The Risk-Incentive Trade-off in Competitive Search Braz Camargo Paula Onuchic Abstract We use the competitive search framework to model a job market with heterogeneous workers in which there is a moral hazard problem in the employer-worker relation. In this setting, we can predict how contracts react to changes in underlying parameters of the market (in particular, the production risk), as well as how the probability of each type of worker being hired responds. Our main finding is that while at the individual level there is a negative risk-incentive trade-off, general equilibrium effects imply that the effect can be positive at the aggregate level. Our results help shed some light on empirical findings on the risk-incentives trade-off. We thank Felipe Iachan and Klênio Barbosa for their comments and suggestions. Braz Camargo gratefully acknowledges support from CNPq. Sao Paulo School of Economics - FGV. Sao Paulo School of Economics - FGV.
Introduction The trade-off between risk and incentives has been a matter addressed by much theoretical and empirical work on contract theory. Traditionally, the agency framework considers a risk-neutral principal proposing take-it-or-leave-it contracts to a risk-averse agent. The choice this employer makes is between providing incentives to the worker by tying pay to performance and rewarding him for bearing risk. In this context, the theory predicts that higher idiosyncratic variance in the worker s output will make principals propose less incentive pay. With higher risk, performance becomes a noisier estimator of the worker s effort, making bonus pay riskier to the agent, who will demand a better compensation for it. Empirically, this relation has been tested in different settings, such as CEO compensation, agricultural contracts and franchising. However, the results have been inconclusive, often pointing towards a positive rather than negative relation between output risk and incentive pay. These puzzling observations have motivated researchers to look for alternatives to risk-sharing to explain the contract choice by principals, such as transaction costs or task delegation as in Prendergast (00). In the present work, we attempt to reconcile the traditional moral-hazard theory with the empirical findings by modeling the market in a general equilibrium setting, rather than in partial equilibrium as is usually done. We use a competitive search equilibrium framework, as in Moen (997) and Shimer (996), with risk averse workers that are heterogeneous in their risk aversion and homogeneous risk neutral firms. In this setting, we can predict how contracts react to changes in the production risk, as well as how the probability of each type of worker being hired responds. The change in probability of employment may affect the distribution of risk aversion among the workers that are hired and, as the less risk averse workers are associated with higher bonus levels than the more risk averse ones, the distribution of bonus levels in the contracts that are actually celebrated in this economy may also shift. In particular, the average level of bonus among the workers that are hired can change. The main finding of our work is that while at the individual level the negative risk-incentive See Holmstrom and Milgrom (987) See Prendergast (00) for a survey of empirical results.
trade-off traditionally predicted holds, general equilibrium effects that change the distribution of types among the workers that are hired imply that the effect over the average bonus can be positive. We find sufficient conditions on the market search frictions and on the distribution of types that guarantee that the overall effect of increasing production risk on the average bonus in this economy either positive or negative. Considering that to the empirical literature the risk aversion of the workers is not observable to the researcher, it is possible that the effect captured empirically may be the overall effect on the average bonus in the economy and not on the bonus of each individual worker. In this case, our work provides reasoning for the positive relation between production risk and incentive pay that is found without contradicting the traditional theory that considers only the impact in partial equilibrium. Literature This work is related to two strands of the literature. First, it relates to research trying to explain the inconsistency between theoretical results and empirical observation regarding the risk-incentive trade-off. Serfes (005, 008) considers that (i) the risk aversion of the workers is not observable or only partially observable to the empirical researcher and (ii) firms and workers are matched assortativelly according to the levels of risk aversion of each worker and production risk of each firm. As riskier firms are matched with lower risk aversion agents, the overall risk-incentive tradeoff may point in the positive direction when empirical researchers do not control for workers risk aversion. Serfes (005) does not propose an equilibrium model, rather assumes that firms and workers match assortativelly to reach the conclusion. As for Serfes (008), it uses the Shapley and Shubik assignment game to motivate the assortative matching. The drive of the inconsistency between empirical and theoretical results proposed by Serfes (005, 008) is closely related to this work. Alternatively to Serfes, we consider a model set in competitive search equilibrium with only worker heterogeneity. An important difference between our work and Serfes (005, 008) is that we can relate our results to characteristics of the market and workers, such as market search frictions and 3
the distribution of risk aversion. The empirical literature on the matter does acknowledge that it is important to control for heterogeneity in unobservables such as risk aversion. However, most works only use proxies for risk aversion of workers, such as wealth and property. Ackerberg and Botticini (00) show that such approach is not enough to solve the endogenous distribution of risk aversion problem. In particular, the CEO compensation literature (Aggarwal and Samwick, 999, for example) attempts to control for the risk aversion heterogeneity by using individual fixed effects. In fact, in the CEO compensation case, it appears that the risk-incentive trade off is negative, as predicted by theory. A second strand of literature to which this work relates is the literature of information frictions in a competitive search framework. Guerrieri, Shimer and Wright (00) highlights the strength of this modeling in being able to predict both intensive (contract types) and extensive margins (probability of contracting). They consider a general environment with heterogeneous agents and adverse selection. Our equilibrium concept is inserted in their framework, except that we consider types to be observable. The scope of our exercise is quite different from that of Guerrieri, Shimer and Wright (00), though. Since we consider a particular problem, as opposed to the general framework proposed by them, we are able to make comparative statics exercises that are not possible in Guerrieri, Shimer and Wright (00). A few other papers proposed extensions of the competitive search framework to an environment with private information. In particular, Moen and Rosen (006), consider an environment with moral hazard. Their model differs from ours mainly in that they consider workers to be all ex-ante homogeneous and ex-post heterogeneous in productivity rather than risk-averseness. In their environment, firms post contracts that tie wage to production in order to screen out the lowproductivity agents. 3 The Model We first describe the environment and then define equilibria. 4
3. Environment We consider an economy set in a competitive search framework. There exists a measure of workers who are heterogeneous in their risk aversion searching for job vacancies and free entry, at cost k > 0 of homogeneous firms searching for employees. One firm has to match with exactly one worker to produce output. Once there is a match, the production faces a standard moral hazard problem. Firms can join the market by paying entry cost and posting a vacancy, specifying and committing to the contract it is offering to workers. Each unemployed worker observes all the contracts posted in the market and directs his search to any of the vacancies he likes. Each agent directs its search to a single vacancy, but could use a mixed strategy to decide which one. If a match is formed, the posted contract determines the payoff to the worker and to the firm involved in the match. All workers and firms left unmatched receive zero payoff. Preferences Firms and workers face two types of decisions. The first type relates to effort and production once they are engaged in a match. The second type regards the searching behavior, choosing between higher probabilities of matching and better matches. We consider these to be decisions of different natures, as the job search decision is a more long term choice, while the effort decision is a more short term on-the-job choice. To stylize this fact, we allow workers to account differently for each type of decision, by allowing u(w, a) to be the utility function with which the workers account for the first type of decisions, while V ( w) is the one used to account for the second type. Hence, the overall utility of a worker that receives a contract ψ is given by V ( w(ψ)), where w(ψ) is the certainty equivalent wage yielded by contract ψ, given by u( w) = E ψ [u(w, a)] with w being income and a costly effort. We impose u(w, a) = e η(w h(a)), where h(a) = a, to follow the standard moral hazard model, meaning workers are risk averse with coefficient of absolute risk-aversion equal to η, and let V (0) = 0, V ( ) > 0, V ( ) 0. Workers are heterogeneous in terms of η, while they do not differ in terms of V. Risk aversion η (0, η] is distributed according to G(η) and types are observable by all. 5
On the other hand, firms make all the decisions when posting a vacancy, as they already post the contract that will conduct the productive relation. Following the standard approach in the literature, we assume that firms are risk neutral. Production and Contracts Output of a match is given by y = a + ξ, where ξ N (0, σ ) is a shock that affects production. Since effort is costly and unobservable, firms post contracts that motivate workers to exhort effort. As usual in the literature, we restrict these to linear contracts, making workers income equal to w = c + by, where c is the fixed part and b the bonus level (power of the contract). Search Technology Firms post vacancies indexed by (c, b, η), specifying the contract it is committing to and the type it searches for. Because the types are observable, the market is segmented into markets for each type of worker. Inside a market, workers compete with each other for the job vacancies and firms compete with each other to haver their vacancies filled. The degree of competition is captured by the ratio of workers to firms, denoted by λ [0, ] and referred to as queue length. There are search frictions in the sense that even if the number of firms and workers in a market is the same (λ = ), there is a positive probability that buyers and sellers are left unmatched. When a firm faces queue length λ, it matches with probability m(λ). Pairwise matching requires that p(λ) = m(λ)/λ is the probability that a worker finds a match. We assume that the number of matches in a market with u unemployed workers and v vacancies is given by a CES matching function M(u, v) = uv(u r + v r ) r, with r > 0, implying m(λ) = M(λ, ) = λ(λ r + ) r and p(λ) = M(λ, )/λ = (λ r + ) r. The idea that relatively more workers make it easier for a firm to fill a vacancy and harder for a worker to find a job is captured by m(λ) strictly increasing and p(λ) strictly decreasing in λ. The parameter r > 0 measures how frictional the search process is; the lower r, the more frictional the process. 3. Equilibrium We follow the concept of Competitive Search Equilibrium proposed in Moen (997) and Shimer (996). Each contract posted in the market for each type of worker forms a submarket. The payoff 6
of each individual firm or worker depends on which submarket they choose to join and on the queue length that is formed in this submarket, which in turn arises from the decisions of other individual firms and workers. When a type η worker directs its search to a vacancy offering contract (c, b) and queue length λ, he gets the following expected payoff: 3 λ is: W η (c, b, λ) = m(λ) λ V ) ( (c + ( ησ ) b + m(λ) ) V (0) () λ For firms, the expected payoff of posting contract (c, b) for type η workers under queue length J η (c, b, λ) = m(λ) [ ( b)b c ] k () Firms post profit maximizing contracts and earn zero profit because of the free entry condition, which also makes firms indifferent between serving each type of worker. Workers direct their search to the submarkets that maximize their expected payoff conditional on the contracts posted in each of the markets, and on the search behavior of other workers. In equilibrium, a single contract will be posted in the market for each type of worker. However, beliefs about the queue lengths are defined for all possible contracts that are actually not offered in equilibrium by the market utility condition that determines that all contracts that are offered must yield the same market utility to the workers of each type. A competitive search equilibrium is composed of the following equilibrium objects: Ψ η R are the contracts offered in equilibrium for type η workers; the equilibrium expected type η worker payoff, denoted by W η R + ; the functions that give the queue length expected for each contract in the market for each type of worker, λ η : R R +. Definition. Ψ η R, Wη R + and λ η : R R +, η (0, η max ], is a competitive search 3 w(c, b) = c + b η σ is the certainty equivalent wage yielded to the type η worker under contract (c,b). The worker chooses to exhort effort a = b as it maximizes the expected payoff: a = argmax a c + ba b ησ a 7
equilibrium if it satisfies:. Profit Maximization: For all ψ η = (c, b) Ψ η and η (0, η max ], (c, b, λ η (c, b)) solve the problem: max ĉ,ˆb,ˆλ J η (ĉ, ˆb, ˆλ) subject to W η (ĉ, ˆb, ˆλ) W η. Optimal Search: For all η (0, η], { } W η = max 0, max W η (c, b, λ η (c, b)), if Ψ η (c,b) Ψ η W η = 0, if Ψ η = 3. Free Entry: J η (c, b, λ η (c, b)) = 0, (c, b) Ψ η, η (0, η max ] Theorem. A competitive search equilibrium exists and is unique. Proof. Proof is in the appendix. 4 Characterization In this section, we characterize the contracts proposed to each type of worker, as well as the queue lengths formed in the market for them. From the definition of equilibrium, the free entry condition implies that firms are indifferent between serving the market for each type of worker among the types that are served. following problem (P η ): A firm that decides to serve the market for type η workers, solves the max c,b,λ m(λ) [ ( b)b c ] k subject to m(λ) λ V (c + ( ησ ) b ) W η (P η ) Lemma shows that, for each type of worker, there is a single level of b η that may solve (P η ). This bonus level is the same equilibrium bonus found in the partial equilibrium models (Holmstrom and Milgrom (987)), and is decreasing both in the risk aversion coefficient and on the market risk σ. 8
Lemma. When a type η worker is hired, the bonus level celebrated in the contract is b η = +ησ, for all η (0, η]. Proof. Holding c and λ constant, b = +ησ solution (c, b, λ ) to (P η ), it must be true that b = +ησ. solves the constrained maximization. Hence, in any The choice of bonus level completely determines the amount of effort the worker will exhort and hence the expected level of production. In turn, this level of production, jointly with η and σ determine the "surplus" to be split between firm and worker, equal to (+ησ ).4 We can now rewrite problem (P η ) as (P η), where the firm is choosing which level of certainty equivalent w to offer to workers. It can either give more surplus to the worker (in terms of the certainty equivalent yielded), which attracts more workers to the vacancy, or keep a bigger portion of it and get a lower probability of matching. Lemma. For each η (0, η], finding (c, b, λ) that solves problem (P η ) is equivalent to finding ( w, λ) that solve the following problem (P η). Proof. Proof is in the appendix. max w,λ [ ] m(λ) ( + ησ ) w subject to m(λ) λ V ( w) W η k (P η) Free entry condition also determines that firms have an outside option payoff of 0, achieved if they do not post any vacancies. Lemma 3 will show that this implies that workers with too high risk aversion are not served by any firms. Lemma 3. No firms direct search to workers of types η > k kσ := η max. Proof. The probability that a firm matches in a market, m is bounded above by ; while no workers direct search to vacancies that provide w < 0. Hence, the payoff a firm achieves by posting a 4 The surplus here refers to the sum of the expected profit to firms and certainty equivalent to workers, rather that wage to workers. The surplus is lower for workers with higher risk aversion for two reasons. First, these workers exhort less effort in equilibrium and hence have a lower expected production of the match; and secondly, as they are more risk averse, the certainty equivalent of receiving an uncertain wage level is relatively lower. 9
vacancy in the market for type η workers is bounded above by (+ησ ) k. This implies that firms can only achieve negative payoffs by serving markets of workers of types η > η max, which determines that no firms will serve such markets. The solution to (P η), along with free entry, define the equilibrium levels of W, w and λ for the workers that do get served. Lemma 4 gives the condition for the equilibrium queue length λ in the market for each type. Lemma 4. For each worker type that is served by firms, η (0, η max ), the equilibrium queue length λ is unique and satisfies the following condition: H(λ) = V ( ) k (+ησ ) m(λ) V ( ) m(λ)r+ k k( m(λ) r ) = 0 (3) (+ησ ) m(λ) Proof. Proof is in the appendix. The following proposition shows that individuals with higher η will be associated with markets with higher equilibrium queue length, hence lower probability of matching to workers. These more risk-averse workers will also receive a lower equilibrium level of w. The intuition here is that more risk-averse workers produce a lower surplus in a match, making them less attractive to firms, that are now willing to search for them only if there is a higher probability of matching. Proposition. λ(η) is increasing and w(η) is decreasing in η. Proof. Proof is in the appendix. Proposition considers how queue length depends on how risk averse workers are regarding their searching behavior. It shows that when workers are more risk averse, they direct their search towards vacancies that guarantee a higher probability of matching, even when this means a lower certainty equivalent level once matched, implying that equilibrium queue length and certainty equivalent are lower for all types. Proposition. Let V be a CARA utility function, that is V ( w) = e ϕ w, with ϕ being the coefficient of risk aversion of workers relative to their searching decisions. For every worker type η (0, η max ), λ(η, ϕ) and w(η, ϕ) are both decreasing in ϕ. 0
Proof. Proof is in the appendix. Proposition 3 shows that, when the searching process is highly frictional, workers and firms cannot find each other, implying that for a firm to have even a very small probability of matching, the queue length must go to infinity, meaning that the measure of firms entering the market goes to zero. The opposite extreme case happens when the market becomes frictionless. In this case, there is perfect coordination in the market, in the sense that firms with vacancies and unemployed workers are always able to find each other. Hence, there is one firm joining the market for each of the unemployed workers existent and both workers and firms match with certainty. In both the extreme scenarios considered, the level of risk aversion η and market risk σ play no role in determining the probability of matching of the workers. Proposition 3. Let the workers be risk-neutral in terms of their searching behavior, V ( w) = w. Then:. When the market is very frictional, r 0, λ(η) + and p(λ(η)) = m(λ(η)) λ(η) 0, η (0, η max ).. When the market becomes frictionless, r +, λ(η) and p(λ(η)) = m(λ(η)) λ(η), η (0, η max ). Proof. Proof is in the appendix. 5 Average Bonus Knowing the distribution of types G in the economy, and the equilibrium values of λ for each of these types, we can write the distribution of types among the hired workers as: F (η) = η 0 η 0 p(λ(ˆη)) dg(ˆη) p(λ(ˆη)) dg(ˆη) Knowing this distribution and that the equilibrium bonus level of each type is given by b(η) =, we see that the average bonus level across the economy is: +ησ
η b avg = df (η) 0 + ησ If there is an increase in the market risk level, σ, there are two effects on b avg. The first effect, that we call Partial Equilibrium Effect, is the decrease in the equilibrium bonus level for each of the worker types. This effect is the one that was found on the traditional partial equilibrium moral hazard models and will always have a negative impact on b avg. The second effect (General Equilibrium Effect) relates to the change in the distribution of types among the hired workers (F ) resulting from an increase in σ and may be a positive or a negative impact on b avg. The equilibrium condition in Lemma 4 implies that, when σ increases, λ(η) also increases, for all types, hence all types of workers now get a lower probability of matching. These probability changes for each of the types have an impact on the distribution F that may in turn have a positive or a negative effect on b avg. For instance, suppose the probability of match has a much stronger decrease for higher types than for lower types, implying a decrease in F in first order stochastic. In this case, as b(η) is decreasing in η, a the decrease in F in first order stochastic (General Equilibrium Effect) would have a positive impact on b avg. It is possible, in certain circumstances, that this effect outweighs the Partial Equilibrium Effect, making the overall impact on b avg of an increase in σ a positive one. 5. Continuous Distribution of Types We now want to assess circumstances under which the average bonus is increasing or decreasing in the market risk level. First we consider the case where the distribution of types in the economy G is a continuous distribution with differentiable density g. Thus far, we have been indexing the distribution of types, as well as the equilibrium levels of bonus and queue length by the risk aversion coefficient η of the workers. However, we can also write η(b) = b bσ being the type of worker that will get bonus b if hired; and use our model to find λ(b), the queue length that forms in the market for workers that gets bonus b. Writing in this form is useful for our purpose as λ(b) does not vary with σ, as can be seen below in the equation that determines λ(b).
H( λ) = V ( b ) k m( λ) V ( r+ m( λ) b ) k k( m( λ) m( λ) r ) = 0 (4) From G, we can write the distribution of workers in the economy that potentially get bonus equal to or less than b [, ) 5 as 6 : G(b) = G( ) ( k kσ G b ) bσ G ( ), with density g(b) = k kσ g ( ) b bσ b σ G ( ) k kσ The distribution of bonus levels among the hired workers in this economy, in turn, can be written as: F (b) = b p(λ(ˆb)) d G(ˆb) p(λ(ˆb)) dg(ˆb) = b p(λ(ˆb)) g(ˆb) dˆb p(λ(ˆb)) g(ˆb) dˆb Now the average bonus level among the hired workers is the expected value of this random variable b with distribution F. b avg = ˆb d F (ˆb) When there is an increase in the production risk σ, there is no change in λ(b), as discussed above. Hence, the workers that are potentially hired under bonus level b are still hired with the same probability p( λ(b)). However, what changes is which type of worker that is now to earn this bonus level, as when σ increases each type is now associated with a lower bonus. This effect changes the distribution F of bonus among the hired workers by changing the distribution G. On the one hand, each type of worker is now associated with a lower bonus, causing an increase in the measure of workers with lower bonus levels. On the other hand, the workers that were served but had really high risk aversion and were associated with really low 5 Workers with bonus lower than b m in := k are those with η > k kσ that are not served by any firms. On the other hand, the bonus level goes to as η goes to 0. 6 G ( ) k kσ is the total measure of workers that potentially get served in this economy, as the free entry condition guarantees that workers with η > k kσ cannot be searched for. G ( ) b bσ is the measure of workers that get bonus higher that b if hired. 3
bonus levels now cease to be served and are out of the market, causing a decrease in the measure of workers with lower bonus levels. As Proposition will show, the overall direction of this effect can be determined by the characteristics of the distribution of types G in this economy. Proposition 4. Let ɛ g (η) := g (η)η g(η) be the elasticity of the density g.. If ɛ g (η) is strictly increasing in η, then an increase in σ causes F to increase in first order stochastic and b avg to increase.. If ɛ g (η) is constant in η, then an increase in σ causes has no effect on F or b avg. 3. If ɛ g (η) is strictly decreasing in η, then an increase in σ causes F to decrease in first order stochastic and b avg to decrease. Proof. Proof is in the appendix. 5. Two Types Proposition 4 shows that, in many cases when G is a continuous distribution of types, the search frictions in the economy are not important in determining the effect of an increase in production risk on the average bonus level among the hired workers. However, turning to a the case where there are only two types of workers, search frictions now become important. There is a measure α of workers with lower risk-aversion, η, and a measure ( α) of workers with higher risk aversion, η. The expression for b avg in this case is: b avg (σ ) = αp(λ η (σ )) ( +η σ ) + ( α)p(λη (σ )) ( +η σ ) αp(λ η (σ )) + ( α)p(λ η (σ )) As the risk σ increases, the bonus level for the two types decrease, pressuring the average bonus downwards. As for the probabilities of matching, they also decrease for the two types, as the equilibrium queue length in both markets, λ η (σ ) and λ η (σ ), are increasing in σ. When the decrease in probability of matching of the high risk aversion type () is stronger than the decrease for the low risk aversion type (), the proportion of type workers increases among the hired 4
workers. As type workers are associated with higher bonus levels, this puts an upward pressure on b avg. This is shown below. Partial Equilibrium Effect {}}{ db avg (σ ) [ αp(λ dσ η (σ )) + ( α)p(λ η (σ )) ][ αp(λ η (σ ))η + ( α)p(λ ] η (σ ))η ( + η σ ) ( + η σ ) [ ][ + α( α) + η σ p(λ + η σ η (σ )) dp(λ η (σ )) p(λ dσ η (σ )) dp(λ ] η (σ )) dσ }{{ } General Equilibrium Effect As discussed in the characterization session, when the market is completely frictionless (r + ), the probability of matching of the workers is not affected by either the risk aversion level η or the production risk σ 7. This explains the next proposition. Proposition 5. When the market is frictionless, r +, the average bonus among the hired workers is decreasing in the production risk level, dbavg(σ ) dσ < 0. Proof. Proof is in the appendix. When the market is perfectly frictionless, the proportion of each type of worker among the hired workers does not change when there is an increase in the risk σ. This implies that the General Equilibrium Effect does not take place and the overall impact of the increase in risk amounts to the Partial Equilibrium Effect, which always points in the negative direction. In case where the market becomes too frictional, r 0, the probability of matching for either type of worker goes to 0, implying that jobs are not formed and contracts are not established and that it makes little sense to consider the average bonus. When there are search frictions, but in a low enough level, General Equilibrium Effects become important. Proposition 6 shows that, if type workers are sufficiently risk averse, the General Equilibrium Effect is positive and outweighs the Partial Equilibrium Effect. Proposition 6. Let r > and hold η constant at η = η. Then there exists η ( η, k kσ ) high enough such that dbavg(σ ) > 0 and η dσ ( η, k ) low enough such that dbavg(σ ) < 0. kσ dσ 7 We are considering here the case where the workers are risk neutral in terms of their search behavior, that is, V ( w) = w. 5
Proof. Proof is in the appendix. Similarly to the results with a continuous distribution of types, in the case with two types and sufficiently low search frictions, the direction of the effect of increasing σ on the average bonus among the hired workers depends on how the types are distributed, particularly in this case on how high is the risk averseness of the type agent. As η becomes high, the probability that type workers are hired becomes more susceptible to changes in σ. This makes the General Equilibrium effect positive and strong, possibly outweighing the Partial Equilibrium Effect and hence implying that the average bonus increases with σ. On the other hand, as η decreases, the difference between the two types becomes lower, making the General Equilibrium effect weaker. In the limit, as the two types become equal, the impact of σ on b avg is only the Partial Equilibrium Effect, which is negative. 6 Conclusion In the presence of workers that are heterogeneous in terms of their risk aversion, which is unobservable to the empirical researcher, estimates of the risk-incentive trade-off may capture not the effect of increasing risk on the bonus level of each individual worker, but the effect on the average bonus across the distribution of types among the hired workers. My model set in a general equilibrium framework allows me to predict the impact of risk on this average bonus, as well as on the bonus of each worker. We find sufficient conditions for the impact on the bonus of each worker to be negative, as predicted by earlier theoretical work, and the effect on the average bonus positive, as sometimes found empirically. References [] Ackerberg, Daniel A., and Maristella Botticini. "Endogenous matching and the empirical determinants of contract form." Journal of Political Economy 0.3 (00): 564-59. [] Aggarwal, Rajesh K., and Andrew A. Samwick. "The Other Side of the Trade-off: The Impact of Risk on Executive Compensation." Journal of Political Economy 07. (999): 65-05. 6
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[5] Shimer, Robert. "Essays in search theory." Diss. Massachusetts Institute of Technology, 996. [6] Shimer, Robert. "Mismatch." The American Economic Review (007): 074-0. 8
Appendix Proofs of Results Proof of Lemma. From Lemma, we know that b that solves (P η ) is b η = +ησ. Using this, (P η ) becomes: Now using w = c + than c and get (P η). max c,λ [ ] ησ m(λ) ( + ησ ) c k (c + ) (5) ησ W η subject to m(λ) λ V ( + ησ ) ησ, we rewrite the problem again as a problem of choosing w rather (+ησ ) Proof of Lemma 4. As m(λ) is a strictly increasing function of λ and m(λ) λ a strictly decreasing one, the constraint in (P η) must be binding. We can then write w in terms of λ and solve the problem for λ. λ (λ min, + ), where λ min is such that (+ησ ) k m(λ min ) = 0. There is an unique solution for λ as lim H(λ) > 0, H (λ) < 0 and lim H(λ) < 0. λ λ min λ + The First Order Condition and Free Entry give, after simple algebra, the condition H(λ) = 0 in the lemma. Proof of Proposition. Increasing η makes H(λ) higher for every level of λ. In turn, ( as H (λ) < ) 0, we conclude that λ(η) increases in η. As λ(η) incrases, it must be the case that V (+ησ k ) m(λ) ( also increases, which will happen when w(η) = (+ησ ) decreases. Hence, w(η) is decreasing in η. k m(λ) V (+ησ ) Proof of Proposition. Using the CARA utility specification for V, the equilibrium condition H(η) = 0 becomes: H(λ) = ϕ exp{ ϕ ( exp { ϕ ( )} k (+ησ ) m(λ) k (+ησ ) m(λ) )} m(λ)r+ k( m(λ) r ) = 0 (6) (7) gives that the higher ϕ, the lower H(λ) is, for any given level of λ(η) and any η, and hence the lower the equilibrium level of λ for any η (0, η max ). 9
Proof of Proposition 3. Using V ( w) = w, the equilibrium condition now becomes: λ r+ (λ r + ) r+ r = m(λ) r+ = k( + ησ ) (7) From this condition, as r 0, in equilibrium m(λ) and λ, implying also that p(λ). In the second case, as r + the probability that a firm finds a match, m(λ), tends to k( + ησ ) and w 0, while λ +, making the probability that a worker finds a match p(λ) = m(λ) λ 0. Proof of Proposition 4. We are going to prove the first statement of Proposition. The second ad third statements can be proved in a similar fashion. Suppose ɛ g (η) is strictly increasing in η. We want to show that F is increasing (in first order stochastic) in σ, implying that b avg is also increasing in σ. We write d g(b) dσ F (b) = b d F (b) dσ p(λ(ˆb)) g(ˆb) dˆb b min, where g(b) = p(λ(ˆb)) g(ˆb) dˆb b g ( ) b bσ b σ G ( ) k kσ p(λ(ˆb)) d g(ˆb) dσ dˆb p(λ(ˆb)) g(ˆb)dˆb p(λ(ˆb)) d g(ˆb) dσ dˆb b p(λ(ˆb)) g(ˆb)dˆb as κ(b) g(b), where κ(b) = d g(b). We claim (and show at the end of this proof) dσ g(b) that κ(b) is strictly increasing in b. Using this, we conclude that, for all b (, ): d F (b) dσ b p(λ(ˆb))κ(ˆb) g(ˆb)dˆb p(λ(ˆb))κ(ˆb) g(ˆb)dˆb b p(λ(ˆb)) g(ˆb)dˆb b p(λ(ˆb)) g(ˆb)dˆb < κ(b) p(λ(ˆb)) g(ˆb)dˆb p(λ(ˆb)) g(ˆb)dˆb b κ(b) p(λ(ˆb)) g(ˆb)dˆb p(λ(ˆb)) g(ˆb)dˆb = 0 0
d F (b) dσ < 0, for all b (, ) It is easy to see that for b = and b =, d F (b) dσ = 0. Hence, F is increasing (in first order stochastic) in σ, implying trivially that b avg = ˆb d F (ˆb) is also increasing in σ. Now we will show that κ(b) is strictly increasing in b. κ(b) = d g(b) dσ g(b) = σ ( b g b σ bσ g ( b ) = σ [ + ɛ g ( b bσ bσ ) ) + σ G ( ) k k kσ bσ kσ G ( ) k ] k kσ kσ As ɛ g (b) is increasing in b, and b bσ decreasing in b, κ(b) is strictly increasing in b. Proof of Proposition 5. As seen above: db avg (σ ) [ p(λ dσ η (σ )) + p(λ η (σ )) ] [ p(λη (σ ))η [ + + η σ + η σ ( + η σ ) + p(λ η (σ ))η ( + η σ ) ][ p(λ η (σ )) dp(λ η (σ )) p(λ dσ η (σ )) dp(λ η ) dσ ] ] where p(λ) = m(λ) λ p( ˆm) = dm(λ η (σ )) dσ dm(λ η (σ )) lim r + dσ ˆm m ( ˆm) and m(λ η(σ )) = [ k( + ησ ) ] +r = kη [ k( + ησ ) ] r +r + r dp(λ η (σ )) = 0 lim = 0 for η {η r + dσ, η } lim [ = r + db avg dσ Also, lim r + p(λ η(σ )) =, for η {η, η } lim [ p(λ(η )) + p(λ(η )) ] [ p(λ(η ))η r + ( + η σ ) + p(λ(η ] ))η ( + η σ ) ] η ( + η σ ) + η ( + η σ ) < 0
Proof of Proposition 6. Let r > and hold η constant at η = η. p(λ η (σ )) = dp(λ η(σ )) dσ = [ ( k( + ησ ) ) ] r r r+ ( )[ kη ( k( + ησ ) ) r r+ r + ] r r ( k( + ησ ) ) r+ First, i will show that there exists η ( η, k kσ ) high enough such that dbavg(σ ) dσ > 0 (ii) (i) p(λ η (σ )) is decreasing in η and lim η k kσ p(λ η (σ )) = 0 dp(λ η (σ )) dp(λ is decreasing in η dσ and lim η(σ )) = η k dσ kσ (i) and (ii) imply that dbavg(σ ) db is increasing in η dσ and lim avg(σ ) = + η k dσ kσ η ( η, k kσ ) such that db avg(σ ) dσ > 0 Now it is easy to see that, as η η, the general equilibrium effect is lowered and dbavg(σ ) dσ decreases. db avg lim η η dσ lim [ p(λ( η )) + p(λ(η )) ] [ p(λ( η )) η η η ( + η σ ) + p(λ(η ))η ( + η σ ) ] = [ p(λ( η )) + p(λ η )) ][ p(λ( η )) η ( + η σ ) + p(λ( η )) η ( + η σ ) < 0 ] η ( η, k kσ ) such that db avg(σ ) dσ < 0 Proof of Theorem - Existence and Uniqueness of Equilibrium. Let (P η ) be the following constrained optimization problem, with W η (c, b, λ) as defined in () and J η (c, b, λ) as defined in (): max c,b,λ W η (c, b, λ) subject to J η (c, b, λ) 0 (P η )
Now consider the larger problem (P ) of solving (P η ) for all η. In step, we establish that any competitive search equilibrium solves the contained optimization problem (P ) and, in step, i show that any allocation that solves this program is a part of an equilibrium. Then we follow to step 3 and show that (P ) has an unique solution. STEP λ η Let {(Ψ η ) η (0, η], ( W η ) η (0, η], (λ η ) η (0, η] } be an equilibrium allocation with (c η, b η) Ψ η and = λ η (c η, b η). We must prove that, for every η (0, η], (c η, b η, λ η) solves the constrained optimization problem (P η ). First, the free entry condition in the equilibrium definition implies that (c η, b η, λ η) satisfies the constraint. Now, We need to show that any other (c η, b η, λ η ) with W η (c η, b η, λ η ) > W η (c η, b η, λ η) must not satisfy the constraint and hence that (c η, b η, λ η) solves the constrained maximization. Assume by way of contradiction that there exists (c η, b η, λ η) such that W η (c η, b η, λ η) > W η (c η, b η, λ η) and J η (c η, b η, λ η)) 0. Then, by continuity of W η, there exists λ η > λ η such that W η (c η, b η, λ η) W η (c η, b η, λ η) and J η (c η, b η, λ η) > 0, because J η is strictly increasing in λ η. Hence, J η (c η, b η, λ η) > J η (c η, b η, λ η) and (c η, b η, λ η) does not satisfy Maximization of Vacancy Value, the first equilibrium condition. (CONTRADICTION) STEP Take (c η, b η, λ η ) η (0, η] that solves the program (P ). Define Γ := {(Ψ η ) η (0, η], ( W η ) η (0, η], ( λ η ) η (0, η] } such that Ψ η = {(c η, b η )} and W η = W η (c η, b η, λ η ) for each η (0, η]. Let λ η (c η, b η) be such that W η (c η, b η, λ η (c η, b η)) = W η for any (c η, b η) R. Condition ( - Optimal Search) of equilibrium is satisfied trivially by Γ. Assume by way of contradiction that condition (3 - Free Entry) is not satisfied, that is, that J η (c η, b η, λ η ) 0 for some η (0, η]. Then either J η (c η, b η, λ η ) < 0 which contradicts the fact that (c η, b η, λ η ) satisfies the constraint, or J η (c η, b η, λ η ) > 0. In this case, continuity of J η implies that there exists λ η < λ η such that J η (c η, b η, λ η) 0 and W η (c η, b η, λ η) > W η (c η, b η, λ η ), which contradicts the fact that 3
(c η, b η, p η ) solves problem (P η ). Now we still need to show that condition (- Profit Maximization) is satisfied. Assume by contradiction that, for some η (0, η], there exists (c η, b η, λ η) such that W η (c η, b η, λ η) W η and J η (c η, b η, λ η) > J η (c η, b η, λ η ). W η being strictly decreasing in λ η and continuity of J η imply that there exists λ η < λ η such that W η (c η, b η, λ η) > W η = W η (c η, b η, p η ) and J η (c η, b η, λ η) J η (c η, b η, λ η ) 0. This contradicts the fact that (c η, b η, λ η ) solves the problem (P η ). STEP 3 Lemmas, and 4 show that problem (P ) has a unique solution. 4