On-the-job search and sorting

Transcription

1 On-the-job search and sorting Pieter A. Gautier Coen N. Teulings Aico van Vuuren July, 27 Abstract We characterize the equilibrium for a general class of search models with two sided heterogeneity and free entry of vacancies for a general contact technology and various wage mechanisms. This class includes models with and without onthe-job search. Besides the well-known congestion externalities, we show that onthe-job search in combination with monopsonistic wage setting without comitment creates a business-stealing externality. In the absence of congestion effects this leads to excessive vacancy creation. Under wage posting with commitment this externality is absent because firms post a higher wage if it is more likely that a future worker is currently employed at a productive match. Finally, allowing for on-the-job search reduces the gap between the (low) estimates of search frictions based on unemployment data and the (high) estimates based on wage dispersion data. Vrije Universiteit Amsterdam, Tinbergen Institute, CEPR, IZA. Address: Department of Economics, Free University Amsterdam, De Boelelaan 115, NL-181 HV Amsterdam, The Netherlands. gautier@tinbergen.nl Netherlands Bureau of Economic Policy Analysis, University of Amsterdam, Tinbergen Institute, CEPR,.CES-IFO, IZA Address: Van Stolkweg 14, NL-2585 JR The Hague, The Netherlands. C.N.Teulings@cpb.nl Vrije Universiteit Amsterdam, Address: Department of Economics, Free University Amsterdam, De Boelelaan 115, NL-181 HV Amsterdam, The Netherlands. vuuren@tinbergen.nl. We thank participants of the conference on Labour market models and the matched employeremployee data in Sandbjerg, the SED in Budapest, World Meeting of the Econometric Society in London the EEA in Amsterdam, participants of seminars in Aarhus and Amsterdam and Maarten Janssen, Dale Mortensen and Robert Shimer, for their useful comments.

2 1 Introduction Two out of every three jobs among young workers end within a year and a large part of those separations reflect job-to-job changes rather than layoffs, see Topel and Ward (1992). 1 This suggests that there are search or information frictions in the labor market that prevent worker types to immediately match with their optimal job type. Any model that aims to give a good description of actual labor market flows should therefore allow for job-to-job transitions and that is what we do in this paper. Specifically, we consider an assignment model with search frictions, on-the-job search, two sided heterogeneity (both for workers and jobs) and free entry of vacancies. Incorporating heterogeneity into a search model is relevant because one of the most important reasons for search is to find the right man for the job. Search frictions frustrate this process and make the assignment of workers to jobs imperfect. The specification of the contact technology between job seekers and firms is general: the Cobb Douglas and the quadratic are special cases of our contact function. For this general class of search models, we address three issues. First, we prove uniqueness of the equilibrium for a number of important special cases. Second, we analyze the efficiency of various wage setting mechanisms, like Nash bargaining, monopsony without commitment of firms on future wage payments, and wage posting as in Burdett and Mortensen (1998). We show that when off- and on-the-job search are equally efficient, firms tend to create too many vacancies due to a business-stealing effect. When opening a vacancy, firms do not internalize the output losses that a job switcher imposes on her previous employer. However, under wage posting, the business-stealing externality is absent. Wage posting allows firms to pay a hiring premium, which exactly offsets the business stealing externality. Third, we show how the cost of search (defined as the utility loss of workers relative to the Walrasian economy) can be estimated either from the unemployment rate or from the wage differentials of identical workers. Gautier and Teulings (26) showed that without on-the-job search, the estimated cost of search based on wage differentials are a lot larger than the estimate based on unemployment data. Hornstein, Krusell and Violante (27) make a similar point, they derive for a general class of search models 1 Only for workers with less than a year of labor market experience the lay-off rate is larger than the job-to-job transition rate. Fallick and Fleischman (24) and Nagypál (25) also give evidence on the importance of on-the-job search. They show that in the US, the rate of job-to-job transitions is twice as high as the rate at which workers move from employment to unemployment. 1

3 that it is impossible to explain the US experience of short average unemployment spells in combination with substantial wage dispersion. They also argue that allowing for on-thejob search decreases the gap between theory and data. On-the-job search can resolve this inconsistency because it makes unemployed workers less choosy about their first job and as a result their reservation wage and the unemployment rate falls. On-the-job search also increases wage differentials for workers with equal skill levels, mainly because unemployed job seekers increase the range of job types that is acceptable for them. Therefore, the ratio of the cost of search and wage differentials is lower with on-the-job search. The implications of on-the-job search for wage bargaining in search models have been addressed recently in Shimer (26). Without on-the-job search, a larger piece of the cake for the worker implies an equivalently smaller piece for the firm. This symmetry breaks down when there is on-the-job search because the expected match duration is increasing in the wage. Hence, firms are willing to pay a no-quit premium for good matches. This makes the set of feasible payoffs of firms and workers possibly non-convex. Shimer shows that a strategic bargaining game in the spirit of Rubinstein (1982) may generate multiple equilibria and that the wage that is the extremum of the product of worker and firm surpluses is always an equilibrium. In this paper we also have potential non-convexities in the set of feasible payoffs. However, when firms have monopsony power, multiplicity is not an issue. The analysis of a model with hierarchical two-sided sorting of worker and job types is a complicated affair. In particular, the lower left and upper right corner of the matching space of worker and job types cause serious analytical problems. Teulings and Gautier (24) approximated the equilibrium for the model without on-the-job search by Taylor series expansions around the Walrasian assignment. This approach turned out not to work when we add on-the-job search. Hence, we take a more convenient approach here. In Gautier et al. (25), we show that the approximated equilibrium in the hierarchical model has roughly the same properties as a circular model in the spirit of Marimon and Zilibotti (1999). 2 Since the circular model is simpler than the hierarchical model, we introduce on-the-job search in the circular model. This paper relates to a number of other papers. Pissarides (1994) also studies on-thejob search in a matching framework. His model differs from ours in at least three ways: (1) he considers identical workers and two job types while we consider a continuum of different 2 The circular model on its turn, is identical to the stochastic matching model of Pissarides (2) with one additional constraint, a zero derivative of the match productivity function at the optimum. 2

4 workers and jobs, (2) our model generalizes the contact technology and allows congestion externalities to be switched on or off, (3) in our model, wages are determined as in Shimer (26) or by wage posting while Pissarides assumes a linear sharing rule. Barlevy (22) has a model that is similar to ours except that in his model, wages are determined by a linear sharing rule, he uses a different contact technology, and his main results are based on simulations where we present analytical solutions. His focus also differs from ours. He makes the important point that the sullying effect of recessions (workers move slower to their optimal job types because of low vacancy creation in recessions) dominates the positive cleansing effect of recessions for realistic parameter values. As in Jovanovic (1979, 1984), our model predicts individual separation probabilities to be decreasing in job tenure because the good matches are the ones that survive. In a companion paper we give empirical evidence for this. Finally, on-the-job search in a bargaining environment is studied in Gautier (22), Moscarini (21, 25), Postel-Vinay and Robin (22) and Cahuc, Postel-Vinay and Robin (26) but these papers either do not consider ex-ante heterogeneity or assume a different bargaining environment. Burdett and Mortensen (1998) show that worker and job heterogeneity is not necessary for job-to-job movements. Mortensen (2) also has wage posting with free entry but in his model, all firms are identical and he does not consider efficiency issues. The paper is organized as follows. Section 2 starts with the assumptions and derives the equilibrium conditions. Section 3 characterizes the equilibrium. In Section 4 we conduct welfare analysis. Section 5 discusses the predictions of the model with respect to the relation between unemployment and wage dispersion and Section 6 concludes. 2 The model The economy that we consider has the following properties. Production There is a continuum of worker types s and job types c; s and c are locations on a circle with circumference 1, so that s = 1 is equivalent to s =, and the same for c. Workers can only produce output when matched to a job. The productivity Y of a match only depends on, the distance between s and c: x(s,c) = min k Z s c + k. Hence: Y = Y (x). We assume that Y (x) is twice differentiable, strict quasi concave and has an interior maximum. Without loss of generality, this maximum is located at x = and 3

5 the value of the maximum is normalized to unity: Y () = 1. Hence, x is a measure of the degree of mismatch of an assignment. These assumptions imply that Y x () =, since x = maximizes Y (x). We consider one of the simplest functional forms that meets those criteria Y (x) = γx2. This functional form can be interpreted as a second order Taylor series expansion of a more general production function. We are interested in non-trivial equilibria where unemployed workers do not accept all jobs. 3 The parameter γ is related to the complexity dispersion parameter discussed in Teulings and Gautier (24: 558) and Teulings (25). Low values of γ imply that worker types are close substitutes. Labor supply We assume that labor supply per s-type is uniformly distributed over the circumference of the circle and that total labor supply equals L. Unemployed workers receive a value of leisure B. Employed workers supply a fixed amount of labor and their pay off is equal to the wage they receive. Labor demand There is free entry of vacancies for all c-types. The flow cost of maintaining a vacancy is equal to K per period. After a vacancy is filled, the firm only pays for the wage of the worker. Job search technology Let m be the total number of contacts between job seekers and vacancies per unit of labour supply, u is the unemployment rate, and v is the total number of vacancies per unit of labor supply. Then, the general expression for the contact technology between job seekers and vacancies reads: ml = µ {[u + ψ (1 u)] L} η {vl} ξ m = λ [u + ψ (1 u)] η v ξ λ = µl η+ξ 1 The parameter ψ, ψ 1, measures the efficiency of on-the-job search relative to search while unemployed; ψ = is the case with no on-the-job search, as analyzed in 3 This requires Y (x) < for at least some x. Since x 1 2, a sufficient condition is γ > 8. 4

6 Teulings and Gautier (24); ψ = 1 is the case where off- and on-the-job search are equally efficient. The parameters η, < η 1 and ξ, < ξ 1 measure the relative contributions of job seekers and vacancies to the contact process; η = ξ is the symmetric case where both inputs yield an equal contribution; η + ξ measures the returns to scale of the contact process; η + ξ = 1 yields the standard matching function with constant returns to scale (CRS), a higher value, η + ξ > 1 yields increasing returns (IRS). The case η = ξ = 1 is the quadratic contact technology which has also been frequently used in the search literature. This case is important since it exhibits no congestion externalities. Hence, our specification of the contact technology encompasses the quadratic and CRS technology as special cases. Teulings and Gautier (24) give a number of motivations why the quadratic technology might be the most adequate assumption in a model with two sided heterogeneity, the main reason being that this technology avoids congestion effects, which have unpleasant implications with two sided heterogeneity. The parameter µ (and hence λ) measures the overall efficiency of the matching process. The limiting case λ yields the Walrasian equilibrium. For the non-crs case, the parameter λ also captures the scale of the labour market. The contact technology is defined for a labor supply normalized to one. A 1 % increase of labour supply for the same number of vacancies per unit of labour supply v would raise the number of contacts per unit of labour supply, m, by η + ξ 1 %. This effect is assumed to be captured by λ. Hence, for IRS, a Walrasian labor market is equivalent to a labor market of infinite size, since both imply λ. The implied contact rates of an (un)employed worker and vacancies respectively read: λ (s,unemployed) job = λvs, λ (s,employed in z) job = ψλvs, λ c unemployed = λus, λ c employed = ψλ (1 u) S, S (u,v) [u + ψ (1 u)] η 1 v ξ 1. (1) We leave out the arguments of S ( ) in what follows. The expression for S reveals the special importance of the quadratic contact technology, since then S = 1 and no longer depends on u and v. Another important special case is when off- and on-the-job search are equally efficient, ψ = 1, since then S = v ξ 1, and does not depend on u so the decomposition of labour supply in employed and unemployed job seekers is irrelevant. 5

7 Job separation Matches between workers and jobs are destroyed at an exogenous rate δ >. Wage setting We focus on two wage setting schemes. First, we assume that wages are determined by efficient bargaining between the worker and the firm where firms cannot commit on future wage payments, see Diamond (1982), Mortensen (1982) and Pissarides (2). Second, we assume that firms can commit on a future wage policy that is contingent on x. In both frameworks, firms pay no-quit premiums, but only in the wage posting model firms can commit ex ante to pay hiring premiums. In bargaining models, where wages are continuously renegotiated, hiring premiums are not credible because they will be immediately eliminated at the moment the worker starts his job. Workers anticipate this, and will therefore not respond to such premiums so firms will not offer them in the first place. In both cases, we assume that a worker must quit his old job before accepting a new one so we do not allow for Bertrand competition between the poaching and incumbent firm as in Postel-Vinay and Robin (22). In the concluding section we briefly discuss the implications of their and Moscarini s (25) wage mechanisms for our results. Golden-growth path We study the economy while it is on a golden-growth path, where the discount rate ρ > is equal to the growth rate of the labor force. This assumption is only made for reasons of tractability: it allows for an analytical solution of an integral which otherwise would have to be computed numerically. Under a quadratic contact technology (which exhibits IRS), the growth of the labor force implies an upward trend in the efficiency of the search process. For the sake of simplicity, we assume that there is an offsetting downward trend in the efficiency of the market so that λ remains constant and the labor market does not continuously become more efficient over time. 4 The implications of this assumption of a golden growth path are equivalent to assuming that the discount rate ρ goes to zero, or more precisely, that the discount rate is much smaller that the job finding and separation rate, ρ δ,λ. This is a common assumption in the wage posting literature, i.e. Burdett and Mortensen (1988). 4 Recall that λ consists of a size-of-the-market part and a search-efficiency part. We assume that the product of both (µl η+ξ 1 ) remains constant over time. 6

8 Distribution of vacancies We assume that vacancies are uniformly distributed over the circumference of the circle: v (c) = v, where v (c) is the number of vacancies per unit of c (that is: the unnormalized density of vacancies at c). In Gautier et al. (25), we prove that this is the case for a similar model without on-the-job search. Here, we show that when we allow for on-thejob search, this uniform distribution of vacancies is an equilibrium, but we cannot prove that no other equilibria exist. However, we conjecture that the uniqueness result carries over to the case with on-the-job search. The intuition is that wages should be high at locations with relatively many vacancies. This is due to both the increase in the outside option of the unemployed workers and the increase in the no-quit premium. The high wages at these locations suggest that the value of a vacancy is relatively low there which violates the assumption of free entry. Hence, a situation where some locations have more vacancies than others cannot be an equilibrium. Since we focus on equilibria where both supply and demand are uniformly distributed, all outcome variables do not depend on either s or c separately, but only on the distance x between them. Hence, we use x as the only argument. For variables depending only on either s or c, like v (c), we drop the argument to simplify notation. 2.1 Equilibrium under wage bargaining Let ˆV E (W) be the asset value of holding a job paying a wage W and let v (W) be the number of vacancies that pay a wage equal to or higher than W. The asset value ˆV E (W) satisfies the following Bellman equation: [ ρˆv E (W) = W + δ V U ˆV ] E (W) + ψλs W [ˆV E (Z) ˆV E (W)] d v (Z), ˆV E W(W) = [ρ + δ + ψλs v (W)] 1, (2) where V U denotes the asset value of unemployment. Throughout the paper, subscripts of functions denote the relevant (partial) derivative. A sufficient condition for equilibrium is that the wage W (x) paid to an s-type worker employed at a job of type c = s ± x 7

9 maximizes the following product: 5 ] [ ] β 1 β W (x) arg max [ˆV E (W) V U Y (x) W. (3) W ρ + δ + ψλs v (W) The first factor in square brackets is the increase in wealth for the worker relative to the status of unemployment, the second factor is the increase in wealth for the firm. The latter is the current income stream Y (x) W divided by a modified discount rate, accounting for the interest rate ρ, the separation rate δ, and the quit rate of the worker to better paid jobs, ψλs v (W). By paying higher wages, firms and workers reduce the probability that the worker quits to a better paying job. This increases the surplus product in the present job. Hence, firms pay a no-quit premium which raises the wage above the simple sharing rule that applies in models without on-the-job search, see also Shimer (26). The vacancy function v ( ) can be interpreted as the wage offer distribution. This distribution has no mass points for the usual reasons, see Burdett and Mortensen (1998). If v ( ) would have a mass point at W, a profitable deviation exists. The surplus product would jump upward by a slight increase in W above W, since all vacancies at the mass point would no longer be able to poach a worker from the job paying this slightly higher wage. This contradicts W being a maximum of the surplus product. Since the surplus product is continuous in W and since both Y (x) W (x) and ˆV E [W (x)] V U are positive (otherwise, either the firm or the worker would not be willing to match), W (x) satisfies the following first-order condition of (3) (by substitution of (2)): ] β [Y (x) W (x)] = (1 β) [ˆV E [W (x)] V U [ρ + δ + ψλs v [W (x)] + ψλs v W [W (x)] [Y (x) W (x)]]. This condition states that the gain from a marginal wage increase for the worker is equal to the cost of that increase for the firm, where both are weighted by their respective bargaining power β and 1 β. Since Y (x) is decreasing in x, so is W (x). Therefore, we can define the number of vacancies and the asset value of a job as functions of x instead of W: V E (x) ˆV E [W (x)], 2vx = v [W (x)]. 5 Since at this stage we cannot rule out that the set of feasible pay offs is non-convex we cannot use the axiomatic Nash bargaining solution. Our solution is consistent with Shimer s alternating offer game where we interpret the β s either as different discount factors or as probabilities to make an offer. 8

10 V E (x) is the asset value of holding a job at distance x from the optimal type c = s. In the second line, we use the uniformity of the distribution of job seekers over s and of vacancies over c, v (c) = v. For a particular worker type s, the number of vacancies at a distance smaller than x is 2vx; the factor 2 comes in because increasing x adds vacancies both to the left and to the right of the optimal type c = s. The uniformity of the distribution of s makes that this expression applies to all s alike. The number of vacancies located at a shorter distance to worker type s than x is therefore equal to vx, and since W (x) is decreasing in x, these vacancies offer a wage higher than W (x). By the chain rule, we obtain: V E x (x) = ˆV E W [W (x)] W x (x), 2v = v W [W (x)]w x (x). (4) Substitution of (4) and the expression for Y (x) in the first-order condition and rearranging terms yields: W x (x) = 2 (1 β) ψλvs [ γx2 W (x) ] [ V E (x) V U] β [ γx2 W (x) ] (1 β) [V (x) V U ] (ρ + δ + 2ψλvSx). (5) The asset value for an unemployed job seeker satisfies the Bellman equation: ρv U = B + 2λvS x [ V E (x) V U] dx, (6) where x is the maximum distance of a job offer that is acceptable for an unemployed job seeker; 2x is also the probability that a job offer is acceptable for an unemployed worker. Similarly, the asset value for an s-type worker holding a job of type c = s ±x satisfies the Bellman equation: ρv E (x) = W(x) + 2ψλvS x [ V E (z) V E (x) ] dz δ [ V E (x) V U]. (7) The disadvantage of writing the asset value this way is that it yields an implicit equation in V E (x). In Appendix A.1 we show that the asset value can also be written in explicit form: V E W(x) (x) = ρ + δ + 2ψλvSx + δ x ρ + δ V U W (z) + 2ψλvS (ρ + δ + 2ψλvSz) 2dz. (8) where the first term is the discounted wage income at the current job. The discount factor consists of a time preference term, ρ, and two terms that take into account the rate at 9

11 which a match ends. This happens either by exogenous shocks, δ, or if the worker finds a better job 2ψλvSx. The number of better job types is given by 2vx (the worker can accept jobs both to the left and to the right of her favorite job type) and the rate at which the worker meets those jobs is ψλ. The second term is the probability to move to the state of unemployment times the properly discounted value of this state and the final term is the probability to find a better job times the discounted expected wage at this better job. For the latter discount factor we have to take into account that both the transition rate and the new state are discounted at rate (ρ + δ + 2ψλvSx) 1. Substitution of this explicit expression for V E (x) in the wage equation yields: (1 β)ψκvs [ W x (x) = γx2 W (x) ] R β [ γx2 W (x) ] (1 β) (1 + ψκvsx)r where R W(x) x 1 + ψκvsx ρv U W (z) + ψκvs (1 + ψκvsz) 2dz, κ 2λ ρ + δ. (9) This is a differential equation in x. Its solution requires an initial condition. At the marginal job, x = x, the surplus from matching is zero and hence neither the worker nor the firm gains from matching. Hence, W (x) = Y (x) = γx2. (1) Substitution of equation (8) in equation (6), and some rearrangement of terms yields: ρv U = x Finally, since ρv E (x) = ρv U, (8) implies: B 1 + κvsx + κvs 1 + κvsx (11) [ W(x) x ] 1 + ψκvsx + ψκvs W (z) (1 + ψκvsz) 2dz dx. = W (x) x 1 + ψκvsx ρv U + ψκvs W (x) (1 + ψκvsx) 2dx. (12) Note that all these relations depend on the composite parameter κ, not on its separate components, ρ, δ, and λ. Let G(x) be the number of individuals of type s who are employed in jobs that are located at greater distances from s than x, and that are therefore less attractive than a 1

12 job at distance x. Hence, G () is total employment of type s, and G (x) =, since there is no employment located at a distance greater than x. Since the density of labor supply is normalized to unity, the rate of unemployment satisfies: u = 1 G (). (13) The equilibrium flow condition for employed workers at distance x or less from their favorite job is: 2λvSx [u + ψg (x)] = (ρ + δ) [G () G (x)]. (14) The left-hand side is the number of people that find such a job, partly from unemployment (the first term in square brackets), partly by mobility from a less attractive job (the second term). The latter number is downweighted by the factor ψ, reflecting the effectiveness of on-the-job search. The right-hand side is the number of people that lose such a job and the growth of the number of people that hold such a job due to the growth of the labor force as a whole: the number of jobs at distance smaller than x, G () G (x), times the separation rate δ plus the growth rate ρ. Mobility within the segment of jobs that are at smaller distance than x, G (x), is irrelevant for this purpose because the disappearance of the old job and the emergence of the new job cancel. Setting x = x yields the equilibrium flow condition for unemployment: 2λvSux = (ρ + δ)g(). Substitution in (13) yields an expression for the rate of unemployment: u = κvsx, (15) and employment at distances greater than x follows from substituting (15) in (14): ( 1 G (x) = κvsx ). (16) 1 + ψκvsx 1 + κvsx By the free entry condition for firms, the option value of a vacancy of type c must be equal to K. Hence, K = 2λS x = κs 1 + ψκvsx 1 + κvsx [u + ψg (x)] γx2 W(x) ρ + δ + 2ψλvSx dx x γx2 W(x) (1 + ψκvsx) 2 dx. (17)

13 The first factor in the integrand on the first line is the effective number of individuals willing to accept a vacancy of type x. It equals the number of unemployed plus the number of workers presently employed at greater distance than x. By the uniform distribution of workers and jobs, the latter number is equal to the number of workers of type s employed in jobs at a greater distance from s than x. The second factor is the value of a filled vacancy. Just as in the wage equation, we discount current revenue Y (x) W (x) by the discount rate plus the separation rate δ plus the quit rate 2ψλvx. The second line follows from substitution of the relations for employment and unemployment. Note that again these relations depend on the composite parameter κ, not on its separate components, ρ, δ, and λ. Definition 1 The bargaining equilibrium consists of the set {W x (x),w (x),ρv U,u,v, x} satisfying (9)-(15) and (17). 2.2 Equilibrium under wage posting When firms can commit to future wage payments, their optimal wage policy W (x) maximizes the value of a vacancy. We allow firms to post a wage conditional on the distance between worker and job type, x. By offering a higher wage, firms can increase their expected labor supply. Equations (6)-(8) and (1)-(16) are not affected by this alternative assumption on wage setting. However, equation (9) does not apply in this case. Let Ĝ [W(x)] G(x) be the number of employed workers who currently earn less than W(x). Then, W (x) is the posted wage, W, for a worker type located at distance x from the vacancy, that maximizes the expected value of the vacancy: ( u + ψĝ W(x) arg max (W) W u + ψ (1 u) 1 ) 1 2 γx2 W ρ + δ + ψλs v(w) [u + ψĝ (W) ] / [u + ψ (1 u)] is the probability that the job seeker accepts the job. Unemployed job seekers will always accept the job, employed job seekers accept the job when their current job pays less than W, the probability that this occurs is increasing in W. The current payoff of hiring a worker at distance x equals γx2 W, which is decreasing in W. Finally, the quit rate ψλs v(w) is decreasing in W, and hence the effect of W on the value of the vacancy via the quit rate is positive. The optimal wage offer W (x) balances these negative and positive effects of W on the value of the vacancy. Taking the first-order condition of (18) and rearranging terms yields: 12 (18)

14 [ W x (x) = 2ψκvS (1 + 2ψκvSx) ] ψg x (x) 1 1 u + ψg (x) 2 γx2 W (x). (19) 1 + 2ψκvSx The first term between square brackets is the no-quit premium, equation (9) is the equivalent under bargaining with β =. The second term is the hiring premium, which is proportional to the elasticity of the labor supply curve, ψg x (x)/[u + ψg (x)]. Definition 2 The wage posting equilibrium consists of the set {W x (x),w (x),ρv U,u,v, x} satisfying (1)-(15), (17), and (19). 3 Characterization of the equilibrium This section characterizes the equilibrium for various cases. As a benchmark, Section 3.1 discusses the case without on-the-job search, ψ =, < β < 1. In this case, it does not matter whether or not firms can commit to future wage payments. Section 3.2 considers the model with on-the-job search where firms have full bargaining power but cannot commit to future wage payments (monopsony without commitment), < ψ 1, β =. This discussion reveals two differences compared to the model without on-the-job search: (i) the Nash bargaining solution is no longer feasible, see Shimer (26) and we can no longer use a simple linear sharing rule of the match surplus since the total surplus is increasing in the wage offer, and (ii) the value of unemployment is not equal to the net discounted value of the reservation wage, since job seekers who accept a job retain (part of) the option of search. We pay special attention to the ψ = 1 case, where on- and offthe-job search are equally efficient. This case is relatively simple because the reservation wage of an unemployed job seeker reduces to W (x) = B because accepting a job does not imply a reduction in the option value of search. The general case < ψ < 1, < β < 1 is hard to analyze analytically because an explicit solution for the wage function W (x) is not available. In section 3.3, we consider wage posting with on-the-job search, where firms can commit to future wage payments. 13

15 3.1 No on-the-job search: ψ =, < β < 1 Without on the job search, equations (9) and (12) simplify considerably: W (x) = β [1 12 ] γx2 + (1 β)ρv U, (2) ρv U = W (x) = γx2. (21) In the first equation we use the fact that W x (x). Hence, since the numerator on the right-hand side of equation (9) equals zero, this equation has a solution only if the denominator is also equal to zero. Rearranging terms yields the first equation above. 6 This equation shows that wages are a simple weighted average of the worker s outside option, ρv U, and the productivity of the job, Y (x), so that workers receive a share β and firms a share (1 β) of the match surplus. This simple linear sharing rule of the match surplus does not hold with on-the-job search because then firms pay wage premiums above this simple sharing rule to prevent the worker from quitting to a better paying firm, as we discuss in the next section. The second equation combines (1) - (12) and implies that the flow value of unemployment, ρv U, is equal to the reservation wage of the unemployed, W (x). Again, this equality does not hold with on-the-job search, since then workers retain a share ψ of the option value of search, so that accepting a job becomes less costly. For future reference, it is convenient to define S as a function of u and x instead of u and v because the equilibrium can then be characterized by a system of 2 equations and 2 unknowns (u and x) only. Ŝ (u,x) S (u,v). An expression for Ŝ (u,x) can be derived by solving equation (15) for v and substitution of the result in the definition of S (u,v) in (1). Proposition 3 For the case ψ =, < β < 1, the equilibrium in u and x for the model is characterized by the solution to the system of equations 3u 3u + 2β (1 u) = B x 2, (22) B 1 γ 2 1 B, 6 Alternatively (2) can be derived from applying a linear sharing rule, see equation (1.18) in Pissarides (2). 14

16 Proof: See Appendix A (1 β)u5/2 Ŝ (u,x) [3u + 2β (1 u)] = 3/2 K, (23) γk K κ (1 B) 3/2. Both (22) and (23) are reduced form asset value equations, the first for an unemployed job seeker, the second for a vacancy. In both equations, we use the equilibrium flow condition for unemployment to eliminate v and the value of the least attractive job acceptable to an unemployed job seeker to eliminate ρv U. Interestingly, when we will allow for on-the-job search in the next section, the structure of those equations remains the same, the lefthand side depends on u and κ and γ only enter through K. The expression B x 2 has a special interpretation. It is the ratio of the output loss in the least paying job and the output loss due to unemployment: B x 2 = Y () Y (x) Y () B. (24) Without search frictions Y (x) = Y () and this ratio equals. The larger search frictions, the larger this ratio. Since B x 2 appears on the right hand side of (22) it relates unemployment to wage dispersion. When workers have no bargaining power, β =, they do not receive part of the productivity surplus above the value of leisure, Y (x) B. Hence, there is no reason for them to be choosy, so that Y (x) = B, or equivalently, B x 2 = 1. For higher values of β, workers get part of the surplus, so that they will reject unattractive job offers. Hence, B x 2 > 1. Note that the interpretation of the right hand side of (24) only holds without on-the-job search. With on-the-job search the right hand side can be far from zero even when search frictions are very low. Proposition 4 For (i) ξ < 1 and (ii) ξ = 1 and K (1 β), there exists a unique equilibrium with a positive supply of vacancies. For (iii) ξ = 1 and K > (1 β), no vacancies are opened and everybody is unemployed. Proof: See Appendix A.3. For the proof we first solve (22) for x to eliminate this variable from (23). The resulting expression (which depends on u only) is monotonic in u. We evaluate it at the upper- and 15

17 lower bound of unemployment, u = and u = 1. Following the seminal paper by Diamond (1982), there has been a lot of discussion on the possibility of multiple equilibria in search models with IRS. The fact that uniqueness is guaranteed even under IRS is therefore an important result. The equilibrium is exactly equivalent to the one in Teulings and Gautier (24) for the hierarchical version of this model. However, the additional complexity of the hierarchical structure made it impossible to prove uniqueness in their model. The symmetry and the lack of corners of the circular model makes it considerably simpler and therefore possible to prove uniqueness here. Cases (ii) and (iii), ξ = 1, are somewhat special, since then S and hence the arrival rate of job seekers do not depend on v. For this reason, it may not be profitable to open a vacancy if entry cost are very high even if v =. For ξ < 1, free entry guarantees a positive stock of vacancies because lim v S =, so that it is always possible to satisfy the zero profit condition for a sufficiently low value of v. This feature also characterizes the equilibria discussed in subsequent sections. The other comparative statics are summarized in Table 1, see Appendix A.4 for the derivation. There are two ambiguities: the first-order derivatives of v with respect to γ and κ. The -sign implies that the first-order derivative is positive for small levels of u and negative for larger values and the -sign implies the opposite situation. For the quadratic contact technology we can derive that for u < 6β 2β, the relationship between 3 2β v and κ is negative, while the relationship between v and γ is positive, and the reverse for u > 6β 2β. The turning point is increasing in β, being equal to for β = and 3 2β equal to for β = 1. In parenthesis, we give the expected sign for realistic values of the unemployment rate. Even for very small levels of β (i.e. 1 percent) the unemployment rate should be over 2% to falsify these expected signs of v and κ and v and γ, which is an irrelevant range. Hence, we take these effects as a benchmark in the subsequent discussion. The comparative statics are easy to understand. A lower degree of substitutability between types of labor (a higher value of γ) increases unemployment and vacancies, since it is more costly to produce at an imperfect match and consequently the maximum acceptable value of x is lower. A high contact rate between vacancies and job seekers κ reduces unemployment and vacancies, since search is more efficient. 7 Workers also increase their reservation wages but not by enough to offset this effect. A higher value of leisure B, makes workers more choosy, thereby reducing the maximum 7 We assume here that the change in κ is driven by a change in µ,ρ or δ and that η and ξ remain constant. 16

18 acceptable distance. This raises unemployment and the number of vacancies. A higher cost of creating vacancies reduces the number of vacancies, raises unemployment and raises the maximum acceptable distance. Table 1: Comparative statics of the model for β > and ψ =. A + sign indicates that the first order derivative is positive u x v γ + (+) κ ( ) B + K + + β + + ξ η 3.2 Monopsony without commitment with on-the-job search: < ψ 1, β = In the model with on-the-job search where employers cannot commit to future wage payments, the differential equation (9) does not have an analytical solution in general. However, for the special case of monopsony without commitment (β = ), the solution is (see Appendix A.5 for a derivation): W(x) = 1 + γ x x ψκsv ψκsvx γx (x 2x) γ1 2 (ψκsv) 2 log ( ) 1 + ψκsvx. (25) 1 + ψκsvx The wage equation (25) is dramatically different from (2) for the case without on-the-job search. In the latter case, a simple linear sharing rule applies where wages are a weighted average of the outside option of the firm and the reservation wage of the worker. Then, the flatness of productivity in the optimal assignment implies the flatness of the wage function at that point: W x () = Y x () =. This characteristic does not carry over to the equilibrium with on-the-job search. The situation is sketched in Figure 1. We draw W (x) both for ψ = (no on-the-job search, β =.5) and ψ = 1 (β = ). The locus of value added Y (x) does not depend on ψ. On-the-job search lowers the reservation wage W (x), since employed workers retain part of the option value of search so that accepting a job is less costly and matching sets become larger. Firms pay no-quit premiums to 17

19 1.2 1 Y (x) without on-the-job search with on-the-job search.8 W(x) s c Figure 1: The wages paid by firms as a function of the levels of x. The parameters used are γ = 12,B =.2,K =.1,κ = 1 and ψ = 1,β = for the case with on-the-job search while ψ =,β =.5 for the case without on-the-job search. avoid that their workers are being poached by other firms. The premium of one firm induces other firms to pay even higher premiums in equilibrium. As a consequence, the wage locus is non-differentiable at the optimal assignment x =, leading to a peak in the locus. Since all firms pay no-quit premiums, there is no net effect on actual quit behavior. So even though firms compete for workers by paying no-quit premiums, the actual mobility pattern is unaffected by these premiums: workers keep on moving towards the optimal assignment, x =. Any vacancy type with a lower x than the present job will be accepted. The remarkable consequence of this argument is that the instantaneous profit margin for an employed worker, Y (x) W (x), does not reach its maximum at the optimal assignment, x = : while Y x () =, W x + () < (and the reverse for the left derivative). Hence, an ε deviation leads to a rise in the surplus. The effect of on-the-job search on W (x) is indeterminate. Since reservation wages are lower with on-the-job search, wages for similar jobs can be either lower or higher: the no-quit premium pushes wages up, the lower reservation wages pulls them down. 18

20 For future reference, it is convenient to define: z(u) ψ 1 u u (26) z can be interpreted as the ratio of (effective) employed and unemployed job seekers in the search pool. Since z is a continuous and monotonically declining function of u with z (1) = and z () =, any positive value of z implies a unique value of u. It turns out that this transformation is convenient for the analysis of the equilibrium. Proposition 5 For the case < ψ 1,β =, the equilibrium values of z and x are the solution to the system of equations: ψz 2 z 2 + (1 ψ) log (1 + z) [log (1 + z) 2z] = B x 2, (27) ( ) ξ+η ξ ψ z 2 +ξ+η ψ + z { [ log(1 + z) 1 + z 1 log(1 + z)] z } ξ ( ) 1 ξ 2 1 B = {z 2 + (1 ψ) log (1 + z) [log (1 + z) 2z]} 1 K. (28) +ξ 2 K Proof: See Appendix A.6. The proposition characterizes the equilibrium as a system of two implicit equations in x and z, similar to the case without on-the-job search. The first relation reflects labor supply: the more jobs are opened, the lower the unemployment rate u (the higher z), the stronger the bargaining position of workers, the higher the choosiness of workers and therefore the lower is x. The second relation reflects labor demand. The system can be solved recursively, by first solving equation (28) for z. Proposition 6 For the limiting case ψ =, u and x converge to their values under monopsony without on-the-job search ψ =,β =. Proof: See Appendix A.7. This result is not trivial because with on-the-job search the standard linear sharing rule does not apply. The proposition shows that in the limit when ψ it does apply. 19

21 Proposition 7 For (i) ξ < 1 and for (ii) ξ = 1 and K ( ) 1 B 1 ξ K 2 equilibrium with a positive supply of vacancies. For (iii) ξ = 1 and K ( 1 B K no vacancies are opened and all workers are unemployed. Proof: See Appendix A there exists an ) 1 ξ> 2 3 2, The proof is analogous to that for Proposition 4 for the case without on-the-job search. The existence proof is based on the continuity of the left hand side of equation (28) with respect to z and its values for its upper- and lower bound, z = and z =. A useful simplification is the case where on- and off-the-job-search are equally efficient, ψ = 1. In that case unemployed workers do not give up any option value of continued search so their reservation wage simply becomes equal to B. Equation (28) then simplifies to ( ) 1+ξ ( 2 1 u 2 +ξ 1 + 1u log u ) ξ ( ) 1 ξ 2 1 B log u 1 = K (29) 1 u 1 u K Proposition 8 For the special case ψ = 1 the equilibrium is unique. Proof: See Appendix A.9. Though we have not been able to find a general proof of monotonicity, the propositions for the upper and lower bound of ψ and plots for intermediate values suggest that the left hand side of equation (28) is monotonic for any combination of parameters. Hence, we conjecture that uniqueness is a general characteristic of this model and that the results should be qualitatively the same. The comparative statics results for the case that ψ = 1 are the same as in the previous section except that x no longer depends on κ,k and B because workers accept all jobs that pay more than B. In addition we are able to show that as long as u < 27% the equilibrium impact of γ on v is positive and the impact of κ on v is negative. Details are in Appendix A Wage posting with on-the-job search: < ψ 1 Like the case of monopsony when firms are unable to commit to future wage payments (β = ), the differential equation for wage setting under wage posting, equation (19) can 2

22 1.2 1 Y (x) Without commitment With commitment W(x) x Figure 2: Wages with and without commitment. Parameter values are the same as in Figure 1. be solved analytically (see Appendix A.11 for a derivation). The solution reads W(x) = γx2 + γx (1 + ψκsvx) 2 ψκsv 1 + ψκsvx γx ψκsv ( ) 2 ( ) 1 + ψκsvx 1 + ψκsvx γ log. ψκsv 1 + ψκsvx (3) Figure 2 plots the wage function for the wage posting equilibrium and for the monopsony without commitment, β =, equilibrium, both for ψ = 1. Because of the additional hiring premia, wages are strictly higher under wage posting. The possibility to commit to a wage increases profits for an individual firm but if all firms have this option, equilibrium profits are lower on average than when firms cannot commit. Hence, commitment makes firms worse off and workers better off because it increases competition. Proposition 9 For the wage posting model with < ψ 1, the equilibrium values of z 21

23 and x are the solution to the system of equations: z 2 { 1 ψ ψ Proof: See Appendix A.12. [ ] } z 1 + z z2 + 6 log(1 + z) 6z + z 2 = B x 2, (31) ( ) ξ+η z 2 +ξ ψ ξ : ψ + z { 2 log(1 + z) + 3z 1+2z z} ξ 1+z { [ 1 ψ 3+z ψ 1+z z2 + 6 log(1 + z) 6z ] = K }1 + z 2 +ξ 2 ( ) 1 ξ 1 B. (32) So, again, the equilibrium is determined by two implicit equations in z and x. Proposition 1 For (i) ξ < 1 and for (ii) ξ = 1 and K ( ) 1 B 1 ξ K 1 3 equilibrium with a positive supply of vacancies. For (iii) ξ = 1 and K ( 1 B K vacancies are opened and all workers are unemployed. Proof: See Appendix A.13. Again, we consider the special case ψ = 1. Equation (32) simplifies to ξ ( u 1 u ) 1+ξ ( 1 u 2 + 2u log u 1 u K there exists an ) 1 ξ > 1, no 3 ) ξ ( ) 1 ξ 1 B = K. (33) K Proposition 11 For ψ = 1 there exists a unique wage posting equilibrium. Proof: See Appendix A.14. Again, we are not able to prove uniqueness of the equilibrium for the general case, but plots for other values of ψ, η and ξ suggest that uniqueness applies generally. The comparative statics for the case of ψ = 1 are qualitatively the same as in Table 1 in Section 3.2. Details are in Appendix A Efficiency In a world with search frictions, output is lower than in a Walrasian world. There are two reasons why search frictions reduce output. First, the constraints imposed by the search technology cause unemployment mismatch and costly vacancy creation. Second, output 22

24 is lost due to inefficient decentralized decision making given the constraints of the search technology. An equilibrium is constrained efficient when this second source of inefficiency is equal to zero. In this section, we present the efficiency properties of the various models and suggest institutional remedies to reduce or even eliminate the inefficiencies if necessary, like changing workers bargaining power β or introducing unemployment insurance, so that agents decisions are better aligned. In a Walrasian world, all workers are assigned to their optimal job where they produce Y () = 1. Since labor supply is normalized to unity, this is equal to total output. We define the cost of search X as the current output in a Walrasian world minus current output in a world with search frictions. Since the frictionless level of output is normalized to one, this absolute cost is also equal to the relative cost. X satisfies X 1 x Y (x)g (x)dx ub + vk. (34) where g(x) G x (x) is the density function of workers who are employed at distance x from the optimal assignment. The first term is Walrasian output, while the second term is actual output. The difference is the cost of suboptimal assignment plus the output loss due to unemployment. However, unemployed job seekers do enjoy leisure, which is captured by the third term. The fourth term is the cost of keeping vacancies open. Using (16) we can derive: κvs 1 + ψκvsx g(x) = (1 + ψκvsx) κvsx. (35) Substitution of (35) together with equation (15) in (34) yields an expression for X as a function of the acceptance rule x and unemployment u, or alternatively using z(u) as defined in (26): [( X = γx ) log (1 + z) ( )] z (z + ψ) z z [ + ψ (1 B) + K z + ψ ψ η z κx ( ) ] 1 η 1/ξ 1 + z. ψ + z This expression depends only on technical constraints, not on decision rules. Hence, a social planner can never do better than maximizing this expression with respect to the available instruments to be discussed later in this section. The subsequent proposition relates this expression for the cost of search to the asset value of unemployment in the decentralized equilibrium. 23 (36)

25 Proposition 12 If u and x satisfy the decentralized equilibria of Proposition 3,5, and 9, then: Proof: See Appendix A X = 1 ρv U. (37) The reason that minimizing X is equivalent o maximizing ρv U is due to the golden growth assumption, which sets the worker s discount rate equal to the growth rate of the labor force. Current output is the result of earlier search effort of older and smaller generations of job seekers. Hence, current output is less than the future output that is to be expected from current search effort. The growth of the economy exactly offsets the discounting of future output by today s job seekers. 9 This feature simplifies our welfare analysis considerably. We can just maximize the asset value of unemployment V U or minimize the cost of search X in the steady state here. This is what we do in the subsequent analysis. Given the constraints imposed on X by the search technology, the market outcome depends on three types of decisions: (1) the number of vacancies v opened by firms, (2) the job acceptance rule x applied by unemployed job seekers, and (3) the job acceptance rule applied by employed workers. The latter decision rule is simple in this model. Employed workers accept any job offer that pays a higher wage than their current job. Since all wage setting rules considered in this paper imply that wages are a decreasing function of the distance, W x (x) <, this implies that workers accept any job that is at shorter distance from the optimal assignment than their current job. This decision rule is clearly efficient, since there is no option value lost by switching to a more productive job, neither for the worker, nor for the firm. This leaves the social planner with two remaining decision variables, x and v. Since equation (15) provides the steady-state relation between v on the one hand and x and u on the other hand, we can just as well use the latter two as the relevant decision variables. The social planner s first best optimum minimizes X with respect to x and u (or the composite variable z). The general case of ψ [, 1] is hard to analyze. Hence, we focus on the two special cases, the case without on-the-job search, ψ =, and the case where on- and off-the-job search are equally efficient. Both cases have direct implications for the value of x. It is useful to discuss these implications a priori. In case ψ =, the reservation wage for 8 The same result can be derived using optimal control techniques. Results are available on request. 9 Again, we obtain exactly the same result when we consider the limit ρ δ,λ, since the effect of discounting vanishes in that case. 24