Matching, Sorting and Wages - CiteSeerX
Matching, Sorting and Wages Jeremy Lise∗
Costas Meghir†
Jean-Marc Robin‡
May 1, 2008 Preliminary and Incomplete
Abstract This paper develops an empirical model of employer-employee matching with search frictions and heterogeneous agents to address the empirical question: How much sorting is there in the labour market with respect to unobserved worker and firm characteristics? The only empirical evidence we currently have on this issue comes from estimates of Mincer type wage regressions which include a fixed worker effect and a fixed firm effect. The correlation between the estimated worker and firm fixed effects can be calculated and, across a surprising number of countries and data sets, has been found to be either zero or negative. This result has been widely interpreted as indicating a lack of sorting in the labour market, which is consistent with a lack of complementarity between workers and firms in production. In this paper we demonstrate that the sign of the correlation between estimated fixed effects is not necessarily informative on the degree of sorting, and depends crucially on the production function. Indeed, we provide examples in which the production function is supermodular, inducing strong positive sorting between workers and firms, and yet the correlation between estimated fixed effects is negative.
∗
University College London and IFS University College London and IFS ‡ Paris School of Economics and University College London †
1
Introduction
This paper develops an empirical model of employer-employee matching with search frictions and heterogeneous agents. 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). However, matching models with heterogeneous workers and firms are a relatively new subject of interest. 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).1 To the best of our knowledge, there has 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 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), and Moscarini (2001) and Sattinger (2003) explore theoretical extensions of Roy models with search frictions. However, few characteristics of workers and jobs are recorded in available data sets. 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. How much sorting is there with respect to these unobserved characteristics? We are aware of only one piece of empirical evidence on that question. Abowd, Kramarz and Margolis (2000) and Abowd, Kramarz, Lengermann and Roux (2003) use French and U.S. matched employeremployee data to estimate a static, linear log wage equation with employer and worker fixed effects (by OLS). They find a small, and if any negative, cross-sectional correlation between firm and worker fixed effects. Abowd, Kramarz, Lengermann and Perez-Duarte (2004) document the distribution of these correlations calculated within industries. In the U.S. 90% of these correlations range between -15% and 5%, and in France between -27% and -5%. The slight shift toward negative numbers is troubling. The authors acknowledge the possible existence of 1
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.
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a negative bias if job-to-job mobility is limited. Nevertheless, Abowd et al. are certainly right to conclude that there is no evidence of positive correlation between person and firm effects. Whether this indicates a lack of positive sorting based on unobservables is a different story. The person and firm effects which are estimated from the linear log wage equation are complicated transformations of the underlying individual-specific, unobserved characteristics. A structural model is thus required to recover the true underlying distribution. The aim of this paper is to develop a structural matching model of heterogeneous workers with frictions and readdress the empirical question raised by Abowd et al. (2004). The model is similar to Shimer and Smith’s but differs in important aspects. Workers and Firms differ in only one, continuous dimension of heterogeneity. Firms’ production function have constant returns to scale in labour, so matches can be described by pairs of workers and employers. Contrary to Shimer and Smith, we make workers and employers essentially different. Workers’ characteristics do not change over time whereas firms’ contributions to match productivity fluctuate. Workers search on the job and may leave their employers to form better matches, or bid up their wage by exploiting outside offers. Finally, we model a process of job creation and destruction.
2
An Overview of the Model
In what follows we provide an intuitive explanation of how the model is setup, and the motivation for each component. We then follow this up with a formal description of the model. In the economy there is a fixed number of individuals and a fixed number of occupations or production lines. Individuals have linear utility in income and the employers managing occupations care about profits. Individuals may be matched with an occupation and thus working, or they may be unemployed job seekers. Occupations on the other hand may be in three different states. First they may be matched with a worker, in which case they are producing. Second, they may be vacant and waiting for a suitable worker. Finally production lines may be inactive, and thus potential entrants in the labour market. Both workers and occupations are heterogeneous with respect to a productivity relevant characteristic; output depends on the characteristic of both sides. 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
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the centre of our model. When an occupation and a worker meet and the total match surplus is positive, the sharing of this surplus is agreed upon based on Nash asymmetric bargaining, where workers and firms have some exogenous bargaining power. When employed workers receive outside offers they either leave their current match or use the outside offer to bid up their current wage.2 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 occupation has zero surplus from entering the market and posting a vacancy. To derive the implications of the model, solve it and ultimately apply it to data we proceed by defining the value functions for the workers and the firms. We use the Nash bargain to define the pay policy of the firm and hence derive how wages depend on heterogeneity and on productivity shocks. We use the free entry condition to determine the distribution of active matches and firms. In what follows we describe formally the model.
3
The Formal Description of the Model
3.1
Setup
Each individual worker is characterised by a permanent productivity relevant characteristic which we denote by x. 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 R ` (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. Occupations are characterised by a productivity parameter y with bounded support y, y . The (stationary) measure of occupation productivity in the population of firms, whether vacant, matched, or inactive is denoted by N (y) and possesses a density n (y). There are N posts in the economy and the (endogenous) measure of vacant posts is V (y) with density v (y). The number of vacancies is denoted by V . The number of inactive posts, i.e. potential posts for which firms have not advertised a vacancy, is I. The endogenous measure of y among these posts is I(y) 2
In parallel work Lentz (2008) considers a model of on-the-job search with endogenous search intensity, where all workers match with any firm when transiting from unemployment and sorting is the results due to the differing return to search by worker type.
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and similarly the density is denoted by i(y). In a given occupation, y fluctuates according to a jump process. δ is the instantaneous arrival rate of jumps and Q (y 0 |y) is the (Markov) transition probability for y. A match between a worker x and a firm y produces a flow of output f (x, y). 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 Z h (x, y) dy + u (x) = ` (x) .
(1)
Similarly we can write an equivalent relationship between the distribution of firm productivities, active matches, unfilled vacancies and inactive firms Z h (x, y) dx + v (y) + i(y) = n (y) .
(2)
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 occupations. We assume that the unemployed workers search for work at a fixed intensity s0 . The search intensity for an employed worker is s1 . 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 M s0 U + s1 L − U , V . It is convenient to define the equilibrium parameter M s0 U + s1 L − U , V κ= . s0 U + s1 L − U V
(3)
Then s0 κv (y) and s1 κv (y) are the rates at which unemployed and employed workers of any type contact vacancies of type y. Symmetrically, s0 κu (x) and s1 κh (x, y 0 ) are the rates at which a firm of any type contacts a worker of type x, either unemployed or currently employed at a firm of type y 0 .
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3.2
Match formation and rent sharing
The value of an (x, y) match to a worker of ability x, matched with an occupation of productivity y, at wage w is denoted as W1 (w, x, y). The value of an (x, y) match to a firm of productivity y matched to a worker of ability x, paying a wage w is denoted as Π1 (w, x, y). Let W0 (x) denote the present value of unemployment for a worker with characteristic x, and let Π0 (y) denote the present value of a vacancy. Define the “surplus” of a match (x, y) as
S (x, y) = Π1 (w, x, y) + W1 (w, x, y) − Π0 (y) − W0 (x) .
(4)
Notice that it may well be that a match (x, y) yields a positive output f (x, y) but the cost of a vacancy exceeds the expected profit. Any expected negative profit implies the same decision: exiting the market, that is pay no vacancy cost and and as a result, have zero chance of hiring a worker. We assume that incumbent employers match outside offers. A negotiation game is then played between the worker and both firms as in Cahuc, Postel-Vinay, and Robin (2006). If a worker x currently paired to a firm y finds an alternative occupation y 0 such that S (x, y 0 ) > S (x, y), the worker moves to the alternative occupation 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, plus a fraction β of the quasi rent: W1∗ x, y 0 |y − W0 (x) = S (x, y) + β S x, y 0 − S (x, y) = βS x, y 0 + (1 − β) S (x, y) .
(5)
Notice that the present value of the new wage contract W1∗ (x, y 0 |y) does not depend on the last wage contracted with the incumbent employer. It depends on the total surplus of the previous match and the total surplus of the current match but not on the particular wage contract in place at the previous match. Next, consider the case where W1 − W0 (x) < S (x, y 0 ) ≤ S (x, y), where W1 > W0 (x) is the 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 up to W1∗ (x, y|y 0 ) − W0 (x), where W1∗ (x, y|y 0 ) − W0 (x) is as in equation (5). 5
If S (x, y 0 ) ≤ W1 − W0 (x), nothing happens. The worker gains nothing from the competition between y and y 0 . When an unemployed worker x finds a vacant occupation y a match is formed if and only if S (x, y) ≥ 0, and the surplus is split according to Nash bargaining: W1∗ (x, y|0) − W0 (x) = βS (x, y) .
(6)
This equivalence result between Nash bargaining and rent splitting is not subject to Shimer’s (2006) critique because the continuation value for workers when the match is destroyed— including when the worker changes occupation through on-the-job search—is not a function of the last negotiated contract. We now denote by M0 (x) the set of occupations y that are acceptable for an unemployed worker: M0 (x) = {y|S (x, y) ≥ 0} . Let also M1 (x, y) be the set of occupations y 0 such that match (x, y 0 ) can be formed and is preferred to an acceptable current match (x, y): M1 (x, y) = y 0 |S x, y 0 > S (x, y) , and let M2 (w, x, y) denote the set of y 0 such that match (x, y 0 ) does not produce a higher surplus than (x, y) but the competition for the worker yields a wage increase: M2 (w, x, y) = {y 0 |S(x, y) > S(x, y 0 ) > W1 (w, x, y) − W0 (x) and S(x, y 0 ) ≥ 0}.
The complement of a set A is denoted A.
3.3
Renegotiation
Consider a worker x employed at a firm y at wage w. If a productivity shock moves y to y 0 such that S (x, y 0 ) < 0, the match is endogenously destroyed. Both the worker and the firm agree that they are better off unmatched. Suppose that a productivity shock moves y to y 0 such that S (x, y 0 ) ≥ 0. The value of the current wage contract is now W1 (w, x, y 0 ). If
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W1 (w, x, y 0 ) − W0 (x) ∈ [0, S (x, y 0 )], neither the worker nor the firm has a credible treat to force renegotiation; both are better off remaining matched at the current wage contract than unmatched. Hence, contracts being renegotiated by mutual agreement only, and the current contract being sustainable, it will not be renegotiated. If, however, W1 (w, x, y 0 ) − W0 (x) < 0 or W1 (w, x, y 0 ) − W0 (x) > S(x, y 0 ), (with S (x, y 0 ) ≥ 0) then either the worker has a credible threat to quit or the firm has a credible threat to fire the employee. In this case a new wage contract is negotiated. The new wage contract is such that it moves the current wage the smallest amount necessary to put it back in the bargaining set. Thus, a new wage w0 is set such that W1 (w0 , x, y 0 ) − W0 (x) = 0 if at the old contract W1 (w, x, y 0 ) − W0 (x) < 0 and W1 (w0 , x, y 0 ) − W0 (x) = S(x, y 0 ) if at the old contract W1 (w, x, y 0 ) − W0 (x) > S(x, y 0 ). Define
y ∈ C (w, x) ⇔ 0 ≤ W1 (w, x, y) − W0 (x) ≤ S (x, y) y ∈ C + (w, x) ⇔ 0 ≤ S (x, y) < W1 (w, x, y) − W0 (x) y 0 ∈ C − (w, x) ⇔ W1 (w, x, y) − W0 (x) < 0 ≤ S (x, y)
Now, consider the case of a feasible match and sustainable contract, y ∈ C (w, x). Upon a productivity shock changing y into y 0 , if y 0 ∈ / M0 (x), the the match is terminated; if y 0 ∈ C (w, x), the contract is not renegotiated and the wage remains equal to w; and if y 0 ∈ C + (w, x) or y 0 ∈ C − (w, x), the contract is renegotiated to a wage w0 as indicated above.
3.4
Value Functions
The next step in solving the model is to characterise the value functions of workers and firms in turn. These define the decision rule for each agent. Let r denote the discount rate. A match is destroyed when it is hit by a productivity shock that changes y into some unacceptable y 0 , or some other adverse shock (at rate ξ). Because it may be welfare improving to reduce equilibrium unemployment, we allow for a tax τ (experience rating) on endogenous separations. We assume that time is discrete with an arbitrarily small unit of time so that two independent events have a negligible chance of occurring simultaneously. 7
Unemployed workers. The present value of unemployment to a worker of type x is W0 (x) that satisfies the option value equation: Z rW0 (x) = b + s0 κ M0 (x)
[W1∗ (x, y|0) − W0 (x)] v(y)dy
Z S (x, y) v(y)dy.
= b + s0 κβ
(7)
M0 (x)
Employed workers. Here, we calculate W1 (w, x, y) the value of a wage contract w. First, consider the case of a sustainable contract, y ∈ C (w, x). After the wage w is paid at the end of the period, the match is exogenously destroyed with probability ξ, a productivity shock changes y into y 0 with probability density δq (y 0 |y), a contact with an alternative employer occurs with probability s1 κV , and nothing happens with probability 1 − ξ − δ − s1 κV . Hence,
W1 (w, x, y) =
W0 (x) w W0 (x) +ξ + δQ M0 (x) |y 1+r 1+r 1+r Z W1 (w, x, y 0 ) +δ q y 0 |y dy 0 1 + r C(w,x) Z Z 0 S (x, y 0 ) + W0 (x) W0 (x) 0 q y |y dy + δ q y 0 |y dy 0 +δ 1+r C + (w,x) C − (w,x) 1 + r Z Z ∗ 0 W1 (x, y |y) W1∗ (x, y|y 0 ) +s1 κ v y 0 dy 0 + s1 κ v y 0 dy 0 1+r 1+r M1 (x,y) M2 (w,x,y) Z W1 (w, x, y) +s1 κ v y 0 dy 0 1+r {y 0 |S(x,y 0 )≤W1 (w,x,y)−W0 (x)} W1 (w, x, y) + 1 − ξ − δ − s1 κV . 1+r
After simplification, this becomes
W1 (w, x, y) − W0 (x) =
Z w − rW0 (x) W1 (w, x, y 0 ) − W0 (x) +δ q y 0 |y dy 0 1+r 1+r C(w,x) Z S (x, y 0 ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) Z βS (x, y) + (1 − β) S (x, y 0 ) +s1 κ v y 0 dy 0 1+r M2 (w,x,y) + [1 − ξ − δ − s1 κV (M1 (x, y) ∪ M2 (w, x, y))]
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(8)
W1 (w, x, y) − W0 (x) . 1+r
Second, suppose that the contract is not sustainable. There are two possible situations, either the current wage contract would assign more than the total surplus to the worker or it would assign less than the outside option to the worker (more than the total surplus to the firm). In either case the contract is renegotiated at the end of the period, unless a productivity shock occurs such that at the new y 0 the contract is sustainable, or the worker receives an outside offer that would lead to a quit or to a wage renegotiation at a sustainable wage contract. Hence, if y∈ / C (w, x), and the worker receives an alternative offer y 0 and W1 (w, x, y) − W0 (x) > S (x, y) then either y 0 ∈ M1 (x, y) and the worker quits and gets W1∗ (x, y 0 |y), or y 0 ∈ / M1 (x, y) and the workers stays and gets S (x, y). If W1 (w, x, y) − W0 (x) < 0 then either y 0 ∈ M1 (x, y) and the worker quits and gets W1∗ (x, y 0 |y), or y 0 ∈ M2 (w, x, y) and the workers stays and gets W1∗ (x, y|y 0 ), or y 0 ∈ / M0 (x) and the worker stays and gets W0 (x). Hence, if W1 (w, x, y) − W0 (x) > S (x, y) then
W1 (w, x, y) − W0 (x) =
Z W1 (w, x, y 0 ) − W0 (x) w − rW0 (x) +δ q y 0 |y dy 0 1+r 1+r C(w,x) Z S (x, y 0 ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) S (x, y) + 1 − ξ − δ − s1 V (M1 (x, y)) 1+r
(9)
and if W1 (w, x, y) − W0 (x) < 0 then Z w − rW0 (x) W1 (w, x, y 0 ) − W0 (x) W1 (w, x, y) − W0 (x) = +δ q y 0 |y dy 0 (10) 1+r 1+r C(w,x) Z 0 S (x, y ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) Z βS (x, y) + (1 − β) S (x, y 0 ) +s1 κ v y 0 dy 0 . 1+r M2 (w,x,y) Combining equations (8), (9), and (10) we can write the Bellman equation in W1 (w, x, y) −
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W0 (x) as Z W1 (w, x, y 0 ) − W0 (x) w − rW0 (x) +δ q y 0 |y dy 0 W1 (w, x, y) − W0 (x) = 1+r 1+r C(w,x) Z 0 S (x, y ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) Z βS (x, y) + (1 − β) S (x, y 0 ) +s1 κ v y 0 dy 0 1+r M2 (w,x,y)
(11)
+ [1 − ξ − δ − s1 κV (M1 (x, y) ∪ M2 (w, x, y))] W1 (w, x, y) − W0 (x) S (x, y) × 1{y∈C(w,x)} + 1{y∈C + (w,x)} 1+r 1+r The Bellman equation defines W1 (w, x, y)−W0 (x) as a fixed point of a contracting operator. A simple iterative algorithm can be used to approximate the fixed point. Let W10 be an initial guess of W1 . Values C 0 (w, x) and M02 (w, x, y) of C (w, x) and M2 (w, x, y) follow from W10 . Then calculate an update of W1 (w, x, y) − W0 (x) using equation (11). This equation allows us to compute the optimal wage contract w given (x, y, y 0 ).
Vacant firms. Consider an open vacancy. Its present value is
(y) rΠopen 0
Z = −c + s0 κ
M−1 0 (y)
(y)] u(x)dx [Π∗1 (x, y|0) − Πopen 0
ZZ + s1 κ M1 (x,y)
∗ Π1 (x, y|y 0 ) − Πopen (y) h x, y 0 dx dy 0 0 Z +δ
Π0 (y 0 ) − Πopen (y) q(y 0 |y)dy 0 , 0
where c is a fixed cost of posting the vacancy; the second term is the flow of benefits from matching with a previously unemployed worker; the third term is the flow of benefits from poaching a worker who is already matched with another firm; the final term reflects the impact of a change in productivity from y to y 0 . Making use of the rent splitting equations: Π∗1 (x, y|0) − Π0 (y) = (1 − β) S (x, y) ≥ 0, Π∗1 (x, y|y 0 ) − Π0 (y) = (1 − β) S (x, y) − S x, y 0 ≥ 0,
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we finally obtain that
(r +
δ) Πopen (y) 0
Z = −c + s0 κ (1 − β)
S (x, y) u(x)dx
M−1 0 (y)
ZZ
+ s1 κ (1 − β)
S (x, y) − S x, y 0
h x, y 0 dx dy 0
M1 (x,y)
Z +δ
Π0 (y 0 )q(y 0 |y)dy 0 . (12)
If the firm 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:
rΠidle 0 (y)
Z h i =δ Π0 (y 0 ) − Πidle (y) q(y 0 |y)dy 0 . 0
The present value of an unmatched job of type y is finally: n o idle (r + δ) Π0 (y) = max (r + δ) Πopen (y) , (r + δ) Π (y) 0 0 Z = δ Π0 (y 0 )q(y 0 |y)dy 0 + max {0, −c + c (y)} .
(13) (14)
with Z c (y) ≡ s0 κ (1 − β)
M−1 0 (y)
S (x, y) u(x)dx ZZ
S (x, y) − S x, y 0 h x, y 0 dxdy 0 . (15)
+ s1 κ (1 − β) M1 (x,y)
and a firm y is inactive whenever open Πidle (y) 0 (y) > Π0
or c > c (y) .
(16)
This condition is like the free entry condition of a standard search-matching model with ex-ante homogeneous firms.
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3.5
The match surplus
We can write the surplus of an (x, y) match as the fixed point defined by
[r + ξ + δ + s1 κV (M1 (x, y)) β] S (x, y) = f (x, y) − rΠ0 (y) − rW0 (x) Z Z − δQ M0 (x) |y τ + δ Π0 (y 0 ) − Π0 (y) q y 0 |y dy 0 S x, y 0 q y 0 |y dy 0 + δ M0 (x)
Z
S x, y 0 v y 0 dy 0 (17)
+ s1 κβ M1 (x,y)
which, using equation (13) becomes
[r + ξ + δ + s1 κV (M1 (x, y)) β] S (x, y) = f (x, y) − rW0 (x) Z − δQ M0 (x) |y τ + δ S x, y 0 q y 0 |y dy 0 − max {0, −c + c (y)} M0 (x)
Z + s1 κβ
S x, y 0 v y 0 dy 0 . (18)
M1 (x,y)
Note that the match surplus is not a function of the current wage contract and is thus not subject to Shimer’s (2006) critique. This is because the Bertrand competition between the two employers at work in the tri-lateral game involving the worker, the incumbent employer and the poaching firm disconnects the final outcome from the worker’s current wage contract.
3.6
Steady-state flow equations.
We now proceed to define the steady state flow equations. The first relates to matches. The total number of matches in the economy will be ZZ L−U =N −V −I =
h (x, y) dxdy.
(19)
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 0 different from y, and the worker may move to unemployment or may keep the job; third, the worker may change job, with probability s1 κV (M1 (x, y))–i.e., a job offer has to be made (at rate s1 κV ) and has to be acceptable (y 0 ∈ M1 (x, y)). On the inflow side, new (x, y) matches are formed when some unemployed or 12
employed workers of type x match with vacant occupations y, or when (x, y 0 ) matches are hit with a productivity shock and exogenously change from (x, y 0 ) 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 ∈ M0 (x) or S (x, y) > 0: Z [δ + ξ + s1 κV (M1 (x, y))] h (x, y) = δ
q(y|y 0 )h x, y 0 dy 0 " Z + s0 u (x) + s1
# 0
h x, y dy
0
κv (y) . (20)
M1 (x,y)
This equation defines the steady-state equilibrium, together with the accounting equations: Z u (x) = ` (x) −
h (x, y) dy,
(21)
Z v (y) + i (y) = n (y) − where v (y) =
(22)
R n (y) − h (x, y) dx if c ≤ c (y) 0
i (y) =
h (x, y) dx,
(23)
if c > c (y) ,
0
if c ≤ c (y)
n (y) −
R
(24)
h (x, y) dx if c > c (y) ,
and the number of vacancies V is equal to the fraction of ys in distribution n (y) −
R
h (x, y) dx
that is such that c ≤ c (y): Z
Z n (y) −
V =
h (x, y) dx dy.
(25)
c≤c(y)
4
Equilibrium
Given values for the exogenous variables: L, N , δ, ξ, s0 , s1 , r, b, c, and β; the distributions of worker and firm types l(x) and n(y); the production function f (x, y); the transition function for productivity dynamics q(y 0 |y); and the matching function M (s0 U +s1 (L−U ), V ), the equilibrium is characterized by knowledge of V , h (x, y) and S (x, y) solving for equations (25), (20) and (17). In these equations, one will substitute for U using equation (19), κ using equation (3), u (x)
13
using equation (21), c (y) using equation (15), v (y) using equation (23), W0 (x) using equation (7).
5
An Illustrative Numerical Example
In order to illustrate the properties and empirical implications of the model we solve and simulate under a particular set of parameters, for several production functions. For the numerical example we set L = 25000, N = 35000 and assume that L(x) = N (y) and are U [0, 1]. We solve using four different production functions: xy, x + y, and 1 − (x − y)2 . The first two production functions are natural representations of complementarity and substitutability. The third is meant to capture the notion that there is a “right man for the job” (Tuelings and Gautier; 2004). The process governing the technological evolution is represented by a Gumbel copula with marginals equal to n(y), q(y 0 |y) = n(y 0 )Gumble(N (y 0 ), N (y)); δ = 0.05; s0 = 1.0; s1 = 0.1; M (s0 U + s1 (L − U ), V ) = α(s0 U + s1 (L − U ))γ V
1−γ
with α = 0.6 and γ = 0.5; r = 0.03; b = 0;
c = 0.1 and β = 0.5. In order to provide some comparability between the simulations, we vary ξ between 0.01 and 0.05 across the simulations to maintain an unemployment rate of roughly 5 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 s1 = 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
14
the matching set M0 (x) alone. The features of the joint distribution of (x, y) matches, h (x, y) are displayed in panels (c) and (d) of the same figures. The process of job accepting; on-the-job search; endogenous and exogenous job destruction; and the decision of which types of jobs to post vacancies for or leave inactive induces the equilibrium distribution of matches. The effect of on-the-job search is evident in that workers initially accept any job in their matching set, and then switch to jobs with a higher surplus. The vertical line that cuts through the distribution (seen prominently for xy at y = 0.375) illustrates the effect of the decision over which types of unmatched jobs to leave inactive and which to post as vacancies. Matches in which the job component, y, falls into 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 0.375. 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.
6
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
log(wit ) = zit β + αi +
J X j=1
15
djit ψ j + uit ,
(26)
where zit are time varying observables of workers, : αi is a worker fixed effect, ψ j is a firm fixed ˆ effect, and : uit is an orthogonal residual. The correlation between α ˆ i and ψ 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 5. In addition we conduct this exercise for the case case in which workers’ bargaining power is set to zero; workers have no bargaining power and wages only rise in response to Bertrand competition between firms competing for the same worker. In Table 1 we compare the equilibrium correlation between x and y in the model to the correlation between estimated worker and firm fixed effects resulting from equation 26. The results of this exercise demonstrate that while the sign of the correlation between the estimated fixed effects correctly picks up the positive/negative sorting in the case of production functions of the form xy or x + y, it is of opposite sign in the cases where there is a best job for each worker. Additionally, even in the case of xy production, where the positive correlation between x and y is picked up by the correlation between the estimated fixed effects, the degree of this correlation bears little resemblance to that actual correlation. This estimated correlation is not necessarily informative on the degree of sorting in the model. Indeed, this suggests the need to estimate the production function in order to answer the question regarding the degree of sorting on unobservables.3 A natural starting point for the production function would be a second order approximation, which would be exact for each of the examples considered here. 3
In addition to this effect, Postel-Vinay and Robin (2006) note that in terms of asymptotics, OLS estimate of β is consistent as i → ∞ for fixed T and OLS estimates of α and ψ are consistent when T → ∞ faster than I and J. In practice, the data contains millions of workers, tens of thousands of firms, and fewer than ten years. Indeed, empirical estimates of sorting which are based on worker and firm fixed effects introduce a negative bias, which will introduce a spurious negative correlation when calculating the correlation between worker and firm fixed effects. This is illustrated as follows; empirically, β and ψ j are estimated from the within transformation log wit − log wi = (xit − xi )β +
J X j (djit − di )ψ j + uit − ui . j=1
This makes it clear that we need to see workers change firm to identify the firm fixed effects ψ j . The worker fixed effects are estimated as J X j ˆ− ˆ . α ˆ i = log wi − xi β di ψ j j=1
Notice, any statistical error affecting the estimate of the firm effect translates directly to the estimate of the worker effect, with a sign reversal. OLS estimates of firm and worker effects are likely to be imprecise and spuriously negatively correlated given short time dimension and limited worker mobility.
16
Table 1: Actual and Estimated Sorting β = 0.5 β=0 ˆ ˆ ) Production Function corr(x, y) corr(ˆ αi , ψ j(i) ) corr(x, y) corr(ˆ αi , ψ j(i) f (x, y) = xy f (x, y) = x + y f (x, y) = 1 − (x − y)2 f (x, y) = 1 + 14 x − (x − y)2 f (x, y) = 1 + 14 y − (x − y)2
7
0.6199 -0.0330 0.6184 0.5774 0.5434
0.1207 -0.0384 -0.0775 -0.0077 0.0462
0.6731 -0.1260 0.5208 0.4707 0.4640
0.2763 -0.1695 -0.0512 -0.0076 -0.0452
Conclusion and further work
Estimation of the model presented here is the subject of current research. The natural type of data to use in the empirical implementation is matched worker and firm data, an avenue we are actively pursuing. One obstacle in this strategy is the need to take a stand on the formation of jobs into firms, and the possibility of interaction between workers within a firm. An additional interesting question is how much we can learn about earnings processes using standard panel data on workers and the restrictions from the model. The model predicts an earning process with lots of heterogeneity, in which the time varying part of earnings is dependent on the permanent component. In addition, job mobility and unemployment durations are dependent on the same underlying permanent component. With an estimated version of the model in hand we will be well placed to evaluate important policy questions, such as employment protection legislation and minimum wages, within a coherent empirical economic model.
17
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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 0 ) match whenever she is contacted by a y 0 and S (x, y 0 ) > 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.
18
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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 0 ) match whenever she is contacted by a y 0 and S (x, y 0 ) > 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.
19
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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 0 ) match whenever she is contacted by a y 0 and S (x, y 0 ) > 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.
20
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Figure 4: Feasible matches with xy production, without on-the-job search, without endogenous job destruction, without vacancy costs, and with an exogenous number of firms set equal to the number of workers. (s1 = 0, δ = 0, c = 0, and N = L)
21
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Figure 5: Output, wages and type distributions, f (x, y) = xy 22
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Figure 6: Output, wages and type distributions, f (x, y) = x + y 23
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Figure 7: Output, wages and type distributions, f (x, y) = 1 − (x − y)2 24
References [1] ABOWD, J. M., KRAMARZ, F., LENGERMANN, P., and S. PEREZ-DUARTE (2005): “Persistent InterIndustry Wage Differences: Rent Sharing and Opportunity Costs"
[2] ABOWD, J. M., KRAMARZ, F., LENGERMANN, P., and ROUX, S. (2003): “Interindustry and firm-size wage differentials in the United States and France,” Cornell University, mimeo.
[3] ABOWD, J. M., F. KRAMARZ and D. N. MARGOLIS (1999) “High Wage Workers and High Wage Firms,” Econometrica, 67, 251–333.
[4] ATAKAN, A. (2006) “Assortative Matching with Explicit Search Costs,” Econometrica, 74, 667–680. [5] BECKER, G. (1973): “A Theory of Marriage: Part I," Journal of Political Economy, 81, 813–846. [6] CAHUC, P., POSTEL-VINAY, F. and ROBIN, J-M. (2006): “Wage Bargaining with On-the-Job Search: Theory and Evidence,” Econometrica, 74: 323–364.
[7] DIAMOND, P. (1982): “Aggregate Demand Management in Search Equilibrium," Journal of Political Economy, 90, 881–894.
[8] HECKMAN, J. and G. SEDLACEK (1985): “Heterogeneity, Aggregation and Market Wage Functions: An Empirical Model of Self-Selection in the Labor Market," Journal of Political Economy, 93(6), 1077–1125.
[9] HECKMAN, J. and B. HONORE (1990): “The Empirical Content of the Roy Model,” Econometrica, 58(5), 1121–1149.
[10] KRUEGER, A. and L. SUMMERS (1988): “Efficiency Wages and the Inter-industry Wage Structure,” Econometrica, 56(2): 259–93.
[11] LU, X., and R. P. McAFFEE (1996): “Matching and Expectations in a Market with Heterogeneous Agents," in Advances in Applied Microeconomics, Vol. 6, ed. by M. Baye, Greenwich, CT: JAI Press, pp. 121–156.
[12] LENTZ, R. (2008): “Sorting in a General Equilibrium On-th-Job Search Model," Wisconsin-Madison, mimeo.
[13] MORTENSEN, D. (1982): “Property Rights and Efficiency in Mating, Racing, and Related Games,” American Economic Review , 72, 968–979.
[14] MOSCARINI, G. (2001): “Excess Worker Reallocation." The Review of Economic Studies, 68, No. 3, pp. 593–612
[15] PISSARIDES, C. (1990): Equilibrium Unemployment Theory. Oxford: Blackwell.
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[16] POSTEL-VINAY, F. and ROBIN, J-M. (2006): “Microeconometric Search-Matching Models and Matched Employer-Employee Data,” Advances in Econometrics, Eighth World Congress, eds. Blundell, Newey and Persson.
[17] SATTINGER . M. (1995): “Search and the Efficient Assignment of Workers to Jobs," International Economic Review , 36, 283–302.
[18] SATTINGER . M. (2003): “A Search Version of the Roy Model." SUNY Albany, mimeo. [19] SHIMER, R. (2005): “The Assignment of Workers to Jobs in an Economy with Coordination Frictions,” Journal of Political Economy, 113(5): 996–1025.
[20] SHIMER, R. (2006): “On-the-job Search and Strategic Bargaining,” European Economic Review , 50(4): 811–830. Also published in Structural Models of Wage and Employment Dynamics, conference volume in honor of Dale Mortensen, edited by Henning Bunzel, Bent J. Christensen, George R. Neumann, and Jean-Marc Robin, Elsevier, 2006.
[21] TEULINGS, C. and GAUTIER, P. (2004): “The Right Man for the Job, ” Review of Economic Studies, 71: 553–580.
26
Costas Meghir†
Jean-Marc Robin‡
May 1, 2008 Preliminary and Incomplete
Abstract This paper develops an empirical model of employer-employee matching with search frictions and heterogeneous agents to address the empirical question: How much sorting is there in the labour market with respect to unobserved worker and firm characteristics? The only empirical evidence we currently have on this issue comes from estimates of Mincer type wage regressions which include a fixed worker effect and a fixed firm effect. The correlation between the estimated worker and firm fixed effects can be calculated and, across a surprising number of countries and data sets, has been found to be either zero or negative. This result has been widely interpreted as indicating a lack of sorting in the labour market, which is consistent with a lack of complementarity between workers and firms in production. In this paper we demonstrate that the sign of the correlation between estimated fixed effects is not necessarily informative on the degree of sorting, and depends crucially on the production function. Indeed, we provide examples in which the production function is supermodular, inducing strong positive sorting between workers and firms, and yet the correlation between estimated fixed effects is negative.
∗
University College London and IFS University College London and IFS ‡ Paris School of Economics and University College London †
1
Introduction
This paper develops an empirical model of employer-employee matching with search frictions and heterogeneous agents. 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). However, matching models with heterogeneous workers and firms are a relatively new subject of interest. 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).1 To the best of our knowledge, there has 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 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), and Moscarini (2001) and Sattinger (2003) explore theoretical extensions of Roy models with search frictions. However, few characteristics of workers and jobs are recorded in available data sets. 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. How much sorting is there with respect to these unobserved characteristics? We are aware of only one piece of empirical evidence on that question. Abowd, Kramarz and Margolis (2000) and Abowd, Kramarz, Lengermann and Roux (2003) use French and U.S. matched employeremployee data to estimate a static, linear log wage equation with employer and worker fixed effects (by OLS). They find a small, and if any negative, cross-sectional correlation between firm and worker fixed effects. Abowd, Kramarz, Lengermann and Perez-Duarte (2004) document the distribution of these correlations calculated within industries. In the U.S. 90% of these correlations range between -15% and 5%, and in France between -27% and -5%. The slight shift toward negative numbers is troubling. The authors acknowledge the possible existence of 1
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.
1
a negative bias if job-to-job mobility is limited. Nevertheless, Abowd et al. are certainly right to conclude that there is no evidence of positive correlation between person and firm effects. Whether this indicates a lack of positive sorting based on unobservables is a different story. The person and firm effects which are estimated from the linear log wage equation are complicated transformations of the underlying individual-specific, unobserved characteristics. A structural model is thus required to recover the true underlying distribution. The aim of this paper is to develop a structural matching model of heterogeneous workers with frictions and readdress the empirical question raised by Abowd et al. (2004). The model is similar to Shimer and Smith’s but differs in important aspects. Workers and Firms differ in only one, continuous dimension of heterogeneity. Firms’ production function have constant returns to scale in labour, so matches can be described by pairs of workers and employers. Contrary to Shimer and Smith, we make workers and employers essentially different. Workers’ characteristics do not change over time whereas firms’ contributions to match productivity fluctuate. Workers search on the job and may leave their employers to form better matches, or bid up their wage by exploiting outside offers. Finally, we model a process of job creation and destruction.
2
An Overview of the Model
In what follows we provide an intuitive explanation of how the model is setup, and the motivation for each component. We then follow this up with a formal description of the model. In the economy there is a fixed number of individuals and a fixed number of occupations or production lines. Individuals have linear utility in income and the employers managing occupations care about profits. Individuals may be matched with an occupation and thus working, or they may be unemployed job seekers. Occupations on the other hand may be in three different states. First they may be matched with a worker, in which case they are producing. Second, they may be vacant and waiting for a suitable worker. Finally production lines may be inactive, and thus potential entrants in the labour market. Both workers and occupations are heterogeneous with respect to a productivity relevant characteristic; output depends on the characteristic of both sides. 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
2
the centre of our model. When an occupation and a worker meet and the total match surplus is positive, the sharing of this surplus is agreed upon based on Nash asymmetric bargaining, where workers and firms have some exogenous bargaining power. When employed workers receive outside offers they either leave their current match or use the outside offer to bid up their current wage.2 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 occupation has zero surplus from entering the market and posting a vacancy. To derive the implications of the model, solve it and ultimately apply it to data we proceed by defining the value functions for the workers and the firms. We use the Nash bargain to define the pay policy of the firm and hence derive how wages depend on heterogeneity and on productivity shocks. We use the free entry condition to determine the distribution of active matches and firms. In what follows we describe formally the model.
3
The Formal Description of the Model
3.1
Setup
Each individual worker is characterised by a permanent productivity relevant characteristic which we denote by x. 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 R ` (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. Occupations are characterised by a productivity parameter y with bounded support y, y . The (stationary) measure of occupation productivity in the population of firms, whether vacant, matched, or inactive is denoted by N (y) and possesses a density n (y). There are N posts in the economy and the (endogenous) measure of vacant posts is V (y) with density v (y). The number of vacancies is denoted by V . The number of inactive posts, i.e. potential posts for which firms have not advertised a vacancy, is I. The endogenous measure of y among these posts is I(y) 2
In parallel work Lentz (2008) considers a model of on-the-job search with endogenous search intensity, where all workers match with any firm when transiting from unemployment and sorting is the results due to the differing return to search by worker type.
3
and similarly the density is denoted by i(y). In a given occupation, y fluctuates according to a jump process. δ is the instantaneous arrival rate of jumps and Q (y 0 |y) is the (Markov) transition probability for y. A match between a worker x and a firm y produces a flow of output f (x, y). 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 Z h (x, y) dy + u (x) = ` (x) .
(1)
Similarly we can write an equivalent relationship between the distribution of firm productivities, active matches, unfilled vacancies and inactive firms Z h (x, y) dx + v (y) + i(y) = n (y) .
(2)
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 occupations. We assume that the unemployed workers search for work at a fixed intensity s0 . The search intensity for an employed worker is s1 . 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 M s0 U + s1 L − U , V . It is convenient to define the equilibrium parameter M s0 U + s1 L − U , V κ= . s0 U + s1 L − U V
(3)
Then s0 κv (y) and s1 κv (y) are the rates at which unemployed and employed workers of any type contact vacancies of type y. Symmetrically, s0 κu (x) and s1 κh (x, y 0 ) are the rates at which a firm of any type contacts a worker of type x, either unemployed or currently employed at a firm of type y 0 .
4
3.2
Match formation and rent sharing
The value of an (x, y) match to a worker of ability x, matched with an occupation of productivity y, at wage w is denoted as W1 (w, x, y). The value of an (x, y) match to a firm of productivity y matched to a worker of ability x, paying a wage w is denoted as Π1 (w, x, y). Let W0 (x) denote the present value of unemployment for a worker with characteristic x, and let Π0 (y) denote the present value of a vacancy. Define the “surplus” of a match (x, y) as
S (x, y) = Π1 (w, x, y) + W1 (w, x, y) − Π0 (y) − W0 (x) .
(4)
Notice that it may well be that a match (x, y) yields a positive output f (x, y) but the cost of a vacancy exceeds the expected profit. Any expected negative profit implies the same decision: exiting the market, that is pay no vacancy cost and and as a result, have zero chance of hiring a worker. We assume that incumbent employers match outside offers. A negotiation game is then played between the worker and both firms as in Cahuc, Postel-Vinay, and Robin (2006). If a worker x currently paired to a firm y finds an alternative occupation y 0 such that S (x, y 0 ) > S (x, y), the worker moves to the alternative occupation 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, plus a fraction β of the quasi rent: W1∗ x, y 0 |y − W0 (x) = S (x, y) + β S x, y 0 − S (x, y) = βS x, y 0 + (1 − β) S (x, y) .
(5)
Notice that the present value of the new wage contract W1∗ (x, y 0 |y) does not depend on the last wage contracted with the incumbent employer. It depends on the total surplus of the previous match and the total surplus of the current match but not on the particular wage contract in place at the previous match. Next, consider the case where W1 − W0 (x) < S (x, y 0 ) ≤ S (x, y), where W1 > W0 (x) is the 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 up to W1∗ (x, y|y 0 ) − W0 (x), where W1∗ (x, y|y 0 ) − W0 (x) is as in equation (5). 5
If S (x, y 0 ) ≤ W1 − W0 (x), nothing happens. The worker gains nothing from the competition between y and y 0 . When an unemployed worker x finds a vacant occupation y a match is formed if and only if S (x, y) ≥ 0, and the surplus is split according to Nash bargaining: W1∗ (x, y|0) − W0 (x) = βS (x, y) .
(6)
This equivalence result between Nash bargaining and rent splitting is not subject to Shimer’s (2006) critique because the continuation value for workers when the match is destroyed— including when the worker changes occupation through on-the-job search—is not a function of the last negotiated contract. We now denote by M0 (x) the set of occupations y that are acceptable for an unemployed worker: M0 (x) = {y|S (x, y) ≥ 0} . Let also M1 (x, y) be the set of occupations y 0 such that match (x, y 0 ) can be formed and is preferred to an acceptable current match (x, y): M1 (x, y) = y 0 |S x, y 0 > S (x, y) , and let M2 (w, x, y) denote the set of y 0 such that match (x, y 0 ) does not produce a higher surplus than (x, y) but the competition for the worker yields a wage increase: M2 (w, x, y) = {y 0 |S(x, y) > S(x, y 0 ) > W1 (w, x, y) − W0 (x) and S(x, y 0 ) ≥ 0}.
The complement of a set A is denoted A.
3.3
Renegotiation
Consider a worker x employed at a firm y at wage w. If a productivity shock moves y to y 0 such that S (x, y 0 ) < 0, the match is endogenously destroyed. Both the worker and the firm agree that they are better off unmatched. Suppose that a productivity shock moves y to y 0 such that S (x, y 0 ) ≥ 0. The value of the current wage contract is now W1 (w, x, y 0 ). If
6
W1 (w, x, y 0 ) − W0 (x) ∈ [0, S (x, y 0 )], neither the worker nor the firm has a credible treat to force renegotiation; both are better off remaining matched at the current wage contract than unmatched. Hence, contracts being renegotiated by mutual agreement only, and the current contract being sustainable, it will not be renegotiated. If, however, W1 (w, x, y 0 ) − W0 (x) < 0 or W1 (w, x, y 0 ) − W0 (x) > S(x, y 0 ), (with S (x, y 0 ) ≥ 0) then either the worker has a credible threat to quit or the firm has a credible threat to fire the employee. In this case a new wage contract is negotiated. The new wage contract is such that it moves the current wage the smallest amount necessary to put it back in the bargaining set. Thus, a new wage w0 is set such that W1 (w0 , x, y 0 ) − W0 (x) = 0 if at the old contract W1 (w, x, y 0 ) − W0 (x) < 0 and W1 (w0 , x, y 0 ) − W0 (x) = S(x, y 0 ) if at the old contract W1 (w, x, y 0 ) − W0 (x) > S(x, y 0 ). Define
y ∈ C (w, x) ⇔ 0 ≤ W1 (w, x, y) − W0 (x) ≤ S (x, y) y ∈ C + (w, x) ⇔ 0 ≤ S (x, y) < W1 (w, x, y) − W0 (x) y 0 ∈ C − (w, x) ⇔ W1 (w, x, y) − W0 (x) < 0 ≤ S (x, y)
Now, consider the case of a feasible match and sustainable contract, y ∈ C (w, x). Upon a productivity shock changing y into y 0 , if y 0 ∈ / M0 (x), the the match is terminated; if y 0 ∈ C (w, x), the contract is not renegotiated and the wage remains equal to w; and if y 0 ∈ C + (w, x) or y 0 ∈ C − (w, x), the contract is renegotiated to a wage w0 as indicated above.
3.4
Value Functions
The next step in solving the model is to characterise the value functions of workers and firms in turn. These define the decision rule for each agent. Let r denote the discount rate. A match is destroyed when it is hit by a productivity shock that changes y into some unacceptable y 0 , or some other adverse shock (at rate ξ). Because it may be welfare improving to reduce equilibrium unemployment, we allow for a tax τ (experience rating) on endogenous separations. We assume that time is discrete with an arbitrarily small unit of time so that two independent events have a negligible chance of occurring simultaneously. 7
Unemployed workers. The present value of unemployment to a worker of type x is W0 (x) that satisfies the option value equation: Z rW0 (x) = b + s0 κ M0 (x)
[W1∗ (x, y|0) − W0 (x)] v(y)dy
Z S (x, y) v(y)dy.
= b + s0 κβ
(7)
M0 (x)
Employed workers. Here, we calculate W1 (w, x, y) the value of a wage contract w. First, consider the case of a sustainable contract, y ∈ C (w, x). After the wage w is paid at the end of the period, the match is exogenously destroyed with probability ξ, a productivity shock changes y into y 0 with probability density δq (y 0 |y), a contact with an alternative employer occurs with probability s1 κV , and nothing happens with probability 1 − ξ − δ − s1 κV . Hence,
W1 (w, x, y) =
W0 (x) w W0 (x) +ξ + δQ M0 (x) |y 1+r 1+r 1+r Z W1 (w, x, y 0 ) +δ q y 0 |y dy 0 1 + r C(w,x) Z Z 0 S (x, y 0 ) + W0 (x) W0 (x) 0 q y |y dy + δ q y 0 |y dy 0 +δ 1+r C + (w,x) C − (w,x) 1 + r Z Z ∗ 0 W1 (x, y |y) W1∗ (x, y|y 0 ) +s1 κ v y 0 dy 0 + s1 κ v y 0 dy 0 1+r 1+r M1 (x,y) M2 (w,x,y) Z W1 (w, x, y) +s1 κ v y 0 dy 0 1+r {y 0 |S(x,y 0 )≤W1 (w,x,y)−W0 (x)} W1 (w, x, y) + 1 − ξ − δ − s1 κV . 1+r
After simplification, this becomes
W1 (w, x, y) − W0 (x) =
Z w − rW0 (x) W1 (w, x, y 0 ) − W0 (x) +δ q y 0 |y dy 0 1+r 1+r C(w,x) Z S (x, y 0 ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) Z βS (x, y) + (1 − β) S (x, y 0 ) +s1 κ v y 0 dy 0 1+r M2 (w,x,y) + [1 − ξ − δ − s1 κV (M1 (x, y) ∪ M2 (w, x, y))]
8
(8)
W1 (w, x, y) − W0 (x) . 1+r
Second, suppose that the contract is not sustainable. There are two possible situations, either the current wage contract would assign more than the total surplus to the worker or it would assign less than the outside option to the worker (more than the total surplus to the firm). In either case the contract is renegotiated at the end of the period, unless a productivity shock occurs such that at the new y 0 the contract is sustainable, or the worker receives an outside offer that would lead to a quit or to a wage renegotiation at a sustainable wage contract. Hence, if y∈ / C (w, x), and the worker receives an alternative offer y 0 and W1 (w, x, y) − W0 (x) > S (x, y) then either y 0 ∈ M1 (x, y) and the worker quits and gets W1∗ (x, y 0 |y), or y 0 ∈ / M1 (x, y) and the workers stays and gets S (x, y). If W1 (w, x, y) − W0 (x) < 0 then either y 0 ∈ M1 (x, y) and the worker quits and gets W1∗ (x, y 0 |y), or y 0 ∈ M2 (w, x, y) and the workers stays and gets W1∗ (x, y|y 0 ), or y 0 ∈ / M0 (x) and the worker stays and gets W0 (x). Hence, if W1 (w, x, y) − W0 (x) > S (x, y) then
W1 (w, x, y) − W0 (x) =
Z W1 (w, x, y 0 ) − W0 (x) w − rW0 (x) +δ q y 0 |y dy 0 1+r 1+r C(w,x) Z S (x, y 0 ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) S (x, y) + 1 − ξ − δ − s1 V (M1 (x, y)) 1+r
(9)
and if W1 (w, x, y) − W0 (x) < 0 then Z w − rW0 (x) W1 (w, x, y 0 ) − W0 (x) W1 (w, x, y) − W0 (x) = +δ q y 0 |y dy 0 (10) 1+r 1+r C(w,x) Z 0 S (x, y ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) Z βS (x, y) + (1 − β) S (x, y 0 ) +s1 κ v y 0 dy 0 . 1+r M2 (w,x,y) Combining equations (8), (9), and (10) we can write the Bellman equation in W1 (w, x, y) −
9
W0 (x) as Z W1 (w, x, y 0 ) − W0 (x) w − rW0 (x) +δ q y 0 |y dy 0 W1 (w, x, y) − W0 (x) = 1+r 1+r C(w,x) Z 0 S (x, y ) +δ q y 0 |y dy 0 C + (w,x) 1 + r Z βS (x, y 0 ) + (1 − β) S (x, y) +s1 κ v y 0 dy 0 1+r M1 (x,y) Z βS (x, y) + (1 − β) S (x, y 0 ) +s1 κ v y 0 dy 0 1+r M2 (w,x,y)
(11)
+ [1 − ξ − δ − s1 κV (M1 (x, y) ∪ M2 (w, x, y))] W1 (w, x, y) − W0 (x) S (x, y) × 1{y∈C(w,x)} + 1{y∈C + (w,x)} 1+r 1+r The Bellman equation defines W1 (w, x, y)−W0 (x) as a fixed point of a contracting operator. A simple iterative algorithm can be used to approximate the fixed point. Let W10 be an initial guess of W1 . Values C 0 (w, x) and M02 (w, x, y) of C (w, x) and M2 (w, x, y) follow from W10 . Then calculate an update of W1 (w, x, y) − W0 (x) using equation (11). This equation allows us to compute the optimal wage contract w given (x, y, y 0 ).
Vacant firms. Consider an open vacancy. Its present value is
(y) rΠopen 0
Z = −c + s0 κ
M−1 0 (y)
(y)] u(x)dx [Π∗1 (x, y|0) − Πopen 0
ZZ + s1 κ M1 (x,y)
∗ Π1 (x, y|y 0 ) − Πopen (y) h x, y 0 dx dy 0 0 Z +δ
Π0 (y 0 ) − Πopen (y) q(y 0 |y)dy 0 , 0
where c is a fixed cost of posting the vacancy; the second term is the flow of benefits from matching with a previously unemployed worker; the third term is the flow of benefits from poaching a worker who is already matched with another firm; the final term reflects the impact of a change in productivity from y to y 0 . Making use of the rent splitting equations: Π∗1 (x, y|0) − Π0 (y) = (1 − β) S (x, y) ≥ 0, Π∗1 (x, y|y 0 ) − Π0 (y) = (1 − β) S (x, y) − S x, y 0 ≥ 0,
10
we finally obtain that
(r +
δ) Πopen (y) 0
Z = −c + s0 κ (1 − β)
S (x, y) u(x)dx
M−1 0 (y)
ZZ
+ s1 κ (1 − β)
S (x, y) − S x, y 0
h x, y 0 dx dy 0
M1 (x,y)
Z +δ
Π0 (y 0 )q(y 0 |y)dy 0 . (12)
If the firm 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:
rΠidle 0 (y)
Z h i =δ Π0 (y 0 ) − Πidle (y) q(y 0 |y)dy 0 . 0
The present value of an unmatched job of type y is finally: n o idle (r + δ) Π0 (y) = max (r + δ) Πopen (y) , (r + δ) Π (y) 0 0 Z = δ Π0 (y 0 )q(y 0 |y)dy 0 + max {0, −c + c (y)} .
(13) (14)
with Z c (y) ≡ s0 κ (1 − β)
M−1 0 (y)
S (x, y) u(x)dx ZZ
S (x, y) − S x, y 0 h x, y 0 dxdy 0 . (15)
+ s1 κ (1 − β) M1 (x,y)
and a firm y is inactive whenever open Πidle (y) 0 (y) > Π0
or c > c (y) .
(16)
This condition is like the free entry condition of a standard search-matching model with ex-ante homogeneous firms.
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3.5
The match surplus
We can write the surplus of an (x, y) match as the fixed point defined by
[r + ξ + δ + s1 κV (M1 (x, y)) β] S (x, y) = f (x, y) − rΠ0 (y) − rW0 (x) Z Z − δQ M0 (x) |y τ + δ Π0 (y 0 ) − Π0 (y) q y 0 |y dy 0 S x, y 0 q y 0 |y dy 0 + δ M0 (x)
Z
S x, y 0 v y 0 dy 0 (17)
+ s1 κβ M1 (x,y)
which, using equation (13) becomes
[r + ξ + δ + s1 κV (M1 (x, y)) β] S (x, y) = f (x, y) − rW0 (x) Z − δQ M0 (x) |y τ + δ S x, y 0 q y 0 |y dy 0 − max {0, −c + c (y)} M0 (x)
Z + s1 κβ
S x, y 0 v y 0 dy 0 . (18)
M1 (x,y)
Note that the match surplus is not a function of the current wage contract and is thus not subject to Shimer’s (2006) critique. This is because the Bertrand competition between the two employers at work in the tri-lateral game involving the worker, the incumbent employer and the poaching firm disconnects the final outcome from the worker’s current wage contract.
3.6
Steady-state flow equations.
We now proceed to define the steady state flow equations. The first relates to matches. The total number of matches in the economy will be ZZ L−U =N −V −I =
h (x, y) dxdy.
(19)
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 0 different from y, and the worker may move to unemployment or may keep the job; third, the worker may change job, with probability s1 κV (M1 (x, y))–i.e., a job offer has to be made (at rate s1 κV ) and has to be acceptable (y 0 ∈ M1 (x, y)). On the inflow side, new (x, y) matches are formed when some unemployed or 12
employed workers of type x match with vacant occupations y, or when (x, y 0 ) matches are hit with a productivity shock and exogenously change from (x, y 0 ) 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 ∈ M0 (x) or S (x, y) > 0: Z [δ + ξ + s1 κV (M1 (x, y))] h (x, y) = δ
q(y|y 0 )h x, y 0 dy 0 " Z + s0 u (x) + s1
# 0
h x, y dy
0
κv (y) . (20)
M1 (x,y)
This equation defines the steady-state equilibrium, together with the accounting equations: Z u (x) = ` (x) −
h (x, y) dy,
(21)
Z v (y) + i (y) = n (y) − where v (y) =
(22)
R n (y) − h (x, y) dx if c ≤ c (y) 0
i (y) =
h (x, y) dx,
(23)
if c > c (y) ,
0
if c ≤ c (y)
n (y) −
R
(24)
h (x, y) dx if c > c (y) ,
and the number of vacancies V is equal to the fraction of ys in distribution n (y) −
R
h (x, y) dx
that is such that c ≤ c (y): Z
Z n (y) −
V =
h (x, y) dx dy.
(25)
c≤c(y)
4
Equilibrium
Given values for the exogenous variables: L, N , δ, ξ, s0 , s1 , r, b, c, and β; the distributions of worker and firm types l(x) and n(y); the production function f (x, y); the transition function for productivity dynamics q(y 0 |y); and the matching function M (s0 U +s1 (L−U ), V ), the equilibrium is characterized by knowledge of V , h (x, y) and S (x, y) solving for equations (25), (20) and (17). In these equations, one will substitute for U using equation (19), κ using equation (3), u (x)
13
using equation (21), c (y) using equation (15), v (y) using equation (23), W0 (x) using equation (7).
5
An Illustrative Numerical Example
In order to illustrate the properties and empirical implications of the model we solve and simulate under a particular set of parameters, for several production functions. For the numerical example we set L = 25000, N = 35000 and assume that L(x) = N (y) and are U [0, 1]. We solve using four different production functions: xy, x + y, and 1 − (x − y)2 . The first two production functions are natural representations of complementarity and substitutability. The third is meant to capture the notion that there is a “right man for the job” (Tuelings and Gautier; 2004). The process governing the technological evolution is represented by a Gumbel copula with marginals equal to n(y), q(y 0 |y) = n(y 0 )Gumble(N (y 0 ), N (y)); δ = 0.05; s0 = 1.0; s1 = 0.1; M (s0 U + s1 (L − U ), V ) = α(s0 U + s1 (L − U ))γ V
1−γ
with α = 0.6 and γ = 0.5; r = 0.03; b = 0;
c = 0.1 and β = 0.5. In order to provide some comparability between the simulations, we vary ξ between 0.01 and 0.05 across the simulations to maintain an unemployment rate of roughly 5 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 s1 = 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
14
the matching set M0 (x) alone. The features of the joint distribution of (x, y) matches, h (x, y) are displayed in panels (c) and (d) of the same figures. The process of job accepting; on-the-job search; endogenous and exogenous job destruction; and the decision of which types of jobs to post vacancies for or leave inactive induces the equilibrium distribution of matches. The effect of on-the-job search is evident in that workers initially accept any job in their matching set, and then switch to jobs with a higher surplus. The vertical line that cuts through the distribution (seen prominently for xy at y = 0.375) illustrates the effect of the decision over which types of unmatched jobs to leave inactive and which to post as vacancies. Matches in which the job component, y, falls into 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 0.375. 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.
6
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
log(wit ) = zit β + αi +
J X j=1
15
djit ψ j + uit ,
(26)
where zit are time varying observables of workers, : αi is a worker fixed effect, ψ j is a firm fixed ˆ effect, and : uit is an orthogonal residual. The correlation between α ˆ i and ψ 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 5. In addition we conduct this exercise for the case case in which workers’ bargaining power is set to zero; workers have no bargaining power and wages only rise in response to Bertrand competition between firms competing for the same worker. In Table 1 we compare the equilibrium correlation between x and y in the model to the correlation between estimated worker and firm fixed effects resulting from equation 26. The results of this exercise demonstrate that while the sign of the correlation between the estimated fixed effects correctly picks up the positive/negative sorting in the case of production functions of the form xy or x + y, it is of opposite sign in the cases where there is a best job for each worker. Additionally, even in the case of xy production, where the positive correlation between x and y is picked up by the correlation between the estimated fixed effects, the degree of this correlation bears little resemblance to that actual correlation. This estimated correlation is not necessarily informative on the degree of sorting in the model. Indeed, this suggests the need to estimate the production function in order to answer the question regarding the degree of sorting on unobservables.3 A natural starting point for the production function would be a second order approximation, which would be exact for each of the examples considered here. 3
In addition to this effect, Postel-Vinay and Robin (2006) note that in terms of asymptotics, OLS estimate of β is consistent as i → ∞ for fixed T and OLS estimates of α and ψ are consistent when T → ∞ faster than I and J. In practice, the data contains millions of workers, tens of thousands of firms, and fewer than ten years. Indeed, empirical estimates of sorting which are based on worker and firm fixed effects introduce a negative bias, which will introduce a spurious negative correlation when calculating the correlation between worker and firm fixed effects. This is illustrated as follows; empirically, β and ψ j are estimated from the within transformation log wit − log wi = (xit − xi )β +
J X j (djit − di )ψ j + uit − ui . j=1
This makes it clear that we need to see workers change firm to identify the firm fixed effects ψ j . The worker fixed effects are estimated as J X j ˆ− ˆ . α ˆ i = log wi − xi β di ψ j j=1
Notice, any statistical error affecting the estimate of the firm effect translates directly to the estimate of the worker effect, with a sign reversal. OLS estimates of firm and worker effects are likely to be imprecise and spuriously negatively correlated given short time dimension and limited worker mobility.
16
Table 1: Actual and Estimated Sorting β = 0.5 β=0 ˆ ˆ ) Production Function corr(x, y) corr(ˆ αi , ψ j(i) ) corr(x, y) corr(ˆ αi , ψ j(i) f (x, y) = xy f (x, y) = x + y f (x, y) = 1 − (x − y)2 f (x, y) = 1 + 14 x − (x − y)2 f (x, y) = 1 + 14 y − (x − y)2
7
0.6199 -0.0330 0.6184 0.5774 0.5434
0.1207 -0.0384 -0.0775 -0.0077 0.0462
0.6731 -0.1260 0.5208 0.4707 0.4640
0.2763 -0.1695 -0.0512 -0.0076 -0.0452
Conclusion and further work
Estimation of the model presented here is the subject of current research. The natural type of data to use in the empirical implementation is matched worker and firm data, an avenue we are actively pursuing. One obstacle in this strategy is the need to take a stand on the formation of jobs into firms, and the possibility of interaction between workers within a firm. An additional interesting question is how much we can learn about earnings processes using standard panel data on workers and the restrictions from the model. The model predicts an earning process with lots of heterogeneity, in which the time varying part of earnings is dependent on the permanent component. In addition, job mobility and unemployment durations are dependent on the same underlying permanent component. With an estimated version of the model in hand we will be well placed to evaluate important policy questions, such as employment protection legislation and minimum wages, within a coherent empirical economic model.
17
0.9
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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 0 ) match whenever she is contacted by a y 0 and S (x, y 0 ) > 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.
18
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(d) Expected job (worker) type by type of worker (job)
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 0 ) match whenever she is contacted by a y 0 and S (x, y 0 ) > 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.
19
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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 0 ) match whenever she is contacted by a y 0 and S (x, y 0 ) > 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.
20
0.9 0.8
Worker Productivity, x
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Figure 4: Feasible matches with xy production, without on-the-job search, without endogenous job destruction, without vacancy costs, and with an exogenous number of firms set equal to the number of workers. (s1 = 0, δ = 0, c = 0, and N = L)
21
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(e) Expected output / wage by worker and job(f) Expected output / wage by worker and job type type
Figure 5: Output, wages and type distributions, f (x, y) = xy 22
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Figure 6: Output, wages and type distributions, f (x, y) = x + y 23
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(e) Expected output / wage by worker and job(f) Expected output / wage by worker and job type type
Figure 7: Output, wages and type distributions, f (x, y) = 1 − (x − y)2 24
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