Job matching when employment contracts suffer from moral hazard

European Economic Review 55 (2011) 964–979

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Job matching when employment contracts suffer from moral hazard$ Dominique Demougin a,b, Carsten Helm c, a b c

EBS University, Department of Governance & Economics, Gustav-Stresemann-Ring 3, 65189 Wiesbaden, Germany University of Paris 2, Paris, France University of Oldenburg, Department of Economics and Law, 26111 Oldenburg, Germany

a r t i c l e in f o

abstract

Article history: Received 23 February 2009 Accepted 25 April 2011 Available online 6 May 2011

We consider a job matching model where the relationships between firms and wealthconstrained workers suffer from moral hazard. Specifically, effort on the job is noncontractible so that parties that are matched negotiate a bonus contract. Higher unemployment benefits affect the workers’ outside option. The latter is improved for low-skilled workers. Hence they receive a larger share of the surplus, which strengthens their effort incentives and increases productivity. Effects are reversed for high-skilled workers. Moreover, raising benefit payments affects the proportion of successful matches, which induces some firms to exit the economy and causes unemployment to increase. & 2011 Elsevier B.V. All rights reserved.

JEL classification: J65 D82 J41 E24 Keywords: Job matching Incentive contracts Unemployment benefits Nash bargaining Moral hazard

1. Introduction Nowadays a substantial proportion of jobs include a performance pay component, and this share has been increasing over the last decades. For example, Lemieux et al. (2009) find for the U.S. labor market that the fraction of workers on performance pay jobs ranges from 30% for craftsmen to 78% for sales workers. The overall incidence of jobs that include a performance pay component has increased from about 38% in the late 1970s to around 45% in the 1990s.1 Often, advances in information and communication technologies that have reduced the costs of monitoring workers are cited as an explanation for this development. This paper integrates performance pay into a job matching model and analyzes the effect of unemployment benefit payments on worker productivity and unemployment levels. Specifically, we consider a stochastic job matching environment with a continuum of workers who differ in their skill level so that they are of heterogeneous productivity (see, e.g. Pissarides, 2000). This productivity is revealed to a firm at the moment it is matched with an unemployed worker, e.g. through assessment centers, job interviews or credentials. Hence labor contracts will depend on productivity, while a

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We are grateful to Stefan Pichler, two anonymous referees and the associate editor, Christopher Flinn, for helpful comments.

 Corresponding author. Tel.: þ49 441 798 4113; fax: þ 49 441 798 4116.

E-mail addresses: [email protected] (D. Demougin), [email protected] (C. Helm). Similar results are reported by Green (2004) for Great Britain and by Kurdelbusch (2002) for major corporations in Germany.

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0014-2921/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.euroecorev.2011.04.002

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worker’s effort during an employment relation is non-contractible, leading to moral hazard. In addition, their wages must include a positive fixed component due to the existence of minimum wages or due to wealth constraints that lead to limited liability. Each firm-worker match bargains over the design of a bonus contract.2 At the negotiation stage a firm’s disagreement point is given by its exit option; hence assuming perfect competition it is zero. In contrast, a worker’s threat point depends on his type, on unemployment benefits and on the unemployment level. In particular, at the contracting stage a higher level of unemployment benefits directly improves the workers’ disagreement point. Hence, for successful matches the respective worker’s share of the total surplus from an employment relationship rises.3 Keeping in mind that moral hazard usually leads to inefficiently low effort levels, raising the worker’s share of surplus induces the negotiating parties to increase the incentive component of the wage contract; hence raising effort efficiency and, thereby, the overall surplus. There are, however, two countervailing effects associated with higher unemployment benefits. First, a firm-worker match which was initially indifferent between agreeing to a contract or breaking up negotiation now prefers the improved outside option. Second, by lowering profits of firms the economy’s labor market equilibrium is affected. In particular, some firms are induced to exit the economy. Both effects work to increase unemployment. This creates a negative indirect effect on the outside option of workers because it takes longer to find a new job. Whether the positive direct effect of raising unemployment benefits or the negative indirect effect dominates depends on the productivity of a worker. In particular, the forgone wage during spells of unemployment is larger for high productivity workers so that their outside option is more affected by the indirect effect. Accordingly, if the effort enhancing direct effect of unemployment benefits dominates, then moral hazard on the job may provide an argument for raising their level. The effects of minimum wages differ substantially from those of unemployment benefits. Minimum wages do not affect a worker’s outside option if negotiations fail and, therefore, leave bargaining power unchanged. However, they impose a lower limit for the fixed component of the wage. Hence, at the contracting level minimum wages tend to reduce bonuses, lowering the power of incentive contracts and, thus, decreasing the level of effort negotiated. This reduces the firms’ expected profit and induces some of them to exit, thus also raising unemployment. Accordingly, we conclude that minimum wages are never welfare enhancing in the context of our model. Obviously, bonus contracts are just one out of several instruments that are used to incentivice workers. Other examples include piece rates, promotions, subjective performance evaluations and deferred compensation (see Prendergast, 1999). Reflecting this variety in a single analytical model is not feasible. We have chosen to focus on bonus contracts because they capture the main idea which drives our results: that higher wages are often associated with higher effort incentives. Furthermore, bonus contracts appear as a suitable modelling device because they are used for different segments of the labor force, ranging from waitresses at the Oktoberfest in Munich to CEOs.4 Our paper contributes to the rich literature on the incentive effects of unemployment benefits (see Holmlund, 1998 and Fredriksson and Holmlund, 2006 for surveys). A part of this literature also finds that benefit payments may raise productivity and unemployment levels. However, the mechanism by which this result occurs is a different one, and many of our modelling choices are driven by the intention to isolate those effects that are original to our paper. For example, while there is a substantial literature that focuses on moral hazard in the search effort of unemployed agents, we analyze effects of moral hazard during an employment relationship. In this respect, the two approaches complement each other. Acemoglu and Shimer (1999, 2000) consider a search model with risk-averse workers. Unemployment insurance encourages workers to take the risk of applying for high wage jobs, and firms respond by creating more capital-intensive, high productivity jobs. Thereby, output is raised, but also the risk of becoming unemployed. Moreover, due to moral hazard workers may respond to higher benefit payments by reducing their search effort. These effects are absent in our paper because workers are risk-neutral, bear no search costs and their productivity does not depend on the firms with which they are matched; hence there is no reason to search for a better match.5 Mortensen (1977) emphasizes the entitlement effect which arises since unemployed people are often not eligible for benefit payments (see also Fredriksson and Holmlund, 2001). Therefore, high unemployment benefits provide an additional incentive to seek employment so as to become entitled to them in the case of a future job loss. Our paper is also related to the literature on efficiency wages since both focus on endogenous work effort. In the standard efficiency wage model, workers that are convicted shirking loose their job. Higher unemployment benefits reduce the associated costs and, therefore, effort incentives (Shapiro and Stiglitz, 1984).6 In our model, jobs are terminated at an exogenous rate that is

2

Bargaining over incentive contracts is also analyzed in Pitchford (1998) and Demougin and Helm (2006). See van der Horst (2003) for empirical evidence that an increase in the replacement rate (the proportion of in-work income that is maintained for somebody becoming unemployed) enables workers to negotiate a higher wage rate. Similarly, there exists evidence for a positive relationship between unemployment benefits and reemployment earnings (e.g. Burgess and Kingston, 1976). 4 Ray Rees from the University of Munich reports that waitresses at the Oktoberfest receive no fixed wage, but 9 percent of the revenue on the beer they sell; hence they have probably the most high-powered incentive contract one can find anywhere (Royal Economic Society Newsletter, Issue 142, 2008). 5 Marimon and Zilibotti (1999) as well as Diamond (1981) also stress the role of unemployment benefits as a ‘‘search subsidy’’ that allows the unemployed to take the time necessary to find a suitable job. 6 However, the opposite result arises when the regulator can pay lower unemployment benefits to agents that have been shirking, as compared to agents that have lost their job for other reasons. In this case, the spread between the utility from shirking and non-shirking increases, which strengthens effort incentives (Goerke, 2000). 3

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independent of effort, but badly performing workers risk not receiving a bonus. Moreover, unemployment benefits may improve a worker’s outside option, thereby leading to a higher bonus and higher effort incentives. Finally, we should mention the substantial literature on minimum wages. A contribution that shares some important elements with our model – such as Nash bargaining of wages and job matching with heterogeneous workers – is Flinn (2006). In this paper, higher minimum wages may enhance welfare because they enable workers to negotiate a higher share of the job surplus. This leads to increased search activity which is beneficial because productivity is match-specific. In our paper, this effect is missing because we do not consider on-the-job search. Moreover, minimum wages do not only affect the size but also the composition of the wage, reducing its incentive component. The remainder of the text is structured as follows. After introducing the basic model (Section 2) and determining the parties’ expected discounted payoffs (Section 3), we analyze contract negotiations for an individual firm/worker match (Section 4). In Sections 5 and 6 we broaden the perspective and determine the general labor market equilibrium. Sections 7 and 8 examine the effect of unemployment benefits and minimum wages on effort incentives and unemployment levels. Section 9 discusses a regulator’s choice of the optimal level of unemployment benefits. Finally, in the concluding Section 10 we discuss some empirical evidence that is related to our results. An appendix contains all proofs. 2. The model We consider an environment populated by a continuum of risk neutral firms and risk neutral workers. The measure of workers is normalized to 1. Firms are free to enter or exit the market. They are identical and employ at most a single worker. Workers differ by their productivity g 2 R þ , with cumulative distribution function GðgÞ. They know their respective productivity which becomes known to the firm at the moment of the match. The instantaneous value of a match also depends on the worker’s instantaneous effort aðtÞ 2 R þ and is given by gfðaðtÞÞ, where fðaðtÞÞ is increasing concave and satisfies the Inada conditions. All workers have the same effort cost function c(a(t)) which is increasing and convex with cð0Þ ¼ c0 ð0Þ ¼ 0. A worker’s instantaneous effort, a(t), and the associated value of the match, gfðaðtÞÞ, are non-verifiable so that labor relationships suffer from a moral hazard problem. However, at every moment in time instantaneous effort generates a contractible signal. That signal is used in order to design a contract that aligns incentives. For the sake of tractability, we restrict attention to contracts without memory. In other words, the instantaneous payment at time t only depends on the signal at time t. From the existing literature, we know that due to risk-neutrality we can restrict attention to an instantaneous binary signal sðtÞ 2 f0,1g (see Milgrom, 1981).7 Moreover, we assume that the distribution of the signal is time-independent. We denote with p(a) the probability of observing the instantaneous favorable signal at t given the worker’s effort at that moment and assume p0 ðaÞ 40,p00 ðaÞ o0.8 Hence, in the remaining all relevant variables are timeindependent and – given our restriction on contracts without memory – we can omit the time index t. In addition, we assume that wage payments to the workers must not fall below a minimum level M 4 0.9 Such a minimum wage may arise from regulation or limited liability of the workers. Due to the structure of the problem, contracts will be binary; the worker always receives a fixed payment F and, in addition, a bonus b when the favorable signal is observed. The restriction requires payments to satisfy the constraints F,F þ b ZM. When a firm and a worker of type g are matched – in the remaining referred to as a ‘‘g-match’’ – they bargain about an employment contract. If negotiations are successful, the worker undertakes effort and the respective instantaneous expected payoffs will be UðgÞ  F þ bpðaÞcðaÞt,

ð1Þ

PðgÞ  gfðaÞFbpðaÞ

ð2Þ

for the worker and the firm. t denotes a uniform lump-sum tax levied on employed workers. Moreover, observe that FðgÞ,bðgÞ and aðgÞ are type specific, but for parsimony we suppress this dependency on g whenever this is possible without confusion. If negotiations fail, the worker remains unemployed and his instantaneous payoff consists only of unemployment benefits s.10 Similarly, the job position within the firm remains vacant, leading to an instantaneous loss of h, e.g. due to advertising and other search costs. Tax revenues are used to finance unemployment benefits. As we focus on stationary states, we require budget balancing; hence us ¼ ð1uÞt,

ð3Þ

where u is the unemployment rate. 7 Specifically, in a risk-neutral agency problem all relevant information from a mechanism design point of view can be summarized by a binary statistic (see, e.g. Kim, 1997). 8 These conditions guarantee that the agent’s problem is well behaved. They are equivalent to considering binary signals satisfying MLRC and CDFC within the class of differentiable signals with constant support (see Rogerson, 1985). 9 Otherwise, first-best contracts are obtainable, as is well known from the literature. 10 For parsimony, we assume that s is independent of the previous employment history and neglect other potential benefits that accrue to the unemployed.

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After having discussed the instantaneous payoffs, we now turn to the matching process. Let Z denote the separation rate of employment relationships, v the vacancy rate (relative to the population of workers) and y  v=u labor market tightness. Therefore, Zð1uÞ measures the exit rate from employment. Moreover, let G^ be the fraction of workers whose type is such that negotiations will fail even if they are matched. Hence by construction uG^ measures the fraction of employable workers within the set of unemployed. Finally, we assume a standard matching function with constant returns to scale and denote by mðyÞ the probability per unit of time of finding a candidate for a vacant job (see Petrongolo and Pissarides, 2001). Similarly, lðyÞ  ymðyÞ denotes the probability ^ is the exit rate from unemployment.11 Altogether, the variation u_ in for an unemployed worker being matched. Hence, lðyÞðuGÞ the unemployed rate is ^ u_ ¼ Zð1uÞlðyÞðuGÞ:

ð4Þ

The stationary unemployment rate follows from u_ ¼ 0. Solving for u yields u¼

Z þ lðyÞG^ : Z þ lðyÞ

ð5Þ

3. Discounted expected payoffs In this section, we derive the workers’ and the firms’ discounted expected payoffs, assuming that all parties correctly anticipate the outcome of contracting. Hence, we take the respective instantaneous expected payoffs of a g-match that arises from successful or failed contract negotiations as given. This determines a worker’s disagreement point in contract negotiations as a function of UðgÞ and s, and a firm’s as a function of PðgÞ and h. In Section 4, we analyze contract negotiations for given disagreement points. Obviously, rationality requires that in equilibrium the outcome of negotiations is equal to what was anticipated. 3.1. Workers’ discounted expected utility Let Ve ðgÞ and Vu ðgÞ denote the discounted expected utility of an employed, respectively, unemployed g-type worker. We have Ve ðgÞ ¼

1 ½UðgÞ dt þð1Z dtÞVe ðgÞ þ Z dtV u ðgÞ, 1 þ rdt

ð6Þ

where r 4 0 is the discount rate and dt a short interval of time. Accordingly, the discounted expected value of being employed is the sum of (i) the present value of the instantaneous expected payoff from employment in dt, (ii) the probability of remaining employed in dt times the value of remaining employed, and (iii) the probability of being separated from the job in dt times the value of being unemployed. Multiplying both sides by 1 þ r dt, cancelling common terms, dividing by dt and rearranging yields Ve ðgÞ ¼

UðgÞ þ ZVu ðgÞ : rþZ

ð7Þ

Similarly, Vu ðgÞ is the sum of the present value of instantaneous unemployment benefit, s, plus the present value of the continuation of the game for a currently unemployed g-type worker. With probability lðyÞ dt the worker is matched and chooses employment if Ve ðgÞ Z Vu ðgÞ. With probability 1lðyÞ dt the worker is not matched and remains unemployed. Hence Vu ðgÞ ¼

1 ½s dt þ lðyÞ dtmaxfVe ðgÞ,Vu ðgÞg þ ð1lðyÞ dtÞVu ðgÞ: 1 þ r dt

ð8Þ

Simplifying and rearranging terms yields

rVu ðgÞ ¼ s þ lðyÞmaxfVe ðgÞVu ðgÞ,0g:

ð9Þ

Lemma 1. The necessary and sufficient conditions that a g-type worker matched with a firm accepts a job contract can be stated equivalently as follows: rVu ðgÞ Z0; (i) Ve ðgÞVu ðgÞ ¼ UðgÞ rþZ (ii) UðgÞ Z s þ lðyÞmaxfVe ðgÞVu ðgÞ,0g; (iii) UðgÞ Z s.

Proof. See Appendix.

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11 Observe the difference to the standard stochastic job matching model, where workers and firms discover the productivity of the match only at the ^ This is larger than the exit moment they are matched (e.g. Pissarides, 2000, Chapter 6). In this case, the exit rate from unemployment would be luð1GÞ. rate in our paper, where a fraction G^ will always remain unemployed, whatever firm they are matched with.

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For employable workers, which are characterized by Ve ðgÞVu ðgÞ Z0, Eqs. (9) and (7) allow us to express rVu ðgÞ as a function of the worker’s instantaneous payoff:

rVu ðgÞ ¼

sðr þ ZÞ þ lðyÞUðgÞ : lðyÞ þ r þ Z

ð10Þ

For non-employable workers, (9) implies rVu ðgÞ ¼ s. 3.2. Firms’ expected discounted profit Let Pe ðgÞ denote the discounted expected profit of a firm that currently employs a g-type worker. We assume free entry and exit so that a firm with a vacant position must have a discounted expected profit of Pv ¼ 0. Symmetric to Eq. (6), Pe ðgÞ can be written as the discounted sum the respective instantaneous payoff, PðgÞ, and the ensuing expected continuation value of the game, yielding

Pe ðgÞ ¼

PðgÞ : rþZ

ð11Þ

Lemma 2. The necessary and sufficient conditions such that a firm matched with a g-type worker accepts a job contract, Pe ðgÞ Z Pv ¼ 0, can be stated equivalently as PðgÞ Z 0. A firm’s expected profits from an empty position is   Z 1 1 h dt þ mðyÞ dt Pv ¼ maxfPe ðgÞ, Pv g dGðgÞ þ ð1mðyÞ dtÞPv , 1 þ r dt 0

ð12Þ

where the integral reflects that the firm does not know with whom it will be matched in the future. Using Pv ¼ 0, the equality simplifies to Z 1 0 ¼ h þ mðyÞ maxfPe ðgÞ,0g dGðgÞ: ð13Þ 0

In equilibrium, this determines the number of firms active in the economy. 4. Negotiations of incentive contracts In this section, we analyze negotiations of g-matches for which there exists a contract that is mutually beneficial, i.e. that satisfies Ve ðgÞVu ðgÞ Z 0 and Pe ðgÞPv Z 0. If at least one of these inequalities is strict, we say that a contract is strictly mutually beneficial. During negotiations the parties take not only the unemployment insurance scheme fs, tg as ~ v. exogenously given, but also their respective disagreement points. We indicate this by tildes, i.e. we write V~ u ðgÞ and P Obviously, in equilibrium the payoff UðgÞ that results from the negotiation of a specific g-match must equal the payoff that determines the worker’s disagreement point via (10). For the negotiation, we use the Nash bargaining solution while restricting the contract fF,bg to satisfy the worker’s incentive and minimum wage constraint. The Nash product for equal bargaining power is ~ vÞ N  ðVe ðgÞV~ u ðgÞÞðPe ðgÞP N¼

1 ðr þ ZÞ2

½UðgÞrV~ u ðgÞPðgÞ,

ð14Þ ð15Þ

~ v ¼ 0. Accordingly, the bargaining problem is equivalent to where the second line follows from Lemma 1(i), Eq. (11) and P one where the parties negotiate over their instantaneous payoffs given a disagreement point ðrV~ u ðgÞ,0Þ.12 If negotiations are successful, the worker faces a contract fF,bg and chooses effort to maximize his payoff. The shape of p(a) and c(a) imply a concave payoff function for the agent. Thus, effort follows from the first-order condition of (1): 0

bp ðaÞ ¼ c0 ðaÞ:

ð16Þ

Using this incentive compatibility requirement, we define the expected bonus which implements effort a as BðaÞ  pðaÞb ¼

c0 ðaÞpðaÞ : p0 ðaÞ

ðICÞ

12 For the case of a static moral hazard framework with an exogenously given outside option, Demougin and Helm (2006) show that the same contract emerges if the concept of bargaining power is analyzed in either of the following three setups: in a standard Principal-Agent model by varying the agent’s outside opportunity, in an alternating offer game, and in a generalized Nash-bargaining game. In a dynamic framework such as the present paper the relation between these different specifications of the bargaining process is more complex because the outside option is endogenous. In particular, from (10) it depends not only on unemployment benefits s but also on the agent’s instantaneous payoff UðgÞ, which varies with the bargaining rule. Nevertheless, we would expect that in all cases changes in s affect the outside option in the same direction, though with different magnitude. Hence, our findings should extent to alternative bargaining rules.

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Observe that B0 ðaÞc0 ðaÞ 4 0.13 Moreover, we assume that the marginal expected bonus to implement effort a is increasing, i.e. B(a) is convex.14 Substituting the instantaneous payoffs from (1) and (2) into the Nash product and using (IC), the negotiated contract of a g-match solves15 max½F þ BðaÞcðaÞtrV~ u ðgÞ½gfðaÞFBðaÞ

ðIÞ

a,F

s:t: F Z M,

ðMWÞ

where (MW) is the minimum wage requirement. The Lagrangian is L ¼ ½F þ BðaÞcðaÞtrV~ u ðgÞ½gfðaÞFBðaÞ þ wðFMÞ,

ð17Þ

where w denotes the multiplier associated with (MW). Differentiating L with respect to F and a yields the following firstorder conditions:

PðgÞ½UðgÞrV~ u ðgÞþ w ¼ 0,

ð18Þ

PðgÞ½B0 ðaÞc0 ðaÞ þ ½UðgÞrV~ u ðgÞ½gf0 ðaÞB0 ðaÞ ¼ 0:

ð19Þ 

For comparison, we define two reference effort levels; the first-best effort level, denoted a , which solves

gf0 ðaÞc0 ðaÞ ¼ 0,

ð20Þ 

and the second-best effort level, denoted a . The latter is the solution to the standard Principal-Agent model where the firm can make a take-it-or-leave-it offer and the worker has an outside option of 0. Hence a maximizes PðgÞ subject to (IC) and (MW), i.e. it solves

gf0 ðaÞB0 ðaÞ ¼ 0:

ð21Þ

Comparing this with the outcome of contract negotiations, which we denote by faN ,F N g, we obtain the following result. 0

0

Proposition 1. For any mutually beneficial contract, a raN ra ; hence gf ðaN Þc0 ðaN Þ Z 0 and gf ðaN ÞB0 ðaN Þ r0. Moreover, if effort is inefficient, then F N ¼ M. Finally, for any contract that is strictly mutually beneficial, either (a) effort is first-best, aN ¼ a , and UðgÞrVu ðgÞ ¼ PðgÞ 4 0, or (b) effort is intermediate, aN 2 ða ,a Þ, and UðgÞrVu ðgÞ 4 PðgÞ 4 0, or (c) effort is second-best, aN ¼ a , and UðgÞrVu ðgÞ 4 PðgÞ ¼ 0. Proof. See Appendix.

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The proposition has a straightforward intuition. The Nash product is maximized if the overall surplus is maximized and divided equally between the parties. This is case (a). However, attaining the first goal of maximizing the overall surplus may require paying out a large bonus to the worker in order to induce first-best effort. Attaining the second goal of dividing this surplus equally between the two parties might then necessitate a fixed payment F oM. The worker’s minimum wage constraint, F ZM, makes such a solution infeasible, thereby introducing a trade-off between both goals. At the optimum the parties set F ¼M and negotiate a reduction of the bonus, trading off the surplus loss against the benefits of a more equal surplus allocation. In the proposition, this is case (b). Finally, consider case (c). By definition, effort a maximizes the firm’s profits (taking into account the incentive and minimum wage constraint). Hence the first-order effect of an increase in effort on the firm’s profit is zero. Moreover, due to concavity of P, the second-order effect is negative. In contrast, raising effort has a positive first-order effect on the worker’s rent due to B0 ðaÞc0 ðaÞ 40. Thus, if it were feasible, the parties would negotiate to increase effort. In other words, aN ¼ a can only maximize the Nash product if the secondary negative impact on the firm’s rent violates its participation, i.e. if PðgÞ ¼ 0. However, the bonus will never be smaller than in the situation where the firm had all bargaining power so that aN Za . Moreover, it cannot be optimal to implement effort above the efficient level, as this would lead to a payoff transfer from the firm to the worker (due to B0 ðaÞ 4 c0 ðaÞ), which could be achieved more efficiently by raising transfer payments F. Hence 0 aN r a . Finally, gf ðaN ÞB0 ðaN Þ r0 reflects that the firm’s marginal rent from inducing effort above the second-best level is negative due to the informational costs. 13

This follows from B0 ðaÞ ¼ c0 ðaÞþ pðaÞ

c00 ðaÞp0 ðaÞp00 ðaÞc0 ðaÞ ½p0 ðaÞ2

,

where the second term is strictly positive by the curvature assumptions. 14 From the foregoing footnote the sign of B00 ðaÞ will depend on the third-order derivatives of c(a) and p(a). 15 From (16) and the curvature assumptions, b 4 0 so that F þ b Z M whenever F Z M.

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Statements (a) to (c) did not consider the situation where the participation constraints for the firm and the worker are binding simultaneously. However, this situation is trivial as we then have two equations in two unknowns (a,F), which fully determines the solution; provided that it exists given the constraint F Z M.16 5. Equilibrium contracts Eq. (10) specifies the disagreement point of a g-worker, Vu ðgÞ, for a given instantaneous expected payoff, UðgÞ, that results from the anticipated outcome of contract negotiations fF N ðgÞ,aN ðgÞg. Conversely, the first-order conditions (18) and (19) determine the contract elements for a given disagreement point. Rationality requires that these two solutions be consistent with each other in equilibrium. In conclusion, equation system (10), (18) and (19) determines for any mutually beneficial g-match the values for Vu ,F, and a that result from contract negotiations. We now analyze how this solution varies with the type of the worker. Intuitively, the effort of a more productive worker is more valuable. Hence negotiations lead to an agreement with more effort implemented by a higher bonus, b, and generating a larger overall surplus. The bargaining context guarantees that some of the additional surplus goes to the worker. Consequently, the instantaneous expected payoff rises in type. The conclusion extends to the disagreement point of the worker which also improves because it accounts for the higher instantaneous payoff in case the worker finds a job in a later period. Proposition 2. For matches for which a mutually beneficial contract exists, more productive agents (i) face a higher bonus b, (ii) exert higher effort a, (iii) have a better disagreement point Vu ðgÞ, and (iv) receive a higher expected instantaneous payoff, UðgÞ, and rent, UðgÞrVu ðgÞ. Proof. See Appendix.

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6. Labor market equilibrium In the foregoing sections, we have analyzed the outcome of contract negotiations in a partial equilibrium framework. Specifically, we examined the bargaining of individual g-matches which took the broader labor market conditions such as the unemployment rate u, the vacancy rate v, and the tax rate t as exogenously given. However, in the general labor market equilibrium these variables are determined endogenously and result from the aggregate outcome of contract negotiations, from the government’s regulatory choice about the level of unemployment benefits s, and from the firms’ decision whether to enter or exit the labor market. Moreover, we need to specify for which g-workers mutually beneficial contracts exist. Starting with the latter, we know from Proposition 2 that more productive matches negotiate a higher effort level and generate a larger overall surplus. Hence it should be easier to negotiate a mutually beneficial contract. The following lemma confirms this intuition. Lemma 3. There exists a critical agent g^ such that contract negotiations are successful if and only if g Z g^ . Proof. See Appendix.

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Moreover, from Proposition 1 the worker’s rent in a successful match is always at least as large as the firm’s rent. Hence the critical productivity, g^ , for which at least one of the parties is indifferent whether to accept the outcome of contract negotiations satisfies

Pðg^ Þ ¼ g^ fðaN ðg^ ÞÞF N ðg^ ÞBðaN ðg^ ÞÞ ¼ 0: This gives us the set of employable workers. Hence, using (11) we can rewrite (13) as Z 1 PðgÞ dGðgÞ, 0 ¼ h þ mðyÞ g^ r þ Z

ð22Þ

ð23Þ

which implicitly yields y. Given y and noting that the fraction of non-employable workers is G^ ¼ Gðg^ Þ, the unemployment rate u follows endogenously from (5). Finally, from y and u we obtain the vacancy rate v, and from the budget balancing requirement (3) the tax t. In conclusion, the labor market equilibrium fu,v, t, g^ ,fFðgÞ,aðgÞ,Vu ðgÞgg Z g^ g follows from the equation system (3), (5), (10), (18), (19), (22) and (23). 7. Participation and incentive effects of unemployment benefits We now analyze how changes in unemployment benefits affect the labor market equilibrium. We begin by identifying the effects on the critical agent g^ and, therefore, on the employability of workers. In the following subsection, we then consider the effects on effort incentives and unemployment. 16 Specifically, adding up the participation constraints we can infer effort aN from gfðaÞcðaÞts ¼ 0 and then solve either of the participation constraints for F.

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7.1. Employability effects The Nash bargaining solution maximizes the product of the worker’s and the firm’s rent in a g-match. Accordingly, as long as only one of the parties is indifferent whether to accept a contract while the other receives a positive rent, the Nash product could be increased by reallocating rent. This can be done by changing either the fixed payment or the induced effort. Remembering that Ua ¼ B0 ðaÞc0 ðaÞ 4 0, both instruments can easily be used to increase the rent of the worker. This explains the result in Proposition 1 that a situation where only the worker receives a zero rent can never arise. By contrast, transfers from the worker to the firm are constrained for two reasons. First, the minimum wage constraint requires F Z M. Second, once effort has been reduced to the second-best level, a , any further reduction lowers not 0 only the worker’s rent but also that of the firm (due to Pa ¼ gf ðaÞB0 ðaÞ 40 for all a o a by concavity of gfðaÞBðaÞ). Therefore, we may have a situation where only the firm receives a zero rent. In this case, maximizing the Nash bargaining product of a g-match requires that both ‘‘transfer instruments’’ are used to the largest feasible extent, i.e. F N ðgÞ ¼ M and aN ðgÞ ¼ a ðgÞ. We denote the associated payoff of the worker by U  ðgÞ  Uða ðgÞ,MÞ. Given that aN Za by Proposition 1 and F Z M, this constitutes a lower bound which a g-worker receives from successful contract negotiations. To analyze the effects of variations in s on employability, it suffices to consider the critical worker g^ (due to Lemma 3). First, suppose that U  ðg^ Þs 40, i.e. in the g^ -match only the firm is indifferent whether to accept the contract while the worker receives a rent. Hence the contract is strictly mutually beneficial and, therefore, fixed at fM,a ðg^ Þg by Proposition 1. Accordingly, the firm’s rent Pðg^ Þ ¼ 0 is not affected by a marginal change in s. Moreover, by continuity the g^ -worker’ s rent remains non-negative for small changes in s. In conclusion, employability does not change, i.e. g^ s ¼ 0 if U  ðg^ Þs 4 0. Second, suppose that U  ðg^ Þs r 0 so that to accept a contract the g^ -worker must receive a payoff U N ðg^ Þ Z U  ðg^ Þ. In such a g^ -match both the worker and the firm receive a rent of zero as otherwise the Nash product could be increased by reallocating rent through adjustments of effort and/or F. Now consider an increase of s. Ceteris paribus it improves the g^ -worker’ s disagreement point rVu ðg^ Þ (by (10)) and, therefore, reduces his rent. Hence one would expect that negotiations of the g^ -match now fail. In the next subsection we show that this results also holds if we account for adjustments of effort and feed-back effects from the general labor market. 7.2. Effort incentives and employment effects We now analyze the effect of variations of unemployment benefits on the effort level that arises from successful contract negotiations, i.e. for workers g Z g^ . From Proposition 1, we can distinguish two cases. First, effort is efficient or second-best. Hence it is defined by (20), respectively, (21) and, therefore, independent of s. Second, contract negotiations lead to an intermediate solution aN 2 ða ,a Þ, thus effort follows from the first-order condition (19) and the disagreement point (10).17 Hence an increase in s which raises the worker’s outside option directly impacts the outcome of negotiation. However, variations in s also affect the general labor market variables u,v, t, and g^ . When evaluating how effort changes in s, we also have to take these indirect effects into account. The standard approach to do so would be to apply the implicit function theorem to the simultaneous equation system that defines the general labor market equilibrium (as specified in Section 6). However, the resulting expressions are cumbersome and difficult to evaluate. Therefore, we adopt a simpler approach which uses only the sign of the effects of s on t and l, rather then their precise functional forms. In particular, substitution of rVu from (10) into (19) yields ðgjMBÞðB0 c0 Þ þ

M þ Bcts ðr þ ZÞðgj0 B0 Þ ¼ 0: r þ Z þ lðyÞ

ð24Þ

This defines equilibrium effort as a function of s and of the other labor market variables which are exogenous to the negotiations of a specific g-match. Of these, t and l are themselves functions of s. Taking this into account, implicit differentiation of (24) and rearranging the denominator yields   dt dl 0 0 þ 1 þ r þUs daðgÞ ds Z þ l ds mðgj B Þ ¼ , ð25Þ ds ðgj0 B0 ÞðB0 c0 Þð1 þ mÞB00 ðUrVu PÞPc00 þ ðUrVu Þgj00 where m  ðr þ ZÞ=ðr þ Z þ lÞ. By Proposition 1 and the curvature assumptions, the denominator is negative.18 Turning to the numerator, if the feed-back effects from the general labor market, ts and ls , were absent, i.e. in a scenario where one would raise s for one specific g-worker only, the numerator would also be negative. Hence effort of this specific worker would rise. By contrast, for a general increase in unemployment benefits daðgÞ=ds4 0 if and only if ðts þ1Þðr þ Z þ lðyÞÞ 4 ðUðgÞsÞls :

ð26Þ

In order to get an intuition, suppose that the unemployment level were initially given. From the budget balancing condition (3) one would find ts þ 1 4 0 so that the left-hand side would become positive. Moreover, a higher s raises 17

Note that the first-order conditions (18) and (19) are block recursive and F N ¼ M for intermediate effort levels. Observe that (24) is not exactly the first-order condition of the Nash product w.r.t. effort because it already incorporates the response of rVu to changes in UðgÞ. Therefore, taking the derivative of (24) w.r.t. effort does not generate a second-order condition. 18

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the worker’s instantaneous payoff from unemployment, making the latter state more attractive. Hence, accounting for the impact of s on the unemployment level should further raise taxes. Turning to the right-hand side of (26), ceteris paribus a higher s worsens the bargaining position of firms. Therefore, some of them will exit the labor market. This reduces the probability of an unemployed worker being matched, i.e. ls o0. Hence the right-hand side should be nonnegative.19 The only term that depends on a worker’s type is the instantaneous payoff UðgÞ, which is increasing in g (by Proposition 2(iv)). In particular, for the worker who is just indifferent between accepting and rejecting an employment contract we have Uðg^ Þs ¼ 0 so that daðg^ Þ=ds4 0.20 Moreover, as we did not impose an upper bound on g, there will also exist no upper bound for UðgÞ.21 Accordingly, by the intermediate value theorem there will be a worker-type, denoted g , such that (26) holds with equality for this worker. In conclusion, for g-workers that currently exert effort aN 2 ða ,a Þ we have daðgÞ=ds 40 for all g o g and daðgÞ=dso 0 for all g 4 g . To understand the intuition behind the effects of s on effort, note that the tax rate rises because unemployment benefits are higher and paid to more workers. Ceteris paribus this reduces the workers’ rent so that their payoff before taxes, F þ BðaÞcðaÞ, should increase in compensation. Moreover, from (10) raising s has two effects on a worker’s disagreement point in contract negotiations. First, there is a direct positive effect because a worker gets a higher payment should he become unemployed. Second, there is an indirect negative effect via lðyÞ because after a worker becomes unemployed the expected time until he will be matched again increases.22 The first effect – the payment of s – is the same for all types. However, the costs of a longer unemployment spell are larger for more productive workers who receive a higher instantaneous payoff UðgÞ in the case of employment. Accordingly, the disagreement point in contract negotiations improves only for low productivity workers. For them, the tax effect and the disagreement point effect both tend to raise the payoff before taxes. If such a low productivity worker exerts intermediate effort, i.e. in cases where (26) determines the sign of daðgÞ=ds, this is best achieved by raising the bonus which induces a higher effort. If he exerts efficient effort, the best instrument is to raise the fixed payment. By contrast, if a worker’s productivity is sufficiently high, then the effect of a worse disagreement point prevails and his negotiated payoff before taxes decreases. For inefficient workers, this leads to a lower bonus and effort level. Turning to employment effects, differentiation of (5) yields gÞ ls Z½Gðg^ Þ1 þ lðyÞdGð ðZ þ lðyÞÞ ds : 2 ðZ þ lðyÞÞ ^

us ¼

ð27Þ

As argued above, higher unemployment benefits induce some firms to leave the labor market and reduce employability. Hence dGðg^ Þ=ds ¼ ðdGðg^ Þ=dg^ Þdg^ =ds Z0. As a consequence, the probability of an unemployed worker being matched falls, i.e. ls o0. From (27) it then follows that unemployment increases in s. Proposition 3. Raising unemployment benefits has the following effects on employability, unemployment and effort of workers g 4 g^ : (1a) (1b) (2) (3a) (3b) (3c)

If so U  ðg^ Þ, then a marginal increase in s has no effect on g^ . Hence dGðg^ Þ=ds ¼ 0 and employability remains unchanged. If s Z U  ðg^ Þ, then an increase in s raises g^ . Hence dGðg^ Þ=ds4 0 and less workers will be employable. Unemployment increases. N If a ðgÞ o aN ðgÞ o a ðgÞ and g o g , then da ðgÞ=ds 4 0. N  N  If a ðgÞ o a ðgÞ o a ðgÞ and g 4 g , then da ðgÞ=ds o 0. N If aN ðgÞ ¼ a ðgÞ or aN ðgÞ ¼ a ðgÞ, then da ðgÞ=ds ¼ 0.

Proof. See Appendix.

&

Result 3 has an interesting immediate implication. Higher effort implies a higher payoff for the worker, and vice versa (due to B0 ðaÞ 4c0 ðaÞÞ. Therefore, the payoff spread between high and low-skilled workers will be decreasing in unemployment benefits. 8. Incentive effects of minimum wages We now contrast the effects of unemployment benefits to those of minimum wages. If effort is efficient or second best, it is not affected by either of the two instruments (see (20) and (21)). However, their effect is quite different for intermediate effort levels. For parsimony, we focus on an economy in which unemployed workers receive no benefit 19

A formal proof of ts þ 1 4 0 and ls o 0 follows below. Remember that we are currently focusing on workers who exert effort aN 4 a so that Uðg^ Þ ¼ s by the analysis in Section 7.1. Moreover, for a given s any negotiated contract with a worker g Z g^ will be mutually beneficial (by Lemma 3). 0 21 Specifically, limg-1 ½PðgÞ þ UðgÞ ¼ limg-1 ½gfðaÞcðaÞt ¼ 1 because gf ðaÞc0 ðaÞ Z 0 and da=dg 4 0. Noting that m and s are the same for all g, also limg-1 ½PðgÞ þ ðUðgÞsÞm ¼ 1. Using (10) and Proposition 1, ðUðgÞsÞm ¼ UðgÞrVu ðgÞZ PðgÞ, which concludes the proof. 22 Specifically, r@Vu ðgÞ=@l ¼ ½UðgÞsm=ðlðyÞþ r þ ZÞ. 20

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payments, i.e. s¼0. Hence one may read this and the preceding section as a comparison where the regulator uses either unemployment benefits or minimum wages as a policy instrument. At the end of this section we will then discuss that taxes, which would be required to finance unemployment benefits s 40, introduce feed-back effects that substantially complicate the analysis of minimum wages. Ceteris paribus, increasing minimum wages, M, raises the worker’s and lowers the firm’s payoff in a g-match. Therefore, it aggravates the unequal surplus distribution that arises for intermediate effort levels (see Proposition 1). Given the constraint F Z M, the only available instrument for a more equal surplus distribution is to lower the bonus b. Hence effort falls. Moreover, as a consequence of the firms’ reduced payoffs, some of them will leave the labor market so that the rate lðyÞ at which workers are matched falls. Hence they stay unemployed for a longer period, which worsens their bargaining position. This reduces their negotiated payoff, implying again a lower bonus and effort. Finally, because the firms’ payoffs fall, less workers will be employable. Together with the longer unemployment spells it follows that unemployment rises. Proposition 4. Assuming s¼0, minimum wages reduce intermediate effort levels aN ðgÞ 2 ða ðgÞ,a ðgÞÞ and raise unemployment. Proof. See Appendix.

&

Obviously, this implies that the optimal level of minimum wages is zero. To see the difference with unemployment benefits, note that both reduce the matching probability of unemployed workers which worsens their bargaining position. However, unemployment benefits also have a positive direct effect on the bargaining position. If this effect dominates, workers are able to negotiate higher wages which are paid, at least partly, in the form of higher bonus payments so that effort increases. By contrast, minimum wages lack this effect and prescribe a higher reliance on fixed payments, which reduces effort incentives. Finally, let us consider a situation with s4 0. A higher minimum wage raises unemployment and hence, the tax must increase. Ceteris paribus, this reduces the workers’ payoffs (see (1)). Accordingly, the surplus distribution becomes more equal and more weight can be put on the bonus instrument for maximizing the Nash bargaining product, thereby enhancing effort efficiency.23 9. Welfare effects of unemployment benefits We now discuss the problem of a benevolent regulator when choosing the level of unemployment benefits s. As in the previous parts of the paper, we only consider the steady state. All workers g Z g^ are currently unemployed with equal probability. Hence the average output of a filled job is Z 1 1 Y YðgÞ dGðgÞ ð28Þ 1Gðg^ Þ g^ where YðgÞ  gjðaðgÞÞcðaðgÞÞ. The share of workers that have a job is 1u. Noting that tax and benefit payments cancel out in the aggregate due to budget balancing, the instantaneous per capita output net of the costs of vacant jobs becomes w ¼ ð1uÞY hv: Differentiation w.r.t. s yields " # Z 1 1u dg^ dYðgÞ da dGðgÞ : gðg^ Þ½Y Yðg^ Þþ ws ¼ us Y hvs þ da ds 1Gðg^ Þ ds g^

ð29Þ

ð30Þ

Accordingly, raising unemployment benefits has three effects on welfare. First, it raises unemployment which reduces welfare. Second, the vacancy rate changes and, therefore, associated costs. Intuitively, given that the matching rate mðyÞ increases in s, one would expect that vacancies fall.24 Third, the average output of a filled job changes. On the one hand, 23 Formally, this can be seen by adjusting Eq. (51) in the appendix, which determines the sign of da=dM. Specifically, when s 40 the numerator must include a term capturing the adjustment of taxes to changes in M and becomes

ðB0 c0 Þ 1tM 

! dl ðUsÞ dM mðgj0 B0 Þ: rþZþl

In general, one would expect that small changes in M induce only small changes in unemployment. Hence tM would be small and the sign of da/dM unaffected. However, we have not imposed any restrictions on the distribution of g. For example, we did not exclude a situation where a substantial mass of workers is clustered in the direct proxomity of g^ . With a higher minimum wage these workers would become unemployed, requiring a substantial increase in the tax rate. If tM is large enough, this could reverse the sign of da/dM. 24 For s ¼ 0 this is easily seen. Writing v ¼ lu=m we obtain vs ¼

ðls uþ lus Þmms lu : m2

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employability falls if s is sufficiently high (by Proposition 3) so that some of the less productive agents will no longer be employed. This raises the average output of a filled job. Formally, this follows from Y Yðg^ Þ 40 as more productive workers exert higher effort (by Proposition 2). On the other hand, effort changes unless it is efficient or second-best. This is the only effect that arises from moral hazard, which has been at the core of this paper. From Proposition 3, it is positive for workers with a low g and negative for those with a high g. Accordingly, if the effort enhancing effect dominates, then moral hazard on the job may provide an argument for raising the level of unemployment benefits. Which of the moral hazard effects dominates is ultimately an empirical matter. However, given the positive relation between effort and productivity, one might conjecture that high-skilled workers more often exert efficient effort that is independent of unemployment benefits. This suggest that the effort enhancing effect on low-skilled workers might be more important. On the other hand, Lemieux et al. (2009) finds that performance pay, which drives the above results, is more prevalent among high-skilled workers.

10. Concluding remarks In this paper we have argued that the integration of performance pay into a job matching model may provide a rationale for unemployment benefits. In particular, unemployment benefits improve the bargaining position of workers whose skills are below a threshold level. This enables them to negotiate higher expected wages which often involve a higher incentive component of the wage, thereby strengthening the workers’ effort efficiency. However, effects are reversed for high-skilled workers. Moreover, higher unemployment benefits reduce the number of firm/worker matches for which mutually beneficial contracts exist and increase the length of unemployment spells, thereby increasing unemployment. The optimal benefit level balances these effects. Performance pay is nowadays an increasingly important component of many wage contracts, but there is little systematic research about its interplay with unemployment benefits and minimum wages. The model has been set-up so as to examine this specific interplay, while neglecting other important mechanisms that are commonly used to explain trends in unemployment, productivity and wages (see, e.g. OECD, 2007). The relative importance of our mechanism as compared to other explanations is ultimately an empirical issue which is beyond the scope of this paper. Nevertheless, our main results should at least be consistent with the empirical evidence, which we now briefly discuss. There is substantial evidence that a rise in unemployment benefits tends to increase unemployment (e.g. Blanchard and Wolfers, 2000; Lalive et al., 2006). Furthermore, Blanchard (2004) shows that in many European countries high benefit payments have led not only to high unemployment levels, but also to a relatively high productivity per hours worked. Using a quantitative model that is calibrated to capture the U.S. labor market for high school graduates, Acemoglu and Shimer (2000) also find a positive effect of unemployment benefits on productivity. These studies do not differentiate between low- and high-skilled workers, for which we predict partly opposing effects. However, for the main implication of this result, that wage inequality should be smaller in countries that have high unemployment benefits, supporting evidence exists. For example, Acemoglu (2003) shows that wage inequality is higher and increased more sharply during the 1980s in the U.S. as compared to continental Europe, where the level of unemployment has remained comparatively high. Moreover, he argues that wage inequality in Denmark and Sweden, i.e. in countries that are known for having high unemployment benefits, is substantially lower than predicted by the traditional explanations. While these findings are consistent with the results from our model, there is less direct evidence regarding the specific mechanism which we have discussed in this paper. Nunziata (2005) and van der Horst (2003) find that an increase in the replacement rate allows workers to negotiate higher wages, but leads to more unemployment. Furthermore, there is an empirical literature which verifies the link between higher bonuses and effort (e.g. Prendergast, 1999; Chiappori and Salani, 2003). There are several possible extensions of the paper. Obviously, distortionary taxation would reduce the optimal level of unemployment benefits. Minimum wages – at least those in the private sector – need not be financed by taxes and, therefore, are less prone to this problem. However, we have shown that they cannot be used as a substitute since they lack the positive effect that unemployment benefits have on effort. Given that we have a repeated moral hazard problem, another interesting extension would be to allow for contracts with memory.25 For instance, by delaying bonus payments the principal may be able to reduce the fixed payment F in future periods below M and still satisfy the limited liability constraint. Over time, this would weaken the trade-off between efficiency and rent, which in our analysis induces the principal to use an inefficiently low bonus. Overall, one would expect (footnote continued) At s ¼ 0, Gðg^ Þ ¼ dGðg^ Þ=ds ¼ 0 (by Proposition 3(1a)) so that substitution from (5) and (27) for u and us yields 

ls u þ lus ¼



ls Z l 1 o0 Zþl Zþl

because ls o 0 from the proof of Proposition 3. There we have also shown that ms 4 0. Upon substitution vs o 0. 25 See Chiappori et al. (1994) for a survey, as well Ohlendorf and Schmitz (forthcoming) for a recent contribution that also considers a framework with risk-neutrality and limited liability.

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that this reduces the agency problem and, therefore, weakens the positive effects of social benefit payments on effort efficiency. Moreover, we have only considered a uniform level of unemployment benefits. If benefits were higher for more skilled workers, this would strengthen their effort enhancing direct effect especially for those workers for which the indirect effect is particularly detrimental. In addition, an increase in the effort of high-skilled workers is particularly rewarding. This would provide a justification for the widely observed dependence of benefit payments on previous earnings. Another extension would be to consider the generalized Nash bargaining solution and, more interestingly, to make the bargaining power coefficient a policy variable that can be set by the social planner. Such an analysis would be particularly interesting if there would be a countervailing effect that balances the current model bias of strengthening the workers’ bargaining power. An immediate idea would be to explicitly introduce capital into the model and discuss the implication of unemployment benefits on investment decisions and growth. Appendix A A.1. Proof of Lemma 1 Acceptance of a contract requires Ve ðgÞ ZVu ðgÞ. Subtracting Vu ðgÞ on both sides of (7) then verifies (i). To verify the equivalence, we proceed as follows: ðiÞ¼)ðiiÞ: Follows by substitution from (9) into (i). ðiiÞ¼)ðiiiÞ: Follows as lðyÞmaxfVe ðgÞVu ðgÞ,0g Z0. ðiiiÞ¼)ðiÞ: Assume UðgÞ Zs, but Ve ðgÞ oVu ðgÞ. The equality in (i) still holds. Hence, UðgÞ o rVu ðgÞ ¼ s þ lðyÞmaxfVe ðgÞVu ðgÞ,0g: However, Ve ðgÞ oVu ðgÞ implies maxfVe ðgÞVu ðgÞ,0g ¼ 0. Hence UðgÞ o s, contradicting the initial assumption.

ð31Þ

&

A.2. Proof of Proposition 1 In contradiction, suppose aN 4 a . Reducing the bonus would reduce effort (by 16) and increase the total rent RðgÞ  PðgÞ þUðgÞrV~ u ðgÞ ¼ gfðaÞcðaÞtrV~ u ðgÞ:

ð32Þ

Ceteris paribus, the worker’s payoff falls because B0 ðaÞc0 ðaÞ 4 0 from the curvature assumptions. Hence the firm’s payoff increases. Fixed payments to the worker, F, can always be increased such that both parties are better off. Hence aN 4 a cannot maximize N . Similarly, suppose aN o a . Then raising the bonus would strictly increase the worker’s payoff (by (16) and 0 0 B ðaÞc0 ðaÞ 4 0) and the firm’s payoff (because gf ðaÞ 4B0 ðaÞ for all a o a by (21) and the curvature assumptions). To see that F N ¼ M whenever effort is inefficiently low, again suppose by contradiction that this were not the case. Increasing the inefficient effort level raises the overall surplus. Moreover, for a given F the worker’s payoff increases due to B0 ðaÞc0 ðaÞ 4 0. As the minimum wage constraint does not bind by assumption, F could be reduced until also the firm is better off. It remains to prove statements (a) to (c). 0

(a) aN ¼ a implies gf ðaN Þc0 ðaN Þ ¼ 0. Substitution of (18) into (19) yields 0

PðgÞ½gf ðaÞc0 ðaÞþ w½gf0 ðaÞB0 ðaÞ ¼ 0:

ð33Þ

Noting that B ðaÞ 4c ðaÞ for all a 4 0, (33) can only hold if w ¼ 0, i.e. the minimum wage constraint is not binding. Hence from (18) and the restriction on strictly mutually beneficial contracts we have UðgÞrV~ u ðgÞ ¼ PðgÞ 40. 0 0 (b) aN 2 ða ,a Þ implies gf ðaN Þc0 ðaN Þ 4 0 and gf ðaN ÞB0 ðaN Þ o 0. Hence (33) can only be satisfied if (i) PðgÞ ¼ w ¼ 0, or (ii) PðgÞ, w 40. However, case (i) is not possible due to (18) and the restriction on strictly mutually beneficial contracts. Using (18), the solution for (ii) follows straightforwardly. 0 (c) aN ¼ a implies gf ðaN ÞB0 ðaN Þ ¼ 0. From (19) and B0 ðaÞ 4 c0 ðaÞ, we obtain PðgÞ ¼ 0. From the restriction on strictly mutually beneficial contracts, UðgÞrV~ u ðgÞ 4 PðgÞ ¼ 0. & 0

0

A.3. Proof of Proposition 2 From (20) and (21) and the curvature assumptions, first-best and second-best effort, a and a , are increasing in g. It remains to consider intermediate effort levels, aN 2 ða ,a Þ, for which F N ¼ M by Proposition 1. Moreover, the first-order conditions (18) and (19) are block recursive. Hence aN and Vu follow from (19) and (10). Substituting rVu from (10) yields UðgÞrVu ðgÞ ¼ m½UðgÞs,

ð34Þ

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where m  ðr þ ZÞ=ðl þ r þ ZÞ. Using this, (19) becomes

PðgÞ½B0 ðaN Þc0 ðaN Þ þ m½UðgÞs½gf0 ðaN ÞB0 ðaN Þ ¼ 0,

ð35Þ

N

where PðgÞ and UðgÞ are evaluated at a . Implicit differentiation, using (34) and factorizing the denominator yields N

0

da fðB0 c0 Þm½Usf ¼ : dg ðgj0 B0 ÞðB0 c0 Þð1 þ mÞB00 ðUrVu PÞPc00 þ ðUrVu Þgj00

ð36Þ

The numerator and denominator are both negative, where the latter follows from UðgÞrVu ðgÞ 4 PðgÞ 4 0 by N Proposition 1(b). In conclusion, we have da =dg 4 0 for all effort levels that can arise in equilibrium (claim ii). From (16), higher effort requires a higher bonus b, which proves (i). Turning to (iv) we distinguish two cases. First, suppose aN o a . From Proposition 1 we have F N ¼ M so that from (1) and the previous results, U 0 ðgÞ ¼ ½B0 ðaN Þc0 ðaN ÞaN g 4 0:

ð37Þ

Second, for aN ¼ a from (1) and (2) UðgÞ þ PðgÞ ¼ gfða Þcða Þt:

ð38Þ

Furthermore, from Proposition 1(a) and (34)

m½UðgÞs ¼ PðgÞ:

ð39Þ

Adding up both equalities and differentiation with respect to g yields U 0 ðgÞ ¼

1 fða Þ 4 0: 1þ m

ð40Þ

Using this, differentiation of (10) w.r.t. g immediately yields (iii). Finally, observe that UðgÞrVu ðgÞ is increasing in UðgÞ from (34). & A.4. Proof of Lemma 3 A match between a firm and a g-type worker fails if it is rejected by (i) only the worker, or (ii) only the firm, or (iii) both. Remembering the criteria such that a contract will be accepted (Lemmas 1 and 2), the first case cannot occur because the worker’s rent is always at least as large as that of the firm (by Proposition 1). Turning to case (ii), from Lemma 2 the critical worker-type is characterized by Pðg^ Þ ¼ 0. By Proposition 1, we get g^ f0 ðaÞ ¼ B0 ðaÞ and F¼M. To see that a mutually beneficial contract remains feasible if we marginally increase g, note that from (2) for successful matches P0 ðg^ Þ ¼ fðaÞ 4 0. Moreover, a worker’s rent, UðgÞrVu ðgÞ, is strictly increasing in g by Proposition 2. Finally, in case (iii), Pðg^ Þ ¼ Uðg^ ÞrVu ðg^ Þ ¼ 0. Solving (19) for PðgÞ and substituting for UðgÞrVu ðgÞ from (34), we obtain 0

PðgÞ ¼ m½UðgÞs

B0 ðaÞgf ðaÞ : B0 ðaÞc0 ðaÞ

ð41Þ 0

To see that P0 ðg^ Þ Z0, note that U 0 ðgÞ 40 by Proposition 2, B0 ðaÞgf ðaÞ Z 0 by Proposition 1, and Uðg^ Þs ¼ 0 by Lemma 1. & A.5. Proof of Proposition 3 To conclude the proof we have to prove the claims that we made after equation (26) and at the end of Section 7.1: (i)

ts þ1 4 0, (ii) ls o0, and (iii) g^ s 4 0 if U  ðg^ Þs r0. (i) Proof of ts þ 1 40. From (3), ts þ 1 ¼

1u þsus ð1uÞ2

:

ð42Þ

Accordingly, at s¼0 we have ts þ 1 40. By contradiction, suppose there is an s such that ts þ 1 r0. Then, by continuity there must be an s such that ts ðsÞ þ 1 ¼ 0. From (42) this can only be satisfied if us ðsÞ o0, which we now show to be wrong. From (27), a sufficient condition for this is that at the benefit level s we have g^ s Z 0 and ls r 0. In Section 7.1, we have already shown that g^ s ¼ 0 if s o U  ðg^ Þ. Now consider the alternative case s ZU  ðg^ Þ so that in the g^ -match both the worker and the firm receive a rent of 0 (see Section 7.1). First, for inefficient effort levels ^ 0 ðaN ÞB0 ðaN Þ r0 by Proposition 1. Hence, if daN =ds4 0, then the change of the firm’s rent in the F N ¼ M and gj g^ -match, N

dPN ðg^ Þ da ðg^ Þ ^ 0 ðaN ðg^ ÞÞB0 ðaN ðg^ ÞÞ ¼ ½gj , ds ds

ð43Þ

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N is non-positive so that g^ s ðsÞ Z 0. Alternatively, suppose da =ds r0. Then at the benefit level s, for which ts þ1 ¼ 0, the change in the g^ -worker’s rent,   N d N da ðg^ Þ dt ½U ðg^ Þs ¼ ½B0 ðaN ðg^ ÞÞc0 ðaN ðg^ ÞÞ  þ1 , ð44Þ ds ds ds

is non-positive so that again g^ s ðsÞ Z 0. Finally, suppose that effort is efficient. Accordingly, effort of the g^ -worker is constant at the level that solves ^ 0 ðaÞ ¼ c0 ðaÞ, but changes in s now affect the fixed payment F. Given that the participation constraint binds for both gj parties, PN ðg^ Þ þU N ðg^ Þs ¼ 0 and d ½PN ðg^ Þ þ U N ðg^ Þs ¼ ð1 þ ts Þ ds

ð45Þ

is zero at the benefit level s; hence g^ s ðsÞ ¼ 0. Next, we show that ls ðsÞ r 0. From the literature (e.g. Cahuc and Zylberberg, 2004) it is well known that d ½ymðyÞ 4 0 dy

and

dmðyÞ o 0: dy

ð46Þ

Moreover, from the chain rule dmðyÞ dmðyÞ dy ¼ ds dy ds

ð47Þ

dlðyÞ d dy ¼ ½ymðyÞ ds dy ds

ð48Þ

and

so that ms and ls have the opposite sign if ys a0, and are both equal to zero else. By contradiction, suppose that ls ðsÞ 40, which implies that ms ðsÞ o 0. Implicit differentiation of (23) yields   R dg^ daðgÞ dFðgÞ  dGðgÞ mðyÞ Pðg^ Þ g^1 ½gf0 ðaðgÞÞB0 ðaðgÞÞ dmðyÞ ds ds ds R1 ¼ : ð49Þ ds g^ PðgÞ dGðgÞ From the above analysis g^ s ðsÞ Z 0. Moreover, for inefficient effort F ¼M by Proposition 1 so that Fs ¼ 0. Alternatively, for efficient effort m½UðgÞs ¼ PðgÞ by Proposition 1 and (34). Total differentiation of both sides w.r.t. s for a ¼ a yields

m½Fs ðts þ 1Þ ¼ Fs :

ð50Þ

At the benefit level s, for which ts þ 1 ¼ 0, this can only be satisfied if Fs ¼ 0. Hence ms ðsÞ o0 requires that daðgÞ=ds o 0 for at least some g-workers. If effort is efficient or second-best, it is defined by (20), respectively, (21), yielding daðgÞ=ds ¼ 0. Alternatively, for intermediate effort levels the sign of daðgÞ=ds follows from (26). The l.h.s. is zero at s and, given the assumption ls ðsÞ 4 0, the r.h.s. is non-positive. Therefore, daðgÞ=dsZ 0 at s and we have a contradiction. It follows that ls ðsÞ r 0. Using this and g^ s ðsÞ Z 0 we have us ðsÞ Z 0 by (27) as we wanted to show. This concludes the proof of ts þ 1 4 0 for all levels of s. (ii) Proof of ls o 0: Above we have already seen that ls r 0 at the benefit level s. This was sufficient to show that ts þ1 4 0, which we will now use to prove the stronger statement that ls o 0 for all levels of s. By contradiction, suppose that ls Z 0 or, equivalently, ms r0, where ms is given by (49). To evaluate this expression, note that Fs Z0 from the above analysis. Specifically, for efficient effort Fs 4 0 from ts þ 14 0 and (50). Moreover, we have to determine the term g^ s . For so U  ðg^ Þ we have g^ s ¼ 0 from the analysis in Section 7.1. Hence, it remains to analyze effort levels s Z U  ðg^ Þ. For the special case of a benefit level s, which leads to ts ðsÞ þ1 ¼ 0, we have already shown above that g^ s ðsÞ Z 0. Using the same proof steps and the result ts þ 1 40, we find that g^ s Z 0 for all levels of  s Z U  ðg^ Þ.26 Moreover, given ts þ1 4 0 and the assumption ls Z 0, from (26) we have da g =ds 40 for all g-workers that exert intermediate effort. Hence, it follows from (49) that ms 40, a contradiction. Therefore, ls o 0. (iii) Proof of g^ s 40 if U  ðg^ Þs r 0. We have just shown that g^ s Z 0. Hence it remains to prove that the inequality is strict if U  ðg^ Þs r0. Remember that for this case the worker and the firm in the g^ -match both receive a rent of zero. Hence 0 Uðg^ Þs ¼ 0 so that from (26) we have daðg^ Þ=ds 40 for intermediate effort levels. Moreover, gf ðaÞB0 ðaÞ o 0 for all a 4a by concavity of gfðaÞBðaÞ. Using this it follows from (43) that the firm’s rent in the g^ -match is strictly decreasing in s so that g^ s 40. Noting that (45) is negative, the same result obtains for efficient effort levels. Finally, suppose that effort is second-best so that daðg^ Þ=ds ¼ 0. Using (44), the g^ -worker’ s rent is strictly decreasing in s; hence also in this case g^ s 4 0. & 26

N

In particular, accounting for ts þ 1 o 0, (44) is still non-positive for da =ds r 0 and (45) becomes negative.

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A.6. Proof of Proposition 4 We adopt the same approach as in Section 7.2. Implicit differentiation of (24) for s ¼ t ¼ 0 and rearranging terms yields   dl ðUsÞ dM mðgj0 B0 Þ ðB0 c0 Þ 1 r þ Z þ l daðgÞ ¼ : ð51Þ dM ðgj0 B0 ÞðB0 c0 Þð1 þ mÞB00 ðUrVu PÞPc00 þðUrVu Þgj00 The denominator is negative so that aM o0 for all g if lM r 0. To see that this is actually the case, we now show that lM 40¼)PM ðgÞ o 0 for all g. However, by (23) the latter implies mM 4 0¼)lM o 0 by the analysis in appendix A5 (simply replace derivatives w.r.t. s by derivatives w.r.t. M). Hence we have a contradiction. In particular, we need to show that for all g that exert intermediate effort

PM ðgÞ ¼ ðgj0 B0 Þ

daðgÞ 1 o0 dM

ð52Þ

if lM 4 0 (if effort is first- or second-best, (52) is trivially satisfied). Substituting from (51), multiplying with the denominator and cancelling common terms yields that PM ðgÞ o 0 iff    ðUsÞlM  1 ð53Þ ðgj0 B0 Þ þ ðB0 c0 Þ ðgj0 B0 Þm 4 B00 ðUrVu PÞPc00 þ ðUrVu Þgj00 : rþZþl The r.h.s. is negative. Noting that B0 c0 4 ðgj0 B0 Þ for inefficient effort, a sufficient condition for PM ðgÞ o 0 is   ðUsÞlM 1 ðgj0 B0 Þðgj0 B0 Þ 4 0, rþZþl which is always satisfied if lM 40. Turning to unemployment effects, from (5) unemployment is decreasing in lðyÞ and increasing in g^ . We have just shown that lM r 0. Next, remember that the critical g^ -match is characterized by either (i) Pðg^ Þ ¼ 0,Uðg^ ÞrVu ðg^ Þ 4 0, or (ii) Pðg^ Þ ¼ Uðg^ ÞrVu ðg^ Þ ¼ 0. In case (i), by Proposition 1 effort follows from g^ f0 ðaÞ ¼ B0 ðaÞ and F ¼M. Due to the binding minimum wage, a marginal change in M reduces Pðg^ Þ so that less workers will be employable. 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