The Joint Effect of Firing Costs on Employment and Productivity in Search and Matching Models

Transcription

1 The Joint Effect of Firing Costs on Employment and Productivity in Search and Matching Models Björn Brügemann October 2006 Abstract In the textbook search and matching model the effect of firing costs on employment is generally ambiguous as both accessions and separations are reduced. Similarly, the effect on average productivity of matches is ambiguous as less productive matches survive while hiring becomes more selective. Thus it would seem that the theory does not generate predictions with respect to the effect of firing costs on employment and productivity. In this paper I show that the theory does generate predictions about the joint response of employment and productivity: while employment may increase or average productivity may rise, it is not possible that both occurs together. That is, firing costs must have a negative effect on at least one of the two outcomes employment and average productivity. Keywords: Employment Protection, Firing Costs, Employment, Productivity. JEL Classification: E24, J63, J65 Department of Economics, Yale University, 28 Hillhouse Avenue, CT bjoern.bruegemann@yale.edu.

2 Much research on employment protection policies has focussed on effects of such policies on the level of employment. Theoretical work has not delivered general predictions on the sign of this effect. In some models employment protection reduces separations and thereby increases employment, in others it reduces hiring and thereby reduces employment, or both effects may be present, making the overall effect ambiguous. Thus the direction of the effect is generally regarded as something to be ascertained empirically. A large number of empirical studies have attacked this question using a variety of approaches, and while the balance points to a negative effect, the evidence is far from overwhelming. 1 Establishing empirically the impact of firing costs on productivity is even more challenging and has only been attempted in a recent paper by Autor, Kerr, and Kugler 2006). They preface their analysis stressing that in theory the sign of this effect is also ambiguous: employment protection may reduce productivity by inducing firms to retain some unproductive worker, but this is potentially offset by firms screening new hires more stringently. The Mortensen-Pissarides search and matching model is one of the most commonly employed theoretical frameworks for studying the effects of firing costs and labor market policies more generally. Within this framework both ambiguities discussed above arise. An increase in firing costs in general reduces both hiring and separations, making the employment effect ambiguous; less productive matches survive while hiring becomes more selective, leading to an ambiguous effect on employment. Thus it would seem that theory makes no predictions regarding the effect of firing costs on employment and productivity, and that this statement also applies to the specific environment of the Mortensen-Pissarides search and matching model. In this paper I show that it is not quite true that this model makes no predictions about the response of employment and productivity to an increase in firing costs. While it is true that it makes no predictions about how the two variables respond individually, the model does restrict their joint behavior in response to an increase in firing costs. I show that while employment may increase or average productivity may rise, it is not possible that both occurs together. Table 1 displays the possible joint responses of employment and productivity to an increase in firing costs. After describing the search and matching model in section 1, in section 3 I show by example that the model can generate each joint outcome indicated affirmatively in Table 1 See Addison and Teixeira 2003) for a survey. 1

3 down Employment up Productivity down yes yes up yes no Table 1: Possible Effects of Firing Cost. 1. Section 4 contains the main theoretical result of the paper, namely that the outcome in the lower right corner of Table 1 is impossible: an increase in firing cost cannot simultaneously increase employment and productivity. Specifically, I will show that if an increase in firing costs increases employment, then it must shift down the distribution of productivity among employed workers in the sense of first order stochastic dominance. Section 5 concludes. 1 Search and Matching Model with Firing Costs Time is discrete. There is a continuum of infinitely lived ex ante identical workers of mass one. At a point in time a worker is either employed or unemployed. The production structure of the economy consists of many firm-worker matches, each composed of one worker and one firm. Timeline. The timing of events within a period is as follows. At the beginning of the period a fraction of workers in existing matches quits exogenously. Then surviving matches receive a new draw of match specific productivity. Next workers unemployed at the end of last period and vacancies posted during last period are matched and each new match receives an initial draw of match specific productivity. This is followed by separation decisions in all matches. Now production takes place in surviving matches. Finally firms decide whether to post vacancies. Preferences. All agents have linear utility with discount factor 1 ρ) 0, 1): the utility of a consumption stream C t is given by t=0 1 ρ)t C t. 2

4 Creation. Maintaining an open vacancy is associated with a cost c per period. If at the end of this period the number of unemployed workers is u and the number of open vacancies is v, then the number of new matches created next period is given by mu,v). The matching function m has constant returns to scale, is continuous, strictly increasing in both arguments, and satisfies mu,v) < min{u,v}. An open vacancy is matched with probability qθ) m 1 θ, 1). The matching probability of an unemployed worker is fθ) m1,θ). The ratio θ = v u is referred to as labor market tightness. To insure existence of equilibrium I assume that lim θ qθ) = 0. Production. The initial productivity of a new match is drawn from a distribution given by the distribution function G 0. Subsequently a match experiences idiosyncratic productivity shocks. In particular, match specific productivity follows a Markov process with state space Y and transition function Q. The process is stochastically monotone: if productivity is high today, it is likely to be high tomorrow; formally y y implies that Qy, ) first order stochastically dominates Qy, ). In addition, I make two standard technical assumptions. First, I assume that the state space is a bounded interval Y = [y min,y max ]. Second, I assume that the transition function satisfies the Feller property. 2 The payoff of non-market activity received by unemployed workers is denoted as z 0. Destruction. There is both exogenous and endogenous destruction. At the beginning of each period a match is destroyed exogenously with probability δ 0, 1). Idiosyncratic shocks to match specific productivity are the source of endogenous destruction. Employment Protection. When dismissing a worker, the firm is bound by statutory employment protection, which is modeled as wasteful firing costs F F R +. After meeting a worker firms can decide not to form a match and walk away without paying the firing costs. Bargaining. For illustrative purposes I will follow the bulk of the literature and use the standard assumption that separations are bilaterally efficient and that at the time of creation the worker and the firm split the value of the match via Nash bargaining with shares β and 1 β, respectively. 2 The Markov process has the Feller property if fz )Qz,dz ) is a bounded and continuous function of z for any bounded and continuous function f. See Stokey and Lucas 1989, p. 220) for a discussion. 3

5 I emphasize, however, that the specific model of wage determination does not play an important role in the results. I will point out those implications of Nash bargaining that the results do rely on. 2 Steady State Equilibrium 2.1 Partial Equilibrium Separation and Match Formation Decision In steady state equilibrium the utility of unemployed workers U is constant. In this section I will study the partial equilibrium separation and creation decision taking as given a level of unemployed utility. In section 2.2 I turn to the determination of the utility of unemployed workers in general equilibrium. In order to study the separation decision, consider a match that has already been created. The separation decision is made to maximize the joint value of the match. Let V y,u,f) be the joint value of a match with productivity y in an environment with firing costs F and constant utility of the unemployed U. It solves the equation { V y,f,u) = max y + δu + 1 ρ δ) } V y,f,u)qy,dy ),U F. 1) where I define δ 1 ρ) δ for notational convenience. The second argument of the maximum operator is the joint payoff if the match dissolves today, given by the utility of unemployment obtained by the worker minus the firing cost liability of the firm. The first argument of the maximum operator is the value of continuing the match. This yields output y this period. With probability δ the match separates exogenously in which case the firm obtains zero it does not have to pay the firing cost) while the worker obtains the utility 1 ρ)u. Taken together this yields the present discounted joint payoff δu. If the worker does not quit the match survives into the next period, receives a new productivity draw y, and once again faces the same decision. The following lemma establishes the properties of the value function V relevant for the analysis here. All proofs not given in the text are collected in the appendix. Lemma 1. a) There exists a unique threshold yf,u) R such that V y,f,u) equals U F for y yf,u) and is strictly increasing in y for y yf,u). The threshold yf,u) is strictly decreasing in F and strictly increasing in U. 4

6 b) Consider F H > F L. The difference V y,f H,U) V y,f L,U) is non-positive. It is bounded below by F L F H. c) Consider U H > U L. The difference V y,f,u H ) V y,f,u L ) is positive. It is bounded above by U H U L, strictly so if y > yf,u H ). Part a) implies that separation occurs if productivity drops strictly below a threshold productivity level yf, U). If productivity equals yf, U) the match is indifferent between separation and continuation. For simplicity I assume that the match continues in this situation. Higher firing costs make splitting up less attractive, while less painful unemployment hastens separations. Part b) considers the comparative statics of the joint value with respect to firing costs. An increase in firing costs reduces the value of the match, but by less than the increase in firing costs as the latter are incurred at some point in the future. Similarly, part c) establishes that an increase in the utility from unemployment increases the joint value, but not by more than the increase in the utility from unemployment. Next I discuss how the joint value of the match is split between the worker and the firm, and the decision whether or not a match is formed. First suppose that a match with productivity y is formed. Then the worker and the firm share the value of the match via Nash bargaining Wy,F,U) U + β [V y,f,u) U], Jy,F,U) 1 β) [V y,f,u) U]. 2) Here Wy,F,U) is the utility of the worker and Jy,F,U) is the value of the firm. The outside opportunity of the worker is unemployed utility U. At the time of match creation the firm can still walk away without paying the firing cost, so its outside opportunity is zero. Thus the surplus of the match is V y,f,u) U, and each of the two parties receives its outside option plus a share of the surplus. Next consider the match formation decision. It is clear from equation 2) that both parties are willing to form the match if the surplus V y,f,u) U is non-negative. Since the joint value is increasing in y it follows that the match is formed if productivity exceeds a threshold y 0 F,U) implicitly defined by the equation V y 0 F,U),F,U) = U. 3) For convenience I assume that a match is formed if the surplus is exactly zero. The match formation threshold has the following properties. 5

7 Lemma 2. a) Positive firing costs F > 0 imply y 0 F,U) > yf,u). b) The threshold y 0 F,U) is weakly increasing in F and strictly increasing in U. Proof. It follows from property a) of Lemma 1 that the separation threshold satisfies V yf,u),f,u) = U F. This implies V yf,u),f,u) < V y 0 F,U),F,U). Since V y,f,u) is strictly increasing in y for y yf) it follows that y 0 F,U) > yf,u). This proves part a). Turning to part b), first consider an increase in firing costs F. This weakly reduces the left hand side of equation 3) and y 0 F,U) must increase weakly to restore equality. For an increase in U property b) of Lemma 1 implies that the right hand side of 3) increases more than the left hand side, so y 0 F,U) must increase strictly to restore equality. Property a) states that as long as firing costs are positive the hiring threshold is more stringent than the separation threshold. This is due to the assumption that at the time of match formation the worker and the firm can still walk away without being subject to firing costs. According to property b) an increase in firing costs for given utility of the unemployed leads to stricter hiring standards. This is because an increase in firing costs reduces the value of the match. An increase in the utility of the unemployed increases the opportunity cost of forming a match more than it increases the value of a match, so again hiring standards become more stringent. Define the ex ante value of the match as ymax V0 F,U) V y,f,u)dg 0 y). 4) y 0 F,U) It is easy to check that it inherits the comparative statics properties of the joint value V established in parts b) and c) of Lemma 1. 3 The ex ante payoff of the firm and the worker are W 0 F,U) U + β [V 0 F,U) U], J 0 F,U) 1 β) [V 0 F,U) U]. 5) 3 The last statement in part b) is not directly applicable to V 0 ; V 0 F,U H ) V 0 F,U H ) is strictly less than U H U L as long as 1 G 0 yf,u H )) > 0. 6

8 2.2 General Equilibrium In a steady state equilibrium the utility of the unemployed U and labor market tightness θ must satisfy the following two conditions. c 1 ρ)qθ)j 0 F,U) with equality if θ > 0, 6) ρu = z + 1 ρ)fθ) [W 0 F,U) U]. 7) Condition 6) is the free entry condition for posting vacancies. The left hand side is the return from posting a vacancy, that is the present discounted value of being matched with a worker next period. In equilibrium it cannot exceed the vacancy cost c and must equal this cost if any vacancies are posted. Condition 7) states that the flow value of unemployment ρu is the sum of the value of non-market activity z and the capital gain from being matched with an employer in the next period. The following lemma establishes existence and uniqueness of the steady state values of labor market tightness θ and unemployed utility U, and how they vary with firing costs F. Lemma 3. a) Existence and Uniqueness) For each level of firing costs F F the conditions 6) 7) have a unique solution U ss F),θ ss F)). b) Comparative Statics) Both U ss F) and θ ss F) are weakly decreasing in F. Proof. See Appendix B for the proof of part a). Turning to part b), first consider the utility of unemployed workers. Suppose that unemployed utility increases in response to an increase in firing costs. Then by equation 6) labor market tightness must fall, since both the increase in firing costs and the increase in the utility of the unemployed reduce the value to the firm of meeting a worker. This drop in labor market tightness reduces the probability that an unemployed worker finds a job. Moreover, both the increase in F and the increase in U reduce the capital gain from finding a job. Thus the right hand side of equation 7) falls, contradicting the increase in U. Labor market tightness must also fall. The only way an increase in labor market tightness could be consistent with the entry condition 6) is for J 0 F,U) to increase. The sharing rule 5) then implies that W 0 F,U) must increase as well. The right hand side of 7) then increases both because of higher labor market tightness and a higher capital gain from finding a job, contradicting the fall in the utility of the unemployed. 7

9 The separation and match formation thresholds as a function of firing costs are given by y ss F) y F,U ss F)) and y ss 0 F) y 0 F,Uss F)). It is clear that the steady state separation threshold is strictly decreasing in firing costs F since both the direct and the indirect effect work in the same direction. In contrast, the direct effect of higher employment protection is to increase the match formation threshold, while the indirect effect through unemployed utility is to lower this threshold. Thus in general the effect of an increase in firing costs on the steady state match formation threshold is ambiguous Productivity Distribution and Employment Take a pair y 0,y) consisting of a match formation threshold and a separation threshold. The following lemma establishes that such a pair induces a unique steady distribution of productivity among employed workers. Lemma 4. If G 0 y 0 ) < 1 a pair y 0,y) with y y 0 induces a unique steady state distribution of productivity among employed workers. Here G 0 y 0 ) is the lefthand limit of G 0 at y 0. The condition G 0 y 0 ) < 1 implies that if a worker and a firm meet they form a match with positive probability. If G 0 y 0 ) = 1 all new matches have productivity strictly below y 0, that is the hiring threshold is so stringent that steady state employment is necessarily zero, in which case there is no meaningful distribution of productivity across employed workers. Let G ss emp y 0,y) denote the unique steady state distribution of productivity among employed workers. Then the steady state destruction probability is given by dy 0,y) δ + 1 δ ) Qy, [y min,y))g ss empdy y 0,y). The first term of the sum is the probability of exogenous destruction. The second term captures endogenous destruction: a match with productivity y that has escaped exogenous destruction is destroyed endogenously if its new productivity draw is below y, which occurs with probability Qy, [y min,y)). 4 In fact, under the assumptions made here once can show that this threshold must be increasing in firing costs. But this property may not generalize to other models of wage determination. Moreover, it is not needed to establish the main results of this paper. Thus it is not emphasized here. 8

10 Steady state employment as a function of the two thresholds and labor market tightness is given by L ss y 0,y,θ) fθ)1 G 0 y 0 )) fθ)1 G 0 y 0 )) + dy 0,y). 8) With slight abuse of notation I write the productivity distribution and employment as a function of firing costs as ) G ss emp F) G ss emp y ss 0 F),yss F), ) L ss F) L ss y ss 0 F),yss F),θ ss F). 3 Possible Joint Responses In this section I begin the analysis of the joint response of employment and productivity to an increase in firing costs. I provide an example for each joint outcome indicated affirmatively in Table 1. This sets the stage for the next section where I show that the only joint outcome that is not possible is for both employment and productivity to increase. 3.1 Employment Up and Productivity Down A special case of the model in which employment increases while productivity declines is obtained by imposing two restrictions on the general model. First, I shut down the hiring margin by letting the initial productivity distribution G 0 be degenerate, so all matches start at a common productivity level y 0. Second, since in general an increase in firing costs reduces recruiting effort of firms, I minimize the negative impact this has on employment by assuming that this recruiting effort plays no productive role in matching, that is mu,v) = µu for µ 0, 1). In this case, as long as matches with productivity y 0 are formed, the job finding rate is constant at µ and not negatively affected by increases in firing costs. It then follows that an increase in firing costs only reduces job destruction, thereby increasing employment. As hiring does not become more selective while destruction is delayed, it also follows that the productivity distribution shifts down. As an illustration, consider the following simple productivity process which I will utilize again to construct the other examples in this section. All matches start with productivity y 0, and with probability γ 0, 1) productivity drops to y 1 < y 0. There are two regions for the 9

11 level of firing costs, one in which low productivity matches are destroyed and one in which they survive. First I compute the value of a new match under the assumption that low productivity matches are destroyed. Later I will check when this is correct. One obtains V y 0,F,U) = y 0 + δu + 1 ρ δ) [ γu F) + 1 γ)v y 0,F,U)]. Solving for V y 0,F,U) yields V y 0,F,U) = y 0 γf + δ + γ)u. 9) ρ + δ + γ where I define γ 1 ρ δ) γ for notational convenience. Substituting into equilibrium condition 7) gives Solving for ρu yields ρu = z + µβ y 0 γf ρu. ρ + δ + γ ρu = ρ + δ + γ)z + µβy 0 γf). 10) ρ + δ + γ + µβ The assumption that low productivity matches are destroyed is correct if y 1 + δu ρ + δ < U F. 11) The left hand side is the value of continuing in the low productivity state, the right hand side is the value of termination. Substituting for U from equation 10), this condition can be written as an upper bound on the low productivity level: y 1 < ρ + δ + γ)z + µβy 0 ρ + δ + γ + µβ ρ + δ + µβ)ρ + δ + γ) F. 12) ρ + δ + γ + µβ Now I choose productivity parameters in the following way. First I set y 0 > z and then I pick y 1 ρ+δ+γ)z+µβy0 ρ+δ+γ+µβ,y 0 ). Let F 1 be the level of firing costs for which 12) holds with equality. The restrictions on the productivity levels insure that F 1 > 0. Now consider an increase in firing costs from zero to F 1. Steady state employment levels are given by L ss 0) = µ δ+γ+µ and L ss F 1 ) = µ. Thus this increase in firing costs strictly increases employment. At zero δ+µ firing costs all matches have high productivity, while at F 1 only a fraction γ γ+δ do. Thus this increase in firing costs is associated with a downward shift of the productivity distribution in the sense of first order stochastic dominance. 10

12 3.2 Employment and Productivity Down In the preceding example I eliminated the negative effect of firing costs on employment by making recruiting effort of firms irrelevant for match creation. In this example I want employment to be decreasing, so I go to the opposite extreme and assume that only vacancies enter the matching function, i.e. mu,v) = µv. Again I start by solving for the equilibrium assuming that low productivity matches are destroyed. Equation 9) for the value of a high productivity match still applies, but now I need to substitute this value into the entry condition 6) in order to obtain an equation for unemployed utility U: Solving for ρu yields c = µ1 β) y 0 γf ρu. ρ + δ + γ ρu = y 0 γf ρ + δ + γ) c. 13) µ1 β) Labor market tightness is obtained by substituting the utility of the unemployed into condition 7): { [ 1 θ ss F) = max 1 β)µ y 0 z γf βµ c ] } ρ + δ + γ), 0 which is valid as long as the assumption of low productivity matches being destroyed is correct. ) ρ+δ+γ Now given the parameters ρ, δ, γ, β, c and µ I pick y 0 z + ρ+δ+γ)c, ρ+δ+γ z + ρ+δ+γ)c ρ+δ 1 β)µ ρ 1 β)µ and set y 1 = 0. The lower bound on y 0 insures that that labor market tightness is strictly positive at zero firing costs. As before the assumption that low productivity matches are destroyed is correct if inequality 11) holds. Substituting from equation 13) into this inequality yields the following upper bound on the low productivity level: [ y 1 < y 0 ρ + δ + γ) c µ1 β) + F 14) ]. 15) Let F 1 be the level of firing costs such that inequality 15) holds with equality. It is straightforward to check that the restrictions on y 0 along with y 1 = 0 implies that F 1 is positive. Now consider an increase in firing costs from zero to F 1. Again this turns a productivity distribution with only high productivity matches into a distribution in which a fraction γ γ+δ of matches has low productivity. It remains to show that employment decreases. Solving equation 15) for F 1 and substi- 11

13 tuting into equation 14) yields { [ θ ss 1 F 1 ) = 1 β)µ βµ ρ+δ)y 0 +γy 1 ρ+δ+γ c z ] } ρ + δ), 0. The lower bound on y 0 along with y 1 = 0 insures that θ ss F 1 ) is strictly positive. Since L ss 0) = µθss 0) µθ ss 0)+δ+γ while Lss F 1 ) = µθss F 1 ) µθ ss F 1 )+δ it follows that employment falls if θss F 1 ) < δ δ+γ θss 0). It is straightforward to verify that the upper bound on y 0 insures that this is satisfied. 3.3 Employment Down and Productivity Up In the preceding examples I considered increases in firing costs that did not make firms more selective in hiring because all matches started at the same level of productivity y 0. Now I assume that initial productivity can take two levels, y H 0 with probability φ and y L 0 < y H 0 with probability 1 φ, letting y 0 denote mean initial productivity y 0 = φy H φ)y L 0. From both levels of initial productivity, a drop to productivity y 1 y L 0 occurs with probability γ. Additionally, I return to the assumption that firm recruiting effort is irrelevant for the production of matches, that is mu,v) = µu. Thus firing costs do not affect labor market tightness. As a consequence an increase in firing costs can reduce employment only by making firms sufficiently more selective in hiring. This will be the case in this example. Suppose matches with low initial productivity are formed. Then it is easy to verify that the utility of the unemployed is once again given by equation 10). The value of a match with low initial productivity is V y0 L,F,U) = yl 0 γf + δ + γ)u. ρ + δ + γ For the assumption that matches with low initial productivity are formed to be correct it must be that V y L 0,F,U) U. This yields the condition y L 0 ρ + δ + γ)z + γf) + µβφyh 0 ρ + δ + γ + φµβ Let F L 0 be the level of firing costs at which this condition is satisfied with equality. For firing costs above F L 0 matches with low initial productivity are formed. As before let F 1 be the level of firing costs for which equation 12) holds with equality. Thus for firing cost levels strictly below F 1 matches with low productivity are destroyed. 12

14 Now pick productivity levels as follows. First choose y0 H > z, then set y0 L = ρ+δ+γ)z+φµβyh 0. ρ+δ+γ+φµβ ) This insures F0 L = 0. Finally pick y 1. This insures F 1 > 0. Now consider z, ρ+δ+γ)z+µβy 0 ρ+δ+γ+µβ an increase in firing costs from zero to any level strictly below F 1. This increase is not enough to keep matches with productivity y 1 from being destroyed, but it keeps matches with initial productivity y L 0 from being formed. Hence this increase in firing costs reduces employment from µ to µφ. In addition, it turns a productivity distribution with a fraction 1 φ) of µ+δ+γ µφ+δ+γ matches with low initial productivity y L 0 into a productivity distribution in which all matches have high initial productivity y H 0. Thus employment falls while the distribution of productivity among employed workers shifts up in the sense of first order stochastic dominance. Remark. The example above demonstrates that it is not true in general that F H > F L implies that for all productivity levels y Y 1 G ss empy F H ) 1 G ss empy F L ), that is it is not true in general that the distribution of productivity across employed workers shifts down in the sense of first order stochastic dominance. However, even if this distribution shifts up, it could still be generally true that F H > F L implies L ss F H ) [ 1 G ss empy F H ) ] L ss F L ) [ 1 G ss empy F L ) ]. That is, while the number of workers with productivity above y may increase as a fraction of employed workers, it could still be the case that it must decrease as a fraction of all workers. For this property to hold, the increase in unemployment must be sufficient to overturn any upward shift in the productivity distribution. The example above has been constructed in such a way that it also violates this weaker property: the increase in firing costs considered in the example increases workers with high initial productivity y H 0 as a fraction of all workers from µφ µ+δ+γ to µφ µφ+δ+γ. 4 Impossibility of Employment and Productivity Up In this section I establish the main result of the paper: an increase in firing costs cannot both increase employment and productivity. More precisely, I show that if an increase in firing costs does not decrease employment, then it must be associated with a downward shift in the productivity distribution in the sense of first order stochastic dominance. 13

15 So far I have proceeded under the standard assumption that wages are determined through Nash bargaining. This particular assumption is not needed for the result of this section to apply. What is needed is the following. Consider an increase in firing costs from F L 0 to F H > F L. Let θ H θ ss F H ), θ L θ ss F L ), y H y ss F H ), y L y ss F L ), y H 0 yss 0 F H ) and y L 0 yss 0 F L ). Under Nash bargaining an increase in firing costs reduces both labor market tightness and the separation threshold, that is θ H θ L and y H y L. In addition, for a given level of firing costs the match formation threshold exceeds the separation threshold, that is y L 0 yl and y H 0 yh. These implications of Nash bargaining are all that is needed, so the result of this section will extend to an alternative model of wage determination as long as it reproduces these implications. So far nothing has been said about how the increase in firing costs affects the match formation threshold. The case in which the match formation threshold falls along with the separation threshold is easily dealt with: in this case the productivity distribution shifts down irrespective of what happens to employment. Lemma 5. Consider y H 0,yH ) with y H 0 y H and y L 0,yL ) with y L 0 yl. If y H 0 y L 0 and y H y L, then G ss empy y H 0,yH ) G ss empy y L 0,yL ) for all y Y. It remains to consider the case in which the match formation threshold increases. As demonstrated in the last example of the previous section, in this case it is possible for the productivity distribution to shift up. However, the following proposition establishes that this is only possible if employment decreases. Let L H L ss F H ) and L L L ss F L ). Proposition 1. Consider y H 0,yH,θ H ) with y H 0 y H and y L 0,yL,θ L ) with y L 0 yl. If y H y L, θ H θ L, and L H L L, then G ss empy y H 0,yH ) G ss empy y L 0,yL ) for all y Y. Proof. The case y H 0 yl 0 has been dealt with in Lemma 1. So suppose yh 0 > yl 0. Let Ḡ 0 y) 1 G 0 y) be the probability that a new matches has productivity strictly higher than y. A quantity that will play an important role in the proof will be the steady steady inflow into productivity levels strictly larger than y as a fraction of employment. It is given by 5 my y 0,y) 1 Lss y 0,y,θ) fθ)ḡ0max{y L ss y 0,y,θ) 0,y}). 16) 5 Although both L ss y 0,y,θ) and fθ) depend on θ, is is easy to check using the formula for steady state employment that the right hand side of equation 16) does not. 14

16 Let Ḡss empy y 0,y) 1 G ss empy y 0,y) be the fraction of workers with productivity strictly above y in steady state. Then for y [y,y max ] Ḡ ss empy y 0,y) = my y 0,y) + 1 δ ) [ y max ] Qy, y,y max ])G ss empdy y 0,y) + Qy min, y,y max ])G ss empy min y 0,y). y min The left hand side is the fraction of employed workers with productivity above y at the time of production this period. It is the sum of two terms. The first term is the mass of workers in new matches with productivity above y as a fraction of employment. The second term is the mass of old matches who survived exogenous destruction and received a productivity draw above y this period. 6 Integration by parts yields Ḡ ss empy y 0,y) = my y 0,y) 1 δ ) [ y ] max G ss empy y 0,y)Qdy, y,y max ])) Qy max, y,y max ])) y min for y [y,y max ]. This can be rewritten as Ḡ ss empy y 0,y) = my y 0,y) + δ) [ ymax ] 17) 1 Qy min, y,y max ]) + Ḡ ss empy y 0,y)Qdy, y, + )) y min for y [y,y max ]. This expression has the following interpretation. If the distribution G ss emp were degenerate with all mass at productivity y min, then exactly a fraction 1 δ ) Qy min, y,y max ]) of matches would survive into next period with productivity above y. This is not the case if not all mass is at y min. In particular, matches with productivity above y min have a higher chance of survival, and the integral y max y min Ḡ ss empy y 0,y)Qdy, y,y max ])) accounts for this fact by adding the increment in the survival chances for matches with productivity above y min. Notice that y H 0 > y L implies Ḡ0max{y H,y}) Ḡ0max{y L,y}) for all y Y. Using equation 16), this fact along with the increase in employment and the decrease in labor market tightness implies my y H 0,yH ) my y L 0,yL ) for all productivity levels y Y. The reason for this is straightforward. The high firing costs economy has fewer unemployed workers who meet firms at a lower rate, so it is clear that 6 The contribution of workers with last period productivity at the lower bound y min is not captured by the integral and must be added separately. Of course this term is zero if there is no mass point at y min, in particular if y > y min. 15

17 the mass of new matches above a certain level productivity levels is lower. Since employment is higher in the high firing costs economy, this holds also when this mass is considered as a fraction of employment. Taking the difference of equation 17) with respect to the two levels of firing costs yields Ḡ ss empy y H 0,yH ) Ḡss empy y L 0,yL ) = my y H 0,yH ) my y L 0,yL ) + 1 δ ) y max [Ḡss ] empy y H 0,yH ) Ḡss empy y L 0,yL ) Qdy, y,y max ]) y min 18) for y [y L,y max ]. Now suppose the distribution under low firing costs does not dominate the distribution under high firing costs. Then over some range the difference between Ḡ ss empy y H 0,yH ) and Ḡss empy y L 0,yL ) is positive. Thus the least upper bound of this gap is strictly positive: Clearly Ḡ ss [Ḡss ] sup empy y H y 0,yH ) Ḡss empy y L 0,yL ) > 0. 19) Y empy y H 0,yH ) Ḡss empy y L 0,yL ) 1 δ ) [Ḡss ] sup empy y H y 0,yH ) Ḡss empy y L 0,yL ) [Qy max, y,y max ]) Qy min, y,y max ])] Y for y [y L,y max ]. To interpret this inequality, notice that the second term on the right hand side of equation 18) gives the contribution of the gaps last period to the gap at productivity level y this period: if the distribution under high firing costs has more mass above y last period, it tends to have more mass above y this period. Under what circumstances is this contribution as large as possible, taken as given the upper bound 19)? This occurs if the gap is constant and equal to the upper bound for all y Y. This corresponds to the extreme case in which last period the distribution with high firing costs is the same as the distribution under low firing costs except for some mass shifted from y min to y max. This maximizes the contribution of this term due to stochastic monotonicity: matches with productivity y min this period are least likely to be above y next period, whereas this is most likely for matches with productivity y max this period. The difference in these probabilities is given by Qy max, y,y max ]) Qy min, y,y max ]). This difference is at most one, which corresponds to the extreme case in which matches with productivity y max this period will be above y next period for sure, while matches with productivity y min this period have no chance to be above 20) 16

18 y next period. Thus Ḡ ss empy y H 0,yH ) Ḡss empy y L 0,yL ) 1 δ ) [Ḡss ] sup empy y H y 0,yH ) Ḡss empy y L 0,yL ) Y 21) for y [y L,y max ]. This immediately generates a contradiction: even when conditions last period are most conducive to creating a gap this period, the gap next period cannot exceed a fraction 1 δ) < 1 of the least upper bound of the gap last period. 7 Thus there cannot be a gap in steady state, that is [Ḡss ] sup empy y H y 0,yH ) Ḡss empy y L 0,yL ) 0 Y which implies G ss empy y H 0,yH ) G ss empy y L 0,yL ) for all y Y. 5 Concluding Remarks I have shown that the standard search and matching model has a testable implication for the joint behavior of steady state employment and productivity in response to an increase in firing costs. A natural extension would be to analyze whether this holds not only in steady state but also throughout the transition from the low firing costs steady state to the high firing costs steady state, and the other way around. This stronger type of prediction would be easier to test empirically. 7 We derived inequality 21) only for y [y L,y max ]. For y [y min,y L ) it is necessarily true that Ḡ ss empy y H 0,yH ) Ḡss empy y L 0,yL ) 0 since Ḡss empy y L 0,yL ) = 1. 17

19 A Proof of Lemma 1 In equilibrium utility from unemployment cannot be lower than the utility from perpetual unemployment U z. Boundedness of the state space Y implies that utility from unemploy- ρ ment cannot exceed some upper bound Ū for any value of firing costs F. Thus it is sufficient to analyze the separation decision for utility from unemployment varying in the set U [U, Ū]. Firing costs are allowed to vary in F = R +. I will establish a more comprehensive result from which Lemma 1 immediately follows. Lemma A. The joint value function V is bounded, continuous, and has the following properties. a) For each F,U) F U there exists a unique threshold yf,u) R such that V y,f,u) equals U F for y yf,u) and is strictly increasing in y for y yf,u). b) Fix U U. Consider F H,F L F with F H > F L. Then yf H,U) < yf L,U). The difference V y,f H,U) V y,f L,U) is non-positive and bounded below by F L F H. c) Fix F F. Consider U H,U L U with U H > U L. Then yf,u H ) > yf,u L ). The difference V y,f,u H ) V y,f,u L ) is non-negative and bounded above by U H U L, strictly so for y > yf,u H ). Proof. Let V be the set of functions V : Y F U R satisfying all the properties stated in the lemma. Let V be the set of functions obtained when the strictly increasing requirement in property a) is replaced by weakly increasing, and the strict requirements in properties b) and c) are replaced by the corresponding weak requirements. Define the operator { } TV )y,f,u) max y + δu + 1 ρ δ) V y,f,u)qy,dy ),U F. I will show that TV) V. The desired result then follows from Corollary 1 to the Contraction Mapping Theorem in Stokey and Lucas 1989) in conjunction with the fact that V is a complete metric space. To verify the claim that TV) V, suppose V V. Then TV is bounded and continuous by Lemma 9.5 in Stokey and Lucas. It remains to verify properties a) c). a) Define CV )y,f,u) y + δu + 1 ρ δ) V y,f,u)qy,dy ). As V is weakly increasing in y and Q is stochastically monotone, it follows that the integral is weakly increasing in y. Thus CV is strictly increasing in y. Set yf,u) equal 18

20 to the unique solution of the equation CV )y,f,u) = U F. Then TV )y,f,u) = U F for y yf,u) and TV )y,f,u) is strictly increasing in y for y yf,u). b) Consider F H,F L F with F H > F L. Since 0 V y,f H,U) V y,f L,U) F L F H for all y Y it follows that 0 CV )y,f H,U) CV )y,f L,U) 1 ρ δ)f L F H ). Since the value of separation drops by F H F L it follows that 0 TV )y,f H,U) TV )y,f L,U) F L F H. Next consider the comparative statics of the separation threshold. As CV )yf L,U),F L,U) = U F L it follows that CV )yf L,U),F H,U) > U F H, so it must be that yf H,U) < yf L,U). c) The proof of property c) proceeds in exactly the same way as the proof of property b). To obtain the additional result that the upper bound U H U L is strict for y yf,u H ), notice that in this case TV )y,f,u H ) TV )y,f,u L ) = CV )y,f,u H ) TC)y,F,U L ) δu H U L ) + 1 ρ δ)u H U L ) < U H U L. B Proof of Lemma 3 Proof of Lemma 3. First consider equation 7) for a given value of θ. The left hand side is strictly increasing in U while the right hand side is weakly decreasing in U. Both are continuous in U. 8 For U = z ρ the right hand side exceeds the left hand side. Since the left hand side is unbounded it follows that there is a unique solution Ûθ). The function Û is continuous. Since an increase in θ strictly increases the right hand side of equation 7) it follows that Ûθ) is strictly increasing. Substituting into the right hand side of equation 6) yields the term 1 ρ)qθ)j 0 F,Ûθ)), which is continuous and strictly decreasing in θ. If it is strictly less than c for θ = 0, then the equilibrium has θ ss F) = 0 and U ss F) = z ρ. Otherwise the assumption that lim θ qθ) = 0 insures that there is a unique value θ ss F) for which this term equals c. Equilibrium utility from unemployment is then given by U ss F) = Ûθss F)). 8 Continuity of the right hand side follows from Theorem 7.38 in Apostol 1974). 19

21 Proof of Lemma 4. In steady state the mass of workers separating equals the mass of workers entering employment. Thus the distribution of productivity across employed workers can be computed from the transition function induced by Q when separated matches are replaced by matches with productivity drawn from G 0. This transition function is given by Q emp y,y y 0,y) 1 δ ) Qy,Y [y,y max ]) + δ + 1 δ ) ) µ0 Y [y Qy, [y min,y)) 0,y max ]). µ 0 [y 0,y max ]) where µ 0 is the probability measure associated with the distribution function G 0. The first term of the sum is the probability of transiting to a productivity level in the set Y by surviving both quits and the separation decision at the beginning of next period. The second term of the sum is the probability of transiting to the set Y via replacement through new matches with a productivity level within that set. It is the product of the destruction rate and the probability of new matches having productivity in Y. Notice that the latter probability is conditional on a new match being formed. Let T emp y 0,y) be the adjoint operator associated with Q emp y 0,y) and let T n emp y 0,y) be the n-fold composition of this operator. 9 To prove the lemma I will show that this transition function satisfies Condition M in Stokey and Lucas 1989, page 348). Set ε = 1 4 δ. Take any Y B where B is the σ- algebra associated with the productivity state space Y.. First suppose µ 0Y [y 0,y max]) µ 0 [y 0,y max]) 1 2. Then Q emp y,y y 0,y) 1 2 δ > ε for all y Y. Next suppose µ 0Y [y 0,y max]) µ 0 [y 0,y max]) < 1 2. Then µ 0 Y c [y 0,y max]) µ 0 [y 0,y max]) > 1 2 and Q empy,y c y 0,y) > 1 2 δ > ε for all y Y. Thus Condition M is satisfied and Theorem in Stokey and Lucas 1989, p. 350) implies that T emp y 0,y) has a unique invariant measure µ ss empy 0,y) and that T n empµ y 0,y) converges strongly to µ ss empy 0,y) as n for any probability measure µ on Y, B). C Proof of Lemma 1. Proof. As a first step I show that Temp y L 0,yL ) dominates Temp y H 0,yH ) according to the definition of dominance in Müller and Stoyan 2002) MS, 2002, p. 180). Using Theorem in MS dominance can be verified by showing that Q emp y, [0,y ] y L 0,yL ) Q emp y, [0,y ] y H 0,yH ) for all y,y Y. For y < y L the desired result follows immediately as Q emp y, [0,y ] y L 0,yL ) = 9 See Stokey and Lucas 1989, pp ) for a definition of the adjoint operator. 20

22 0. Next consider the case y L y < y L. Then 0 Q emp y, [0,y ] y L 0,yL ) = 1 δ ) Qy, [y L,y ]) Finally suppose y y L. Here it is helpful to notice that 0 Thus it is enough to show that 1 δ ) Qy, [y L,y ]) + µ 0 [y L 0,y ]) µ 0 [y L,y 0 max]) µ 0[y H 0,y ]) µ 0 [y H,y 0 max]). 1 δ ) Qy, [y H,y ]) Q emp y, [0,y ] y H 0,yH ). δ + 1 δ ) ) µ0 [y H Qy, [0,y L 0 )),y ]) µ 0 [y H,y 0 max]) 1 δ ) Qy, [y H,y ]) + δ + 1 δ ) ) µ0 [y H Qy, [0,y H 0 )),y ]) µ 0 [y H,y 0 max]). Simplifying, this condition reduces to Qy, [y H,y L )) µ 0[y H 0,y ]) µ 0 [y H 0,y max]) Qy, [yh,y L )) which is satisfied. Now let µ be a probability measure on Y, B). By Theorem in MS T n empµ y L 0,yL ) FSD T n empµ y H 0,yH ) for all n 0. Since first order stochastic dominance is closed with respect to strong convergence, it follows that µ ss emp y L 0,yL ) FSD µ ss emp y H 0,yH ). References Addison, John T., and Paulino Teixeira The Economics of Employment Protection. Journal of Labor Research 24 1): Apostol, Tom M Mathematical Analysis. Addison-Wesley. Autor, David H., William R. Kerr, and Adriana D. Kugler Do Employment Protections Reduce Productivity? Evidence from U.S. States. Mimeo. Müller, Alfred, and Dietrich Stoyan Comparison Methods for Stochastic Models and Risks. Wiley. Stokey, Nancy L., and Robert E. Lucas Recursive Methods in Economic Dynamics. Cambridge and London: Harvard University Press. 21