it is motivated by the idea of a wage schedule, in which the firm commits to pay w1 the first worker hired, w2 to the second, and so on. In a static model there is no notion of ‘order of arrival,’ so I formalize this with lotteries.20 In each case I assume that all posted wages weakly exceed b so that workers accept the offered wages. I restrict myself to allowing firms to post only a single contract, in each case. If a firm posts multiple contracts, the queue lengths at each are related to each other in a complex manner that does not share the simple recursive determination which applies under wage posting (where, as in Section 4, the firm always hires applicants to low wages preferentially, and hires applicants to higher wages only if it does not receive enough applicants to low wages). Allowing for this makes the notation cumbersome without affecting the results I prove here. To define competitive search equilibrium in this environment, first note that if m applicants arrive at a firm that has posted contract (H, W ), then firm profit maximization requires the firm to hire some number of workers h which maximizes y(h) − W (m, h) over H(m). To economize on notation, I will define equilibrium only for the case of contracts for which this maximizer is unique for all m; in this case, denote the firm’s optimal choice of hiring by h(m, H, W ). The utility of workers from applying to a firm offering the contract (H, W ) if the expected queue length at such a firm is q > 0 is therefore given by ∞ ∞ 1 X q m e−q 1 X q m e−q [W (m, h(m, H, W )) + (m − h(m, H, W ))b] = b+ [W (m, h(m, H, W )) − h(m, H, W )b] . q m=0 m! q m=0 m!
In equilibrium, if the market utility is U , the queue length associated with (H, W ) can therefore be determined 20 Thanks
to Marek Pycia for suggesting the lottery interpretation.
21
analogously to (9) such that21
(17)
q(U − b) =
∞ X
[W (m, h(m, H, W )) − h(m, H, W )b]
m=0
Firm profit is π(H, W, U ) =
P∞
m=0
q m e−q . m!
1 m −q q e , where q solves (17). [y(h(m, H, W )) − W (m, h(m, H, W ))] m!
Denote by C the set of all possible contracts (H, W ) firms can post. A competitive search equilibrium with contracts is a set D ⊆ C, a measure φ on D, and a market utility U such that (i) conditional on U , if (H, W ) ∈ R D, then (H, W ) maximizes π(H, W, U ) over C; and (ii) the resource constraints D q(H, W ) dφ(H, W ) = µ and φ(D) = ν are satisfied. I can now state the main result of this section. Proposition 7. Under contracting environments (A), (B), and (C), there exists an efficient competitive search equilibrium. ¯ In environment (A), efficiency is obtained by firms posting a wage and committing to hire up to h workers at that wage. In environments (B) and (C), efficient equilibria can occur if the contract is chosen to ensure that the surplus is shared between firm and applicants appropriately and that firms face the correct incentives to hire additional workers at the margin. In environment (B) this occurs if the wage is equal to workers’ outside option, b, and the application fee is equal to U ∗ − b. In environment (C), it occurs ¯ with the differences if the wages (w1 , w2 , . . .) are chosen so that wh ≤ y(h) − y(h − 1) for each h ≤ h, [(y(h) − y(h − 1)] − wh chosen so as to split the surplus in the appropriate ratio between firms and workers.22 In view of the efficiency of contracting environments (A), (B), and (C), it is perhaps necessary to justify why wage posting is considered the benchmark. The first reason is that wage posting is the case most frequently considered in the competitive search literature. More fundamentally, in augmented versions of the model, contract environments (A), (B), or (C) may not be optimal. For example, suppose that workers are identical ex ante, but when an applicant arrives at a firm, a match-specific productivity shock is drawn, so that the marginal product of the applicant is either the normal marginal product (given by y(h) − y(h − 1) if 21 To guarantee a unique solution for q, assume that 1 [W (m, h(m, H, W )) − h(m, H, W )b] is weakly decreasing in m = 1, 2, . . . m and not a constant function. This condition is satisfied by environments (A) and (B), and also by environment (C) under the additional restriction that w1 ≥ w2 ≥ . . . ≥ wN . Under this assumption on (H,P W ), the right side of (17) is strictly decreasing 1 m −q in q, so that there is a unique q satisfying (17). This is because ψ : q 7→ 1q ∞ is a strictly decreasing m=1 mφ(m) m! q e P∞ 1 m−1 e−q and differentiate to obtain φ(m) q function whenever φ(·) is strictly decreasing. Rewrite as ψ(q) = m=1 (m−1)! P 1 m−1 e−q , which is negative because φ(·) is decreasing. ψ 0 (q) = ∞ m=1 [φ(m + 1) − φ(m)] (m−1)! q 22 It is interesting that while the expected value of an application is in equilibrium the same in each of the efficient environments (A), (B), and (C), the distributions of realized payments differ substantially. In environments (B), all applicants receive the same risk-free payment (allowing for the fact that applicants who are not hired earn unemployment income b); the firm bears all the risk. In environment (C), the realization of the worker’s payment from the set {w1 , w2 , . . .} is stochastic at the time of application. In environment (A), all hired workers receive the same payment, but the unemployed receive b < w. It is natural to conjecture that there may be no efficient equilibrium under environments (A) and (C) if workers are not risk neutral; verifying this is left for further research.
22
the firm employs h−1 other productive workers), or large and negative (so that employing the worker reduces the value of the firm’s production). The realization of the productivity shock for each applicant is observable only to the firm, before it makes its hiring decisions. Then it is easy to verify that a contract taking the form of an unconditional hiring commitment, as in environment (A), cannot be optimal. Even abstracting from this issue, the commitment requirement of environment (A)—that firms can commit to hiring an arbitrarily large number of workers at the posted wage even if it is not ex post profitable to do so—seems unreasonably strong; moreover, such commitments are not frequently observed empirically. Environment (C) requires that firms can treat workers differently ex post despite the fact that they are identical and applied to the same contract; non-discrimination legislation, fairness concerns or internal wage structure policies may prevent this. It is also unclear whether contracts of the form optimal in environment (C) are explicitly seen in real labor markets. In addition, unless each worker’s wage is public information, then there is a moral hazard problem that the firm would like to pay all hired workers the minimal wage draw that can arise in the wage lottery. Finally, paying workers an ‘application bounty’ as in environment (B) is unattractive since it may generate applications from workers uninterested in the job who simply hope to receive the application payment.
6.2
Single Wage Per Firm
In the benchmark model, firms can post multiple wages and hire multiple ex ante identical workers at the same time at different wages. It is optimal for a firm to do this in the model in cases where the market utility lies below, but close to, y(h) − y(h − 1) for some h (that is, the marginal product of some worker). In this case the number of workers the firm can attract by posting a wage below y(h) − y(h − 1) is bounded: queue lengths will be low at wages only slightly above the market utility. The firm may therefore prefer to post multiple wages, one wage between the market utility and y(h) − y(h − 1) in order to attract a few workers relatively cheaply, as well as a higher wage which attracts additional workers at a higher cost. Bewley (1999, ch. 9) argues that fairness concerns often lead to a common wage for new hires. If such a concern applies, firms might only be able to post a single wage, rather than many. Under this constraint on the wage-posting technology, the main results of the paper still apply (with the obvious exception that there will no longer be wage dispersion in equilibrium). A competitive search equilibrium exists, but is not in general constrained efficient. However, if firms can post contracts as in Section 6.1, efficiency can be restored. The interested reader is referred to an earlier version of this paper (Hawkins, 2006) for details.
23
6.3
Entry
I assumed in this paper that there are a fixed number of firms and workers in the economy. Constrained efficiency is then an undemanding criterion: all firms should post the same wage tuple and hiring should be efficient. (The first requirement guarantees that applicants are in expectation distributed uniformly across firms, which is efficient because of decreasing returns.) It is interesting to consider endogenizing the number of firms in the economy. To do this, augment the model with an additional ex ante stage, in which there are a large number of potential firms each of which could enter the labor market on payment of an entry cost k > 0.23 As usual, because ex post competition among firms tends to increase posted wages and reduce the profits of other firms, entry occurs until the expected profits a firm makes in the labor market equal k. Because the equilibrium is not in general constrained efficient even without entry, it is not surprising that it is also in general not constrained efficient when entry is endogenized. In particular, if the entry cost k is sufficiently low, then in any equilibrium, enough firms enter to drive market tightness µ/ν low enough that the inefficiency described by Proposition 5 applies.24 However, with endogenous entry, the equilibrium can be constrained inefficient even when hiring is ex post efficient. Recall from Proposition 4 that ex post ¯ − y(h ¯ − 1). If the constraint efficient hiring requires that applications only go to wages no greater than y(h) ¯ − y(h ¯ − 1) in order to allow the firm to hire up to h ¯ workers binds, that the wage must not exceed y(h) the equilibrium can be constrained inefficient. The reason is that although hiring is efficient ex post, firms make higher profits (conditional on ν) than they would if they were able to post a higher wage while still ¯ workers. These higher profits then drive excessive entry. committing to hire up to h ¯ − y(h ¯ − 1), then it is straightforward to adapt the proof of If hiring is ex post efficient and w < y(h) constrained efficiency of entry in Acemoglu and Shimer (1999b) to establish the corresponding result for my model. Thus, the efficiency result for the standard competitive search model carries over to the environment of this paper in this cases where the hiring inefficiency I studied does not arise. A similar efficiency result also applies if contracts are posted, as in Section 6.1.
6.4
Returns to Scale
A key assumption of the model I studied is that there are decreasing returns to labor in production.25 Production is nontrivial to incorporate into search models in general and into the directed search framework in particular. Here I consider the role of my assumption of decreasing returns to labor and how the current 23 The entry cost can be thought of as the cost of the capital necessary for the firm to produce, as in Acemoglu and Shimer (1999b, 2000). It can also be thought of as the cost of vacancy posting, which is otherwise not modeled in this paper. 24 I establish this and other results stated in this section more formally in Hawkins (2006), at least for the case where firms post a single wage. 25 The discussion in this section was suggested by two anonymous referees.
24
paper compares to other treatments in the literature. As a point of departure, first consider the standard random search framework with bargained wages. This is the subject of a large literature in the tradition of Diamond (1982), Mortensen (1982), and Pissarides (1984). As in models of directed search, the benchmark is the case where matching is one-to-one: a firm can productively employ only a single worker. This can be thought of as the limiting case of extreme decreasing returns. Many results are known in this case: for example, Hosios (1990) showed that the equilibrium is constrained efficient provided workers’ bargaining power equals the unemployment elasticity of the matching function. A peculiarity of the random search environment is that allowing for a production function that exhibits constant returns to labor often generates very similar results. For example, holding market tightness constant, when a worker bargains with a firm which can employ only that one worker at marginal product A, the resulting wage is the same as the worker would receive if she instead bargained with a firm with the constant returns to scale production function y(h) = Ah. For more details, see Pissarides (2000, Section 2.7). However, in the random search model, the intermediate case where there are strictly decreasing returns to labor in production, but the marginal product of the second worker remains strictly above the outside option, exhibits qualitatively different behavior. Smith (1999) showed that the equilibrium is in general not constrained efficient when firms bargain continuously with their workforce: additional hiring by the firm drives down the marginal product of labor, which reduces wages, and this incentive ensures that the equilibrium is not constrained efficient because firms over-hire.26 The role of wages in a model of competitive search of course differs from that in a model of random search with bargaining. In the benchmark competitive search model (Moen, 1997; Shimer, 1996), firms again wish to hire only one worker. In order for wages to direct workers’ search, firms must commit to them ex ante. If a nondegenerate distribution of wages is posted, workers trade off the higher value of getting a job with a higher wage with the understanding that they will get this job with a lower probability. The lower probability arises because even though more workers apply, the firm still wishes to hire only one of them. Now suppose that there are constant returns to labor in production instead. Then such a tradeoff does not arise and there is no coordination friction. A firm with production function y(h) = Ah that posts a wage less than A is willing to hire arbitrarily many workers at this wage. In equilibrium all workers simply apply to the firm posting the highest wage of all (or, if there are several such firms, they apply at random to one of them). If there are at least two firms in the economy, the unique competitive search equilibrium in this environment is that all firms post the wage A and make zero profits, as in the Bertrand paradox.27 26 See also Stole and Zwiebel (1996a,b) who examine the effect of over-hiring in a static model; Wolinsky (2000), who also embeds the model of Stole and Zwiebel in an equilibrium framework; and Hawkins (2011a), who shows that commitment to long-term contracts ameliorates this effect. 27 If there is only one firm, of course, it posts the wage w = b and makes positive profits. If entry was allowed for (so that there are finitely many potential entrant firms each of whom can enter at cost c > 0), the only symmetric equilibrium then
25
The current paper studies an intermediate case, lying between the extreme decreasing returns of the oneworker-per-firm benchmark and the degenerate case of constant returns. I showed that, even when the entry margin is ignored and the number of firms and workers is fixed, the equilibrium is in general not constrained efficient. As has been seen, the reason is that the same wages which direct workers’ search also play an allocative role in determining ex post hiring decisions. This tradeoff arises in neither of the extreme cases: when only one worker is needed for production, firms never post a wage above the product of that single worker, and when there are constant returns, again firms never post a wage above the constant marginal product of labor. In either case, doing so would lead to no hiring and zero profits. In the intermediate case studied here, the same logic implies that a firm will never post a wage greater than y(1), but this is not sufficient for efficient hiring when the marginal product of additional workers beyond the first is strictly less than y(1) and strictly greater than workers’ outside option. One special case where the production function exhibits decreasing returns to scale but the equilibrium under wage posting is still constrained efficient is when y(h) = A min{h, h0 } for some h0 ≥ 2. In this case, which is the one studied by Burdett et al. (2001) and Lester (2010), there are constant returns up to the point where the firm has hired h0 workers, at which point the marginal product of labor falls to zero. In this case a single wage less than A is consistent with efficient hiring ex post, and the equilibrium will be constrained efficient under wage posting. Finally, note that the reason the equilibrium is inefficient when there are decreasing returns to labor in production is of a very different flavor in my model (with directed search) from the reason for inefficiency under random search with bargained wages. In the current paper, inefficiency arises because one instrument (the wage) is not sufficient to solve two problems (attracting the right number of workers and ensuring the firm hires efficiently once applicants have arrived). In the random search models mentioned above, inefficiency arises because when wages are bargained, the firm has an incentive to overhire in order to drive down the wages of already-hired workers. The two mechanisms are completely unrelated.
6.5
Vacancies
The hiring process in my model proceeds by firms posting arbitrarily many wages at zero cost. Firms are then able to hire as many workers as they choose from among their applicant pool. Both the fact that the firm is not limited to hiring only one worker per posted vacancy and the zero cost of posting vacancies are not completely standard in the literature. I argued in Section 6.3 that making vacancy posting costly is takes the form that all firms enter with positive probability determined so that the probability a firm is alone in the market ex post and therefore earns a profit is just sufficient to allow it to break even ex ante. The equilibrium of the model with a fixed number of firms is constrained efficient, but with endogenous entry this result no longer applies since there is a positive probability of no entry by any firms and a positive probability of wasteful entry by more than one firm.
26
straightforward if entry is made endogenous. In this section I argue the absence of capacity constraints in hiring is the natural assumption to make in the environment of the current paper. In the directed search literature, the posting of a vacancy is sometimes taken to constrain the firm to hire at most one worker at the associated posted wage. However, if firms only wish to hire one worker in total, the constraint does not bind. When firms wish to hire multiple workers the assumption that a vacancy entails a capacity constraint does not seem compelling. Workers are identical. If a plumbing firm wants to hire two plumbers, the assumption of a capacity constraint of one worker hired per vacancy posted requires that it post at least two vacancies. However, if it posts two vacancies, say with the same posted wage, it can happen that two unemployed plumbers apply to the first posted vacancy, and none to the second. It seems unreasonable to assume that the firm’s recruitment department cannot overcome this coordination failure, which is now internal to the firm, by offering the job posted as the second vacancy—which is, after all, identical—to one of the plumbers who applied to the first vacancy.28 It seems all the more unreasonable since I assume that wage posting does not imply a commitment to hire a worker at the posted wage. If firms’ recruitment departments can optimize over their entire applicant pool in order to hire the applicants to low wages preferentially, it seems odd if they cannot solve the coordination problem discussed in this paragraph. Posting two vacancies with capacity one is then the same as posting one vacancy with capacity two. Moreover, even if the firm can only post vacancies that allow it to hire only one worker per vacancy, but can post such vacancies at low cost, the coordination problem discussed in the previous paragraph can still be overcome. To do so, the firm simply posts more copies of the same vacancy. (In the example above, rather than posting two vacancies for plumbers, the firm should post a very large number all offering the same wage.) The total expected queue length across all these vacancies remains bounded as the number of postings becomes large, because it is still true that the firm is willing to hire no more than h workers if the posted wage w exceeds y(h + 1) − y(h): thus, the original coordination friction from the benchmark model still applies. However, as the number of posted vacancies becomes large, the probability that multiple applicants apply to the same vacancy becomes small. As the firm posts infinitely many vacancies, the effect is the same as that of posting wage w in the benchmark model, that is, of being willing to hire up to h applicants at this wage. In summary, the only setting in which it would be reasonable for vacancies to entail capacity constraints would be if the firm’s recruitment department cannot transfer identical applicants across identical posted vacancies internally, and if posting multiple copies of the same recruitment advertisement is expensive. These conditions do not seem empirically relevant. This is particularly so in view of evidence from the Job 28 If workers were not identical—for example, if the first vacancy is for a plumber and the second for an administrative assistant—this argument does not apply, but all workers were assumed perfect substitutes in production.
27
Openings and Labor Turnover Survey (JOLTS) of the Bureau of Labor Statistics on the number of hired workers per posted vacancy. Davis et al. (2010) report using the JOLTS establishment-level data that in the period between December 2000 and December 2006, the fastest-growing firms hire more than six workers in a month for each posted vacancy at the end of the previous month; the average across all firms is 1.3. While time aggregation is no doubt important here—the fastest-growing firms almost tautologically fill vacancies quickly, including vacancies posted during the same month the hire occurs—the evidence is suggestive that firms hire multiple workers per posted vacancy routinely, as in my model.
7
Additional Related Literature
In this section I briefly discuss how this paper relates to other work not described already. For comments on how the paper relates to other models of frictional wage dispersion, the reader is referred to Section 4.3. A discussion of the relation of the paper to several benchmark models of directed and random search appears in Section 6 and especially in Section 6.4. First, and most obviously, the current paper is relevant to work studying models of frictional labor markets in which the production technology exhibits decreasing returns to labor. Two important recent papers of this type using directed search are Kaas and Kircher (2011) and Schaal (2010).29 The focus of both papers differs substantially from the current work. Kaas and Kircher investigate whether multi-worker firm models help understand cyclical fluctuations in the labor market. In their model the equilibrium is constrained efficient despite the fact that firms post a dynamic generalization of wage contracts. The reason for the discrepancy with the current paper is that they lack the realistic feature of the current paper that employment takes on integer values only. In their continuous-employment model, the inefficiency I focus on under wage posting does not arise. Employment is also a continuous variable in Schaal (2010), which studies the effects of increases in firm-level uncertainty on the aggregate labor market and allows in addition for on-the-job search. Lester (2010) and Tan (2012) both consider models in which firms wish to hire at most two workers and seek to account for empirical size-wage correlations. In neither paper does the issue that wages may be a binding constraint on the firm’s ex post hiring that is the focus of the current paper arise: Lester assumes that y(2) = 2y(1), while Tan allows the production function to be locally convex (y(2) ≥ 2y(1)). See also Geromichalos (2010), who does not restrict to wage posting but does allow for production functions that are not globally concave. Second, the paper is also related to a group of papers that find that competitive equilibrium with wage 29 In addition to these directed search papers, there is also a literature with multi-worker firms and random search. As noted in Section 6.4, Stole and Zwiebel (1996a,b), Smith (1999), Wolinsky (2000), and Hawkins (2011a) study the efficiency properties of these models, while Cooper et al. (2007), Fujita and Nakajima (2009), Elsby and Michaels (2010), Acemoglu and Hawkins (2010), and Hawkins (2011b) use them in an applied setting to understand cyclical fluctuations.
28
posting does not always decentralize the constrained efficient allocation. Closest to the current paper is Inderst (2005), where firms wish to hire only one worker and workers are ex ante heterogeneous in productivity. When firms are unable to post wage contracts contingent on worker type, the equilibrium is constrained inefficient. My paper differs from his both in that I study an environment with multi-worker firms, and in that the equilibrium in my model is inefficient even when workers are identical. (Workers are heterogeneous in productivity in my model ex post, rather than ex ante: the marginal product of a worker who is the sole applicant to a particular firm is greater than that of an applicant to a firm which has many other applicants.) Guerrieri (2008) and Menzio and Shi (2010) also demonstrate situations in which wage posting does not suffice for efficiency of equilibrium under directed search. In both these models, the reason for inefficiency arises from a dual role played by wages, which must direct workers’ search and also play another role, as in this paper, but that other role is not the same as here (where the wage plays an allocational role of determining the firm’s hiring decision). Guerrieri studies a case where wages determine whether the applicant (who has private information on the disutility of labor) is willing to accept the job. Menzio and Shi study a case where wages determine not only where unemployed workers apply, but also where employed workers look for new jobs. Beyond the case of wage posting, other modifications of the benchmark directed search model can also lead to inefficient equilibria. Examples include allowing for multiple applications, as for example in Albrecht et al. (2006), or for asymmetric information, as in Guerrieri et al. (2010).
8
Conclusion
Important recent progress has been made in understanding directed search models of labor markets in the realistic case in which firms’ production functions exhibit decreasing returns to labor. However, the desire for tractability frequently leads authors to make the simplifying assumption that employment at the firm level is a continuous variable. This paper has showed that this assumption is by no means innocuous. In my model in which firms more realistically employ only whole numbers of employees, the equilibrium is no longer constrained efficient in general; firms may post multiple wages; and there can be wage dispersion even though all firms and all workers are ex ante identical. The paper, however, by no means exhausts the implications for directed search and for labor economics more generally of the setting I consider. Most importantly, a proper understanding of the quantitative importance of the features I highlight can only come when a dynamic version of the model is studied more deeply. To do so requires understanding the relation between the time-consuming process of recruitment and the time-consuming process of production. Does hiring occur sequentially, or are many workers hired in parallel? If a firm hires multiple workers at the same time, are the negotiations independent or interdependent? Are
29
the contracts of existing workers contingent on the firm’s hiring behavior? The results of the present paper suggest that this would be a fruitful area both for empirical and theoretical research.
Appendix Proof of Lemma 1. The case where either qk = 0 or qk+1 = 0 is trivial. Therefore, suppose that when the firm posts w that both qk and qk+1 are strictly positive. According to (9), h(wk )−1
qk (U − b) = (wk − b)
X n=0
∞ Qnk−1 e−Qk−1 X q m e−qk min{m, h(wk ) − n} k n! m! m=1
and h(wk+1 )−1
qk+1 (U − b) = (wk+1 − b)
X n=0
∞ q m e−qk+1 (Qk−1 + qk )n e−(Qk−1 +qk ) X min{m, h(wk+1 ) − n} k+1 . n! m! m=1
According to the hypothesis of the lemma, h(wk ) = h(wk+1 ) = h. Next, observe that (18) ∞ h−1 ∞ Qnk−1 e−Qk−1 X q m e−qk+1 q m e−qk X (Qk−1 + qk )n e−(Qk−1 +qk ) X min{m, h − n} k + min{m, h − n} k+1 n! m! n! m! n=0 m=1 n=0 m=1 h−1 X
=
∞ Qnk−1 e−Qk−1 X (qk + qk+1 )m e−(qk +qk+1 ) min{m, h − n} n! m! m=1 n=0 h−1 X
This is intuitive: the left side of (18) is the expected number of hires associated with two wages wk < wk+1 in the interval Ih that attract respectively queue lengths of qk and qk+1 , while the right side is the expected number of hires associated with a single wage in this interval that attracts queue length qk + qk+1 . Because the firm is willing to hire up to h applicants from wages up to and including wk+1 , it does not matter for the expected number of applicants whether the total queue length qk + qk+1 is split in two smaller queues or not. To establish (18), use the binomial theorem to write the second term on the right side as
(19)
h−1 n XX n=0 l=0
∞ Qlk−1 qkn−l e−Qk−1 e−qk X q m e−qk+1 min{m, h − n} k+1 , l!(n − l)! m! m=1
and reverse the order of summation and relabel the indices to rewrite this as
(20)
∞ X q i−n e−qk X Qnk−1 e−Qk−1 h−1 q m e−qk+1 k min{m, h − i} k+1 . n! (i − n)! m=1 m! n=0 i=n h−1 X
30
Comparing coefficients of terms in Qnk−1 , it suffices to show that for each n ∈ {0, 1, . . . , h − 1}, (21) ∞ X
min{m, h−n}
m=1
∞ ∞ h−1 X q m e−qk+1 (qk + qk+1 )m e−(qk +qk +1) q k e−qk X qki−n e−qk X = + min{m, h−i} k+1 . min{m, h−n} m! m! (i − n)! m=1 m! m=1 i=n
Arguing analogously to (6), rewrite the right side of (21) as " # h−1 h−i−1 m X e−qk+1 qk+1 qkm e−qk X qki−n e−qk + h−i− h−n− (h − i − m) (h − n − m) m! (i − n)! m! m=0 m=0 i=n " # h−n−j−1 h−n−1 h−n−1 m X X X q j e−qk qk+1 e−qk+1 qkm e−qk k =h−n− (h − n − m) + h−n−j− (h − n − j − m) m! j! m! m=0 m=0 j=0 h−n−1 X
=h−n−
h−n−1 X j=0
=h−n−
qkj e−qk j!
h−n−j−1 X
(h − n − j − m)
m=0
h−n−1 s X X
(h − n − s)
s=0
=h−n−
j s−j −qk −qk+1 s! e j!(s−j)! qk qk+1 e
s!
j=0
h−n−1 X
(h − n − s)
s=0
m qk+1 e−qk+1 m!
∞ X (qk + qk+1 )s e−(qk +qk+1 ) (qk + qk+1 )m e−(qk +qk +1) = min{m, h − n} . s! m! m=1
Here the first equality follows by writing j = i − n, the second by canceling terms, the third by grouping terms differently and summing over s = j + m first, the fourth by the binomial theorem, and the last by the same argument used to generate the first expression. This establishes (18). The remainder of the proof is straightforward. Add (19) and (20) and use (18) and the fact that wk < wk+1 to observe that
(qk + qk+1 )(U − b) > (wk − b)
∞ Qnk−1 e−Qk−1 X (qk + qk+1 )m e−(qk +qk+1 ) min{m, h − n} n! m! n=0 m=1 h−1 X
and
(qk + qk+1 )(U − b) < (wk+1 − b)
∞ Qnk−1 e−Qk−1 X (qk + qk+1 )m e−(qk +qk+1 ) min{m, h − n} . n! m! n=0 m=1 h−1 X
By continuity, there exists w0 ∈ (wk , wk+1 ) such that
(22)
(qk + qk+1 )(U − b) = (w0 − b)
∞ Qnk−1 e−Qk−1 X (qk + qk+1 )m e−(qk +qk+1 ) min{m, h − n} . n! m! n=0 m=1 h−1 X
A firm posting w0 = (w1 , . . . , wk−1 , w0 , wk+2 , . . . , wN ) will attract queues q0 = (q1 , . . . , qk−1 , qk +qk+1 , qk+2 , . . . , qN ). 31
(22) establishes that the queue for wage w0 is qk + qk+1 . For j < k, the fact that the queue for wage wj is qj follows directly from the assumption that w and q satisfied (9), since changing higher wages does not affect the attractiveness of a given posted wage. For j > k, the fact that wage w0 attracts queue length qk + qk+1 means that Qj−1 is unchanged, so that again the fact that the queue length for wage wj is qj follows from (22). The fact that the firm’s profits are unchanged follows from the observation that the expected wage payment is unchanged, as is expected hiring. Proof of Proposition 1. The proof consists of two parts. I first show that there exists a maximizer of the function w 7→ π(w, U ) for each U . I then apply the Theorem of the Maximum to complete the proof. Establishing the existence of the maximizer π(w, U ) would be trivial if the intervals Ih were closed and therefore compact. Since w 7→ π(w, U ) is continuous, it would be immediate that for any increasing ¯ there would exist πS (U ) ≡ max{π(w, U ) | w = subsequence S = (s1 < · · · < sk ) of elements of {1, 2, . . . , h}, (w1 , . . . , wk ), w1 ∈ Is1 , . . . , wk ∈ Isk }. Then the existence of π(w, U ) = maxS πS (U ) would follow since the set of sequences S is finite. However, because Ih is open at its left limit (because a firm posting w = y(h + 1) − y(h) is willing to hire up to h + 1 workers at that wage, while a firm posting any strictly higher wage is willing to hire only at most h workers), this argument does not directly apply. To prove existence, I therefore extend each interval Ih to include an additional wage w ˆh at its left limit, use the previous argument to show that a (possibly larger) maximum exists when firms can post tuples of ˆh for any wages from the set [0, ∞) ∪ {w ˆ1 , . . . , w ˆh¯ }, and then show that it is not optimal to post any wage w h, so that the maximum is attained by a wage tuple w from the non-augmented set. For each h, then, define the closure of the interval Ih as I¯h = w ˆh ∪ Ih , where posting the wage w ˆh means that the firm commits to pay each applicant y(h+1)−y(h), but also to hire only max{0, h−m} applicants to w ˆh if the firm receives m applicants to wages belonging to the set [0, y(h) − y(h − 1)] ∪ {w ˆh+1 , w ˆh+2 , . . . , w ˆh¯ }. One can think of the wage w ˆh as infinitesimally greater than y(h + 1) − y(h): w ˆh lies strictly above y(h + 1) − y(h), so that the firm is willing to hire at most h applicants to wage w ˆh , and strictly below all wages w > y(h + 1) − y(h), so that applicants to w ˆh are hired preferentially to all applicants to such wages. I will ¯
use this ordering on [0, ∞) ∪ {w ˆh }hh=1 below. It is clear how to extend the construction of the queue lengths according to (7), (8), and (9) to this extended wage posting environment. It is intuitive that posting a wage w ˆh is not optimal. Posting y(h + 1) − y(h) instead would allow the firm to be able to hire an additional worker in the state of the world where it received at most h applicants to wages strictly below w ˆh and at least h + 1 applicants to these wages together with w ˆh . However, this deviation might not be profitable because the queue length to y(h + 1) − y(h) would be strictly higher than that to w ˆh , and this affects the queue lengths at wages greater than w ˆh . Dealing with this issue requires
32
replacing w ˆh not with y(h + 1) − y(h) but with a strictly lower wage. ¯
More precisely, let w be a wage tuple including some elements of {w ˆh }hh=1 , and without loss of generality assume that each element of w receives a positive queue length. Let w ˆk be the smallest wage in w that ¯
belongs to {w ˆh }hh=1 , and denote the associated queue length by qˆ. Also, let ω be the largest element of w that is less than w ˆk (or, if no such wage exists, let ω = b). I claim that there exists w0 with ω < w0 < y(h+1)−y(h) such that the queue length associated with w0 is also equal to qˆ. If such a w0 exists, define w0 by replacing w ˆk with w0 in w. Then by Lemma 2 π(w0 , U ) > π(w, U ). To establish the existence of such a w0 , let Q be the total queue length associated with wages no higher than ω. According to (9), the queue length attracted by a wage w ∈ (ω, y(h + 1) − y(h)] is the unique nonnegative solution for q(w) to h(w)−1
(23)
X
q(w)(U − b) = (w − b)
n=0
∞ 1 n −Q X 1 Q e min{m, h(w) − n} q(w)m e−q(w) . n! m! m=1
Then observe that the function w 7→ q(w) has the following properties: (i) limw→ω+ q(w) = 0; (ii) q(y(h + 1) − y(h)) > qˆ; (iii) q(·) is continuous on any interval of the form Ik ; and (iv) q(y(k + 1) − y(k)) ≥ limw→(y(k+1)−y(k))+ q(w) for each k. Properties (ii) and (iv) follow immediately from the observation that in (23), q(w) is strictly increasing in h(w). Property (iii) is clear. The only non-obvious property is (i), in the case when ω > b. (The case ω = b is obvious: sufficiently low wages get no applicants.) If ω > b, (i) states that if two wages very close together are posted and the lower one receives a positive measure of applicants, then the higher wage gets no applicants (intuitively because the benefit of the higher wage is outweighed by the fact that applicants to the lower wage are hired first). I now establish property (i). Assume that ω > b, and that ω attracts a positive queue length; denote the queue length attracted by wages strictly below ω by Q0 and the queue length attracted by ω as q0 , with Q0 + q0 = Q. Again apply (9) to observe that h(ω)−1
(24)
q0 (U − b) = (ω − b)
X n=0
h(ω)−n X 1 n −Q0 q0m e−q0 Q e h(ω) − n − (h(ω) − n − m) . n! 0 m! m=0
To see that limw→ω+ q(w) = 0, it suffices to show that h(ω)−1
(25)
(ω − b)
X n=0
1 (Q0 + q0 )n e−(Q0 +q0 ) < U − b. n!
33
Multiply (25) by q0 > 0 and combine with (24) to see that it is sufficient to establish that h(ω)−1
(26)
q0
X n=0
h(ω)−n h(ω)−1 m −q0 X X 1 q e 1 . (h(ω) − n − m) 0 (Q0 + q0 )n e−(Q0 +q0 ) < Qn0 e−Q0 h(ω) − n − n! n! m! m=0 n=0
Ph(ω)−1 1 n −Q0 Ph(ω)−n−1 1 m −q0 Rewrite the left side of (26) as n=0 n! Q0 e q m=0 to see that it is also sufficient to m! q0 e Ph(ω)−n−1 1 m −q0 Ph(ω)−n q m e−q0 show for each n that q m=0 < h(ω) − n − m=0 (h(ω) − n − m) 0 m! . This follows from m! q0 e the observation that X q m e−q0 0 0 < (h(ω) − n) 1 − m! m=0
h(ω)−n
h(ω)−n
= (h(ω) − n) −
X
(h(ω) − n − m)
h(ω)−n X q m e−q0 q0m e−q0 − m 0 m! m! m=1
(h(ω) − n − m)
q0m e−q0 − q0 m!
m=0 h(ω)−n
= (h(ω) − n) −
X m=0
h(ω)−n−1
X m=0
q0m e−q0 . m!
The combination of properties (i)-(iv) establishes that as w increases from ω to y(h + 1) − y(h), q(w) is continuous except at wages of the form y(k + 1) − y(k), at which it jumps downwards; at such wages q(·) is left-continuous but not right-continuous. It follows that the function q˜(·) defined by q˜(w) = max{q(w0 ) | ω ≤ w0 ≤ w} is well-defined and continuous on [ω, y(h + 1) − y(h)]. Moreover, q˜(ω) = 0 and q˜(y(h + 1) − y(h)) ≥ q(y(h + 1) − y(h)), which is in turn greater than qˆ by property (ii). It follows by the intermediate value theorem that there is a w00 ∈ (ω, y(h + 1) − y(h)) such that q˜(w00 ) = qˆ. The existence of w0 ∈ (ω, w00 ] such that q(w0 ) = qˆ follows immediately. Now that I have established that there exists w that maximizes π(w, U ) for each U , the remainder Pk of the proof follows simply. Define a function ψ(w1 , . . . , wk ; U ) = i=1 qi , where q1 , . . . , qk are defined ¯ then ψ by (9). Choose any increasing subsequence S = (s1 < · · · < sk ) of elements of {1, 2, . . . , h}; is a continuous function when its domain is restricted to the set {(w1 , . . . , wk ) | w1 ∈ I¯s1 , . . . , wk ∈ I¯sk } (the topology on I¯s1 × · · · × Isk is the product of the standard topology on each interval I¯h , regarding w ˆh as the left limit of the interval). In addition, by the Theorem of the Maximum it follows that the correspondence ΓS : U 7→ {ψ(w1 , . . . , wk ; U ) | (w1 , . . . , wk ) ∈ arg maxw∈S π(w, U )} is upper hemicontinuous; therefore, so too is the correspondence Γ : U 7→ {ψ(w1 , . . . , wk ; U ) | (w1 , . . . , wk ) ∈ arg maxw π(w, U )}, where the maximization is not restricted to a single S. Moreover, it is clear that lim inf U →b+ Γ(U ) = +∞ (as applications become very cheap, the firm optimally uses a very large number), while lim supU →y(1)− Γ(U ) = 0. It follows by upper hemicontinuity of Γ and the intermediate value theorem that for any positive µ and ν,
34
either there exists U > b such that µ/ν ∈ Γ(U ), or else there exists U > b such that there are q 1 , q 2 ∈ Γ(U ) with q 1 < µ/ν < q 2 . In the first case it is now immediate from the definition of equilibrium that there exists an equilibrium where all firms post some wage tuple w belonging to arg maxw π(w, U ) for which ψ(w) = µ/ν. In the second case, write µ/ν = αq 1 + (1 − α)q 2 for α ∈ (0, 1); then there is an equilibrium in which measure αν firms post a wage tuple w1 and measure (1 − α)ν firms post a wage tuple w2 , with w1 , w2 ∈ arg maxw π(w, U ) and such that ψ(w1 ) = q 1 and ψ(w2 ) = q 2 . This completes the proof. Proof of Lemma 2. Immediate from the fact that for each k, the function Q 7→ 1 − increasing in Q. To see this observe that the derivative is
1 k−1 −Q e (k−1)! Q
Pk−1
1 m −Q m=0 m! Q e
is
> 0.
0 Proof of Proposition 2. Let w0 = (w10 , . . . , wk0 , wk+1 , . . . , wN ) be a wage tuple satisfying the hypotheses of 0 the Proposition, and denote the associated queue length tuple by q = (q10 , . . . , qN ). According to (9), qk is
a continuous and strictly increasing function of wk on the interval Ih = (y(h + 1) − y(h), y(h) − y(h − 1)], holding other variables constant; similarly, qk+1 is a continuous and strictly increasing function of wk+1 on Ih0 . Use the implicit function theorem to define a function ψ on a neighborhood of wk0 which maps a wage wk to that value of wk+1 that holds constant the resulting value of qk + qk+1 associated with the tuple 0 0 0 (w10 , . . . , wk−1 , wk , ψ(wk ), wk+2 , . . . , wN ). From (9), queue lengths other than qk and qk+1 are unchanged. If
wk > wk0 , then qk is strictly greater than qk0 by the argument in footnote 12. It follows from Lemma 2 that 0 the new wage tuple generates a strictly higher profit than w0 . (One can show that also wk+1 > wk+1 but I
omit this argument since it is not needed to establish the Proposition.) Proof of Proposition 3. It is immediate from Proposition 2 that profit-maximizing wage tuples take the form either of a single wage, or else of a tuple of the form (y(2) − y(1), w), with w ∈ (y(2) − y(1), y(1)). First consider the case where U ≥ y(2) − y(1). In this case no wages low enough to allow the hiring of a second worker will attract any applicants. Therefore any profit-maximizing wage tuple in a simple equilibrium consists only of a single wage w ∈ (U, y(1)). In this case the problem reduces to the standard competitive search setup of Moen (1997) and Shimer (1996). The firm’s profit from posting such a wage is
(27)
π(w, U ) = [1 − e−q ](y(1) − w)
where q is defined implicitly by the specialization of (9) to this environment, namely that
(28)
q(U − b) = [1 − e−q ](w − b).
35
Substitute from (28) into (27) to eliminate w, which reduces the firm’s profit maximization problem to
max[1 − e−q ](y(1) − b) − q(U − b)
(29)
q
subject to 0 ≤ q ≤ q1M , where q1M is the queue length arising from making w as large as possible, that is, the M
solution to q1M (U − b) = [1 − e−q1 ](y(1) − b). (29) defines a single-peaked function of q ∈ [0, ∞). Posting either w = U or w = y(1) (that is, choosing q = 0 or q = q1M ) generates zero profits, in the first case since no workers apply and in the second case since the firm makes no profits on any hire, while posting any wage strictly between these limits generates strictly positive profits. Therefore the solution to the first-order condition, given by q1∗ = log(y(1) − b) − log(U − b), gives the optimal wage-posting strategy, which is the same for all firms (given U ). Now consider the case where U < y(2) − y(1). In this case, posting a single wage w ∈ (y(2) − y(1), y(1)] cannot be optimal. To see this, first observe that posting w = y(1) generates zero profits and is dominated by, for example, posting w = y(2) − y(1). Next, if w < y(1), then posting the tuple (U, w) instead of the singleton w generates the same profit (because the wage U attracts no applicants, so that the queue at w is unchanged). Now increase U slightly and then increase w so that the total queue length to the wage tuple remains unchanged. Lemma 2 ensures that this increases profits. Together with Proposition 2, this argument ensures that any optimal wage-posting strategy consists either of a single wage w ∈ (U, y(2) − y(1)], or of a wage pair of the form (y(2) − y(1), w), with w ∈ (y(2) − y(1), y(1)]. I consider each case in turn. The firm’s profit from posting single wage w ∈ (U, y(2) − y(1)] is
(30)
π(w, U ) = [1 − e−q ](y(1) − w) + [1 − e−q − qe−q ](y(2) − w)
where q solves
q(U − b) = (w − b)[2 − 2e−q − qe−q ].
(31)
Eliminate w in (30) using (31) to write the firm’s problem as
(32)
max[1 − e−q ](y(1) − b) + [1 − e−q − qe−q ](y(2) − b) − q(U − b) q
subject to 0 ≤ q ≤ q2M , where q2M is defined implicitly by
(33)
M
M
q2M (U − b) = (y(2) − y(1) − b)[2 − 2e−q2 − q2M e−q2 ].
36
(32) defines a single-peaked function of q on [0, ∞); also, the profits from posting any wage no greater than U are zero, while the profits from posting any higher wage are strictly positive. It follows that profits are maximized either at q = q2M or at the q2∗ ∈ (0, q2M ) which satisfies the first-order condition ∗
∗
e−q2 (y(1) − b) + q2∗ e−q2 (y(2) − b) = U − b.
(34)
∗
M
Note that in this case, it follows that e−q2 (y(1) − b) < U − b, so a fortiori also e−q2 (y(1) − b) < U − b. This implies that if an interior solution is optimal conditional on posting a single wage in the interval (U, y(2) − y(1)], then a firm that instead posted a wage pair (y(2) − y(1), w) with w ∈ (y(2) − y(1), y(1)] would receive no applicants to w. Even if w were as high as y(1), it would still not be attractive, given the number of applicants (given by q2M ) that are attracted by y(2) − y(1). Also, note that if q2∗ is feasible, then since it is the unique optimum, all firms choose the same wage. The last possibility is that the firm posts a wage pair of the form (y(2) − y(1), w), with w ∈ (y(2) − y(1), y(1)). As observed in the previous paragraph, this is not optimal if q2∗ < q2M . However, if q2∗ ≥ q2M , then the optimal single wage in the interval (U, y(2) − y(1)] for the firm to post was y(2) − y(1). In this case, posting a wage tuple (y(2) − y(1), w) generates strictly higher profits if and only if w attracts a strictly positive queue length (and w < y(1)). The existence of such a w is guaranteed if it would be the case that if the firm posted (y(2) − y(1), y(1)), the wage y(1) would attract a strictly positive queue. This occurs if M
e−q2 (y(1) − b) > U − b.
(35)
If this condition holds, then the firm’s problem is given by maximizing (14) subject to (15). The q1 defined in (15) is equal to the q2M defined in (33). Substituting from (15) into (14) to eliminate w generates an equivalent representation: the firm solves
(36)
h i M M M max y(1) − b + [1 − e−q2 − q2M e−q2 ](y(2) − b) − e−(q2 +q) (y(1) − b) − (q2M + q)(U − b) q
M
M
subject to 0 ≤ q ≤ q3M , where q3M is defined implicitly by q3M (U − b) = e−q2 [1 − e−q3 ](y(1) − b). The M
maximand in (36) is a single-peaked function of q on [0, ∞) and takes the value [1 − e−q2 ](y(1) − b) + [1 − M
M
e−q2 − q2M e−q2 ](y(2) − b) − q2M (U − b) at both q = 0 and q = q3M (where w = y(1) and so the firm makes no profits from hiring a second worker). Under (35), greater profits than this are possible by posting any wage slightly below y(1). Therefore the maximum profit in this case occurs at the unconstrained optimal queue length, which the first-order condition for (36) shows is given by q3∗ = −q2M + log(y(1) − b) − log(U − b). Because the solution for q3∗ is unique, all firms post the same wage tuple in this case. 37
To summarize, I have shown that if U ≥ y(2) − y(1), the optimal strategy for the firm is to post a single wage in (U, y(1)), and all firms post the same wage. If U < y(2) − y(1), then if q2∗ < q2M , the optimal strategy for the firm is to post a single wage in (U, y(2) − y(1)), and all firms post the same wage; if q2∗ ≥ q2M and (35) does not hold, then the optimal strategy of the firm is to post the single wage y(2) − y(1); while finally if q2∗ > q2M and (35) holds, then the optimal strategy is to post a wage tuple of the form (y(2) − y(1), w) for some w ∈ (y(2) − y(1), y(1)), and all firms post the same tuple. To complete the proof, I need to establish that there are U1∗ ≤ U2∗ < y(2) − y(1) such that if U < U1∗ , then q2∗ < q2M ; if U ∈ [U1∗ , U2∗ ] then q2∗ ≥ q2M and (35) does not hold; if U ∈ (U2∗ , y(2) − y(1)) then q2∗ ≥ q2M and (35) holds. This set of conditions is equivalent to q2∗ < q2M iff U < U1∗ and (35) holds iff U2∗ < U < y(2) − y(1). Let U1∗ be the value of U such that q2∗ = q2M . To see that q2∗ < q2M if U < U1∗ , it suffices to show that the wage w(q2∗ ) which generates a queue length of q2∗ , obtained by substituting q = q2∗ into (31), is increasing in U . (This is because q2∗ < q2M if and only if w(q2∗ ) < y(2) − y(1).) Differentiate (34) to see that 1 1 ∂q2∗ = − −q∗ =− . ∗ ∗ ∗ −q −q 2 2 ∂U e (y(1) − b) + (q2 − 1)e (y(2) − b) U − b − e 2 (y(2) − b) The second equality comes from substituting from (34). Because y(2) < y(1), (34) implies that (1 + ∗
q2∗ )e−q2 (y(2) − b) < U − b, so that
(37)
0>
∂q2∗ 1 1 1 1 + q2∗ >− =− . 1 ∂U U − b 1 − 1+q∗ U − b q2∗ 2
Next, differentiate (31) with respect to U to see that h h ∗ h ii ∂q ∗ i ∂w(q ∗ ) ∗ ∗ ∗ 2 2 = q2∗ + U − b − (w − b) e−q2 + q2∗ e−q2 2 − 2e−q2 − q2∗ e−q2 ∂U ∂U ∗ ∗ ∗ 2 − 2e−q2 − 2q2∗ e−q2 − q2∗ 2 e−q2 ∂q ∗ = q2∗ + (U − b) 2 ∗ ∗ ∗ −q −q ∂U 2 − 2e 2 − q2 e 2 ∗
>
q2∗
∗
∗
2 − 2e−q2 − 2q2∗ e−q2 − q2∗ 2 e−q2 1 + q2∗ − . ∗ ∗ q2∗ 2 − 2e−q2 − q2∗ e−q2
The second equality follows by substituting from (31) and the last line by substituting from (37). It is easy to verify numerically that the expression on the right side of the previous inequality is always strictly positive for q2∗ > 0, as is the coefficient on the left side. This establishes the claim. Let U2∗ be the value of U such that the value of q2M defined by (33) satisfies q2M = log(y(1)−b)−log(U −b), M
or equivalently e−q2 (y(1) − b) = U − b. To see that (35) holds if and only if U > U2∗ , first differentiate the
38
definition of q2M in (33) with respect to U to obtain that M
(38)
2
M
q2M [1 − e−q2 − 12 q2M e−q2 ] ∂q2M q2M 1 = − = −qM M 2 −q M U − b , M M M M −q −q ∂U [e 2 + q2M eq2 ](y(2) − y(1) − b) − (U − b) 2 2 2 1−e − q2 e − q2 e
where the last equality comes from substituting for y(2) − y(1) − b from (33) and rearranging. It suffices to M ∂q M e−q2 (y(1)−b) ∂ show that ∂U > 0, or equivalently that −(U − b) ∂U2 − 1 > 0. (38) implies that U −b M
(39)
∂q M q2M − 1 + e−q2 . − (U − b) 2 − 1 = 2 M M M ∂U 1 − e−q2 − q2M e−q2 − q2M e−q2
Both numerator and denominator of (39) are strictly positive for q2M > 0, as can readily be verified (the M
numerator is zero at q2M = 0 and has derivative 1 − e−q2 > 0 for q2M > 0; the denominator can be written M P∞ 1 Mj e−q2 j=3 j! q2 ). This completes the proof. Proof of Proposition 4. It is immediate from the argument given in the main text that the equilibrium is ¯ − y(h ¯ − 1). To constrained inefficient if a positive measure of applicants apply to a wage greater than y(h) establish the converse, since hiring is ex post efficient it is sufficient to show that the equilibrium is symmetric (so that applications are distributed evenly across firms). To prove this, observe that if a firm posts a wage ¯ − y(h ¯ − 1)], this wage w must maximize profits subject to (9); that is, w solves w ∈ [b, y(h)
(40)
max
¯ ¯ w∈[b,y(h)−y( h−1)]
¯ ¯ − b) − Ph−1 subject to h(w j=0
¯ − hw ¯ − y(h)
¯ h−1 X j=0
q j e−q ¯ ¯ − j)w] [y(h) − y(j) − (h j!
1 j −q ¯ (h j! q e
− j)(w − b) = q(U − b). Substitute from the constraint to eliminate ¯ q j e−q ¯ ¯ ¯ − hb ¯ − Ph−1 w, so that (40) is equivalent to maximizing −q(U − b) + y(h) j=0 j! [y(h) − y(j) − (h − j)b] for q ∈ [0, q¯]. The second derivative with respect to q of this expression is given by ¯ h−2 ¯ X q j eq ∂2π q h−1 e−q ¯ ¯ − 1) − b , = [y(j + 2) − 2y(j + 1) + y(j)] − y(h) − y(h 2 ¯ ∂q j! (h − 1)! j=0
¯ − y(h ¯ − 1) − b ≥ 0. Thus which is strictly negative because of the strict concavity of y(·) and because y(h) the optimum for q, and therefore for w, is unique. Proof of Proposition 5. It suffices to prove that if µ/ν is sufficiently small, then the market utility is no less ¯ − y(h ¯ − 1). In this case, all wages that attract positive queues must strictly exceed y(h) ¯ − y(h ¯ − 1); than y(h) this ensures the equilibrium is not constrained efficient according to Proposition 4. However, in the proof of Proposition 6, I will make use of the stronger result that if µ/ν is sufficiently small, then the market utility
39
exceeds y(2) − y(1). I therefore prove this stronger result. Therefore, suppose that there is a simple equilibrium with market utility U < y(2) − y(1). Since all firms maximize profits, it must be that among the profit-maximizing wage tuples, at least one attracts a queue length no greater than µ/ν, or else the labor market would not clear. If a firm attracts a total queue length bounded above by q = µ/ν, then from (13) its profit is bounded above by what it would produce if it always ¯ paid b to each of these workers, and did not make any hired all workers it received, up to a maximum of h, ¯ 1 j −q ¯ Ph−1 ¯ ¯ further payments to workers. Thus, its profit is bounded above by y(h)− [y(h)−y(j)−( h−j)b]. j=0 j! q e As q → 0, this expression converges to zero. However, this is inconsistent with profit maximization, since the firm could generate strictly higher profits by posting a wage greater than y(2)−y(1). Suppose the firm posts the wage 12 y(2) = 21 [y(1)+(y(2)−y(1))] > y(2) − y(1). The queue length attracted by posting 12 y(2), according to (9), satisfies 1q [1 − e−q ]( 12 y(2) − b) = U − b, and so is bounded below by the solution for q to the equation 1q [1 − e−q ]( 12 y(2) − b) = y(2) − y(1) − b because U < y(2) − y(1). The profit from posting 21 y(2), given by [1 − e−q ](y(1) − 12 y(2)), is then bounded away from zero by a bound that does not depend on µ/ν or U . This contradiction completes the proof. Proof of Proposition 6. I first prove part (a). As observed in the proof of Proposition 5, if µ/ν is sufficiently small, then U > y(2) − y(1). It follows that in a simple equilibrium, each firm posts a single wage at which it is willing to hire a single worker. In this range, the firm’s problem takes the same form as one already studied in the proof of Proposition 3, that of maximizing (27) subject to (28). As observed there, the wage the firm posts is such as to generate a queue length of log(y(1) − b) − log(U − b). This queue length is the same for each firm. Market clearing then requires that U be such that this queue length equal µ/ν (so that U = b + (y(1) − b) exp(−µ/ν)). Total output is Y e (µ, ν) = µb + ν(1 − exp(−µ/ν))(y(1) − b). Social welfare Y ∗ (µ, ν) is given by (3), so that hP ¯
i P∞ m −q e ¯ − hb ¯ qm e−q − (1 − e−q )(y(1) − b) − mb] q m! + m=h+1 y(h) ¯ m! ∆(µ, ν) = Ph¯ P∞ q m e−q ¯ − hb ¯ qm e−q + m=h+1 y(h) ¯ m=0 [y(m) − mb] m! m! 2 2 q(y(1) − b) + O(q ) − q(y(1) − b) + O(q ) = = O(q). q(y(1) − b) + O(q 2 ) h m=0 [y(m)
To prove part (b), first observe that no firm ever pays a worker a wage exceeding y(1), and no firm ever ¯ workers. Thus, the total income of workers is bounded above by ν hy(1) ¯ ¯ = employs more than h + (µ − ν h)b ¯ ¯ ν h[y(1) − b)] + µb, so that the market utility is bounded above by b + µν h[y(1) − b]. Since this expression converges to b as ν → 0, it follows that as ν → ∞, the market utility U → b. Next, the profit generated by a firm whose lowest posted wage is w is bounded above by maxh {y(h)−hw}, which is the realized profit when the firm receives many applicants to w. This expression is clearly decreasing 40
¯ ¯ ¯ ¯ ¯ ¯ in w; for w > y(h)−y( h−1), it is therefore bounded above by y(h−1)−( h−1)[y( h)−y( h−1)]. This bound is ¯ ¯ strictly less than the maximum profit attained by a firm posting a lower wage w ∈ (b, y(h)−y( h−1)], which is ¯ hw. ¯ However, for U small enough, the profit obtained by a firm posting any fixed w ∈ (b, y(h)−y( ¯ ¯ y(h)− h−1)] ¯ − hw. ¯ converges to y(h) This is because the queue length a firm whose lowest posted wage is w will attract to that wage becomes arbitrarily large as U → b. To see this, observe that in the notation of footnote 12, (9) shows that the queue length attracted by w satisfies [U − b]/[w − b] = 1q φh¯ (q). Because φh¯ (·) is strictly decreasing and continuous with limq→∞ φh¯ (q) = 0, it follows that q → ∞ as U → b. As q becomes large, no higher wages posted in addition to w will attract any applicants, so that firm profit is given by ¯ 1 m −q ¯ − hw ¯ − Ph−1 ¯ ¯ ¯ − hw ¯ as q → ∞. It follows that as U → b, the → y(h) y(h) m=0 [y(h) − y(m) − (h − m)w] m! q e ¯ − y(h ¯ − 1)]. In this case the optimal posted wage tuple for a firm consists only of a single wage w ∈ (b, y(h) equilibrium is constrained efficient according to Proposition 4. Proof of Proposition 7. Define U ∗ , the shadow value to the planner of an additional worker, by
(41)
U∗ − b =
¯ h−1 X
[y(m + 1) − y(m) − b]
m=0
µ m −µ e ν ν m!
.
I first prove a preliminary Lemma. Lemma 3. Suppose there exists a contract (H, W ) ∈ C and a queue length q > 0 such that for each m ∈ N, ¯ and such that (17) is satisfied for q = µ/ν and U = U ∗ . Then there is an h(m, H, W ) = min{m, h}, equilibrium in which all firms post (H, W ), and this equilibrium is constrained efficient. ˆ W ˆ ) satisfy Proof of Lemma 3. The profits of a firm posting any contract (H,
ˆ W ˆ , U ∗) = π(H,
∞ h i m −ˆq X ˆ W ˆ )) − W ˆ (m, h(m, H, ˆ W ˆ )) qˆ e y(h(m, H, m! m=0
= −ˆ q (U ∗ − b) +
∞ h X
ˆ W ˆ )) − h(m, H, ˆ W ˆ )b y(h(m, H,
m=0 ∞ X
i qˆm e−ˆq m!
qˆm e−ˆq ¯ − min{m, h}b ¯ y(min{m, h}) m! m=0 m − µ ∞ X µν e ν µ ∗ ¯ ¯ ≤ (U − b) + y(min{m, h}) − min{m, h}b ν m! m=0
≤ qˆ(U ∗ − b) +
Here qˆ is determined according to (17), which establishes the second equality. The second-last line is a singlepeaked function of qˆ; taking the first order condition shows that it is maximized when qˆ = µ/ν (the first-order condition coincides with (41)), which establishes the last line. However, posting the contract (H, W ) means that both inequalities hold with equality, since (17) guarantees that the queue length associated with this
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contract is µ/ν. This also implies that the resource constraint holds when all firms post this contract, completing the proof that this allocation is an equilibrium and therefore the proof of Lemma 3. Lemma 3 is very intuitive: if a contract allows for efficient hiring and for a worker’s expected income to equal U ∗ , then there is no way a deviant firm can generate greater profits by posting an alternative contract when workers must be paid U ∗ in expectation. Using the Lemma, the proof of Proposition 7 is ¯ workers and choose the straightforward. For environment (A), the firm needs to commit to hire min{m, h} h i ¯ µ m −µ 1 ¯ − Ph−1 ¯ wage to satisfy (w−b) h e ν = µν (U ∗ −b), so that (17) is satisfied for q = µ/ν and m=0 (h − m) m! ν U = U ∗ . For environment (B), the firm needs to commit to an application fee of U ∗ − b and a wage payment to each hired worker of b. For environment (C), the construction is given in the main text. It is easy to verify in the first two cases that
1 m
[W (m, h(m, H, W )) − h(m, H, W )b] is weakly decreasing in m = 1, 2, . . .,
as required in the definition of equilibrium; in the last case this property holds if w1 ≥ . . . ≥ wh¯ .
References Acemoglu, D. and W. B. Hawkins, “Equilibrium Unemployment in a Generalized Search Model,” unpublished, September 2010. Acemoglu, D. and R. Shimer, “Efficient Unemployment Insurance,” Journal of Political Economy 107 (October 1999a), 893–928. ———, “Holdups and Efficiency with Search Frictions,” International Economic Review 40 (November 1999b), 827–49. ———, “Wage and Technology Dispersion,” Review of Economic Studies 67 (October 2000), 585–607. Albrecht, J., P. A. Gautier and S. Vroman, “Equilibrium Directed Search with Multiple Applications,” Review of Economic Studies 73 (October 2006), 869–891. Albrecht, J. W. and B. Axell, “An Equilibrium Model of Search Unemployment,” Journal of Political Economy 92 (October 1984), 824–40. Bewley, T. F., Why Wages Don’t Fall During A Recession (Cambridge, MA: Harvard University Press, 1999). Burdett, K. and K. L. Judd, “Equilibrium Price Dispersion,” Econometrica 51 (July 1983), 955–69.
42
Burdett, K. and D. T. Mortensen, “Wage Differentials, Employer Size, and Unemployment,” International Economic Review 39 (May 1998), 257–73. Burdett, K., S. Shi and R. Wright, “Pricing and Matching with Frictions,” Journal of Political Economy 109 (October 2001), 1060–1085. Cooper, R., J. Haltiwanger and J. L. Willis, “Search frictions: Matching aggregate and establishment observations,” Journal of Monetary Economics 54 (September 2007), 56–78. Davis, S. J., R. J. Faberman and J. C. Haltiwanger, “The Establishment-Level Behavior of Vacancies and Hiring,” Working Paper 16265, National Bureau of Economic Research, August 2010. Diamond, P. A., “Wage Determination and Efficiency in Search Equilibrium,” Review of Economic Studies 49 (April 1982), 217–27. Elsby, M. and R. Michaels, “Marginal Jobs, Heterogeneous Firms, and Unemployment Flows,” unpublished, January 2010. Fujita, S. and M. Nakajima, “Worker Flows and Job Flows: A Quantitative Investigation,” Working Paper 09-33, Federal Reserve Bank of Philadelphia, November 2009. Gaumont, D., M. Schindler and R. Wright, “Alternative theories of wage dispersion,” European Economic Review 50 (May 2006), 831–848. Geromichalos, A., “Directed Search and Optimal Production,” Working Paper 10-16, UC Davis, August 2010. Guerrieri, V., “Inefficient Unemployment Dynamics under Asymmetric Information,” Journal of Political Economy 116 (October 2008), 667–708. Guerrieri, V., R. Shimer and R. Wright, “Adverse Selection in Competitive Search Equilibrium,” Econometrica 78 (November 2010), 1823–1862. Hawkins, W. B., “Competitive Search, Efficiency, and Multi-worker Firms,” Ph.D. diss., MIT, Chapter 1, July 2006. ———, “Bargaining with Commitment between Workers and Large Firms,” unpublished, May 2011a. ———, “Do Large-Firm Bargaining Models Amplify and Propagate Aggregate Productivity Shocks?,” unpublished, July 2011b.
43
Hosios, A. J., “On the Efficiency of Matching and Related Models of Search and Unemployment,” Review of Economic Studies 57 (April 1990), 279–298. Inderst, R., “Competitive search markets with heterogeneous workers,” European Economic Review 49 (August 2005), 1525–1542. Kaas, L. and P. Kircher, “Efficient Firm Dynamics in a Frictional Labor Market,” unpublished, January 2011. Lester, B., “Directed search with multi-vacancy firms,” Journal of Economic Theory 145 (November 2010), 2108–2132. Menzio, G. and S. Shi, “Block recursive equilibria for stochastic models of search on the job,” Journal of Economic Theory 145 (July 2010), 1453–1494. Moen, E. R., “Competitive Search Equilibrium,” Journal of Political Economy 105 (April 1997), 385–411. ´n, “Does Poaching Distort Training?,” Review of Economic Studies 71 (October Moen, E. R. and ˚ A. Rose 2004), 1143–1162. Montgomery, J. D., “Equilibrium Wage Dispersion and Interindustry Wage Differentials,” Quarterly Journal of Economics 106 (February 1991), 163–79. Mortensen, D. T., “Property Rights and Efficiency in Mating, Racing, and Related Games,” American Economic Review 72 (December 1982), 968–79. Peters, M., “Ex Ante Price Offers in Matching Games: Non-steady States,” Econometrica 59 (September 1991), 1425–54. Pissarides, C. A., “Efficient Job Rejection,” Economic Journal 94 (Supplement: Conference Papers 1984), 97–108. ———, Equilibrium Unemployment Theory, second edition (Cambridge, MA: MIT Press, 2000). Schaal, E., “Uncertainty, Productivity and Unemployment during the Great Recession,” unpublished, December 2010. Shimer, R., Contracts in a Frictional Labor Market, Ph.D. thesis, MIT (1996), chapter 1. ———, “The Assignment of Workers to Jobs in an Economy with Coordination Frictions,” Journal of Political Economy 113 (October 2005), 996–1025.
44
Smith, E., “Search, Concave Production and Optimal Firm Size,” Review of Economic Dynamics 2 (April 1999), 456–471. Stole, L. A. and J. Zwiebel, “Intra-firm Bargaining under Non-binding Contracts,” Review of Economic Studies 63 (July 1996a), 375–410. ———, “Organizational Design and Technology Choice under Intrafirm Bargaining,” American Economic Review 86 (March 1996b), 195–222. Tan, S., “Directed Search and Firm Size,” International Economic Review (February 2012), forthcoming. Wolinsky, A., “A Theory of the Firm with Non-Binding Employment Contracts,” Econometrica 68 (July 2000), 875–910.
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