Journal of Economic Theory 145 (2010) 1325–1353 www.elsevier.com/locate/jet

Stock–flow matching ✩ Ehsan Ebrahimy, Robert Shimer ∗ Department of Economics, University of Chicago, Chicago, IL, United States Received 11 March 2009; final version received 12 February 2010; accepted 17 February 2010 Available online 20 February 2010

Abstract We develop the implications of the stock–flow matching model for unemployment, vacancies, and worker flows. Workers and jobs are heterogeneous, so most worker–job pairs cannot profitably match, leading to the coexistence of unemployment and vacancies. Productivity shocks cause fluctuations in the number of jobs, which in turn cause fluctuations in other labor market variables. We derive exact expressions for employment and for worker transition rates in a finite economy and analyze their limiting behavior in a large economy. A calibrated version of the model is consistent with the observed co-movement and volatility of labor market variables. © 2010 Elsevier Inc. All rights reserved. JEL classification: E24; E32; J63; J64 Keywords: Unemployment; Vacancies; Worker flows; Aggregation; Business cycles; Labor market; Beveridge curve; Matching function

1. Introduction This paper develops and quantifies the implications of the stock–flow matching model [3,5,29] for labor market outcomes. Workers and jobs are heterogeneous, so most worker–job pairs cannot profitably match, leading to the coexistence of unemployed workers and job vacancies. Ag✩ We are grateful for comments from an anonymous referee and associate editor, seminar participants at Brown University, the Federal Reserve Banks of Minneapolis, New York, Philadelphia, and St. Louis, the 2006 SED Annual Meetings, the 2006 NBER Summer Institute, the 2007 UCSB–LAEF Conference on Trading Frictions in Asset Markets, and the 2009 JET Symposium on Search at Yale. Shimer’s research is supported by a grant from the National Science Foundation. * Corresponding author. E-mail addresses: [email protected] (E. Ebrahimy), [email protected] (R. Shimer).

0022-0531/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jet.2010.02.012

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gregate productivity shocks affect the number of jobs that firms create, which in turn cause fluctuations in unemployment, job vacancies, and worker flows. We derive exact expressions for these variables in an economy with finitely-many workers and jobs; prove convergence to a well-behaved limit when the number of workers and jobs is large and the probability that any particular worker–job pair can match is small; and quantitatively analyze the behavior of unemployment, vacancies, and worker flows in the large economy limit. We consider an economy where at any point in time, there are L workers and M jobs, with L constant and exogenous and M time-varying and dependent on firms’ job creation decision. Each worker–job pair can either match productively or cannot match. The probability that any particular match is unproductive is independent across workers and jobs and denoted by x. Moreover, it is costly for a firm to learn whether its job has a productive match with a particular worker, so match quality is an inspection good. We argue that if the inspection cost C is neither too large nor too small, it is optimal for firms with vacant jobs to inspect unemployed workers until they find a productive match, but it is not optimal for a vacant job to inspect an employed worker. If inspection costs are too high, firms with vacant jobs will not inspect unemployed workers; if costs are too low, they will inspect employed workers. Idiosyncratic productivity heterogeneity and moderate inspection costs are the key frictions in our economy. To understand how matching works, suppose that at some point in time, E workers are productively employed in a job, while each of the V = M − E vacant jobs knows that it cannot productively match with any of the U = L − E unemployed workers. Now consider a firm that creates a new job. It then proceeds to inspect each of the unemployed workers until it finds a productive match. If it fails to find one, the job becomes vacant. Similarly, when an idiosyncratic shock causes a filled job to exit the labor market, vacant jobs that have not yet inspected the newly-unemployed worker pay the inspection cost to see if they can productively match with her. If none can, she joins the stock of unemployed workers. Thus the inflow of newly unemployed workers matches with the stock of vacant jobs and symmetrically the stock of unemployed workers matches with the inflow of new jobs, the essence of the stock–flow matching model. Because of the idiosyncracies in matching, the number of employed workers is a random variable, even conditional on the total number of workers and jobs. We derive an exact formula for the distribution of the number of employed workers as a function of the current number of workers and jobs and the probability that any worker–job match is productive. We also derive an exact formula for the probability that the entry of a new job leads to an unemployed worker finding a job and for the probability that the exit of a job leads to an employed worker becoming unemployed. We then consider a sequence of economies in which the expected number of workers who can productively match with a job, α ≡ L(1 − x), and the expected cost of finding a good match, c ≡ C/(1 − x), are constant, but each of the components varies. More precisely, we take the limit of our finite-agent economy as the number of workers L converges to infinity, the matching probability 1 − x converges to 0, and the inspection cost C converges to 0, all at appropriate rates. We prove that in a large economy the employment rate is deterministic and depends only on the contemporaneous ratio of the number of jobs to workers, m ≡ M/L, and the parameter α. Similarly, the probability that a job exiting causes an employed worker to become unemployed and the probability that a job entering causes an unemployed worker to become employed are functions of m and α. We also argue that there is a range of values for the inspection cost c such that it is optimal for a vacant job to inspect a worker if and only if she is unemployed. Finally, we quantitatively examine how the economy responds to aggregate productivity shocks. The calibrated model generates two robust features of the U.S. labor market: the negative

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correlation between unemployment and vacancies at business cycle frequencies (the Beveridge curve) and the positive correlation between the rate that unemployed workers find jobs and the vacancy–unemployment (v–u) ratio (the reduced-form matching function). In the model and in the data, vacancies are slightly more volatile than unemployment and the two variables are strongly negatively correlated. The model predicts that a ten percent increase in the v–u ratio should be associated with a two percent increase in the job finding rate. In particular, the elasticity of the model-generated reduced-form matching function is virtually constant. Empirically the elasticity is constant but closer to 0.3. The calibrated model explains sixty percent of the volatility in the job finding rate, three-quarters of the volatility in the v–u ratio, and almost all of the volatility in the separation rate of employed workers into unemployment in response to small productivity shocks. These numbers are much larger than the corresponding values that [25] found in a search and matching model based on [22]. Previous research on stock–flow matching models has focused either on how wages are set [3,29] or on the empirical consequences of the model’s implication that the number of matches depends on both the stock and inflow of unemployed workers and job vacancies. For example, [4,5,28] stress that the matching function [22] might not represent a structural relationship; see also [13]. Most of these papers assume that all matches last forever.1 This simplifies the exposition of the model by obviating the need to compute the probability that a worker can take a new job when she loses her old job, but it also limits the possibility of using the model to examine the labor market. In particular, these papers have not derived the distribution of employment or the transition rate from employment to unemployment and back in the finite economy, nor the limiting behavior in the large economy; and they have not shown that stock–flow matching is quantitatively consistent with the empirical Beveridge curve and reduced-form matching function. This paper also contributes to the search [15] and matching [20,22,23] literature and especially to recent attempts to evaluate the matching model’s ability to explain the business cycle properties of unemployment, vacancies, and worker flows [e.g. [10,25]]. Our approach here abandons two key assumptions in the matching model. Rather than posit the existence of a stable matching function, we derive a matching process explicitly from the microeconomic heterogeneity. And rather than make particular assumptions about wage determination, we focus on the solution to a social planner’s problem. Our approach significantly improves the ability of the model to explain the cyclical behavior of labor markets. This paper is most closely related to [26]. In that paper, workers and jobs are located in distinct labor markets, corresponding to occupations or geographic locations. Any worker can take any job in her labor market but no job in another market, and each market clears with wages determined competitively. By assumption, the allocation of workers and jobs to labor markets is random, and so there are unemployed workers in some labor markets and vacancies in others. That paper does not explain why workers do not move to take the available jobs, nor why firms do not create jobs in markets where workers are available. One contribution of the current paper is to sidestep this exogenous mobility restriction. Our assumption that matching is idiosyncratic 1 A recent exception is [2], which assumes that each job has a constant returns to scale production technology and so is willing to hire any suitable worker. In this environment, the evolution of the stock of suitable jobs for each worker is independent of the behavior of all other workers and so is straightforward to calculate. Unfortunately for our purposes, while their model has a clear notion of unemployment, the stock of vacancies is not well defined. Moreover, it has quantitative difficulties in simultaneously explaining the transition rates between employment and unemployment and from one job to another. Our model comes closer to matching these facts.

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makes it plausible that it may be hard for an individual to alter her ability to match simply by moving. Technically, one can think of matching possibilities both in this paper and in [26] as an L × M matrix of zeros and ones, with each entry representing whether a particular match is productive. In both papers, firms do not know the realized structure of this matching matrix until after they create a job, although this paper is more explicit about the reason: inspecting workers is costly. The key difference between the two models is that in this paper the entries in the matching matrix are independent, while in [26] the matching matrix is block diagonal: if workers i and i  are both productive in some job j , then worker i can take job j  if and only if worker i  can. One can envision a range of alternative correlation structures that encompasses these two extremes.2 The stock–flow matching model in this paper delivers more amplification of productivity shocks than the mismatch model in [26], but that is entirely a consequence of the inspection costs. Introducing an analogous cost into [26] would deliver results that are nearly indistinguishable from the ones in this paper.3 This suggests to us that these results may be more general than either particular model. Still, we argue in the conclusion that the models have distinct microeconomic predictions, e.g. for duration dependence in unemployed workers’ job finding rate. In any case, the frictions analyzed in search models, mismatch models, and stock–flow matching models are complementary. A more comprehensive model might recognize that workers in distinct labor markets are poor substitutes; that not every worker can take every job within a labor market, as in this paper; and that switching labor markets or locating a suitable job within a labor market may require time-consuming search. We leave an empirical exploration of the relative importance of these various frictions for future research. The next section describes our model. Section 3 characterizes the planner’s solution in the finite economy. Section 4 considers the behavior of the limiting economy with many workers and a small chance that each job can match with any worker. Section 5 calibrates the limiting model and evaluates its quantitative performance. Section 6 concludes. 2. Model 2.1. Economic agents and preferences We study a continuous time, infinite horizon model. At any point in time t, there are a finite number L infinitely-lived workers and a countable number of infinitely-lived firms. All agents are risk-neutral and discount the future at rate r. While the number of workers is exogenous, the number of jobs M(t) is determined by firms’ job creation decision and by the exogenous exit of jobs. Each firm may create at most one job during the entire time horizon by paying a sunk cost k > 0, while each existing job is hit by an idiosyncratic productivity shock with arrival rate δ, independent across jobs and over time, forcing it to exit. The firm is inactive once its job exits. 2 [26] assumes a law of large numbers in an economy with a continuum of workers, jobs, and labor markets. A technical contribution of this paper is to characterize the limiting behavior of a sequence of finite economies, proving convergence and establishing the speed of convergence. 3 [19] shows how turnover costs that do not vary cyclically amplify productivity shocks in a matching model. Their logic carries over to the current framework. See also [27]. Indeed, inspection costs are countercyclical, since there are few workers to inspect during booms.

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2.2. Idiosyncratic heterogeneity Any worker–job pair can be either productive or unproductive. We let x ∈ (0, 1) denote the probability that a worker–job pair is unproductive and assume that the realization of this random variable is independent across worker–job pairs and is fixed as long as the job survives. A single (unemployed) worker produces z > 0 units of the numeraire homogeneous consumption good at home, while a single (vacant) job produces nothing. A productive worker–job pair can jointly generate a flow p(t) > z units of the same good if matched at time t, while an unproductive pair produces nothing and hence will not match. These stark assumptions give a concrete notion of unemployment and vacancies. Time variation in p(t), which we discuss below in Section 2.5, is the impulse for aggregate fluctuations. 2.3. Information At time t, all firms costlessly observe the history of productivity shocks p(τ ) and the history of the number of jobs M(τ ) for all τ  t, H t ≡ {p(τ ), M(τ )}tτ =−∞ . The strategy of a firm that has not yet created a job is a stopping rule, the optimal time to create a job, contingent on this information. Once a firm pays the cost k to create a job, it immediately observes all workers’ employment status and unemployment duration. Similarly, it observes whether other active jobs are filled and the duration of vacancies. The firm continues to costlessly monitor these outcomes as long as its job remains active. Only one information friction remains: the firm does not initially know whether it has a productive match with any particular worker. Matches are an inspection good, so the firm can learn about the quality of one of its potential matches by paying a cost C. Inspection is instantaneous, which means that the firm can immediately turn around and inspect more workers if it wants. The strategy of a firm with an active job is whether to inspect a worker, and if so which worker, and whether to hire a worker if a suitable one is available. Of course, this strategy must be measurable with respect to the firm’s information set, including workers’ employment status and the quality of each of the matches it has previously inspected. We introduce the inspection cost C to motivate our restriction, discussed further below, that firms with vacant jobs only inspect unemployed workers. If the inspection cost is too high, vacant jobs will be unwilling to inspect unemployed workers; if it is too low, they will inspect employed workers. We argue that for an intermediate range of C, a vacant job will inspect any unemployed worker but will never inspect an employed worker. We prove by construction in Section 4.3 that this range is nonempty in the large economy limit. 2.4. Social planner Rather than work directly in a decentralized economy, we consider the problem of a hypothetical social planner whose objective is to maximize the present value of output net of job creation and inspection costs. This sidesteps difficult issues about wage determination in this environment. Still, we are interested in understanding whether a decentralized economy can achieve the social optimum under some wage determination mechanism (see Section 4.5). To ensure that this is an interesting question, we impose the same information constraints on the social planner as arise in the decentralized economy. The planner must decide whether to create a new job only as a function of the history of productivity and the number of jobs, H t . He instructs firms with

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active jobs whether to inspect or hire workers, and if so which one to inspect or hire, as a function of the firm’s information set. Note that we assume firms always follow the planner’s instructions, even though the planner cannot directly observe their behavior. The implicit assumption that there are no incentive or enforcement constraints can only help the social planner and so strengthens the decentralization results. We also assume that the planner cannot circumvent the firms’ information constraint, for example by asking a firm with an active job to report the number of unemployed workers. The availability of this information would in general make the planner better off and so we impose this constraint to make the planner’s problem more comparable to the decentralized economy.4 2.5. Aggregate shock We focus on a single type of aggregate shock, fluctuations in aggregate productivity p(t). Our analytical results extend to fluctuations in other parameters, as long as aggregate shocks affect the labor market only through the job creation margin. Assume      (1) p(t) = py(t) = exp y(t) + 1 − exp y(t) p, where p is the minimum level of productivity, sufficiently high to ensure that firms with vacant jobs always find it profitable to inspect unemployed workers, and y(t) lies on a finite grid:   y ∈ Y ≡ −ν, −(ν − 1), . . . , 0, . . . , (ν − 1), ν .  > 0 is the step size and 2ν + 1  3 is the number of grid points. A shock hits y according to a Poisson process with arrival rate λ. The new value y  is either one grid point above or below y: 1  y y + , 2 (1 − ν ),  y = with probability y 1 y − 2 (1 + ν ). The probability that y  = y +  is smaller when y is larger, falling from 1 at y = −ν to 0 at y = ν. This implies y tends to revert to its mean of zero. Indeed, [25] shows that one can represent the stochastic process for y as dy = −γ y dt + σ db,

(2) √ where γ ≡ λ/ν measures the speed of mean reversion and σ ≡ λ is the instantaneous standard deviation.5 To save on notation, let Ep Xp denote the expected value of an arbitrary statecontingent variable X following the next aggregate shock, conditional on the current state p. 3. Characterization This section characterizes the solution to the planner’s problem under the restriction that vacant jobs always inspect unemployed workers and never inspect employed workers, while filled 4 The plannner’s decision to create a job would in general depend on productivity, the number of active jobs, and the number of employed workers. It is straightforward to construct examples in which this last piece of information is useful to the planner. 5 Suppose one changes the three parameters of the stochastic process, the step size, arrival rate of shocks, and number √ of steps, from (, λ, ν) to ( ε, λε , νε ) for any ε > 0. This does not change either γ or σ , but as ε → 0, y converges to an Ornstein–Uhlenbeck process.

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jobs never inspect any worker. We offer some justification for this restriction in Section 3.1. We start by describing the state of the economy and the planner’s policy determining which job inspects which worker in which order. Using that, we construct an algorithm that tells us which worker is matched to which job at any point in time conditional on the state of the economy. We use our algorithm to establish a form of history-independence: the distribution of employment at time t conditional on the history of productivity and the number of jobs until time t, H t , in fact depends only on the number of jobs at time t, M(t); and similarly for the probability that the entry of a job causes an unemployed worker to become employed and the probability that the exit of a job causes an employed worker to become unemployed. This allows us to reduce the payoff-relevant state space to just the current values of productivity and the number of jobs, (p(t), M(t)). Finally, we use these mechanical relationships to describe firms’ decision to create a job as a function of p(t) and M(t) alone. 3.1. State and policy We first describe the initial state of the economy at time t. There are M(t) jobs and E(t)  min{L, M(t)} employed workers, each in a productive match with one job. Each firm with a filled job knows it has a productive match with its employee and may know that it has an unproductive match with some of the other workers in the economy, but is otherwise uninformed about its matching possibilities. Each of the V (t) = M(t) − E(t) vacant jobs knows that it does not have a productive match with any of the U (t) = L − E(t) unemployed workers. Vacant jobs may also know that they have an unproductive match with some of the other workers in the economy, but are otherwise uninformed about their matching possibilities. We argue that an economy starting from this initial condition replicates this situation as firms create jobs, jobs stochastically end, and jobs inspect available workers.6 In particular, suppose the planner uses the following policy, which we justify below: Definition 1. The planner’s policy is to instruct firms to inspect workers as follows: • Immediately after a firm creates a job, it inspects the unemployed workers sequentially, starting with the one who has been unemployed the longest. If it has a productive match with one of the workers, it stops the inspections and hires the worker. Otherwise the job remains vacant. • Immediately after a filled job exits, vacant jobs that have not yet inspected the previouslyemployed worker do so sequentially, starting with the job that has been open the longest. If the worker has a productive match with one of the jobs, the inspections stop and the worker is hired. Otherwise the worker becomes unemployed. This policy ensure that a firm is matched with a worker if and only if it knows it has a productive match with the worker. A vacant firm knows that it cannot productively match with any of the unemployed workers. Thus entry and exit of jobs replicates the initial state of the economy. 6 Note that we do not attempt to solve the planner’s policy with an arbitrary initial condition, but we conjecture that the economy will converge to this situation from an arbitrary starting point.

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To execute this policy, the planner needs a complete ordering over jobs based on the time that they were created. If two or more jobs were created at the same instant, as may happen after a positive productivity shock, we assume the planner determines the order of the jobs randomly. The outcome of this randomization affects the probability that each job is subsequently filled but does not affect the value of the planner’s objective function. It remains to prove that this policy is optimal. The policy has three components. First, when a firm creates a job, it inspects the workers sequentially, starting with the one who has been unemployed the longest. One can formally prove that this does not affect the value of the planner’s policy, and so we make this tie-breaking assumption for expositional convenience. Second, when a worker becomes unemployed, the firm that has had a vacant job for the longest inspects the worker first. This is not a normalization, but rather is at least locally optimal. To understand why, observe that if two jobs j and j  are vacant and job j is older than job j  , then job j has unsuccessfully inspected every worker that job j  has inspected, but possibly not vice versa. It follows that job j is less likely to have a productive match among the entire population of workers. Since one of these workers may later become unemployed, generating an opportunity for the job to match, the planner has an incentive to fill job j in preference to job j  when both choices are feasible. Finally, and most importantly, the planner instructs a vacant firm never to inspect an employed worker and instructs a firm with a filled job never to inspect any worker. If the inspection cost were too small, such inspections might be optimal. For example, suppose there are two workers and one job, which happens to be held by worker 1. A new job enters and does not have a productive match with worker 2. If the new job could productively match with worker 1 and the old job could productively match with worker 2, reallocating workers and jobs would create more output. With costly information acquisition, learning about this possibility may be suboptimal, even though inspecting unemployed workers may be optimal. To understand why, note that it costs C for the new job to inspect worker 2, generating a match with probability 1 − x. Now consider the costs and benefits of the new job inspecting worker 1. First note that, under the proposed policy, a firm never knows that it can productively match with more than one worker, so the old job either knows it cannot match with worker 2 or it is uncertain about the possibility. If the old job cannot match with worker 2, then sending the new job to inspect worker 1 costs C and cannot possibly create an additional match. On the other hand, suppose the old job is uncertain and the new job inspects worker 1 at cost C. If the inspection is successful, with probability 1 − x, the old job inspects worker 2 at cost C. The total expected cost from following this policy is (2 − x)C, while a new job is created only if both inspections are successful, with probability (1 − x)2 . Letting W denote the value of a filled job, the net benefit from a vacant job inspecting an unemployed worker is (1 − x)W − C, while the net benefit from a vacant job inspecting an employed worker 2 is (1 − x)2 W − (2 − x)C. For C ∈ ( (1−x) 2−x W, (1 − x)W ), a nonempty interval, the first is positive and the second is negative.7 Of course, many other configurations are possible, beyond this simple two-job, two-worker example. We defer a precise statement of the conditions under which the proposed policy is optimal until Section 4.3, where we study the limiting economy with many workers and jobs.

7 This calculation assumes that the new job knows that the old job has not yet inspected worker 2. If there is uncertainty about this, the net benefit from the new job inspecting worker 1 is smaller.

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3.2. A matching algorithm Given the planner’s policy, we next describe an algorithm that computes the matching pattern at any point in time t. To execute this algorithm, we order workers based on the amount of time since they were last displaced from a job and order jobs based on the time that they were created. Definition 2. The matching algorithm takes a set of L workers and M(t) jobs and iteratively assigns workers to jobs. Let ja denote the ath-oldest job. 1. Set a = 1 and start with all the workers unmatched. 2. If job ja can productively match with at least one of the workers whom the algorithm has not yet matched, assign it the one who has experienced the longest time since her last displacement. Otherwise job ja is vacant. 3. Increase a by 1. If a  M(t), repeat step 2. We stress that this is not a description of how matching happens in the economy, but rather a computational algorithm that allows us at any point in time to determine who matches with whom. It is useful because it provides a simple way of establishing some important characteristics of the planner’s policy. Proposition 1. At any time t, the assignment of workers to jobs under the planner’s policy is the same as the one given by the matching algorithm. The proof is in Appendix A. 3.3. History independence We are interested in analyzing how a few summary statistics depend on current and past labor t ) denote the distribution of employment at time t conditional on ˜ market conditions. Let φ(E|H the history of the productivity and the number of jobs H t ; let Π˜ UE (E t , H t ) denote the probability that the entry of a job at time t leads to an unemployed worker finding a job, conditional on H t and on the history of employment E t ≡ {E(τ )}tτ =−∞ ; and let Π˜ jEU (E t , H t ) denote the probability that the exit of job j at time t leads to an employed worker becoming unemployed conditional on the same variables. The following result simplifies the history-dependence of these objects: Lemma 1. For any histories H t and E t ,         φ˜ E|H t = φ E|M(t) , Π˜ UE E t , H t = Π UE E(t), M(t) ,     Π˜ jEU E t , H t = ΠjEU E(t), M(t) .

and

In words, each of these functions depends only on the current state. t ). The matching ˜ Proof of Lemma 1. Consider the distribution of the employment rate φ(E|H algorithm constructs the realized matching pattern just from knowledge of the current number of jobs, the ordering of workers and jobs, and the ability of each job–worker pair to match. Since each possible ordering is equally likely and the ability of each job–worker pair to match is a

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binomial random variable, the probability distribution of the employment rate depends on the current number of jobs alone. The proof of the other results is similar. 2 3.4. Employment distribution We now use the matching algorithm to find the probability distribution over the number of employed workers given the number of jobs. Proposition 2. The probability that there are E ∈ {0, 1, 2, . . .} employed workers when there are M jobs is φ(E|M) = x (L−E)(M−E)

E−1  i=0

(1 − x L−i )(1 − x M−i ) . 1 − x i+1

(3)

The proof is in Appendix A. Note that φ(E|M) = 0 for all E > M. Proposition 2 provides a precise characterization of the distribution of employment conditional on the current number of jobs. [12] calls φ the “absorption distribution” and describes several environments where it may arise, including an unrelated birth–death process. Her characterization of the properties of this distribution is critical for many of our results in Section 4. 3.5. Worker flows We next use the matching algorithm to find the probability that the entry of a job allows an unemployed worker to find a job. Proposition 3. The probability that the entry of a new job leads to an unemployed worker finding a job when there are already E employed workers and M jobs is Π UE (E, M) = 1 − x L−E ,

(4)

independent of the number of jobs M. Proof of Proposition 3. The entry of a new job leads to an unemployed worker finding a job if the new job can match with one of the unemployed workers, with probability 1 − x U , where U = L − E is the number of unemployed workers. 2 The expression for the probability that the exit of a job causes an employed to become unemployed is almost as simple but more cumbersome to derive. Proposition 4. The probability that the exit of job j causes an employed worker to become unemployed when there are already E employed workers and M jobs is ΠjEU (E, M) =

x −E − 1 . x −M − 1

(5)

The proof is in Appendix A. We stress that ΠjEU (E, M) does not depend on which job exits and hereafter suppress its dependence on j . When a newer job exits, it is more likely to be vacant. When an older job

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exits, it is more likely that it is filled by a worker who can immediately move to another job. Perhaps surprisingly, these effects offset, so the probability that an employed worker becomes unemployed when a job exits does not change with the age of the job. One can verify algebraically that Π UE (E −1, M −1)φ(E −1|M −1) = Π EU (E, M)φ(E|M). This is a statement that worker flows balance: the left-hand side is the probability that there are E − 1 employed workers when there are M − 1 jobs and the entry of the Mth job leads to an unemployed worker finding a job. The right-hand side is the probability that there are E employed workers when there are M jobs and the exit of one of the jobs leads to an employed worker becoming unemployed. Similarly, ∞

Π EU (E, M)φ(E|M) =

E=0

∞   E φ(E|M) − φ(E|M − 1) , E=0

so the expected decrease in employment when one of M jobs exits is just the expected difference in employment between an economy with M and M − 1 jobs. 3.6. Job creation decision We finally consider the planner’s decision to create a new job. Let Wp (M) denote the expected present value of net output per worker when current productivity is p and the current number of jobs is M. We can represent the planner’s policy recursively as

M   ˜ p (M) − Wp (M) rWp (M) = φ(E|M) pE + z(L − E) + λ Ep W E=0

˜ p (M − 1) − Wp (M) − + δM W

where



E C − Π EU (E, M) , 1−x M

M C ˜ p (M) = max Wp (M), W ˜ p (M + 1) − k − W φ(E|M)Π UE (E, M) . 1−x

(6)



(7)

E=0

We start by explaining Eq. (7), which shows the planner’s decision to create a new job. The ˜ p (M) captures that decision. If the planner does not create a new job, auxiliary value function W his value is Wp (M). Otherwise, he pays the creation cost k and some inspection costs. When there are L − E unemployed workers, the inspection cost is C if the first inspection succeeds, with probability 1 − x; 2C if the first inspection fails but the second succeeds, with probability x(1 − x), etc. In addition, if all L − E inspections fail, with probability x L−E , the inspection cost is C(L − E). In total, the expected inspection cost is

L−E−1  C i L−E Π UE (E, M), (i + 1)x (1 − x) + (L − E)x C = 1−x i=0

where we obtain the equality by simplifying the summation and using Eq. (4). Multiplying by the probability that E workers are employed and summing, we arrive at Eq. (7). It is worth noting an alternative derivation of the expected inspection costs. Suppose that the planner anticipates that, when he creates a new job, it is immediately filled with some probability π . To obtain this probability, the job has to inspect on average π/(1 − x) firms, since

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each inspection succeeds only with probability 1 − x. The expected inspection cost is therefore πC/(1 − x). In this case, the entry of a new job leads to an unemployed worker become emUE (E, M), giving the expected inspection costs ployed with probability π = M E=0 φ(E|M)Π in Eq. (7). Now consider Eq. (6), which describes the value function of a planner who has decided not to increase the number of jobs. We stress that, although this choice is available at each instant, it is only optimal to create new jobs at the moments after either productivity changes or a job exits. Eq. (6) is a standard continuous time Bellman equation. The planner’s flow value is obtained by integrating across the unknown number of employed workers, E with probability φ(E|M). Three terms follow. First, total output is pE + z(L − E). Second, an aggregate shock hits at ˜ p (M) − Wp (M); the tilde indicates rate λ, leading to a capital gain with expected value Ep W that the planner may want to exercise his option to increase the number of jobs at this juncture. Finally, a job exits at rate δM. This has two consequences. First, there are now M − 1 jobs ˜ p (M − 1) − Wp (M), again with the option rather than M jobs, causing a capital loss of W of increasing the number of jobs. Second, firms may incur some cost of inspecting additional workers, here equal in expectation to (C/(1 − x))(E/M − Π EU (E, M)) conditional on current unemployment E. This is easiest to see using the second derivation above. Now π = E/M − Π EU (E, M) is the probability that when a job exits, an employed worker switches jobs. This is the difference between the probability that the job is filled and the probability that the exit of a job causes a worker to become unemployed. Then the expected inspection costs is πC/(1 − x). Associated with these Bellman equations is an optimal policy: for each p, there are values ˜ p (M) = Wp (M) and the planner does not create new jobs; and values of M of M such that W ˜ such that Wp (M) > Wp (M) and the planner immediately creates at least one new job. One would expect this policy to be described by state-contingent thresholds Mp∗ so the planner creates no new jobs if M  Mp∗ and productivity is p and otherwise immediately increases the number of jobs to Mp∗ . 4. Large economy Although it is possible to work in an economy with a finite number of workers and jobs, it is computationally cumbersome. We show in this section that the finite economy has a relatively simple limit as it grows large. Moreover, we find numerically that convergence to that limit is rapid. Our approach parallels the previous section: we first describe the mechanics of employment and worker flows conditional on the ratio of jobs to workers and then turn to the determination of the number of jobs. Let m ≡ M/L denote the number of jobs per worker at some point in time, α ≡ L(1 − x) denote the expected number of workers with whom a job can productively match, and c ≡ C/ (1 − x) denote the expected inspection costs to create a job. We focus on the limiting behavior of the economy as L converges to infinity holding α and c fixed (so the probability a match is unproductive, x, converges to 1 and the cost of inspecting a worker, C, converges to 0), for arbitrary values of the endogenous variable m. 4.1. Employment Proposition 2 provides an exact expression for the probability that there are E employed workers when there are L workers and M jobs. This distribution has a simple limit in the large economy.

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Table 1 Convergence. L

1000

10,000

100,000

∞

U/L

5.180% (0.344) 2.348% (0.354)

5.220% (0.109) 2.389% (0.112)

5.224% (0.035) 2.393% (0.036)

5.224% (0) 2.394% (0)

Π UE

0.6367 (0.0244)

0.6370 (0.0077)

0.6371 (0.0024)

0.6371 (0)

Π EU

0.6412 (0.0431)

0.6375 (0.0135)

0.6371 (0.0043)

0.6371 (0)

V /L

Rows show unemployment rate, vacancy rate, probability entry causes employment, and probability exit causes unemployment (standard deviations in parenthesis) with M = 0.971L and x = 1 − 19.4/L and various values of L.

Proposition 5. Fix α = L(1 − x). For given m = M/L, consider the limit as the number of workers L converges to infinity. The fraction of workers who are employed, E/L, converges in mean square to e(m) = 1 + m −

  1 log exp α + exp(αm) − 1 . α

(8)

The proof is in Appendix A. Manipulation of Eq. (8) gives the unemployment rate (unemployment divided by unemployment plus employment) and vacancy rate (vacancies divided by vacancies plus employment) as well:   1 (9) u(m) = 1 − e(m) = log exp α + exp(αm) − 1 − m, α

   e(m) 1 1 v(m) = 1 − = log exp α + exp(αm) − 1 − 1 . (10) m m α This implicitly defines the unemployment rate as decreasing in m and the vacancy rate as increasing in m, and hence the vacancy rate as a decreasing function of the unemployment rate for any α, a theoretical Beveridge curve. The first two rows in Table 1 show the rapid convergence of the unemployment rate and vacancy rate when the job–worker ratio is fixed at 0.971 and there are on average α = 19.4 suitable workers per job. We choose these values because they imply unemployment and vacancy rates of 5.2 and 2.4 percent, respectively, in the limiting economy, the recent average values in the U.S. economy.8 In the finite economy, U/L and V /L are random variables, and so the numbers in parenthesis show how the standard deviation falls as the economy increases in size. To obtain some intuition for Eq. (8), we can work with a version of the matching algorithm in an economy with a continuum of agents. Order the workers i ∈ [0, 1] according to the amount of time since they last lost a job. Similarly order the jobs according to the amount of time since they entered, with job 0 the oldest. Then match jobs to workers sequentially, giving job 0 the opportunity to match first. Since there are 1 ≡ u(0) workers available, job 0 has a match with probability 1 − exp(−αu(0)). Proceeding sequentially, when job m has the opportunity to match, 8 We discuss the unemployment and vacancy data in Section 5.

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there are u(m) available workers and so she has a match with probability 1 − exp(−αu(m)). This suggests u (m) = −(1 − exp(−αu(m))). The solution to this differential equation gives Eq. (9). Although this intuition obtains the correct solution, we believe our convergence results are still important. First, it may be easier to understand the mechanics and the matching procedure in the finite case. Second, by starting from a finite economy, we can check how quickly the economy converges to its limiting values. Third, there are standard measurability issues in the continuum agents economy. The intuition in the previous paragraph not only assumes the existence of the function u(m), but also its derivative. To ensure the existence of such a function, the set of all available workers for any job m must be measurable for an appropriate and well-defined measure; proving this is difficult, especially when the set of all available workers for job m is defined in a stochastically sequential manner. Moreover even if we succeed in constructing a well-defined function u(m), we need to appeal to a law of large numbers in the continuum agents economy to obtain the expression for u (m). A traditional law of large numbers does not exist [7,11], and standard remedies require assumptions like stochastic independence that are either false or not simple to verify in our context [see [1], for a recent discussion of these issues]. For example, [30] requires convergence in mean squared, a property that we establish only by explicitly taking limits of the discrete model. More importantly, many of these proposed solutions are built on a sequence of discrete measures, the limit of which has analogous features to the continuum. While such an approach may yield a sensible analog of the law of large numbers, it is unclear whether there are measures that would allow us to differentiate u(m). Still, our hope is that this paper may justify the use of continuum agent models in similar contexts. 4.2. Worker flows We also obtain simple limits for the probability that the entry of a job leads to an unemployed worker finding work and that the exit of a job leads to an employed worker losing her job. Proposition 6. Fix α = L(1 − x). For given m = M/L, consider the limit as the number of workers L converges to infinity. The probability that the entry of a new job leads to an unemployed worker finding a job and the probability that the exit of an old job leads to an employed worker becoming unemployed both converge in mean square to π(m) =

exp α − 1 . exp α + exp(αm) − 1

(11)

The proof is in Appendix A. The last two rows of Table 1 show the rapid convergence of Π UE and Π EU to π . Note that in a large economy, the probability that an entrant hires a worker equals the increase in the employment rate from the entry of a single job, π(m) = e (m); we can confirm this directly by differentiating Eq. (8). Symmetrically, this must also equal the probability that a job exiting leads to a worker becoming unemployed. 4.3. Inspection costs In the large economy, we can bound the inspection cost c to ensure that the planner does not instruct vacant jobs to inspect employed workers. The planner has the strongest incentive to undertake such an inspection when a firm has been vacant for a long time, and hence has learned that

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it cannot match with almost any worker. To be precise, suppose that the firm has inspected all but ε of the workers, all of whom are employed. We consider a one-time deviation, where the planner instructs such a firm to inspect the remaining workers. Since the deviation is small, it does not affect the subsequent matching opportunities of workers and firms.9 It produces a productive match with probability αε and so the expected cost of inspecting these workers is cαε. If the inspections fail, the firm possibly saves the cost of inspecting those workers at a later date, assuming it is still vacant when the workers’ current employers exit. The expected undiscounted savings is 12 cαε, where the 12 accounts for the fact that the vacant firm may exit before the current employer, while the discounted savings is worth less. Thus the net cost of the inspections is at least 12 cαε. On the other hand, if the inspection succeeds, with probability αε, a filled job is released to inspect the unemployed workers. We know that this released job can match with at least one worker (the one it had formerly employed) and in expected value can match with at most another α workers, as would be the case if it had not yet inspected any of the other workers. It follows that the released job is no better than 1 + 1/α new jobs. In general the value of a released job may be lower. For example, the released job may know that it cannot match with some workers; and the 1 + 1/α new jobs can hire 1 + 1/α workers, while the released job can hire at most one. This provides an upper bound for the value of this released job, the cost k(1 + 1/α) of creating this many new jobs. The benefit of inspecting a filled job is bounded above by the product of this and the probability that the inspection succeeds. Putting this together, a sufficient condition to ensure that it is unprofitable to inspect an employed worker is 12 cαε  k(1 + α)ε or 2(1 + α) k. (12) α We impose this inequality in our numerical work but emphasize the more general point that inspection costs may explain why vacant jobs do not inspect employed workers.10 On the other hand, a sufficient condition to ensure that the planner finds it profitable for a vacant job to inspect an unemployed is c

p − z  c(r + δ).

(13)

This implies that even in the worst possible state, the net gain from employing a worker, p − z, exceeds the annuitized cost of inspecting the worker. Since productivity is mean reverting, this condition could be weakened as well. 4.4. Job creation decision The planner’s job creation decision is slightly simpler in a large economy. Now let Wp (m) denote the expected value of output net of job creation and inspection costs when the firm– worker ratio is m. Taking limits of Eqs. (6) and (7) and using π(m) = e (m), this satisfies      rWp (m) = max pe(m) + z 1 − e(m) + g Wp (m) − k − cπ(m) − δmWp (m) g0     − δc e(m) − mπ(m) + λ Ep Wp (m) − Wp (m) . (14) 9 We are unable to characterize global deviations because we cannot compute their implications for subsequent matching distributions. We therefore focus on the best local deviation from the proposed allocation. 10 Another possibility, particularly relevant in a decentralized economy, is that firms are reluctant to engage in competition for workers.

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Here g is the gross increase in the number of jobs per worker. The flow value of the planner can be divided into four terms. First is current gross output, p for each of the e(m) employed workers and z for each of the 1 − e(m) unemployed workers. Second is the total cost of creating jobs at gross rate g. Each new job increases m, raising the value of the state variable. From this we subtract the direct cost k of creating the job and the expected inspection costs, which are proportional to the resulting increase in employment. Third is the change in the value that comes from jobs exiting at rate δ. Fourth is the inspection cost from worker turnover. An employed worker loses her job at rate δe(m), while the exit of a job causes a worker to become unemployed at rate δmπ(m). The difference between these numbers is the rate that an employed worker switches jobs, which incurs the inspection cost c. The final term reflects the possibility of an aggregate shock. The first order condition for the gross amount of job creation conditional on (p, m) is   Wp (m)  k + cπ(m), and gp (m) Wp (m) − k − cπ(m) = 0, (15) gp (m)  0, where we use e (m) = π(m) to simplify the expression slightly. That is, whenever the marginal value of a job is smaller than the cost of creating the job and inspecting the unemployed workers, k + cπ(m), gross job creation is zero and conversely, if some jobs are being created, the marginal value of a job must equal its cost. The envelope condition is    (r + δ)Wp (m) = (p − z)π(m) + gp (m) − δm Wp (m) − cπ  (m)   + λ Ep Wp  (m) − Wp (m) , (16) where we again use e (m) = π(m). Combining these conditions, we can define productivity-contingent thresholds m∗p such that if m > m∗p , no new jobs are created, gp (m) = 0, so     (r + δ)Wp (m) = (p − z)π(m) − δm Wp (m) − cπ  (m) + λ Ep Wp  (m) − Wp (m) . (17) m = m∗p ,

Wp (m) = k + cπ(m),

gp (m) = δm and so the envelope condition On the other hand, if reduces to          (r + δ)k = p − z − c(r + δ) π m∗p + λ Ep Wp  m∗p − k − cπ m∗p . (18) In steady state, this implies that the cost of creating a new job k must be equal to the present value of the output of a filled job, p/(r + δ), minus the present value of the worker’s opportunity cost, z/(r + δ), minus the expected inspection cost from a new job c, all multiplied by the probability that a new job is immediately filled, π(m). Finally, if m < m∗p , entry immediately drives m up to m∗p so Wp (m) = k + cπ(m). We close by providing a constructive proof of the existence and uniqueness of a solution to the planner’s problem. Proposition 7. There is a unique solution to the planner’s problem. In it, the targets m∗p are increasing. The proof is in Appendix A. Lower current productivity, which presages lower future productivity, reduces the revenue from a filled job. The optimal response is to reduce the number of jobs. This raises the share jobs filled by a worker who would otherwise be unmatched, π(m), leaving the expected return to creating a job equal to zero.

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4.5. Decentralization We conclude this section by suggesting one way that a market may decentralize the social planner’s problem. We suppose that, while firms must pay the entry cost k, workers bear the inspection cost c. When a worker is employed, she earns a wage that is a weighted average of her productivity p and the sum of her value of leisure z and the annuitized screening cost c(r + δ). The probability she receives the high wage p is equal to the probability that she could find a job among the vacant firms conditional on all available information. If she succeeded in finding such a job, the two firms would be forced into Bertrand competition, driving the wage up to p. Otherwise she receives the lower wage z + c(r + δ). This ensures that the expected flow profitability of a job is (p −z−c(r +δ))π(m), where π(m) is the probability that a job is filled by a worker with no opportunities among the vacant firms, rather than being either vacant or filled by a worker with at least one alternative employment possibility. Critically, the expected flow profitability of a job is independent of its age, since π(m) is independent of age (Proposition 4). Newer jobs are more likely to be vacant, while older jobs are more likely to be filled, but by a worker who is paid a higher wage because she is more likely to find another job among the vacant firms, if she would look for one. This means firms do not have an incentive to try to enter before (or after) their competitors, as might be a concern following a positive productivity shock. This wage is a version of the Mortensen rule [17]: in order for firms to have the proper incentive to create jobs, they must compensate the worker only for the value of her time and the inspection cost when the worker would otherwise be unemployed, and must pay the worker her full marginal product if the worker would otherwise have a job opportunity. According to the model, a matched worker and firm do not know about the worker’s other employment opportunities, but the since both are risk neutral, the worker can simply be paid according to the expected value of this calculation. Let Jp (m) denote the value of creating a job when current productivity is p and the current number of jobs per worker is m. As in the planner’s problem, there is a productivity-contingent target for the number of jobs m∗p . When productivity is p and m > m∗p , the value of a job is smaller than the cost of creating a new one, Jp (m) < k, and the value evolves according to a standard continuous time Bellman equation:   rJp (m) = p − z − c(r + δ) π(m) − δJp (m) − δmJp (m)   + λ Ep Jp (m) − Jp (m) . (19) The flow value of a job is given by the sum of four terms. The first is the expected flow of profits, p − z − c(r + δ) if the job is filled by a worker with no other employment opportunities and zero otherwise. The second term is the risk the job is destroyed, at rate δ. The third is the rate of change in the stock of jobs, −δm, multiplied by the effect this has on the firm’s value. Finally, a capital gain can change the level of productivity from p to p  . When productivity is p and m = m∗p , job destruction is exactly offset by job creation, keeping m constant and Jp (m∗p ) = k. In this case, the Bellman equation satisfies         rk = p − z − c(r + δ) π m∗p − δk + λ Ep Jp m∗p − k . (20) The expression is basically unchanged, except the stock of jobs is now constant. Finally, if productivity is p and m < m∗p , firms immediately create jobs to bring m up to m∗p . In this case, Jp (m) = k as well. Using these Bellman equations, we prove the following result:

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Proposition 8. The decentralized equilibrium is unique and coincides with the solution to the planner’s problem. The proof in Appendix A verifies that Jp (m) = Wp (m) − cπ(m) and shows equivalence between the equations governing the decentralized equilibrium and the planner’s problem. 5. Quantitative evaluation We now calibrate the large economy model to quantify the cyclical behavior of unemployment, vacancies, and worker flows. We first discuss our choice of the parameter α which governs the level of unemployment and vacancies. We then calibrate the remaining parameters and simulate a stochastic version of the model economy. 5.1. Beveridge curve We start by examining the model-generated unemployment and vacancy rates, given by Eqs. (9) and (10). For a given value of α, these equations implicitly define the vacancy rate as a decreasing function of the unemployment rate. We compare this with U.S. data on unemployment and job vacancies. The Bureau of Labor Statistics (BLS) uses the Current Population Survey (CPS) to measure the unemployment rate each month. The ratio of unemployment to the sum of unemployment and employment is the unemployment rate. Since December 2000, the BLS has measured job vacancies using the JOLTS. This is the most reliable time series for vacancies in the U.S. According to the BLS, “A job opening requires that 1) a specific position exists, 2) work could start within 30 days, and 3) the employer is actively recruiting from outside of the establishment to fill the position. Included are full-time, part-time, permanent, temporary, and short-term openings. Active recruiting means that the establishment is engaged in current efforts to fill the opening, such as advertising in newspapers or on the Internet, posting help-wanted signs, accepting applications, or using similar methods.”11 We measure the vacancy rate as the ratio of vacancies to vacancies plus employment. The dots in Fig. 1 show the strong negative correlation between unemployment and vacancies over this time period, the empirical Beveridge curve. From December 2000 to November 2007, the unemployment and vacancy rates averaged 5.2 percent and 2.4 percent, respectively. Inverting Eqs. (9) and (10), this is consistent with α = 19.4 and m = 0.971. Now hold α fixed and consider how variation in m, in response to productivity shocks according to the model, affects unemployment and vacancies; this is the line in Fig. 1. We stress that the model cannot produce points off this line unless α changes. The fit of the model to the data is excellent. That the level of the model-generated Beveridge curve fits the data reflects the choice of α, but that the slope and curvature of the model-generated Beveridge curve also fits the data comes from the structure of the model. Fig. 1 is virtually indistinguishable from Fig. 1 in [26]. The similar results in the mismatch and stock–flow models suggests to us that the Beveridge curve may simply be an aggregation phenomenon; however, there is no mathematical sense in which one model nests the other. [18] provides a third example of this aggregation result. 11 See BLS news release, July 30, 2002, available at http://www.bls.gov/jlt/jlt_nr1.pdf.

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Fig. 1. Beveridge curve. The dots show U.S. monthly data from December 2000 to November 2007. The unemployment rate is measured by the BLS from the CPS. The vacancy rate is measured by the BLS from the JOLTS. The line shows the model generated Beveridge curve with α = 19.4 and m varying from 0.952 to 0.998.

5.2. Calibration This model is parameterized by 9 numbers: the average number of matches per job α, the job termination rate δ, the discount rate r, the value of leisure z, the cost of creating a job k, and the four parameters of the stochastic process for productivity: the lower bound p, the number of steps ν, the arrival rate of shocks λ, and the step size . We keep α fixed at 19.4 and calibrate the remaining parameters of the model to match salient facts about the U.S. economy. The model is in continuous time and so we normalize a time period to represent a quarter. We set the quarterly discount rate to r = 0.012 and let the job termination rate be δ = 0.161. We choose this latter value to ensure a quarterly separation rate to unemployment of 0.105 in the deterministic steady state with m = 0.971 jobs per worker, consistent with average value reported in [25]. The productivity process in Eq. (1) is centered around 1, a normalization. We set the value of leisure to z = 0.4. As in the search model, this is a critical parameter for the volatility of (r+δ)k , aggregate productivity [9]. The lower bound on productivity is p = z + c(r + δ) + 1−exp(−α) the lowest value which ensures that, even in the worst possible state, the unemployment rate stays between 0 and 1; see Eq. (26), which then implies π(m∗−∞ ) = 1 − exp(−α) and hence m∗−∞ = 0 by Eq. (11). This satisfies Eq. (13) as well, and so ensures that vacant jobs always inspect unemployed workers. We let ν = 1000, λ = 90, and  = 0.0129. This implies a mean reversion parameter of γ = 0.09 and a standard deviation of σ = 0.122 for the latent variable y (see Eq. (2)). We choose these values to match the standard deviation and autocorrelation of detrended productivity in U.S. data. If we change ν, λ, and  without altering γ and σ , the results are scarcely affected. Finally, we assume that Eq. (12) binds, so c = 2(1+α) α k. We set k = 0.943 and c = 1.984 to generate a 5.2 percent unemployment rate in the deterministic steady state, which matches the mean unemployment rate both during the post-war period and during the seven year period when vacancy data are available. We do not have a good justification for this choice of k and c, beyond ensuring that Eq. (12) is satisfied. A higher value of c and a correspondingly lower value of k would generate more volatility in employment. This is because the inspection cost is effectively countercyclical, rising when jobs contact more workers. This amplifies aggregate fluctuations.

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Table 2 Summary statistics, quarterly U.S. data, 1951 to 2003. U Standard deviation Quarterly autocorrelation

0.190 0.936

U V V /U Correlation matrix f s p

V

V /U

f

s

p

0.202 0.940

0.382 0.941

0.118 0.908

0.075 0.733

0.020 0.878

1 – –

−0.894 1 –

−0.971 0.975 1

−0.949 0.897 0.948

0.709 −0.684 −0.715

−0.408 0.364 0.396

– – –

– – –

– – –

1 – –

−0.574 1 –

0.396 −0.524 1

Seasonally adjusted unemployment U is constructed by the BLS from the Current Population Survey (CPS). The seasonally adjusted help-wanted advertising index V is constructed by the Conference Board. The job finding rate f and separation rate s are constructed from seasonally adjusted employment, unemployment, and short-term unemployment, all computed by the BLS from the CPS. See [25] for details. U , V , f , and s are quarterly averages of monthly series. Average labor productivity p is seasonally adjusted real average output per person in the non-farm business sector, constructed by the Bureau of Labor Statistics (BLS) from the National Income and Product Accounts and the Current Employment Statistics. All variables are reported in logs as deviations from an HP trend with smoothing parameter 105 .

To characterize the equilibrium, we first compute the targets m∗p for each of the 2ν + 1 states following the procedure in the proof of Proposition 7. We then choose an initial value for p(0) and m(0) and select the timing of the first shock t, an exponentially-distributed random variable with mean 1/λ. We compute the number of unemployed workers who find jobs and the number of employed workers who lose jobs during the interval [0, t]: if m(0) > m∗p(0) , there is a time interval when no new jobs are created; and if m(0) < m∗p(0) , m∗p(0) − m(0) jobs immediately enter and u(m(0)) − u(m∗p(0) ) unemployed workers find work. We next compute the number of jobs at time t: if m(0)  exp(δt)m∗p(0) , m(t) = m∗p(0) ; otherwise, m(t) = exp(−δt)m(0) as the number of jobs decays with exits. Finally, we choose the next value of p(t) as described in Section 2.5 and repeat. At the end of each month (1/3 of a period), we record unemployment, vacancies, cumulative matches and separations, and productivity. We measure the job finding rate f for unemployed workers as the ratio of the number of matches during a month to the number of unemployed workers at the start of the month; if the number of jobs were constant at m during the month, this would equal δmπ(m)/u(m). We similarly measure the separation rate s as the number of workers who separate to unemployment divided by the number of employed workers at the start of the month; if the number of jobs were constant at m during the month, this would equal δmπ(m)/e(m). We throw away the first 25,000 years of data to remove the effect of initial conditions. Every subsequent 53 years of model-generated data gives one sample. We take quarterly averages of monthly data and express all variables as log deviation from an HP filter with parameter 105 , the same low frequency filter that we use on U.S. data. We create 10,000 samples and report model moments and the cross-sample standard deviation of those moments. We compare these results with the U.S. data reported in [25, Table 1] and repeated here in Table 2 for convenience. 5.3. Results Table 3 summarizes the model generated data. The last column shows the driving force, labor productivity. By construction, we match the standard deviation and quarterly autocorrelation in U.S. data. The remaining numbers are driven by the structure of the model.

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Table 3 Model generated data (and standard errors). U

V

V /U

f

s

p

0.128 (0.017) 0.876 (0.031)

0.163 (0.022) 0.876 (0.031)

0.291 (0.039) 0.876 (0.031)

0.070 (0.008) 0.739 (0.060)

0.071 (0.011) 0.882 (0.029)

0.020 (0.003) 0.873 (0.032)

U

1

−0.996 (0.002)

−0.999 (0.001)

−0.870 (0.028)

0.992 (0.002)

−0.992 (0.004)

V

–

1

V /U

–

–

0.999 (0.000) 1

0.874 (0.029) 0.873 (0.029)

−0.982 (0.006) −0.987 (0.004)

0.977 (0.011) 0.984 (0.007)

Correlation matrix f

–

–

–

1

s

–

–

–

–

−0.897 (0.020) 1

0.853 (0.028) −0.993 (0.002)

p

–

–

–

–

–

1

Standard deviation Quarterly autocorrelation

Results from simulations of the benchmark model. See the text for details.

The first two columns show unemployment and vacancies. Both of these variables only depend on the contemporaneous number of jobs. Thus the model generates a nearly-perfect negative correlation between them, stronger than the empirical correlation of −0.89. The model also explains 81 percent of the observed volatility in vacancies and 67 percent of the observed volatility in unemployment. The theoretical autocorrelations of the two variables are about equal, consistent with the empirical evidence. The third column shows the v–u ratio, which [25] argues is a key cyclical variable. The model generates 76 percent of its observed volatility. The fourth column shows that the model produces 59 percent of the observed volatility in the job finding rate; however, the model fails to generate a sufficiently strong autocorrelation in this variable. The empirical autocorrelation is 0.91, while the theoretical correlation is significantly lower at 0.74. This low autocorrelation is intrinsic to the structure of the model: the job finding probability fluctuates with the inflow rate of new jobs, i.e. in response to changes in the number of jobs. In contrast, vacancies and unemployment depend on the stock of jobs. This leads to a correlation between the job finding probability and both the level and change in the v–u ratio. [4] argue that this offers a way to test the stock–flow matching model; however, in U.S. data the correlation in levels is remarkably strong. One possible way to reduce the gap between model and data would be to make the marginal cost of job creation increasing in gross job creation; this should dampen the sharp transitory fluctuations in the job finding probability. Despite this, the model generates a “reduced-form matching function”—a relationship between the job finding probability and the v–u ratio—that is similar to the one in U.S. data. Empirically, a one percent increase in the v–u ratio is associated with a 0.28 percent increase in the job finding probability. The corresponding theoretical elasticity is about 0.21. Moreover, one can test for a constant elasticity by regressing the log job finding probability on the log v–u ratio and its square. The quadratic term is insignificant at conventional confidence levels in the data and significant only 1.6 percent of the time in our simulations of the model. The fifth column shows that the model generates 95 percent of the observed volatility in the separation rate into unemployment even though there are no fluctuations in the job termination

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rate δ. The flip side of this is that the model produces a strongly procyclical job-to-job transition rate, consistent with the facts reported in [6].12 All of these labor market outcomes are significantly more volatile in this model than in [25] (a search and matching model) or [26] (a mismatch model). There are several reasons for this. First, as we noted before, screening costs raise aggregate volatility by squeezing the surplus from a match, p − z − c(r + δ). If we set c = 0 and ignore the planner’s incentive to send vacant jobs to screen employed workers, the results from this model and from [26] are nearly indistinguishable. Second, [26, p. 1085] explains that aggregate volatility is higher in the mismatch model than in the search and matching model for two reasons. As in this paper, the mismatch model assumes that there is a sunk cost of job creation k, rather than the flow cost of maintaining a job in the search and matching model. Introducing a sunk cost into the search and matching model closes almost half the gap between it and the mismatch model. The remaining gap is explained by the different microeconomic structures of the two models. Because the calibrated model has only one shock, most of the correlations are close to one in absolute value. Moreover, a one shock model probably should not be able to explain all the volatility in vacancies and unemployment; there must be other shocks in the data, e.g. to the cost of investment goods k [8]. [10,19,24] propose evaluating one shock models by examining the standard deviation of the projection of the detrended v–u ratio on detrended productivity. The projection in the data is actually smaller than the projection in the model. Similarly, the model over-explains fluctuations in the separation rate. 6. Conclusion This paper develops a stock–flow matching model where frictions arise because it is costly to learn which worker–job matches are productive. We have derived explicit expressions for the distribution of the unemployment rate, for the probability that a job entering the labor market causes an unemployed worker to find a job, and for the probability that a job exiting the labor market causes an employed worker to become unemployed. These have simple limits in a large economy, which we use to quantify the model’s implications for cyclical fluctuations in unemployment, vacancies, and worker flows. The quantitative results are broadly consistent with U.S. data and are similar to those in the mismatch model [26], which suggests that the possibility of a more general approximate aggregation result. Although the stock–flow and mismatch models predict similar aggregate behavior, there are important differences in the nature of unemployment and vacancies between the two models. The mismatch model develops a theory of the coexistence of unemployed steel workers and vacant jobs in nursing, or of unemployed workers in one state and job openings in another. This means that the long-term unemployed are typically located in markets where jobs are scarce, and so the mismatch model predicts that the probability of finding a job should fall with unemployment duration. In contrast, the stock–flow model explains why unemployment and vacancies can coexist within a well-defined labor market. Except for tie-breaking rules that determine which worker a firm hires when it can hire more than one, the model predicts that the probability of finding a job should be constant during an unemployment spell, conditional on the values of α and m. A large literature examines why the job finding probability declines with unemployment duration. In their survey, [16] find little evidence for duration dependence in the job finding rate after 12 We do not report the job-to-job transition rate here because data limitations allow us to construct the series only after 1994. Including this series in the table would severely reduce the length of all the other time series.

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controlling for observed heterogeneity, and even less evidence after controlling parametrically for unobserved heterogeneity. Arguably this is consistent with the predictions of the stock–flow matching model and inconsistent with the mismatch model. The stock–flow matching model can be used to examine other labor market issues. For example, suppose match productivity takes on many values, so workers and jobs must choose a productivity threshold for accepting a partner. In such an environment, labor market policies like unemployment benefits may raise the threshold, reducing the average number of acceptable workers per job, analogous to the parameter α in this paper, and shifting the Beveridge curve away from the origin. This is consistent with evidence in some European countries since 1960 [see, for example, [21]]. The stock–flow approach also pertains to other markets. [3] label the agents in their model “buyers” and “sellers” and discuss the real estate market. [14] examines the taxicab market in a related model. Idiosyncratic heterogeneity is likely also important in the marriage market. The stock–flow matching model can also capture markets where idiosyncratic heterogeneity is less important; the appropriate value of α is larger and the equilibrium unmatched rates smaller. Our analysis may provide a set of tools that will prove useful in studying these problems as well. Appendix A. Omitted proofs Proof of Proposition 1. Suppose that for some a ∈ {1, . . . , M(t)}, we have proved that assignment of workers to jobs {j1 , . . . , ja−1 } is identical in the decentralized equilibrium and the matching algorithm at time t. This is trivially true for a = 1. We prove that the assignment of a worker or a vacancy to job ja is identical in the decentralized equilibrium and the matching algorithm at time t as well, and so establish the result by induction. Suppose that in the decentralized equilibrium, job ja is filled by worker i. By construction, this is a productive match. Now consider any other worker i  . It is impossible that i  has a productive match with ja and was unemployed for more time than i when i and ja matched, for then ja would have matched with i  . The remaining possibilities are that i  does not have a productive match with ja , or that i  has a productive match but was already matched to some ja  when worker i and job ja matched, or that i  has a productive match but was unemployed for less time than i when i and ja matched. We consider each of these possibilities in turn and show that in each case, the matching algorithm does not assign i  to ja . First, suppose i  does not have a productive match with ja . Then the matching algorithm trivially does not assign i  to ja . Second, suppose i  has a productive match with ja but was matched with some job j  when i and ja matched. This implies that job j  is older than job ja , for otherwise i  would have matched with ja in the decentralized equilibrium. If job j  is still open at time t, the induction step implies that the matching algorithm also assigns i  to j  and hence does not assign i  to ja . If job j  is closed by time t, the time since displacement is shorter for i  than for i, and hence again the algorithm does not assign i  to ja . Third, suppose i  has a productive match with ja but was unemployed for less time than i when i and ja matched. Then at time t, the elapsed time since displacement is again shorter for i  than for i and so the algorithm does not assign i  to ja . Finally, the matching algorithm assigns workers to jobs {j1 , . . . , ja−1 } exactly as in the decentralized equilibrium, and in particular leaves worker i unmatched. Since job ja has a productive match with i, the algorithm must assign some worker to job ja ; and since it does not assign any other worker i  , it must assign worker i.

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Alternatively, suppose job ja is vacant at time t. Then any worker i  either does not have a productive match with ja or has a productive match but was already matched to some ja  when job ja entered. Repeating the same arguments shows that the matching algorithm does not assign i  to ja , and hence leaves ja vacant. This completes the induction step and hence the proof. 2 Proof of Proposition 2. We can trivially prove that φ(0|0) = 1 and φ(E|0) = 0 for any E > 0, consistent with Eq. (3). Moreover, φ(0|M) = x LM for any M, since there are no employed workers only if all worker–job pairs are unproductive. Again this is consistent with Eq. (3). We now proceed by induction. Suppose that we have proved Eq. (3) for some M − 1  0 and all E  0 and want to establish it for M and some E  1. Using the matching algorithm and the induction step, the probability that there are E − 1 matches among the M − 1 oldest jobs is φ(E − 1|M − 1). Conditional on E − 1 matches among those jobs, the probability the newest job is matched is 1 − x L−E+1 , leaving us with E matches among the M jobs. The other way to attain E matches among the M jobs is if there are E matches among the M − 1 oldest jobs, with probability φ(E|M − 1), and the newest job is unmatched, with probability x L−E . Putting this together,   φ(E|M) = 1 − x L−E+1 φ(E − 1|M − 1) + x L−E φ(E|M − 1). Expanding φ(E − 1|M − 1) and φ(E|M − 1) using the induction step and simplifying yields Eq. (3). 2 Proof of Proposition 4. It is trivial that ΠjEU (0, M) = 0 for all j and M, since there are no employed workers. We can also directly characterize the probability that the exit of the newest job, call it job M, causes a worker to become unemployed; this happens if and only if the job is filled: EU (E, M) = ΠM

(1 − x L−E+1 )φ(E − 1|M − 1) . (1 − x L−E+1 )φ(E − 1|M − 1) + x L−E φ(E|M − 1)

(21)

To understand this, partition the configurations with E employed workers and M jobs in two. First, job M is filled, so without the newest job there would be E − 1 employed workers, with probability φ(E − 1|M − 1). Conditional on this, job M is filled with probability 1 − x L−E+1 . Second, job M is vacant, so without the newest job there would be E employed workers, with probability φ(E|M − 1). Conditional on this, job M is vacant with probability x L−E . The relatively likelihood of the former configuration gives us the probability the newest job is filled, explaining Eq. (21). Next, we can verify directly using Eq. (3) that for all M  1 and 1  E  min{M, L},   x −E − 1 φ(E|M) 1 − x L−E+1 φ(E − 1|M − 1) = −M x −1 x −M − x −E x L−E φ(E|M − 1) = φ(E|M). x −M − 1

and

Substituting these into Eq. (21) and simplifying verifies Eq. (5) for j = M. Now use induction to complete the proof. Suppose we have proved Eq. (5) for arbitrary M − 1  1 and all E ∈ {0, 1, . . . , M − 1} and j ∈ {0, 1, . . . , M − 1}. We extend it to M and all E ∈ {0, 1, . . . , M} and j ∈ {0, 1, . . . , M − 1} and hence establish the result by induction. We start with the following recursive equation:

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EU ΠjEU (E, M) = ΠjEU (E − 1, M − 1)ΠM (E, M)   EU + ΠjEU (E, M − 1)x 1 − ΠM (E, M) .

(22)

To see this, again partition the configurations with E employed workers and M jobs in two. EU (E, M). According to the matching algorithm—which First, job M is filled, with probability ΠM matches workers and jobs identically to the decentralized equilibrium by Proposition 1—the exit of job j when there are E employed workers, M jobs, and job M is filled causes a worker to become employed exactly it when would have with E −1 employed workers and M −1 jobs, with probability ΠjEU (E − 1, M − 1). In the second configuration, job M is vacant, with probability EU (E, M). Proposition 1 implies that conditional on this, the exit of job j leads to a worker 1 − ΠM becoming unemployed if it would have led to a worker becoming unemployed with E employed workers and M − 1 jobs, with probability ΠjEU (E, M − 1), and the unemployed worker cannot match with job M, with probability x. Now plug the known formulae for ΠjEU (E − 1, M − 1), EU (E, M) and Π EU (E, M − 1) into Eq. (22) and simplify to establish Eq. (5). This completes ΠM j the induction step and the proof. 2 Proof of Proposition 5. We break the proof into three steps to improve readability. Step 1. [12, Eq. (13)] proves that for any L, M, and x, the expected number of employed workers solves ∞ ∞ E−1 L−i )(1 − x M−i ) i=0 (1 − x Eφ(E|M) = . (23) 1 − xE E=0 E=1  Using x = 1 − α/L and M = mL, the employment rate is e(m) ≡ limL→∞ ∞ E=1 BE (L), where E−1 (1 − (1 − α/L)L−i )(1 − (1 − α/L)mL−i ) . (24) BE (L) ≡ i=0 L(1 − (1 − α/L)E ) Define B¯ E ≡ lim BE (L) =

E−1 i=0

limL→∞ (1 − (1 − α/L)L−i ) limL→∞ (1 − (1 − α/L)mL−i )

L→∞

=

α limL→∞

1−(1−α/L)E 1−(1−α/L)

((1 − exp(−α))(1 − exp(−αm)))E , αE

where the last equation uses the fact that for fixed i, limL→∞ (1 − α/L)L−i = exp(−α) and E limL→∞ (1 − α/L)mL−i = exp(−αm); and for fixed E, limL→∞ 1−(1−α/L) 1−(1−α/L) = E. This implies ∞ E=1

    1 B¯ E = − log 1 − 1 − exp(−α) 1 − exp(−αm) α =1+m−

∞

  1 log exp α + exp(αm) − 1 , α

aE E=1 E = − log(1 − a) E=1 limL→∞ BE (L), that is, if

since  ∞

for any a < 1. Eq. (8) follows if limL→∞ we can switch the order of limits.

∞

E=1 BE (L)

=

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Step 2. To prove that we can switch the order of limits, note that for all n  0,  ∞    n ∞ ∞ ∞ n         BE (L)  BE (L) +  BE (L) − B¯ E − B¯ E + B¯ E .      E=1

E=1

E=n+1

E=n+1

E=1

(25)

E=1

We prove that for any ε > 0, there exists an n and an L¯ such that each of the terms on the ¯ right-hand side is smaller than ε/3 for all L > L.  ¯ ¯ Start with the first term. Since BE  0 and ∞ ¯ 1 such that E=1 BE is finite, there exists an n ∞ ¯ E=n+1 BE < ε/3 for all n  n1 . Next look at the second term. For any i  0, E  1, and L > α, (1 − α/L)L−i  (1 − α/L)L , implies BE (L)  (1 − α/L)mL−i  (1 − α/L)mL , and (1 − α/L)E  1 − α/L. Then Eq. (24) zE /α, where z ≡ (1 − (1 − α/L)L )(1 − (1 − α/L)mL ) ∈ (0, 1). It follows that ∞ E=n+1 BE (L)  zn+1 /α(1 − z). In particular, for fixed n, lim

L→∞

∞

((1 − exp(−α))(1 − exp(−αm)))n+1 zn+1 = . L→∞ α(1 − z) α(1 − (1 − exp(−α))(1 − exp(−αm)))

BE (L)  lim

E=n+1

The last expression is smaller than ε/6 for all n  n2 . Fix any n  max{n1 , n2 } in the rest of the proof; there exists an L¯ 2 such that for all L  L¯ 2 , the second term in Eq. (25) is smaller than ε/3. Now turn to the last term in Eq. (25). For the given value of n, the last term is smaller than ε/3 for all L > L¯ 3 since BE (L) → B¯ E for all E. Let L¯ = max{L¯ 2 , L¯ 3 } to complete this step and prove that the expected employment rate is e(m) in a large economy. Step 3. We finally prove that the variance of E/L converges to zero. [12, Eq. (14)] also proves that for any L, M, and x,

E−1   ∞ ∞ E−1 L−i )(1 − x M−i )  1 i=0 (1 − x . E(E − 1)φ(E|M) = 2 1 − xi 1 − xE E=0

E=1

i=1

Also note that ∞ 2 ∞ E E(E − 1) lim φ(E|M) = lim φ(E|M), L→∞ L→∞ L L2 E=0 E=0  E since Step 1 implies limL→∞ ∞ E=0 L2 φ(E|M) = 0. Thus ∞ 2 E lim φ(E|M) L→∞ L E=0

E−1   ∞ E−1 L−i )(1 − x mL−i )  2 1−x i=0 (1 − x = lim 2 . L→∞ α 1 − xi (1 − x E )/(1 − x) E=1

i=1

Again replacing x = 1 − α/L, switching the order of limits using an argument analogous to Step 2, and taking the same limits as before, we get

E−1 

 ∞ ∞ 2 E ((1 − exp(−α))(1 − exp(−αm)))E 2 1 lim φ(E|M) = 2 L→∞ L i E α E=0

E=1

i=1

   2 1 = 2 log 1 − 1 − exp(−α) 1 − exp(−αm) = e(m)2 , α

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 E−1 1 a E 1 2 where the second line uses ∞ E=1 ( i=1 i )( E ) = 2 log(1 − a) for any a < 1. We have proved that

2 ∞ 2 ∞ E E lim φ(E|M) = lim φ(E|M) , L→∞ L→∞ L L E=0

E=0

and hence the limiting variance of E/L is zero. Finally, convergence of the expected value of E/L to e(m) and of the variance of E/L to zero implies mean square convergence of E/L to e(m). 2 Proof of Proposition 6. Consider a sequence of economies indexed by the number of workers L with L converging to infinity. In an economy with L workers, there are M = mL jobs, x = 1 − α/L probability of a pair being unproductive, and a random number E(L) employed workers, with distribution given by Eq. (3). Then Eq. (4) implies  

     α L 1−E(L)/L UE Π E(L) = 1 − 1− → 1 − exp −α 1 − e(m) , L since Proposition 5 proves E(L)/L converges to e(m) in mean square. Replace e(m) using Eq. (8) and simplify to get Eq. (11). Similarly, Eq. (5) implies   ((1 − Lα )−L )E(L)/L − 1 exp(αe(m)) − 1 → . Π EU E(L), mL = (1 − Lα )−mL − 1 exp(αm) − 1 Algebraic simplification again yields Eq. (11).

2

Proof of Proposition 7. We start by constructing the unique solution to the planner’s first order condition that has increasing targets. The last paragraph proves that there is no other solution. Start with the smallest value p = p−ν with associated target m∗p−ν . Following an aggregate shock, productivity increases by one step with certainty and so the target number of job increases to m∗p−(ν−1) > m∗p−ν . If m = m∗p−ν , the marginal value of a job is k + cπ(m∗p−ν ) both before and after the shock, Wp −ν (m∗p−ν ) = Wp −(ν−1) (m∗p−ν ) = k + cπ(m∗p−ν ). Then evaluating Eq. (18) at p = p−ν and m = m∗p−ν gives     (r + δ)k = p−ν − z − c(r + δ) π m∗p−ν . (26) This uniquely defines m∗p−ν since π is a decreasing function, as can be confirmed directly from Eq. (11). We now proceed by induction. Suppose that for some y > −ν, y ∈ Y , we have shown that the targets m∗py  are increasing and we have computed Wp y  (m∗py− ) for all y  < y, y  ∈ Y . For m ∈ [m∗py− , m∗py ] and y  < y, Eq. (17) implies   (r + δ)Wp y  (m) = (py  − z)π(m) − δm Wpy  (m) − cπ  (m)

  λ y   + 1+ Wpy  − (m) − Wp y  (m) 2 ν

  λ y   Wpy  + (m) − Wp y  (m) . (27) + 1− 2 ν

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In addition, Wp y (m) = k + cπ(m) for m ∈ [m∗py− , m∗py ]. This is a system of ν + y/ differential equations in m with the same number of terminal conditions from the previous induction steps and so we can compute Wp y  (m), m ∈ [m∗py− , m∗py ] for all y  < y, y  ∈ Y . The only catch is that we do not yet know m∗py . To compute it, evaluate Eq. (18) at py and m = m∗py :     (r + δ)k = py − z − c(r + δ) π m∗py

     λ y   + (28) 1+ Wpy− m∗py − k − cπ m∗py , 2 ν where we use Wp y+ (m∗py ) = k + cπ(m∗py ) to eliminate the term coming from a positive shock. This uniquely defines m∗py since both π and Wp y− are decreasing. To complete the induction argument, suppose Eq. (28) defines m∗py  m∗py− . Then         (29) py − z − c(r + δ) π m∗py = (r + δ)k < py− − z − c(r + δ) π m∗py− . The equality uses Wp y− (m∗py ) = k + cπ(m∗py ) whenever m∗py  m∗py− . The inequality uses Eq. (27) evaluated at y  = y −  and m = m∗py− , but drops the capital gain terms; those are all negative-valued since m∗py  m∗py− (by assumption in this paragraph) and m∗py−2 < m∗py− (from the induction assumption). Since py > py− , Eq. (29) implies π(m∗py ) < π(m∗py− ) or equivalently m∗py > m∗py− , a contradiction. Finally, suppose there were a solution to the planner’s problem with m∗py  m∗py− for some y ∈ Y . Focus on the largest such y, so either m∗py < m∗py+ or y = ν, in which case productivity can only decline from py . Analogous to the reasoning behind Eq. (29), we find         py − z − c(r + δ) π m∗py = (r + δ)k  py− − z − c(r + δ) π m∗py− , since a productivity shock when p = py and m = m∗py does not affect the marginal value of a job (the target goes up), while a productivity shock when p = py− and m = m∗py− may reduce the marginal value of a job. The inequalities imply m∗py > m∗py− , a contradiction. 2 Proof of Proposition 8. We claim that Jp (m) = Wp (m) − cπ(m) for all p and m. Given this conjecture, it is straightforward to verify that Eqs. (17) and (19) are identical, as are Eqs. (18) and (20). Moreover, the requirement that Wp (m) = k + cπ(m) when m < m∗p implies Jp (m) = Wp (m) − cπ(m) there as well. Since there is a unique solution to these first order conditions of the planner’s problem, there is a unique solution to these equations as well and hence a unique equilibrium. 2 References [1] N.I. Al-Najjar, Aggregation and the law of large numbers in large economies, Games Econ. Behav. 47 (1) (2004) 1–35. [2] C. Carrillo-Tudela, E. Smith, Wage dispersion and wage dynamics within and across firms, Mimeo, 2007. [3] M.G. Coles, A. Muthoo, Strategic bargaining and competitive bidding in a dynamic market equilibrium, Rev. Econ. Stud. 65 (2) (1998) 235–260. [4] M.G. Coles, B. Petrongolo, A test between unemployment theories using matching data, Mimeo, July, 2003. [5] M.G. Coles, E. Smith, Marketplaces and matching, Int. Econ. Rev. 39 (1) (1998) 239–254. [6] B. Fallick, C. Fleischman, Employer-to-employer flows in the U.S. labor market: The complete picture of Gross worker flows, in: Federal Reserve Board, Finance and Economics Discussion Series, Working Paper 2004-34, 2004.

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