Competition and Moral Hazard

Competition and Moral Hazard Shingo Ishiguro Graduate School of Economics Osaka University February 2004 Abstract This paper investigates the equilibrium consequences of a contractual market with moral hazard where multiple principals compete each other to offer incentive contracts to agents who choose unobservable ex post actions. We employ the approach of directed search and provide the robust predictions for the limit equilibrium as competition becomes sufficiently intense. In particular we show that simple contracts emerge as equilibrium outcomes in highly competitive environments: First, when the trade off between incentive and risk sharing causes the moral hazard problem on the side of agents, the full insurance contract, which makes the payments to agents constant across all state realizations, can be a unique limit symmetric equilibrium as the number of competing principals becomes sufficiently large. Thus the equilibrium contract can be uniquely characterized by the low-powered incentive scheme in the limit. On the other hand, when agents are risk neutral but the moral hazard problem stems only from their limited wealth, the first best contract which attains the maximum social welfare can be a unique limit equilibrium. JEL Classification Numbers: D80, D82 Keywords: Competition, Directed Search, Limited Liability, Moral Hazard, Risk Sharing, Unique Limit Equilibrium I am grateful to seminar participants at Contract Theory Workshop (CTW) and Kobe University for useful comments and discussions. I would also like to thank Hideshi Itoh, Hideo Konishi, Tadashi Sekiguchi and Hideo Suehiro for valuable comments. Of course all remaining errors are my own responsibility. I also acknowledge a financial support from Grant in Aid for Young Scientists (No.15730119) by Japan Society for the Promotion of Science. Correspondence: Shingo Ishiguro, Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560 0043, Japan. Phone: +81-6-6850-5220. Fax: +81-6-6850-5256. e-mail: ishiguro@econ.osaka-u.ac.jp 1

1 Introduction This paper studies the equilibrium consequences of a contractual market with moral hazard where multiple principals compete each other by offering incentive contracts to attract agents who choose unobservable ex post actions after contracts are signed. In particular we will address the issues of what incentive contract emerges in the limit equilibrium as the number of competing principals becomes sufficiently large and of how the efficiency loss of the market caused by the moral hazard problem tends to be decreased or increased as market competition is sufficiently intense. In this attempt we will provide the robust predictions for the limit equilibrium. In particular we will show that simple contracts emerge as equilibrium outcomes in highly competitive environments. There are many relevant examples about the contractual markets with two features mentioned above: the moral hazard problem on the side of agents who accept contracts and the competition among principals who offer contracts. For example consider a labor market where many firms compete each other to attract workers by offering wage contracts. The workers may choose unobservable efforts after they are hired. In loan markets lenders offer loan contracts to borrowers who choose some unobservable actions (e.g., project choices) after contracts are signed. In insurance markets insurance companies are engaged in competition for attracting insured agents whose ex post actions affect their probabilities of facing accidents. The equilibrium contract forms emerged in those contractual markets depend on how many principals and agents participate in the market and how they find trading partners as well as what agency problems arise in each principal agent relationship. These are in contrast with the standard bilaterally monopoly agency model in which one principal is assumed to contract with agents. Our model is based on the so called directed search model in which principals simultaneously offer contracts and then each agent independently selects one principal to visit, given an offered contract profile. Since we suppose that each principal needs at most one agent, some agents may be rationed in equilibrium. Our focus is then on the symmetric subgame perfect Nash equilibria (SSPE) in which all principals offer the same contract and all agents play the same mixed strategy to visit each principal with equal probability, given an equilibrium contract profile. It is known that this class of equilibria has some desirable properties such as continuity and robustness to small perturbation of the model, which are the reasons we will confine our attention to these equilibria. 1 Our main result is that whether or not the SSPE contract becomes high powered and mitigates the moral hazard problem when the number of competing principals becomes large enough depends on the source of 1 See Burdett, Shi and Wright (2001) for more detailed discussion on this issue. 2

the efficiency loss by the moral hazard problem. If the efficiency loss stems from the standard trade off between incentives and risk sharing, the limit equilibrium becomes the full insurance contract which makes the payment to agents constant across all state realizations. This in turn implies that the efficiency loss of the market may be increased by tighter competition. However, if the moral hazard problem is due to only the limited wealth of risk neutral agents, the limit equilibrium becomes the first best contract which attains the maximum social welfare of the market. The moral hazard problem can be thus fully resolved in the limit. The basic intuition behind the above two contrast results is as follows: A highly competitive market may force the competing principals to offer a contract which ensures a large equilibrium rent to the agents. Such high rent in turn changes the equilibrium action choice of the agents in the different ways, depending on whether or not risk sharing matters: All hired agents are motivated to choose the least costly action in the limit when they are risk averse while they are induced to choose the first best action in the limit when they are risk neutral but wealth constrained. In the former case, it becomes too costly for the principals to impose any risk on the risk averse agents when they must be guaranteed a large equilibrium rent. Thus intense competition among principals results in lower powered incentives. However, in the latter case, large equilibrium rent helps to make the limited liability constraint slack and hence derive the correct action choice from the risk neutral agents. This effect can work to resolve the moral hazard problem by limited liability. Recently several papers have focused on competing mechanism design (see Inderst (2001), McAfee (1993) and Peters (1997, 2001)). These papers have mainly analyzed the issues of adverse selection where private information arises ex ante before mechanisms are offered. McAfee (1993) examined a dynamic model in which many sellers compete each other to offer general price quantity schedules to buyers who have private information about their own values on the goods and showed that holding the second price auction by each seller can be a stationary equilibrium. 2 Inderst (2001) considered a matching model in which agents have ex ante private information and both principals and agents have bargaining powers to make contract offers. Inderst showed that the distortions by adverse selection disappear when the market frictions become sufficiently low. The common feature of these papers is to uncover the fact that some simple mechanism like the second price auction becomes an equilibrium in competitive environments, although more complex mechanisms are derived in a bilaterally contractual model (with one principal and one agent). 3 As in 2 Peters (2001) extends the model of McAfee (1993) to allow agents private information to be correlated. 3 See also Armstrong and Vickers (2001) and Rochet and Stole (2002) for related issues in oligopoly models with price discrimination. 3

these papers, we also show that some simple contract like the full insurance contract emerges as a limit equilibrium. However, our results are different from theirs. First, we address the issue of moral hazard but not adverse selection which has been extensively analyzed in the literature. Second, we show the importance of risk attitude and limited wealth, both of which play the central role to determine whether or not equilibrium contract becomes high powered and hence market efficiency is enhanced when market is highly competitive. Third, and this is a main motivation of the paper, we provide the robust predictions for the limit equilibrium by showing the uniqueness of SSPE. Our efficiency result in the presence of risk neutral agents with limited liability is also related to Inderst (2001) in the sense that the endogenization of agent s reservation utility 4 is verified to be the key to eliminate the distortions by information asymmetry between contracting parties. However, our model has the different structure from Inderst: We adopt the directed search model with the timing that principals make ex ante contract offers before they match with agents, while Inderst examines a matching model with adverse selection in which ex post contract offer is assumed to make after a principal and an agent matched. Second, we suppose that full the bargaining power to make contract offer is allocated to the side of principals, while in Inderst s model both sides of principals and agents are allowed to make contract offers with positive probability. 5 Following the above preceded literature, we assume the existence of contract constraint, which ensures that each agent can contract with at most one principal. There is also another strand of the models to deal with multiple principals, by allowing each agent to accept more than one contract. This is called the problem of common agency, in which case some externality arises among competing principals. These studies include, for example, Kahn and Mookherjee (1998), Parlour and Rajan (2001) and Segal (1999). The most suitable case for our model with contract constraint to be applied is the labor market where it is naturally assumed that each worker can contract with and work under at most one firm. Of course, our analysis is applied to other cases where some transaction costs prevent agents from contracting with multiple principals simultaneously. Furthermore, we assume that each principal can contract with only fixed number of agents (at most one in our model). This is also justified when there exist some transaction costs or when the principals have inelastic demands. For example, in loan markets each lender finds it too costly to contract with multiple borrowers because of high monitoring and bankruptcy costs. 6 In labor markets each firm may 4 The reservation utility we mentioned here is referred to as the value each agent expects to obtain if he or she switched the trading partner. 5 As noted in the paper, Inderst s result does not hold if these two assumptions are dropped. 6 See Parlour and Rajan (2001) for this issue. 4

have inelastic labor demand due to capacity constraint for its production. Both the contract constraint and the assumption that each principal can contract with a fixed number of agents endogenously determine the bargaining power of the principals relative to the agents through competition among them. The remaining sections are organized as follows. In Section 2 we set up the model. In Section 3 we will consider the case that agents are risk averse and the moral hazard problem arises due to the trade off between incentives and risk sharing. In this case we will show that the full insurance contract tends to be a unique limit equilibrium. In Section 4 we will move to the case that agents are risk neutral but protected by limited liability. As opposed to the case of risk averse agents, the first best contract becomes a unique limit equilibrium. Section 5 concludes the paper. 2 The Model 2.1 Contractual Market with Moral Hazard We consider a contractual market where n (n 2) principals compete each other to offer incentive contracts to m (m 1) agents. All principals have the identical preference and each of them has a capacity constraint that she 7 needs at most one agent to generate her profits. Each agent can also contract with at most one principal. The agent hired by some principal takes an action a A after a contract is signed. This action gives the principal who hires that agent a stochastic return y Y R, where Y is some compact subset of real space. We denote y min Y and assume y > 0. The returns y are independently distributed across different hired agents. Moreover, no production externalities exist so that the return each hired agent yields depends only on his action but not other agents actions. We will denote by E[ a] the expectation with respect to y conditional on the agent s action a. Each principal is risk neutral and obtains the profits by hiring an agent, y w, where w is the payment to the agent. Every agent has the same von Neumann and Morgensern utility function U(w, a) u(w) c(a) on his income w and action a. We assume that u is continuously differentiable, increasing and concave, i.e., u > 0 and u 0. Let a denote the least costly action, i.e., a arg min a A c(a) and assume c(a) >c(a) for all a a. Byφ we will denote the inverse function of u. φ > 0 and φ 0 follow from the assumptions on u. We also make the assumption that φ( ) <E[y a] < 7 If there exists unlimited capacity, then we can easily see that the Bertrand competition among the principals drives their equilibrium payoff down to the reservation payoff as long as n 2. Then the change of the number of competing principals does not affect the equilibrium contract at all. 5

φ(+ ). 8 The reservation payoffs of the principals and the agents are normalized to zero. Any agent can always guarantee this payoff by exiting from the market. Any principal can also ensure her reservation payoff by offering the null contract (nothing). The game will proceed with the following timing: Stage 1 In the first stage the principals simultaneously offer incentive contracts to the agents. Each principal is allowed to offer the null contract which guarantees the reservation payoff to herself. Stage 2 In the second stage the contracts offered in Stage 1 are publicly observable. Then each agent independently visits one principal, from whom he wants to accept the offered contract. In this stage any agent can exit from the market to obtain the reservation payoff. Stage 3 Each principal who has succeeded in attracting at least one agent will select one and only one agent and sign the contract with that agent. If more than one agent go to some principal, then the principal is assumed to select one agent randomly with equal probability. Stage 4 Finally, in each principal agent relationship the selected agent chooses an action a A, the final return y Y is realized and the payment to the agent is made according to the signed contract. In this stage the action each agent has taken is observable only to himself, but not to all principals and other agents. Throughout the paper we will assume that only the realizations of y Y each hired agent yields are verifiable. This means that agents action choice is subject to the moral hazard problem. This assumption and independency of the realized returns across different hired agents imply that the incentive contract a principal offers depends solely on the realization of the final return y generated by the agent who is hired by that principal. The contract thus must specify the payments to the agent w(y) contingent on the return he will yield y Y. Let also u(y) u(w(y)) denote the corresponding utility payment according to w(y). The equilibrium concept to be used is the subgame perfect Nash equilibrium. In particular we will confine our attention to the symmetric subgame perfect Nash equilibrium (SSPE) where all principals offer the same incentive contract and all agents use the same strategy. The details of the strategies taken by the principals and the agents will be described below. 8 This is satisfied when φ varies over (, + ) and E[y a] < +. 6

2.2 Selection Strategy and Continuation Equilibrium We define the strategies of the principals and the agents at each stage of the game as follows: Stage 1 (Contract Offer of Principals): Principal i will offer a contract C i {a i,u i (y)} which specifies the action to be taken a i by the hired agent and the utility payment u i (y) depending on the realization of y. Letting denote the null contract, we will add to the above contract offer, i.e., C i = {a i,u i (y)}. Recall that offering C i = gives principal i zero payoff. Moreover, we will here assume that possible contracts are restricted to be anonymous in that they do not depend on the identities of the agents. Let C be the set of all possible contracts each principal can offer. Stage 2 (Selection Strategy of Agents): Each agent chooses the probability to visit each principal, given a profile of contracts offered in Stage 1, C (C 1,...C n ). The selection strategy used by agent j is defined by the mapping z j : C n [0, 1] n, where z j (z 1j,..., z nj ) and i z ij 1. The interpretation of z ij is that agent j selects principal i to visit with probability z ij. Note also that z ij = 0 for all i when agent j decides to exit from the market. The agent who is selected by principal i will obtain the rent U i E[u i (y) a i ] C(a i ) in Stage 4, because the contract C i is incentive compatible and hence the agent will choose the specified action a i. Assuming that a contract profile C was offered in Stage 1 and other agents than j use the selection strategy z j (z 1,..., z j 1, z j+1,..., z m ), the expected payoff of agent j who uses the selection strategy z j is given by n EU(z j) z ij Prob{j is hired by principal i; z j}u i (1) i=1 where Prob{j is hired by principal i; z j } denotes the probability that agent j is selected by principal i when other agents use the selection strategies z j. With this probability agent j is hired by principal i and will obtain the rent U i after he chooses the specified action a i and the payment u i (y) is made according to the realization of y in Stage 4. Note also that agent j obtains the reservation payoff (zero) if he is not hired at all. We will see below how the probability Prob{ }is determined. Principal i obtains the following expected payoff when she offers a contract C i in Stage 1: EV (C i ) (1 Π m j=1(1 z ij ))E[y φ(u i (y)) a i ]. (2) In the above expression principal i can obtain the expected payoff E[y φ(u i (y)) a i ], which will be ensured provided the hired agent takes the specified action a i and φ(u i (y)) is paid according to the realization of y in Stage 7

4, if and only if at least one agent visits her, which occurs with probability (1 Π m j=1 (1 z ij)). First we define the continuation equilibrium in the subgame after a contract profile C is given. Definition. The continuation equilibrium consists of the selection strategy profile z =(z 1,.., z m ) and the action choice by hired agents as follows: (i) Given C, agent j chooses z j to maximize his own expected payoff (1), taking as given the selection strategy used by other agents z j, and (ii) After all agents have made their selection decisions, the agent who was hired by principal i chooses an action a A to maximize ex post expected payoff E[u i (y) a] c(a), provided he has accepted the contract C i. Let ẑ(c) (ẑ 1,..., ẑ m ) denote a selection strategy profile in the continuation equilibrium for a given contract profile C. Then, by taking into account this continuation equilibrium outcome, the expected payoff of principal i (2) is reduced to EV (C i )=(1 Π m j=1(1 ẑ ij (C)))E[y φ(u i (y)) a i ]. (3) Then we define the SSPE (symmetric subgame perfect Nash equilibrium) as follows: Definition. A contract Ĉ = {â, û(y)} constitutes a SSPE if the following conditions hold: (i) Each principal maximizes her expected payoff (3) by offering the contract Ĉ in Stage 1, taking as given the strategy of other principals offering the same contract Ĉ, and (ii) Given Ĉ (Ĉ,..., Ĉ), every agent selects any principal with equal probability in the continuation equilibrium: ẑ ij =1/n for all i and all j. Remark. There may be also asymmetric continuation equilibria followed the offer of the same contract Ĉ in Stage 1, where the agents select the principals with different probabilities. We will rule out such asymmetric equilibria in order to make the analysis simple and derive clear cut results. 9 To see what contract becomes a SSPE, it is useful to consider how a unilateral deviation by one principal changes the continuation equilibrium outcomes. Suppose that all but one principal, say i, offer the same incentive 9 The approach to focus only on symmetric equilibria is standard in the existing literature. See for example McAfee (1993). See also Burdett, Shi and Wright (2001) for more discussion on the robustness feature of this class of symmetric equilibria. 8

contract Ĉ {â, û(y)}. On the other hand, principal i offers a different contract C = {a, u(y)} Ĉ. If such deviation is not profitable, then the contract Ĉ becomes a SSPE. We will denote by Û the equilibrium rent every hired agent obtains in a SSPE in which Ĉ is offered: Û E[û(y) â] c(â). (4) For the time being suppose that Û > 0. We will see below that this is actually the case occurred in any SSPE. Let us consider the subgame in Stage 2 after principal i makes a deviation contract offer. To derive the continuation equilibrium in the subgame after unilateral deviation occurs, we will confine our attention to the symmetric continuation equilibrium in which all agents use the same strategy to select the principals: When it is not optimal to exit from the market, every agent selects the deviating principal i with probability z [0, 1] and any other principal with the same remaining probability (1 z)/(n 1). This restriction gives us some desirable properties of the continuation equilibrium and hence makes the analysis of SSPE much simpler. As we will see below, the selection probability z determined in the symmetric continuation equilibrium as defined above is continuous in the rent the deviating principal offers: It can be taken close to the equilibrium selection probability 1/n when such rent becomes close to the equilibrium one. Moreover, z will be verified to be increasing in the ratio between the rent the deviating principal offers U and the equilibrium one Û, i.e., U/Û. Thus the deviating principal can continuously raise the probability of being selected by at least one agent in the continuation equilibrium by offering higher rent than the equilibrium one. Given the above selection strategy, the probability that the agent who selects the deviating principal with certainty can be hired by that principal is given by ( ) m 1 m 1 G(z) z k (1 z) m k 1 1 k k +1. (5) k=0 This probability is calculated by using the symmetric property that all other agents select the deviating principal with the same probability z and one agent is hired by that principal with equal probability 1/(k + 1) when k other agents go to that principal. On the other hand, the probability that the agent who selects other principal than the deviating principal with certainty can be hired is given by G((1 z)/(n 1)). Thus the selection probability z is determined in the symmetric continuation equilibrium when 0 <z<1, as follows: ( ) 1 z G(z)U = G Û (6) n 1 9

where U E[u(y) a] c(a) denotes the rent the deviating principal offers under a deviation contract C. The above equality means that every agent is indifferent for selecting between the deviating principal and any other principal, provided that all other agents select the deviating principal with probability z and any other principal with the remaining probability (1 z)/(n 1). The left hand (resp. right hand) side means the expected payoff obtained by selecting the deviating principal (resp. non deviating principal) with certainty. When (6) holds, the agent s expected payoff (1) is rewritten as EU = G(z)U = G((1 z)/(n 1))Û. The following lemma is useful to derive the continuation equilibrium. Lemma 1. (i) For any z 0, G(z) can be rewritten as or (ii) G < 0, and (iii) G > 0. Proof. See Appendix. G(z)= 1 (1 z)m mz (7) m 1 G(z) =(1/m) (1 z) k, (8) k=0 By defining the following function ( ) 1 z f(z) G /G(z), (9) n 1 (6) can be then rewritten as U = f(z)û. (10) The following lemma shows that f is increasing and convex. This ensures that the symmetric continuation equilibrium z followed unilateral deviation is uniquely determined by f(z) =U/Û, and it is increasing in U/Û. Thus the deviating principal can increase her probability of being selected by the agents by offering higher rent U than the equilibrium one Û. Lemma 2. (i) f > 0, (ii) f > 0 and (iii) f(1/n) =1. Proof. See Appendix. The properties of Lemma 2 will be exploited below for characterizing SSPE. 10

3 Equilibrium Contracts with Risk Averse Agents 3.1 Characterization of SSPE This section will characterize SSPE when agents are risk averse, u < 0 and so φ > 0. We will first consider the benchmark case where only one (monopoly) principal exists in the market and has full the bargaining power to make a contract offer to the agents. Since a principal needs at most one agent to be hired, this is the standard agency problem with moral hazard. The monopoly principal thus solves the following problem by offering a contract {a, u(y)}: (MP) subject to max a,u(y) E[y a] E[φ(u(y)) a] a arg max a A E[u(y) a ] c(a ) E[u(y) a] c(a) 0 (IC) (IR) where (IC) means the incentive compatibility constraint that the hired agent optimally chooses the action a specified in the contract. (IR) guarantees that the agent obtains at least his reservation utility, zero. Let C M {a M,u M (y)} be the optimal monopoly contract to solve the above problem (MP). We make the following assumption which avoids the trivial result that the principal offers the null contract C = even when she is a monopolist in the market. Assumption 1. E[y φ(u M (y)) a M ] > 0. We will now turn to the case of multiple principals and characterize SSPE. First notice that every SSPE never contains the offer of the null contract C =. To see this suppose that it becomes a SSPE for all principals to offer C = in Stage 1. Then all principals and agents obtain the reservation payoff, zero. But then some principal deviates from the equilibrium strategy of offering the null contract to the offer of a contract C = {a M,u M (y)+ɛ} where ɛ>0 is small. Thus the deviating principal yields a positive rent ɛ>0 to the hired agent. It then becomes a dominant strategy for any agent to select the deviating principal with certainty. 10 This deviation hence 10 If the agent is hired by the deviating principal, he can obtain a positive rent ɛ>0 and thus be strictly better off, and even if this is not the case he is not worse off. Since the first case occurs with some positive probability, selecting the deviating principal becomes a dominant strategy for each agent. 11

makes the deviating principal strictly better off by Assumption 1. Thus we can suppose that every SSPE contract Ĉ must not be the null contract (Ĉ ). Moreover, the equilibrium payoff of the principals must be greater than the reservation payoff, zero, in any SSPE. To see this suppose that the equilibrium payoff of the principals becomes zero in some SSPE. Suppose also that some principal deviates from the equilibrium contract Ĉ = {â, û(y)} to the offer of a contract C = {â, û(y) ɛ} where ɛ>0 is small. 11 Thus the deviation contract gives the agent who accepted this a smaller rent than the equilibrium one by ɛ>0, which in turn means that the deviating principal can increase her payoff from zero when she succeeds in attracting at least one agent. Taking a small enough ɛ, the probability of the deviating principal being selected by at least one agent z in the continuation equilibrium can be sufficiently close to the equilibrium one 1/n. Thus the deviating principal can obtain a positive expected payoff by such deviation. This argument is based on the continuity property of the symmetric continuation equilibrium we have defined above. The following result states that any SSPE is characterized by a solution to the problem as if a single principal contracted with a single agent as in the standard bilaterally monopoly model. Proposition 1. Suppose that Ĉ = {â, û(y)} is a SSPE and every hired agent obtains the equilibrium rent Û. Then Ĉ must be a solution to the following program (EP): (EP) max E[y a] E[φ(u(y)) a] a,u(y) subject to (IC) and E[u(y) a] c(a) Û. (IR ) Proof. See Appendix. Proposition 1 shows the condition any SSPE must possess, implying that the equilibrium contract must solve the simple problem (EP) as if we were in the single principal case where the reservation utility of the agent is given by Û. The intuition of this proposition follows from its proof: For Ĉ to be a SSPE, it must not be profitable for any principal to offer other contracts satisfying (IC) and (IR ) than Ĉ giving the rent Û, provided that all other principals offer the same contract Ĉ. If a different feasible contract C Ĉ attains a higher value of the objective function in the program (EP), some principal makes a deviation to offer C instead of Ĉ, because C gives the hired agent at least the equilibrium rent Û and thus the probability of the deviating principal being selected cannot be less than the equilibrium one 11 Note that the above argument shows Ĉ. 12

1/n, which in turn implies the strict improvement of the deviating principal s payoff. The equilibrium rent Û can be here viewed as the reservation utility of the agents when the deviating principal makes a contract offer (Of course Û itself will be endogenously determined in the market competition). Proposition 1 will greatly simplify the following analysis. Notice also that Û>0must be satisfied in any SSPE. This is easily seen because if Û = 0 in some SSPE there exists some principal who deviates from the equilibrium strategy to offer a modified contract which gives the hired agent a small positive rent. Then it becomes a dominant strategy for each agent to select such deviating principal with certainty. This creates a discrete jump of the deviating principal s expected payoff, which breaks the supposed SSPE. 12 Thus we will hereafter suppose that Û>0. Anticipating the selection strategy used in the symmetric continuation equilibrium after the unilateral deviation, the deviating principal offers a contract C = {a, u(y)} to solve the following program: (DP) max 0 z 1,a A,u(y),U 0 (1 (1 z)m ){E[y a] E[φ(u(y)) a]} subject to a arg max a A E[u(y) a ] C(a ), U = f(z)û, U = E[u(y) a] c(a). (IC) (ID) (R) Several comments on the above problem are in order. First note that at least one agent goes to the deviating principal with probability (1 (1 z) m ), in which case the deviating principal obtains the expected payoff E[y a] E[φ(u(y)) a] and nothing otherwise respectively. This yields the expected payoff function of the deviating principal given in (DP). Second, in the program (DP) the deviating principal is assumed to control directly the probability z [0, 1] for herself to be selected by the agents. This is because choosing the contract {a, u(y)} determines the agent s rent U, which in turn 12 For example consider a deviation contract such that C = {û(y) +ɛ, â} by adding a small constant ɛ > 0 to the equilibrium contract Ĉ = {û(y), â}. Suppose also that the equilibrium contract Ĉ gives all hired agents zero rent Û = 0. Then Proposition 1 tells us that such SSPE must solve the problem (EP) with Û = 0. However, then (EP) is reduced to the monopoly problem (MP) and thus Ĉ = C M. Thus every principal obtains the positive expected payoff (1 (1 1/n) m )E[y φ(u M (y)) a M ] > 0 by Assumption 1. However, the above deviation contract still implements the same action and gives a hired agent some positive rent ɛ>0. Letting ɛ +0 the deviating principal can thus increase her expected payoff in a discontinuous way, because the probability that she is selected by at least one agent becomes one instead of the equilibrium probability (1 (1 1/n) m ). 13

determines the selection probability z in the continuation equilibrium by the indifference condition U = f(z)û. Thus choosing z subject to U = f(z)û is equivalent to choosing the rent U. 13 Thus any SSPE can be derived by finding the value of equilibrium rent Û such that the deviating principal optimally chooses z =1/n in the above program (DP). 3.2 Market Competition and Low Powered Incentive We now turn to the problem of what contract can be a SSPE when the market competition becomes sufficiently intense as the number of the principals goes to infinity, given the number of the agents m. Let U P be the agent s utility to satisfy E[y a] φ(u P + c(a))=0. (11) Since φ is increasing and φ( ) <E[y a] <φ(+ ), such U P exists. We will also impose the following assumption. uniquely Assumption 2. φ (c(a)) < sup a a E[y a] E[y a] c(a) c(a) <φ (U P + c(a)). (12) Intuitively, the first half of Assumption 2 shows the necessary condition to avoid the trivial case that implementing the least costly action can be optimal even in the monopoly principal case. If the reverse inequality holds in the first half, it is then verified that implementing the least costly action a becomes the optimal solution to the problem (MP). Note also that E[y a] φ(c(a)+û) is the maximum payoff attained in (EP) for a given action a A. 14 The last half of Assumption 2 then says that the marginal return of this payoff by the increase of action from the least costly one, which corresponds to the second term of Assumption 2, can be less than its marginal cost, which corresponds to the final term, when the agent s rent is given by U P. Thus Assumption 2 guarantees the existence of the agent s rent, for above which the principal optimally implements the least costly action. 13 We can here restrict our attention to z (0, 1] such that z is determined by the indifference condition U = f(z)û. No generality is lost by such restriction: First, inducing z = 0 is weakly dominated by the offer of the null contract. Second, suppose that the deviating principal induces the selection strategy z = 1 such that every agent selects her with probability one. This case occurs when U f(1)û. However, if this inequality is strict, i.e., U > f(1)û, the deviating principal can slightly reduce the rent U while keeping U>f(1)Û and still inducing z = 1. Thus we can assume that U = f(1)û. 14 This is actually the payoff the principal can attain in (EP) without (IC). 14

Assumption 2 ensures that U p > 0 because φ > 0. Under Assumption 2 we can also find the value U>0 such that sup a a E[y a] E[y a] c(a) c(a) = φ (U + c(a)). (13) It then follows that U<U P. Let A (U) be the set of solutions to maximize the principal s first best payoff: A (U) arg max E[y a] φ(u + c(a)), (14) a A for given U. Then we show the following result. Lemma 3. U U, A (U) ={a} under Assumption 2. Proof. Fix U U where U is given by equation (13). Then, the following inequality holds for any a a: E[y a] E[y a] φ (U + c(a))[c(a) c(a)] < φ(u + c(a)) φ(u + c(a)) where the first inequality follows from Assumption 2 and c(a) >c(a) for any a a, and the final inequality from the strict convexity of φ respectively. The above inequality shows that A (U) ={a} for any U U. Q.E.D. By Lemma 3 we will see that the equilibrium contract must ensure constant payment across all realizations of returns. Let C denote such contract, called the full insurance contract, and define as follows: C {a,u} where u c(a)+û for the rent Û giving to the agent. Thus C specifies the least costly action to be taken and the corresponding fixed payment u, which covers both the action cost and the rent, to be made for all realizations of y. Then we can show the following result. Lemma 4. (i) The full insurance contract C uniquely solves the program (EP) when Û U. (ii) A SPPE must be the full insurance contract C when its equilibrium rent Û belongs to the interval [U,U p]. Proof. (i) Fix some Û U. Then we obtain E[y a] E[φ(u(y)) a] E[y a] φ(e[u(y) a]) E[y a] φ(û + c(a)) 15

for any feasible contract {a, u(y)} which satisfies (IC) and (IR ) in (EP). The first inequality follows from Jensen s inequality and φ > 0, and the second inequality from (IR ) respectively. Then, by Lemma 3, the least costly action a maximizes the last expression of the above inequality for Û U. Since the full insurance contract C where u c(a)+û for all y Y can attain this upper bound, the unique optimal solution to (EP) must be C. (ii) Suppose that the equilibrium rent Û is given by Û [U,U p] in a SSPE. Then by Proposition 1 and the above proof of (i) the SSPE must be the full insurance contract C. Note here that the offer of the null contract is not optimal strategy for the principal. Indeed this is the case because by definition of Û [U,U P] we have E[y a] φ(û + c(a)) > 0. Also, since Û>U>0, any agent does not exit from the market as well. Q.E.D. When the reservation utility Û is large, under Assumption 2 the principal finds it optimal to implement the least costly action in the program (EP) via the full insurance contract. Proposition 1 then implies that any SSPE must be the full insurance contract for sufficiently large Û. The following lemma is useful. Lemma 5. ˆn such that G (1/n)/(n 1) is increasing in n for n ˆn. Proof. See Appendix. Then the next proposition shows that the equilibrium rent Û belongs to the interval [U,U P ] when n becomes large enough and hence there exists a unique SSPE characterized by the full insurance contract C. Proposition 2. There exists some n such that for all n n a unique SSPE is given by the full insurance contract C. Proof. See Appendix. The above proposition implies that the equilibrium contract tends to be the lowest-powered in the sense that all hired agents are fully insured when market competition becomes sufficiently intense. The proof of this result consists of two parts: existence and uniqueness. First we will show that there exists a SPPE in which the full insurance contract C is offered when the number of competing principals becomes sufficiently large. This can be shown by using the fact that the equilibrium rent can be large enough when n is sufficiently large: Each principal has stronger incentive to offer a high rent as the number of competing principals becomes larger, provided all other (many) principals offer the same high rent. Thus competition forces the principals to offer a high rent, which is in turn means that the equilibrium contract becomes the full insurance 16

contract due to Lemma 4. This result holds even when implementing the least costly action is not optimal in the monopoly principal case (see the problem (MP)). The difference between the case of multiple principals and the monopoly case is that in the former the reservation utility of the agent Û (which we call the equilibrium rent) is endogenously determined but fixed to be zero in the latter. It is helpful to compare the problem (MP) with (EP): Setting Û = 0 in the problem (EP), the problem (EP) is reduced to the problem (MP). But Û itself is determined by the market competition. This endogenization of reservation utility in turn affects the formation of equilibrium contract via competition among the principals. Second we will show that every SSPE must be the full insurance contract C as the number of competing principals tends to be large enough. The strategy to prove this result is as follows: First we take any SSPE contract Ĉ = {â, û(y)} and the corresponding equilibrium rent Û, and consider a possible deviation by some principal such that a contract C = {â, û(y) + δ(z)} is offered. Here the constant term δ(z) is determined by Ûf(z) =E[û(y) â] c(â)+δ(z). By definition of Û, δ(1/n) = 0. Thus the deviating principal can induce the selection probability z chosen in the continuation equilibrium by adjusting the rent U such that U = f(z)û, while implementing the same action â as taken in the supposed SSPE. This gives her the following deviation payoff: (1 (1 z) m )E[y φ(û(y)+δ(z)) â]. For such deviation not to be profitable, maximizing this with respect to z must yield z = 1/n. The equilibrium rent satisfies this first order condition. We will then find the lower bound of the equilibrium rents, which in turn tends to be greater than U when n. Thus it follows from Lemma 4 that any SSPE must be the full insurance contract when n is sufficiently large. 15 One might think that sufficiently large rent drives the equilibrium payoff of the principals down to their reservation payoff, zero. This does not however occur by the construction of equilibrium rent Û(n) in the proof (See equation (A1) in Appendix.) Indeed the equilibrium payoff of the principals is verified to be always positive for any n and approach to zero (the reservation payoff) when n +. Put differently the equilibrium rent Û(n) approaches to the upper bound U p when n +. Since the principals have the capacity constraint that each of them can contract with at most one agent, they can still keep the bargaining powers relative to the agents even in the limit as the number of them goes to infinity. 16 Without capacity 15 This uniqueness result follows by keeping our assumption that continuation equilibrium after a unilateral deviation by some principal is symmetric in that all agents select the deviating principal with the same probability. 16 This may be understood by an analogy to the Cournot oligopoly in which n symmetric 17

constraint, the equilibrium payoff of the principals is always driven down to zero. 3.3 Competition and Market Efficiency Our next concern is to explore the welfare implications about the limit equilibrium shown above. In particular we will consider whether or not the efficiency loss caused by the moral hazard problem is decreased by more intense competition (when n ). To evaluate such the efficiency loss, we adopt the Benthamian type welfare function, which is defined as the unweighted sum of all parties ex ante payoffs, and use it for measuring the contractual market welfare. In a SSPE in which Ĉ = {â, û(y)} is offered every principal obtains the following expected payoff: ˆV (1 (1 (1/n)) m ){E[y â] E[φ(û(y)) â]}. (15) On the other hand, in the SSPE each agent obtains the equilibrium rent Û with probability G(1/n) and nothing with the remaining probability respectively. Thus ex ante expected payoff each agent obtains is given by G(1/n)Û. Since G(1/n) =(1/m)n[1 (1 (1/n)) m ] by Lemma 1 (i), the market welfare (the unweighted sum of ex ante payoffs of all principals and agents) is then given by Ŵ n ˆV + mg(1/n)û = n(1 (1 (1/n)) m ){E[y â] E[φ(û(y)) â]+û}. (16) Notice here that n(1 (1 1/n) m ) is the expected number of matched pairs of the principal and the agent. As a reference case, we define the first best welfare as the upper bound for the equilibrium welfares. This is defined by the maximized total expected payoffs of all principals and agents: W FB max n(1 (1 a,u(y) (1/n))m ){E[y a] E[φ(u(y)) a]+e[u(y) a] c(a)}. (17) Since the agents are risk averse, the first best contract requires full insurance that u(y) =const. for all y Y, and the first best action a FB is defined as a FB arg max a A E[y a] c(a). Thus the first best welfare is given by W FB = n(1 (1 (1/n)) m ){E[y a FB ] φ(u FB )+u FB c(a FB )} (18) firms sell a homogenous good and have the same constant marginal cost with no fixed cost: The equilibrium profit of the firm in that market is always positive for any n and tends to be zero when n +. The price setting power of each firm can be negligible when the number of competing firms becomes large enough but cannot be exactly zero even in the limit. 18

where u FB satisfies φ (u FB )=1. We will use the following measure for the efficiency loss of the market equilibrium relative to the maximum social welfare: L W FB Ŵ = E[y afb ] φ(u FB )+u FB c(a FB ). (19) E[y â] E[φ(û(y)) â]+û Note that L 1. This measure captures the pure welfare loss which comes only from the moral hazard problem on the side of agents, by eliminating the direct effect of the numbers of the principals and the agents (note that the term n(1 (1 (1/n)) m ) appeared in both W FB and Ŵ is canceled out in the expression of L). Thus the number of competing principals matters for the efficiency loss only through the effect that it changes the equilibrium contract to be offered and hence the action choice of agents. Since we are concerned with the problem of whether or not more intense competition can reduce the welfare loss by the moral hazard problem, L becomes an appropriate measure of evaluating such efficiency loss. We will hereafter use the notation L = L(n) by expressing the dependency of L on the number of principals n explicitly. By using Proposition 2, we can then show the following welfare result. Proposition 3. Suppose that φ (c(a)) 1 holds. Then L(n ) >L(n ) for any n >n max{n, ˆn}, where n (resp. ˆn) is the number given in Proposition 2 (resp. Lemma 5). Proof. Suppose that n max{n, ˆn} where n (resp. ˆn) is given in Proposition 2 (resp. Lemma 5). From Proposition 2 we know that the full insurance contract C becomes a SSPE when n n. In such SSPE the equilibrium rent Û(n) is also increasing in n because of Lemma 5 and equation (A1) (see Appendix). Then the proposition will hold if we show that Ŵ is decreasing in Û when n max{n, ˆn}. Note here that when n n, Ŵ is given by Ŵ = E[y a] φ(û + c(a)) + Û (20) due to Proposition 2. Differentiating this with respect to Û we get dŵ dû = φ (Û + c(a)) + 1 < φ (c(a)) + 1 0 (21) where the first inequality follows from Û(n) > 0 because Û(n) > U > 0 when n n, and the second inequality from the condition stated in the proposition respectively. Q.E.D. 19

The above result shows that when competition is sufficiently intense a higher degree of competition among the principals gives rise to a higher efficiency loss measured by L(n). This is because the tighter competition forces the competing principals to offer the full insurance contract with larger equilibrium rent. Under the full insurance contract, whether the larger rent improves the social welfare defined by (20) depends on the slope of the utility function u. One unit increase of the rent can raise the utility of the agent by one unit but it decreases the utility of the principals by φ (U + c(a)) when the full insurance contract is offered. Thus, if the latter effect dominates the former as we assumed in Proposition 3, total welfare is decreased by the increase of the agent s rent. 4 Equilibrium Contracts with Limited Liability We have so far assumed that the agents are risk averse and they are not wealth constrained. This assumption is the key factor to derive the above limit theorem that every SSPE contract tends to be low-powered and hence result in the lower welfare. As an alternative way, in order to focus on the moral hazard issue we may resort to the model in which agents are risk neutral but wealth constrained. In this setting the moral hazard occurs due to the limitation of heavy punishment but not risk sharing consideration. The different sources to cause the moral hazard have the different effects on the equilibrium contract in the limit, as we will see below. In this section we will consider a situation where every agent is risk neutral but protected by limited liability so that the payment to him must be non negative for all realizations of the return y Y. Specifically we will assume that the utility function of each agent takes a form U(w, e)= w c(a) and that w 0 must be satisfied in any equilibrium. 4.1 Benchmark Case: Monopoly Principal We first define the monopoly principal problem where a single principal exists in the market and has full the bargaining power to offer a contract. In this case the monopoly principal will offer a contract {a, w(y)} to solve the following program: (MPL) subject to max a A,w(y) E[y w(y) a] a arg max a A E[w(y) a ] c(a ) E[w(y) a] c(a) 0 (ICL) (IRL) w(y) 0 for any y Y (LL) 20

where the last constraint (LL) means the limited liability constraint that the payment w(y) must be non negative for all realization of y Y. In the presence of risk neutral agent the first best action is defined to maximize the total payoffs of a pair of the principal and the agent: a FB arg max E[y a] c(a). (22) a A The first best payoff the principal can attain is then given by E[y a FB ] c(a FB ). This maximum payoff can be achieved in the program (MPL) if the so called selling the store scheme w(y) y + t 0, where t 0 c(a FB ) E[y a FB ], satisfies (LL), i.e., y + t 0 0. Otherwise, the action choice would be distorted. The selling the store scheme gives the agent full the marginal return and thus induces the first best action, together with a fixed transfer to be paid by the agent whatever returns realized. Let C M l {a M l,w M l (y)} denote the optimal monopoly contract to solve the problem (MPL). To avoid the trivial result that the principal offers the null contract even in the monopoly situation, we will make the following assumption. Assumption 3. E[y w M l (y) a M l ] > 0. 4.2 Characterization of SSPE Now we will examine the case of multiple principals and characterize SSPE. By a similar argument to Proposition 1, we can show that any SSPE contract must solve a simple agency program as if a single principal contracted with a single agent. Proposition 4. Suppose that Ĉ {â, ŵ(y)} is a contract offered in a SSPE with the equilibrium rent Û E[û(y) â] c(â). Then Ĉ must be a solution to the following program (EPL): (EPL) max E[y w(y) a] a A,w(y) subject to (ICL), (LL) and E[w(y) a] c(a) Û (IRL ) To see what contracts are the solutions to the problem (ELP), we define the first best contract as C FB {a FB,y+ t(û)}, (23) 21