Matching, Sorting and Wages

We develop an empirical search-matching model which is suitable for analyzing the wage, employment and welfare impact of regulation in a labor market with heterogeneous workers and jobs. To achieve this we develop an equilibrium model of wage determination and employment which extends the current literature on equilibrium wage determination with matching and provides a bridge between some of the most prominent macro models and microeconometric research. The model incorporates productivity shocks, long-term contracts, on-the-job search and counter-offers. Importantly, the model allows for the possibility of assortative matching between workers and jobs due to complementarities between worker and job characteristics. We use the model to estimate the potential gain from optimal regulation and we consider the potential gains and redistributive impacts from optimal unemployment insurance policy. Here optimal policy is defined as that which maximizes total output and home production, accounting for the various constraints that arise from search frictions. The model is estimated on the NLSY using the method of moments. (Copyright: Elsevier)


Introduction
Our aim is to develop and estimate a search-matching model with productivity shocks so as to analyze policy interventions in a labour market with heterogeneous agents. To achieve this we need an equilibrium model of wage determination and employment, which is consistent with key empirical facts. As such our model extends the current literature on equilibrium wage determination with matching and provides a bridge between some of the most prominent macro models and microeconometric research.
The empirical work on wage dynamics has offered important insights on how wages evolve over time for individuals. Some of this work has emphasised the stochastic evolution of wages 1 , while other work has considered carefully wage growth over the lifecycle including the role of experience, tenure and job mobility. 2 Most of this work is essentially reduced form, in that there is little explanation of how the stochastic components arise or how wages are determined and the match surplus, if it exists, gets shared between workers and firms. As a result these empirical studies offer rich descriptions of wages, but do not allow us to assess the impact of policies in all but the most restrictive labour market frameworks. Essentially we do not have a theoretical framework that can explain the empirical facts and justify the complex statistical models of wages fitted to the data. In parallel a rich literature has developed on equilibrium wage distributions with some degree of heterogeneity in workers and/or jobs. However, this literature is not capable of accounting for the rich wage dynamics and for a number of key issues such as sorting; it does not provide a framework that is rich enough to explain what we observe about wages. Our aim in this paper is to take the first steps in bridging this gap offering a model of wages with stochastic shocks, which is consistent with equilibrium wage determination, when jobs and workers are heterogeneous and there are search costs.
Thus the key ingredients of our model are: Workers differ from each other according to a productivity relevant characteristic. Firms are also heterogeneous and their productivity is subject to possibly persistent shocks -this will lead to stochastic shocks to wages. The production function allows for complementarity between worker and job characteristics, leading to the possibility of sorting in the labour market. Jobs can decide whether to remain completely idle, or post a vacancy. They hire any worker who leads to a positive surplus -this accounts of course for the option value of keeping the vacancy. Finally, there are search costs: workers receive job offers while unemployed or employed at exogenously given rates (that may differ across these states). We solve this model and derive the implied dynamics of wages and the cross sectional wage distribution, as well as the distribution of matches.
Our model offers both an empirical framework for understanding wage determination and as a result offers a way for evaluating the impact of labour market regulation, such as the minimum wages or restrictions on firing. In a search framework protecting workers from being fired can have ambiguous effects on employment. Our framework will allow this effect to be quantified.
But, as important, it allows us to analyse the effect of regulation on the distribution of wages and profits and thus showing who pays and who benefits from such a policy in this non-competitive environment.
Our paper draws from the literature on matching and assignment models (Sattinger, 1993) as well as on the literature on equilibrium wage distributions in a search environment (Mortensen and Pissarides, 1995) and Burdett and Mortensen, 1998). Matching models of the labour market have become standard in the macroeconomic literature since the seminal works of Diamond (1982), Mortensen (1982) and Pissarides (1990). Moreover, it is now well understood that search models can give rise to wage dispersion even if workers are homogeneous (see Burdett and Judd, 1983 and Burdett and Mortensen, 1998). However, matching models with heterogeneous workers and jobs is a relatively new topic of interest fueled by the need to understand dispersion of wages of similar individuals. In general, workers differ by the numbers of years of education and experience, and jobs differ by the type of industry. There is thus an enormous amount of differences between workers and between jobs that are not accounted for by observables in the data. Marriage models with heterogeneous agents in a frictional environment are studied in Sattinger (1995), Lu and McAffee (1996), Shimer and Smith (2000), and Atakan (2006). 3 To the best of our knowledge, there have not yet been any empirical applications of assignment models with transferable utility in a frictional environment with heterogeneous agents.
There is a large body of empirical evidence showing that wages differ across industries, thus 3 Sattinger develops a framework but does not prove the existence of an equilibrium. Lu and McAffee prove the existence for a particular production function (f (x, y) = xy). Shimer and Smith prove the existence of an equilibrium in a more general setup and derive sufficient conditions for assortative matching. Atakan shows that Becker's (1973) complementarity condition for positive sorting is sufficient if there exist explicit search costs.
indicating that a matching process is at work in the economy (see for example Krueger and Summers, 1988). Static, competitive equilibrium models of sorting (Roy models) have been estimated by Heckman and Sedlacek (1985) and Heckman and Honore (1990) In the U.S. 90% of these correlations range between -15% and 5%, and in France between -27% and -5%. These negative numbers, although hard to interpret, offer prima faciae evidence of no positive sorting. However, the evidence based on the log-linear decomposition used by AKM should not be interpreted as evidence that there is no sorting: the person and firm effects which are estimated from the linear log wage equation are complicated transformations of the underlying individual-specific, unobserved productivity-relevant characteristics; A structural model is thus required to recover the true underlying joint distribution of characteristics. Abowd et al. present some evidence that a matching model inspired from Shimer (2005) could both generate sorting on unobservables and the sort of empirical regularity that they find. More recently, Melo (2008) has proposed a matching model, extending Lu and McAffee (1996) and Shimer and Smith (2000) to allow for on-the-job search, that also produces the same prediction.
Our framework allows us to investigate empirically whether sorting actually is important in practice.
In many ways, our model is similar to Mortensen and Pissarides's (1990) model. Workers and jobs meet at random at a frequency that depends on some matching function and productivity shocks are responsible for endogenous lay-offs. As in equilibrium search models, on-the-job search generates both job-to-job mobility and wage dispersion, and we follow Postel-Vinay and Turon's (2007) extension of Postel-Vinay and Robin's (2002) sequential auction model to model offers and counter offers and contract renegotiation upon productivity shocks.

An Overview of the Model
In the economy there is a fixed number of individuals and a fixed number of jobs or production lines. Individuals may be matched with a job and thus working, or they may be unemployed job seekers. Jobs on the other hand may be in three different states. First they may be matched with a worker, in which case output is produced. Second, they may be vacant and waiting for a suitable worker to turn up. Finally jobs may be inactive, and thus potential entrants in the labour market. Individuals all have different levels of human capital, indexed by x. Jobs on the other hand also differ from each other according to some productivity relevant characteristic y. Output depends on the characteristics of both sides with possible complementarities. Crucially though, productivity follows a first order Markov process, which leads to the value of the match changing, with consequences for wage dynamics, worker mobility, job creation, and job destruction that are at the centre of our model. Individuals maximise their discounted income over an infinite horizon; jobs maximise profits.
When a job and a worker meet and the total match surplus is positive, the worker is hired and thus the match is formed. At this point the worker is paid a wage consistent with the reservation value of the best alternative option: an unemployed person is paid a wage to make the value of employment equal to value of unemployment; a worker who receives an outside offer either gets a pay increase to match the value of the alternative offer, or moves to the new job, always depending on the comparison of the two jobs. 4 A further important feature is that a shock to job productivity may trigger a wage renegotiation. This will happen if under then new productivity the match surplus remains positive but the wage is too high under the new conditions. On the other hand there is no incentive to renegotiate when there is a shock increasing the surplus, but as we shall show this will increase the value of being employed in this job because of the prospect of future wage increases. Finally we close the model by a free entry condition: all production lines, whether active or not have a productivity relevant parameter, which they know. This determines whether they will want to enter the market and post a vacancy. The marginal job has zero surplus from entering the market and posting a vacancy.

Setup
Each individual worker is characterised by a permanent productivity relevant characteristic which we denote by x. The agents all observe x; whether the econometrician observes this or not is an empirical identification issue, which we will deal with later. We assume that x has bounded support defined by x ∈ [x, x]. The measure of worker heterogeneity in the population is L (x) and we assume it possesses density (x). There are L = dL individuals of which U are unemployed. We also denote by U (x) the (endogenous) measure of x among the unemployed.
Jobs are characterised by a productivity parameter y with bounded support y, y . The In a given job, y fluctuates according to a jump process. δ is the instantaneous arrival rate of jumps and Q (y |y) is the (Markov) transition probability for y.
A match between a worker x and a job y produces a flow of output f (x, y); this allows for the possibility that x and y are complementary in production, implying that sorting will increase total output.
We denote the measure of existing matches by H (x, y) with density: h (x, y). We can relate the density of individual productivities to the density of active matches as well as the density of productivities for the unemployed by Similarly we can write an equivalent relationship between the distribution of job productivities, active matches, unfilled vacancies and inactive jobs In both cases the relationship is essentially an accounting identity. Finally, matches can end both endogenously, as we characterise later, and exogenously. We denote by ξ the instantaneous rate of exogenous job destruction.
We now discuss the process by which workers get to know about vacant jobs. We assume that the unemployed workers search for work at a fixed intensity s 0 . The search intensity for an employed worker is s 1 . The process of search leads to a total number of meetings, that as usual depends on the number of posted vacancies as well as on the number of total searchers in the economy, weighted by their search intensities. This matching function is denoted by Then s 0 κv (y) and s 1 κv (y) are the rates at which unemployed and employed workers of any type contact vacancies of type y. Symmetrically, s 0 κu (x) and s 1 κh (x, y ) are the rates at which a job of any type contacts a worker of type x, either unemployed or currently employed at a job of type y .

Match formation and rent sharing
The value of an (x, y) match to a worker of ability x, matched with a job of productivity y, at wage w is denoted as W 1 (w, x, y). The value of an (x, y) match to a job of productivity y matched to a worker of ability x, paying a wage w is denoted as Π 1 (w, x, y). Let W 0 (x) denote the present value of unemployment for a worker with characteristic x, and let Π 0 (y) denote the present value of a vacancy. Define the "surplus" of a match (x, y) as Feasible matches (x, y) are such that S (x, y) ≥ 0. By convention, production is better than inactivity.
We assume that incumbent employers match outside offers. A negotiation game is then played between the worker and both jobs as in Postel-Vinay and Robin (2002). If a worker x, currently paired to a job y such that S (x, y) ≥ 0, finds an alternative job y such that S (x, y ) > S (x, y), the worker moves to the alternative job and the new employer signs with the worker a contract that is worth the value of the total surplus of the previous (x, y) match: Thus, the present value of the new wage contract W * 1 (x, y |y) does not depend on the last wage contracted with the incumbent employer, but only on the total surplus of the previous match.
Next, consider the case where value to the worker of the current wage contract that x and y have agreed upon. The worker uses the external offer to obtain a wage rise, increasing the value of being employed at this job (over and above the unemployment value) to W * is as defined in equation (5).
, the worker has nothing to gain from the competition between y and y and the wage does not change.
When an unemployed worker x finds a vacant job y a match is formed if and only if S (x, y) ≥ 0, and the job offers a wage contract that exactly compensates the worker for her opportunity cost of employment, defined by: It is worth noting that in this context this can imply negative wages: the worker may be willing to pay to work for a job of sufficiently high productivity because she anticipates future wage increases. The continuation value for workers when the match is destroyed, by the worker moving to unemployment or an alternative job is not a function of the last negotiated contract. This allows us to define matching sets, i.e. sets of acceptable job/productivity levels that make a match preferable to the current state. Specifically: is the set of jobs y such that match (x, y) is feasible: • M 1 (x, y) is the set of jobs y such that match (x, y ) can be formed and is preferred to the current match (x, y): • M 2 (w, x, y) is the set of y such that match (x, y ) does not produce a higher surplus than (x, y) but the competition for the worker yields a wage increase: In what follows we will denote by A the complement of a set A.

Renegotiation
Wages can only be renegotiated by mutual agreement. This will happen either when a suitable outside offer is made, or because a productivity shock reduced the value of the surplus sufficiently.
Specifically, a productivity shock changes y to y . If y is such that S (x, y ) < 0, the match is endogenously destroyed. The worker becomes unemployed and the job will either post a vacancy, or perhaps become idle and not seek to fill the position again. Suppose now that S (x, y ) ≥ 0. The value of the current wage contract becomes , neither the worker nor the job has a credible threat to force renegotiation because both are better off with the current wage w being paid to the worker than walking away from the match. In this case there will be no renegotiation. If, however, It is convenient for notational reasons below to define the sets of y that are consistent with a particular wage contract x, giving a positive share to both parties, and the two cases where the resulting share of the job or the worker respectively is negative. Thus define: • C (w, x) is the set of productivities y such that contract w is sustainable and thus no renegotiation takes place.: is the set of jobs y such that contract w gives the worker more than the whole surplus : is the set of jobs y such that contract w gives the job more than the whole surplus:

Value Functions
The value functions of the agents have been kept implicit up to now. The next step in solving the model is thus to characterise the value functions of workers and jobs. These define the decision rules for each agent. proceed by assuming that time is discrete with an arbitrarily small unit of time so that two independent events have a negligible chance of occurring simultaneously. The discount rate is denoted by r.
Unemployed workers. Unemployed workers are always assumed to be available for work at a suitable wage rate. Thus the present value of unemployment to a worker of type x is W 0 (x), which satisfies the option value equation: The second equality follows from our assumption that a worker when first employed always gets paid her reservation wage, which means that whatever job they meet W * 1 (x, y|0) = W 0 (x) (see where c is a per-period cost of keeping a vacancy open. In (11)  The job keeps all the surplus in a new match and has to pay workers it poaches at least the surplus in the job they were poached from. This gives which after substitution in (11) we obtain S (x, y) − S x, y h x, y dx dy + δ Π 0 (y )q(y |y)dy . (12) If the job decides to remain inactive, then it does not pay the cost of posting vacancies and has no chance of meeting a worker. Its present value only depends on future productivity draws, which may lead them into posting a vacancy: Note that a match (x, y) may yield a positive output f (x, y) but the cost of a vacancy exceeds the expected profit. Thus it is possible that a job which looses a worker to another job, just "closes down" rather than posting a new vacancy, until its potential productivity y increases.
Combining the two ways in which a job may be unmatched, the present value of an unmatched job of type y is obtained by: with S (x, y) − S x, y h x, y dxdy . (14) and a job y is inactive whenever This condition is like the free entry condition of a standard search-matching model with ex-ante homogeneous jobs.
Employed workers We need to define the value to a worker of type x who is matched with a job with productivity y and is currently paid w. This needs to be defined even when the match is not sustainable so as to deal with the case where a productivity shock arrives, making the current level of pay off equilibrium. We denote this value by W 1 (w, x, y).
The timing of events is assumed to be as follows: after the wage w is paid at the end of the period. Then one of the three events may happen: the match may be exogenously destroyed with probability ξ, or a productivity shock arrives, with probability δ, changing y into is the probability that the contacting job has productivity such that it can induce a wage renegotiation or poach the worker. If none of these events happen, which occurs with probability , it is optimal to terminate the match if S (x, y) < 0, and it is optimal to renegotiate the wage if y ∈ C + (w, x) ∪ C − (w, x). Otherwise the match continues with the same wage w. The above can be formalised in the value function S (x, y ) + W 0 (x) 1 + r q y |y dy + s 1 κ After simplification, this becomes S (x, y ) 1 + r q y |y dy S (x, y ) 1 + r v y dy The Bellman equation defines W 1 (w, x, y)−W 0 (x) as a fixed point of a contracting operator.
A simple iterative algorithm can be used to approximate the fixed point. Let W 0 1 be an initial guess of W 1 . Values C 0 (w, x), M 0 2 (w, x, y), C (w, x) and M 2 (w, x, y) follow from the initial value W 0 1 . Then calculate an update of W 1 (w, x, y) − W 0 (x) using the previous equations.
Knowledge of the value function W 1 (w, x, y) will eventually allow us to compute the optimal wage contract w given (x, y, y ).
Producing jobs. Similarly, the corresponding Bellman equation defining Π 1 (w, x, y) is given below, again for all combinations of (w, x, y) even when they do not represent an equilibrium.

The match surplus
Summing equations (16), and (17) and using equations (10), (13), we can write the surplus of any (x, y) match as the fixed point defined by S (x, y ) 1 + r q y , y dy The important point to note from this expression is that the surplus of an (x, y) match never depends on the wage; the Bertrand competition between the two jobs for the worker ensures

Steady-state flow equations.
To solve for equilibrium we need to define the steady state flow equations.
The total number of matches in the economy will be Existing matches, characterised by the pair (x, y), can be destroyed for a number of reasons.
First, there is exogenous job destruction, at rate ξ; second, with probability δ, the job component of match productivity changes to some value y different from y, and the worker may move to unemployment or may keep the job; third, the worker may change job, with probability s 1 κV (M 1 (x, y)) -i.e., a job offer has to be made (at rate s 1 κV ) and has to be acceptable (y ∈ M 1 (x, y)). On the inflow side, new (x, y) matches are formed when some unemployed or employed workers of type x match with vacant jobs y, or when (x, y ) matches are hit with a productivity shock and exogenously change from (x, y ) to (x, y). In a steady state all these must balance leaving the match distribution unchanged. Thus formally we have for all (x, y) such that the match is acceptable , i.e. y ∈ M 0 (x) or S (x, y) > 0: This equation defines the steady-state equilibrium, together with the accounting equations for the workers: and the jobs v (y) + i (y) = n (y) − h (x, y) dx, Noting that vacancies are posted only if costs are low enough (given productivity) we get that the density of vacancies is and the density of inactive jobs is given by The total number of vacancies V is thus obtained as

Equilibrium
In equilibrium all agents follow their optimal strategy and the steady state flow equations defined In these equations, we substitute for U using equation (19), κ using equation (3), u (x) using equation (21), c (y) using equation (14), and v (y) using equation (23).

Policy instruments
Our model provides a way of evaluating the employment and distributional impact of labour market regulation. It is ideally suited for understanding who pays and who benefits from such policies. Labor market regulation can consist of payments for the unemployed, increasing b(x),which is already in the model. We also introduce three further policy instruments: experience rating, minimum wages, and severance pay. We model experience rating as a tax on endogenous separations. This can be accomplished by subtracting the term δQ(M 0 (x)|y)τ 1+r from equations (17) and (18).
Incorporating a minimum wage puts a constraint on the ability of workers and jobs to make transfers. As a result, the condition for match feasibility will depend on the match surplus being high enough to cover a minimum wage contract, and still provide positive surplus to the job. Given this constraint, an (x, y) match is feasible if and only if S (x, y) ≥ 0 and Π 1 (w, x, y) ≥ Π 0 (y). The matching sets defined in 7, 8 and 9 are modified to incorporate the constraint become The key practical difficulty is that the matching set for the unemployed M c 0 (x) depends both on whether the surplus is positive and on the value of filling the vacancy at the minimum wage.
Before S(x, y) > 0 was sufficient to determine the feasible matches. For all matches that are feasible subject to the minimum wage constraint, the wage is determined as where w * solves (16) with either (5) or (6)  provide some comparability between the simulations, we calibrate α, γ, and ξ to obtain an unemployment rate of 5 percent, monthly job loss rate of 2 percent, montly job changing rate of 4 percent and a monthly job finding rate (by the unemployed) of 38 percent.
The key aspects of the equilibrium surplus function, S(x, y), are represented by the matching sets in panels (a) and (b) of Figures 1, 2, and 3. In panel (a) we plot the equilibrium matching set, that is all feasible matches: {(x, y)|S(x, y) ≥ 0}. The asymmetry of the matching set is the result of on-the-job search and endogenous match destruction resulting from technology shocks.
By way of comparison, if we set s 1 = 0, δ = 0, c = 0, and N = L we have the environment studied in Shimer and Smith (2000), and replicate their Figure 1 (for the production function xy) here as Figure 4. For production functions displaying complementarity, matches between high and low types do not occur in equilibrium, while with substitutability it is matches among low types that do not occur.
In panel (b) of figures 1-3 we illustrate the preferred matches, those that a worker would leave her current match for. The contour lines here mark the contours of the surplus function, with "hotter" colours indicating a higher match surplus. For a given worker x, and reading horizontally across the job types y, the process of on-the-job search will cause workers to move to matches with a higher surplus, increasing the degree of sorting above what is induces from the matching set M 0 (x) alone.
The features of the joint distribution of (x, y) matches, h (x, y) are displayed in panels (c) and ( this region due to a shock produce sufficient surplus that they are not endogenously destroyed. However, it the match was exogenously destroyed, the job would become inactive, rather than posted as a vacancy. The expected flow output associated with such a y is not sufficient to cover the expected posting costs required to obtain a new worker. In the illustration of xy, all matches begin with a y greater than 0.425. In panel (d) we plot the average worker (job) type matched to a given job (worker) type, further illustrating the positive sorting in the cases of xy and 1 − (x − y) 2 production and the negative sorting in the case of x + y production.
In figures (5), (6), and (7) we plot the equilibrium distributions of matched and unmatched workers and jobs and the relationships between wages, output and the underlying types. The general observation is the that distributions of the observables, wages and output, as well as the relationship between these variables and the underlying types varies markedly across production technologies.
5 What might we learn about sorting from wages?
As discussed in the introduction, Abowd et al (1999) use a simple empirical measure of sorting that can be obtained by estimating a log-wage equation in which wages are a linear function of a worker fixed effect, a firm fixed effect, and an orthogonal worker-firm effect where z it are time varying observables of workers, : α i is a worker fixed effect, ψ j is a firm fixed effect, and : u it is an orthogonal residual. The correlation betweenα i andψ j(i) in a given match is taken as an estimate of the degree of sorting.
To asses the degree to which the correlation between these estimated fixed effects is informative on the degree of sorting on type, we conduct this exercise for each of the numerical examples considered in Section 4. 5 In Table 1 Figure 1: The production function is xy. The green area in panel (a) represent all the feasible matches, that is all pairs of (x, y) such that S(x, y) ≥ 0. In panel (b) we plot the contour lines of S (x, y). The "hotter" colours represent higher values of S(x, y). A worker of type x will leave an (x, y) to form an (x, y ) match whenever she is contacted by a y and S (x, y ) > S (x, y). In panel (c) we plot the joint distribution of matches, h (x, y), with "hotter" colours indicating more matches. In panel (d) we plot the average type of job (worker) that a worker (job) of a given type matches with.  Figure 2: The production function is x + y. The green area in panel (a) represent all the feasible matches, that is all pairs of (x, y) such that S(x, y) ≥ 0. In panel (b) we plot the contour lines of S (x, y). The "hotter" colours represent higher values of S(x, y). A worker of type x will leave an (x, y) to form an (x, y ) match whenever she is contacted by a y and S (x, y ) > S (x, y). In panel (c) we plot the joint distribution of matches, h (x, y), with "hotter" colours indicating more matches. In panel (d) we plot the average type of job (worker) that a worker (job) of a given type matches with. (d) Expected job (worker) type by type of worker (job) Figure 3: The production function is xy. The green area in panel (a) represent all the feasible matches, that is all pairs of (x, y) such that S(x, y) ≥ 0. In panel (b) we plot the contour lines of S (x, y). The "hotter" colours represent higher values of S(x, y). A worker of type x will leave an (x, y) to form an (x, y ) match whenever she is contacted by a y and S (x, y ) > S (x, y). In panel (c) we plot the joint distribution of matches, h (x, y), with "hotter" colours indicating more matches. In panel (d) we plot the average type of job (worker) that a worker (job) of a given type matches with.