GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.

23, 75]96 Ž1998.

GA970607

The Interaction of Implicit and Explicit Contracts in Repeated Agency David G. Pearce Department of Economics, Yale Uni¨ ersity, New Ha¨ en, Connecticut 06520

Ennio Stacchetti* Department of Economics, Uni¨ ersity of Michigan, Ann Arbor, Michigan 48109-1220 Received September 12, 1993

In a repeated principal]agent model in which the agent’s actions are observable to the principal but not verifiable in court, the agent’s incentives derive both from salary payments based on verifiable signals and from implicit promises by the principal of bonuses for good behavior. Explicit short-term contracts are designed to enhance the effectiveness of the infinite-horizon implicit contract between principal and agent. In a constrained-efficient equilibrium, bonuses smooth the consumption path of the risk-averse agent by moving in the opposite direction from salaries, total consumption, and expected discounted utility for the rest of the game. Journal of Economic Literature Classification Numbers: C7, C73, D8. Q 1998 Academic Press

1. INTRODUCTION This paper investigates the relationship between two ways to sustain cooperation in repeated principal]agent models. Typically a principal can make commitments to an agent by offering him a legally enforceable contract which specifies payments contingent on information available to the courts. If the principal can observe more than the publicly verifiable information, implicit self-enforcing agreements between principal and agent that supplement the terms of the explicit contract may be mutually beneficial. We show how explicit contracts are designed to support constrained efficient equilibria of the agency supergame, emphasizing the role * To whom correspondence should be addressed. E-mail: [email protected]. We are grateful for the financial support of the National Science Foundation and the Center for Economic Policy Research at Stanford University. Our thanks go to the anonymous referees for their helpful comments. 75 0899-8256r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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of voluntary supplemental payments from the principal to the agent to provide the appropriate incentives for both parties in a second-best manner. Although it is usually realistic to assume that the principal has information Žrelated to the agent’s action. beyond that which is verifiable in a court of law, the distinction played little role in classical agency theory. The reason is that in a static setting, unverifiable observations are useless, barring certain ‘‘revelation schemes’’ discussed later. In indefinitely repeated relationships, however, such information is vital to implicit agreements which exploit the multiplicity of equilibria in the supergame to make both players’ payoffs depend on this information. For example, an agent whose actions are observed directly by the principal expects that by failing to exert the implicitly promised amount of effort, he would adversely affect his future payoffs, beyond the indirect effects it might have on the compensation guaranteed by the explicit contract he holds. Selecting an institutional setting in which to explore these ideas involves specifying the kinds of contracts that will be enforced and the opportunities open to the agent. Should a court enforce a contract having lotteries as contingent payments, for example? Over what period are contracts valid and are they binding on both parties? How much access does the agent have to capital markets or storage technologies? The modelling is fairly traditional in these respects. The explicit contracts considered are one-sided commitments requiring the principal to pay the agent deterministic amounts contingent on the realizations of publicly observable random variables Žtaken for simplicity to be gross profits.. The agent cannot borrow, lend, or save, but can in any period elect to abandon the principal in favor of earning a fixed salary elsewhere. The agent’s action is observable to the principal but not to the courts. Only single-period contracts are enforceable. We find that on the equilibrium path of an optimal equilibrium, the compensation actually received by the agent usually differs from that which he is promised ex ante by his legal contract. At the end of each period the agent receives a ‘‘bonus’’ that varies inversely with the compensation promised by the explicit contract and with the agent’s expected payoff from the remainder of the equilibrium. This transfers some risk from the risk-averse agent to the risk-neutral principal without diminishing the agent’s incentives to take the correct action. There is a large body of work on infinitely repeated principal]agent problems beginning with Rubinstein Ž1979. and Radner Ž1985.; among those that studied implicit contracts Žbut not alongside explicit contracts . are Bull Ž1987. and Spear and Srivastava Ž1987.. Bull Ž1987. showed that it is possible for an infinitely lived firm employing overlapping generations of finitely lived workers to maintain a reputation for rewarding workers

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according to their effort levels. Spear and Srivastava Ž1987. used dynamic programming techniques to characterize the constrained optimal implicit contracts that arise in infinite-horizon principal]agent problems. The literature on incomplete contracts Žearly contributions to which included Green and Laffont Ž1987., Grossman and Hart Ž1986., Hart and Moore Ž1988., Huberman and Kahn Ž1986., and Tirole Ž1986.. emphasizes the distinction between observability and verifiability of information. The analysis is quite different, however; because these papers consider two- or three-period games, questions of optimal long-run relationships do not arise. The equilibria that we study are unusual in that at the beginning of each period t, both parties can predict perfectly the total compensation the agent will receive at the end of the period Žas a function of the verifiable signal realization in t ., and yet the ultimate terms of compensation cannot be specified in the original explicit contract governing period t. It is interesting to note that this is not an environment to which the equivalence results of Fudenberg, Holmstrom, and Milgrom Ž1990. and ¨ Ž . Malcomson and Spinnewyn 1988 for short- and long-term contracts apply. When checking incentive constraints in an implicit contract in an infinite-horizon game after some history, both the equilibrium continuation values for players and the severest equilibrium punishments associated with the continuation contract are important. These cannot simultaneously be simulated by a monetary transfer of the kind used in proving the equivalence in finite-horizon agency games. In their paper on incomplete contracts, Hart and Moore Ž1988. considered game forms in which the terms of the explicit contract depend upon messages sent simultaneously by the two contracting parties to the courts; this is a powerful device having its origins in the literature on the implementation of social choice rules Žsee, for example, Maskin Ž1977... Even in a one-shot moral hazard problem in which the principal Žbut not the courts. could observe the agent’s action, there are revelation schemes that have fully efficient equilibria. We disallow such contracts on the grounds that they are not likely to be legally enforceable.1 We suggest one theoretical justification for Žbut probably not an ‘‘explanation of’’. courts’ unwillingness in practice to enforce these revelation contracts. Suppose that provisions such as the following were deemed enforceable: two parties independently send the court the message ‘‘conformed’’ or ‘‘did not conform,’’ and both are assessed enormous penalties if the messages disagree. This amounts to writing the contracting parties a judicial blank check. While the provision could be used to ensure that an employee exert 1 Similarly, in a breach of contract suit, the court typically will not award damages at the level specified in the contracts if this exceeds the actual harm caused by breach. See Schwartz Ž1990..

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the appropriate amount of effort and that the principal fully insure him, it could equally well support an implicit agreement that the agent will be compensated for illegal activities such as committing perjury or murder. Two papers that come closest to our work require discussion here. MacLeod and Malcomson Ž1989. studied the role of implicit contracts in a world in which explicit contracts can depend only on where the worker has been employed Žin particular, output realizations are not verifiable.. Their model differs radically from the standard agency model in that there is no uncertainty and no risk aversion. Independently of our work, Baker, Gibbons, and Murphy Ž1994. investigated the combined use of subjective and objective performance measures in implicit and explicit contracts. The institutional setting is richer than ours; it allows the principal to observe the agent’s effort only imperfectly, for example. They did not allow implicit rewards Žbonuses. to depend on realized explicit rewards Žsalaries .; this interaction is the main focus of our attention. Less closely related to our work, but also emphasizing the consequences of observable but nonverifiable information, is a paper by Bernheim and Whinston Ž1997., who showed in a finite-horizon setting that contractual incompleteness is often an optimal response to nonverifiability. Section 2 presents the underlying agency problem, and Section 3 characterizes the constrained efficient equilibria of the supergame. Brief concluding remarks are found in Section 4.

2. THE MODEL In our model a principal and a single agent interact repeatedly, but the explicit contract held by the agent in any particular period commits the principal to Žoutput-contingent. payments in that period only. Such a restriction is relevant if long-term contracts are either unenforceable or prohibitively costly. Information and Payoffs in the Component Game The agent chooses from the finite set of available actions A s  a0 , a1 , . . . , a n4 . The principal observes the choice, but the court sees only whether or not a0 is chosen. Whereas a1 , . . . , a n are alternative actions the agent may take while working for the principal Žperhaps effort levels or degrees of care., a0 represents ‘‘nonparticipation.’’ If the agent selects a0 , he works for some outside employer at the reservation salary r ) 0. Define R [ R j  y`4 . The agent’s von Neumann]Morgenstern utility function

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u: Rq= A ª R takes the form u Ž c, a i . s U Ž c . y d i , where c is his monetary compensation or consumption and d i is the disutility associated with taking action a i . Each action induces a probability distribution over verifiable outcomes, which are assumed for simplicity to be finite in number. Without essential loss of generality, the outcomes are taken to be the gross profits of the principal. Let P s p 1 , . . . , pm 4 be the space of gross profits and let pi k s P wp k < a i x, i s 0, . . . , n, k s 1, . . . , m, where P wp k < a i x denotes the probability of p k conditional on a i . Assumptions: ŽA1. U is differentiable, strictly concave, and strictly increasing. ŽA2. pi k ) 0, i s 1, . . . , n, k s 1, . . . , m. ŽA3. Ý m ks 1 pi k p k is minimized when i s 0. Normalizations: m

d 0 s 0 and

Ý

p 0 k p k s 0.

ks1

The agent’s sole sources of money are the principal and the reservation salary. The strict concavity of U reflects his risk aversion. The principal is risk-neutral, caring only about the actuarial value of her profits, net of the payments to the agent; she faces no bankruptcy constraints. Assumption ŽA3. guarantees that it is better for the principal to have an agent working for her unpaid, than to have nonparticipation. In the absence of ŽA3., it would have been natural to complicate the agency game by allowing the principal to deny the agent the opportunity of working for her Žat any wage.. The discounted Žaverage. value to the agent of an infinite sequence Ž c t , a t .4 of compensations and actions is 1yd

d

`

Ý d t UŽ ct . y dt

,

ts1

where d t is the disutility corresponding to action a t and d g Ž0, 1. is the discount factor. Similarly, the principal’s value of the stream Žp t , c t .4 of

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outcomes and compensation is 1yd

d

`

Ý d t wp t y c t x . ts1

Explicit Contracts At the beginning of each period t, t s 1, 2, . . . , the principal offers the agent a legally enforceable contract Žcalled an explicit contract. making payment contingent on the outcome andror the agent’s participation. mq 1 Thus an explicit contract is a vector s g Rq , where s0 s payment to the agent if he takes action a0 Ž doesn’t participate . , sk s payment guaranteed to the agent when he participates and outcome k arises, k s 1, . . . , m. The guaranteed payments sk are called salaries and are received by the agent at the end of the period once output is realized. The principal may then choose to supplement the salary with some nonnegative bonus b. The bonus in period t may depend on k and on the entire history of play to that point. Indeed, the advantage of using a bonus Žwhich could, after all, have been subsumed in the salary. is that it can be denied to the agent if he fails to take the proper action. In a given equilibrium, when the agent has chosen the right action, the principal is expected to give a particular bk G 0 when outcome k arises. The agent’s total compensation when p k occurs is c k [ sk q bk . One can show that no purpose is served by considering two-sided gift-giving: as we explain later, for any equilibrium in which the agent voluntarily returns some of his salary in some contingency, there exists another equilibrium with the same actions and patterns of consumption in which this does not occur. It is clearly redundant to allow for a bonus corresponding to nonparticipation; such a bonus could be incorporated into the salary. Any equilibrium of the agency supergame draws continuation payoffs for the two players from the equilibrium value set Ždefined formally below.. We shall see in the next section that the structure of an optimal equilibrium is greatly simplified if this value set is convex: its upper boundary is then a concave function, so the principal must sacrifice greater amounts of her own utility for each successive increment in the agent’s utility. To guarantee that the equilibrium value set is convex, it is sufficient to introduce a ‘‘public randomization device’’ as follows: at the beginning of each period, before the contract is offered, the players commonly observe the realization of a random variable uniformly distributed on the interval

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w0, 1x. The random variables are independent Žacross time.. A strategy for the agent in the supergame specifies for each t the action to be taken by the agent in period t as a measurable function of the entire history of the game until that time, including realizations of the randomization devices and gross profits, the contracts offered by the principal in periods 1 to t inclusive, the agent’s previous choices of action, and the bonuses paid. A strategy for the principal specifies for each t the explicit contract to be offered as a measurable function of everything that has happened in the first t y 1 periods and the outcome of the period t randomization device, and specifies for each t the bonus to be paid as a measurable function of the history, including the period t values of the randomization device, the contract offered, the action taken, and the realization of gross profits. A pair E of strategies for the agent and the principal, respectively, induces a probability distribution over streams of actions, outcomes, and compensations, and hence yields a pair n Ž E . s Ž nAŽ E ., n P Ž E .. of expected values for the respective players.2 We are interested in the pure strategy subgame perfect equilibria 3 of the repeated game, which will usually simply be called equilibria. There is no need to employ a stronger equilibrium refinement because the game is one of perfect information. Denote by G the infinitely repeated game we have described. The equilibrium value set is V [  n Ž E . < E is a subgame perfect equilibrium of G 4 ; R 2 .

ˆ the Žinfinite-horizon. subgame of G that follows any We denote by G realization of the public randomizing device in period 1 and let Vˆ be the ˆ Notice that the equilibria of G are probability equilibrium value set of G. ˆ and hence V s co V. ˆ It is convenient to distributions over equilibria of G, abuse notation by letting n Ž E . represent the payoff pair associated with E ˆ rather than of G. By Caratheodory’s even when E is a profile of G Theorem, any particular value in V ; R 2 can be expressed as a convex ˆ therefore, without loss of combination of no more than three points in V; generality we henceforth restrict attention to equilibria which randomize ˆ at the beginning of any period. over at most three equilibria of G Theorem 5 of the Appendix establishes the existence of an equilibrium ˆ Žand hence of G, since the probability distribution over equilibria of of G 2

For some strategy profiles a player’s expected value may not be finite, but such profiles cannot arise in equilibrium. 3 It can be shown that for these games, by introducing public randomization devices before each move of each player Žrather than just at the beginning of periods., one renders redundant the consideration of mixed strategies. This would have been cumbersome but would not have changed the nature of our results.

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FIGURE 1

ˆ can be taken to be degenerate. and the compactness of the equilibrium G value set V. The following notation is used extensively in the analysis below and is illustrated in Fig. 1. Let u [ min  u < Ž u, ¨ . g V 4 u [ max  u < Ž u, ¨ . g V 4 ¨ [ min  ¨ < Ž u, ¨ . g V 4 ¨ [ max  ¨ < Ž u, ¨ . g V 4

u P [ max  u < Ž u, ¨ . g V 4 . For each u g w u, u x define f Ž u. [ max ¨ <Ž u, ¨ . g V 4 . The set upperŽ V . [ Ž u, f Ž u..< u g w u, u x4 is called the upper frontier of V. Since V is convex, f : w u, u x ª R is concave. The pair Ž u, ¨ . g V is Pareto efficient in V if for all Ž u9, ¨ 9. g V distinct from Ž u, ¨ ., either u9 - u or ¨ 9 - ¨ . An equilibˆ is efficient relati¨ e to V if n Ž E . is Pareto efficient rium E of either G or G in V.

IMPLICIT AND EXPLICIT CONTRACTS

83

3. CONSTRAINED EFFICIENCY IN THE AGENCY SUPERGAME Our analysis is in the spirit of dynamic programming and, in addition, owes much to Abreu Ž1988.. It is useful to view the value to a player of an ˆ as being the sum Ž1 y d .Žfirst period payoff. q d Žexpected equilibrium of G . future value , where we mean by ‘‘expected future value’’ the player’s payoff in equilibrium from the beginning of the second period onward, evaluated at the beginning of the second period. A similar decomposition is often a convenient way to check the equilibrium incentive constraints associated with a player’s first-period choices. When the principal, for example, considers not paying Žor ‘‘seizing’’. the bonus in period 1, she weighs the immediate savings against the resulting change in the expected future value. Similarly, in determining whether to deviate from the action prescribed by an implicit agreement, the agent takes into account the immediate gain, if any Žincluding the change in the bonuses. and the change in the future payoff. The incentives to stay on the equilibrium path are strongest when a deviation is followed by the worst possible equilibrium for the deviating party in the resulting subgame. Thus we restrict attention without loss of generality to equilibria in which the principal’s expected future value if she seizes the bonus in any period is ¨ , her payoff from offering the wrong contract does not exceed ¨ from the beginning of the current period onward,4 and a deviating agent has expected future values u and receives a bonus of zero. As noted earlier, allowing for two-sided gift-giving serves no purpose. In an equilibrium in which the principal and agent simultaneously exchange positive gifts after some history, no incentive constraint is tightened Žand some are loosened. if the gifts are reduced by the same amount until one of them is zero. If it is the agent’s gift that remains positive, it cannot exceed his salary Žconsumption is constrained to be nonnegative.; reduce the salary by the amount of the gift and change the agent’s gift to zero. Incentives for conforming to the equilibrium are preserved and equilibrium payoffs are unchanged. ˆ with some desired To demonstrate the existence of an equilibrium of G properties, it is often easier to specify the equilibrium path in the first period Žthe contract s offered, the agent’s action a i , and the bonuses bk , k s 1, . . . , m. and expected future values Ž u k , ¨ k . g V, k s 0, . . . , m, where

4

By the definition of ¨ , there cannot exist a deviation to some contract s* such that the principal’s payoff in the worst equilibrium of the ensuing subgame exceeds ¨ . However, if the contract she offers uniformly promises extravagant salaries, for example, the principal’s worst equilibrium continuation value could fall short of ¨ .

84

PEARCE AND STACCHETTI

Ž u 0 , ¨ 0 . is the expected value following nonparticipation.5 Let b s Ž b1 , . . . , bm .. If the players’ incentive constraints are satisfied in period 1 Žwhen they believe that conformity results in the future values Ž u k , ¨ k . and deviations are met with the severest ‘‘punishments’’ described above., then ˆ with first-period it is easy to check that there exists an equilibrium of G ˆ we path Ž s, a i , b . and continuation values Ž u k , ¨ k .. For any profile E of G, denote the components of the first-period path induced by E by

s Ž E . [ the contract offered a Ž E . [ the action taken by the agent b Ž E . [ the vector of bonuses b k Ž E . , k s 1, . . . , m. Our goal is to characterize the Žconstrained. efficient equilibria of the agency supergame. For d very near 1 or 0, this is quite simple. If players are extremely impatient, their future payoffs are of little concern to them, so self-enforcing agreements collapse: the solution coincides in every period with that of the static agency problem. If instead players are sufficiently patient, the folk theorems of Fudenberg and Maskin Ž1986. and Fudenberg, Levine, and Maskin Ž1994. Žsee especially Section 9. imply that ˆ without the first-best payoffs can be approximated in equilibria of G, recourse to explicit contracts. We are interested in the intermediate cases in which some implicit cooperation can be sustained, but incentives pose a substantial constraint. While the results presented below appear to characterize only the first period of an efficient equilibrium, they hold at every point on the equilibrium path. This follows from the fact that an efficient equilibrium induces, after any t-period equilibrium history, a continuation equilibrium that is itself efficient: after such a history, when an inefficient continuation equilibrium is replaced by a Pareto-superior equilibrium Žwithout changing what is specified after any alternative history., incentives to conform to equilibrium play are improved Žor at worst unchanged.. Although efficient equilibria induce further efficient equilibria on subgames following equilibrium histories, this is not true, for example, for subgames following a deviation in which the principal has offered the wrong contract. This raises the question of whether the subgame perfect equilibria under study are ‘‘renegotation-proof.’’ The answer depends upon how one extends the alternative definitions of renegotiation-proofness suggested by Bernheim and Ray Ž1989., Farrell and Maskin Ž1989., and Pearce Ž1987. to repeated games having multiple stages within each period. In our opinion there is currently no fully satisfactory solution 5 Although the expected future value following nonparticipation could, in some equilibrium, depend on current realized profits, only the average Ž u 0 , ¨ 0 . of these expected values is relevant.

IMPLICIT AND EXPLICIT CONTRACTS

85

concept available for these games. In any case, the equilibria we study are efficient within the class of subgame perfect equilibria; no further restrictions are considered. A central feature of optimal implicit agreements is the way in which the agent’s rewards are divided among salaries, bonuses, and expected future values. Considerations of efficiency place some powerful restrictions on the pattern of rewards following any t-period history h, as long as players are not so patient that incentive compatibility is consistent with the agent’s receiving a constant compensation Žregardless of realized profits. in period t q 1 following h. Notice that if we could ignore the principal’s temptation to seize the bonus, it would always be useful to decrease some salary and increase the corresponding bonus by the same amount, leaving players’ payoffs unchanged, but strengthening the agent’s incentive to take the appropriate action Žsince one of the bonuses he might lose by cheating is now larger.. This strict incentive compatibility would allow an adjustment of salaries that would decrease the variation in compensation received in the period in question, shifting risk from the agent to the principal. Taking the principal’s incentives into account, one sees that a bonus can be increased only until it equals the wedge between the principal’s expected future value after paying the bonus, and the worst punishment value ¨ she ˆ is inefficient can be given for seizing the bonus. Thus, an equilibrium of G if the bonuses are not equal to the appropriate ‘‘wedges’’; Theorem 1 makes this precise.

ˆ Let s [ s Ž E ., ai [ a Ž E ., THEOREM 1. Let E be an equilibrium of G. b [ b Ž E ., and Ž u k , ¨ k . g V, k s 0, . . . , m, be its expected future ¨ alues. Suppose Ži.

i / 0;

Žii.

there exists l such that bl / w drŽ1 y d .xŽ ¨ l y ¨ . and sl ) 0;

Žiii.

Consumption c k [ sk q bk is not constant in k.

Then there exists an equilibrium E* that Pareto dominates E: nAŽ E*. ) nAŽ E . and n P Ž E*. s n P Ž E .. Proof. Note that bk F w drŽ1 y d .xŽ ¨ k y ¨ . for each k s 1, . . . , m. Otherwise, for some k, seizing the bonus bk would be a profitable deviation for the principal: the adverse effect on his expected future equilibrium payoff is at most wŽ drŽ1 y d .xŽ ¨ k y ¨ .. Hence, Žii. implies bl - w drŽ1 y d .xŽ ¨ l y ¨ .. We first construct an equilibrium E9 as follows: s Ž E9. [ s9, a Ž E9. [ a i , b Ž E9. [ b9, and the continuation profiles Žfrom the second period on-

86

PEARCE AND STACCHETTI

ward. are equilibria of G with values Ž u k , ¨ k ., k s 0, . . . , m, where sXk [

½

sk sl y «

if k / l if k s l

bXk [

½

bk bl q «

if k / l if k s l

and « is any number in Ž0, min sl , w drŽ1 y d .xŽ ¨ l y ¨ . y bl 4.. It is easily checked that E9 is indeed an equilibrium and that n Ž E9. s n Ž E .. However, in E9 the agent strictly prefers a i to any other action a j with j / 0: his payoff from choosing a i is the same as in E, whereas deviating to a j entails a greater loss in bonus when p l occurs, and pjl ) 0. Note that cXk s sXk q bXk s c k for each k s 1, . . . , m. Let m

w[

Ý

m

pi k U Ž c k .

and

ks1

c[

Ý

pi k c k .

ks1

Define the contract s* by sUk q bXk s Ž 1 y l . c k q l c

k s 1, . . . , m,

where l g Ž0, 1. is to be determined below. The consumption distribution cUk [ sUk q bUk , k s 1, . . . , m, is ‘‘smoother’’ than c: the movement from c* to c is a mean-preserving spread. Consider a profile E* such that s Ž E*. s s*, a Ž E*. s a i , b Ž E*. s b9, and with continuation profiles that are equilibria with values Ž u k , ¨ k ., k s 0, . . . , m. We claim that E* is an equilibrium. Since in the first period of E9 the agent strictly prefers a i to any other action a j , j / 0, for sufficiently small l ) 0, this remains true in E* Žrecall that the action set A is finite.. By assumption, the first period compensation c of E is not constant; therefore, UŽ c . ) w and m

w* [

Ý ks1

pi k U Ž cUk . ) Ž 1 y l .

m

Ý

pi k U Ž c k . q lU Ž c . ) w.

ks1

Hence nAŽ E*. ) nAŽ E9. s nAŽ E .. Incentives to pay the first-period bonuses are the same as in the equilibrium E9, because for each k, the size of the bonus bXk and the wedge ¨ k y ¨ are the same in E* and E9. Since U Ž . Ž . Ž . c* [ Ý m ks 1 pi k c k s c, we have n P E* s n P E9 s n P E , and the principal’s payoff from conforming and deviating, respectively, are not changed in the transition from E9 to E*. Thus E* is an equilibrium with the required properties. Q.E.D. In the notation of Theorem 1, if ¨ l ) ¨ and c l ) 0, the bonus bl can be assumed Žstrictly . positive without loss of generality even if condition Žiii. does not hold. If bl is zero, sl must be positive; sl can be decreased slightly and bl correspondingly increased without violating incentive constraints.

IMPLICIT AND EXPLICIT CONTRACTS

87

One way to guarantee that compensation is nonzero in equilibrium is to impose the following restriction on the agent’s utility function, which will be in force for the remainder of this section. Assumption ŽA4.. lim c ª 0 UŽ c . s y`. DEFINITION. Let C be the inverse of the utility function U. Ž C can be viewed as a cost function.. If any salary offered on the equilibrium path were zero, the agent’s incentive constraints in the corresponding contingency would be slack: taking the wrong action results with positive probability in a consumption of zero, which has utility y`. Consequently, the salary could be increased slightly and the bonus reduced by an equal amount, while maintaining the correct incentives. Without loss of generality, then, we confine attention to equilibria in which all salaries offered on the equilibrium path are positive. Assumption ŽA4. also guarantees that one can alter any equilibrium by lowering the agent’s first-period salaries Žleaving bonuses unchanged. in such a way that his utility of consumption is lower uniformly, until his expected utility for the entire equilibrium is driven down to his reservation level. This produces a new equilibrium with utility UŽ r . y a0 . Threatening the agent with contract termination has the same utility consequence for him, and hence nothing would be gained in our model by considering contract termination explicitly. We simplify the statements of Proposition 1 and Theorem 2 below by assuming that the function f defined in Fig. 1 is differentiable Žat the relevant point.; the discussion following the proof of Theorem 2 indicates the nature of the results in the absence of differentiability. Proposition 1 provides another link between current and future rewards, essentially stating that the marginal cost to the principal of increasing the agent’s utility in some contingency should be the same whether she gives extra compensation today or forgoes some profit tomorrow Žmoving clockwise along the efficient frontier of V .. The result is similar to Proposition 1 of Rogerson Ž1985..

ˆ efficient relati¨ e to V. Let PROPOSITION 1. Let E be an equilibrium of G, s [ s Ž E ., a i [ a Ž E ., and b [ b Ž E ., and suppose that i / 0 and E has continuation ¨ alues Ž u k , ¨ k . g V, k s 0, . . . , m. Let c k [ sk q bk , k s 1, . . . , m. Then for each k s 1, . . . , m for which u P - u k - u, C9 Ž U Ž c k . . s yf 9 Ž u k . . Proof. Let Ž u, ¨ . [ ¨ Ž E .. Since E is efficient, ¨ s f Ž u., ¨ k s f Ž u k ., and u P - u k - u for each k s 1, . . . , m. Let l be such that u - u l - u. Then since ¨ l s f Ž u l . and f is concave, ¨ l ) ¨ . Therefore, it is unrestrictive to assume bl ) 0.

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PEARCE AND STACCHETTI

By contradiction, assume that C9ŽUŽ c .. ) yf 9Ž u. Žthe case C9ŽUŽ c .. yf 9Ž u. is similar 6 .. Let D be a small positive number. Let uUl solve the equation

Ž 1 y d . U Ž sl q bUl . q d uUl s Ž 1 y d . U Ž sl q bl . q d u l

Ž 1.

and define bUk [

½

bk bl y D

if k / l if k s l

uUk [

½

uk uUl

if k / l if k s l

and ¨ Uk [ f Ž uUk ., k s 0, . . . , m. Let E* be a profile with s Ž E*. s s, a Ž E*. s a i , b Ž E*. s b*, and equilibrium continuation profiles with values Ž uUk , ¨ Uk .. Define Ul [ UŽ c l . and UlU [ UŽ sl q bUl .. We have ¨ Ul s f Ž uUl . f f Ž u l . q f 9 Ž u l . Ž uUl y u l .

s ¨l q f ¨l q s ¨l q

1yd

d 1yd

d

f 9 Ž u l . Ž Ul y UlU . f 9 Ž u l . U9 Ž c l . D

1 y d f 9Ž ul .

d

Ž from 1 .

C9 Ž Ul .

D ) ¨l y

1yd

d

D

Ž by assumption .

The expected values to the principal of deviating in the first period of E* are the same as they were in E; thus she has no incentives to deviate from E* if her payoff from conformity is no lower in E* than in E. This is the case because

d ¨ Ul y Ž 1 y d . bUl ) d ¨ l y

ž

1yd

d

D y Ž 1 y d . Ž bl y D .

/

s d ¨ l y Ž 1 y d . bl , so n P Ž E*. ) n P Ž E .. The agent has the same incentives in E* as in E to take action a i : Eq. Ž1. implies that action a i is equally lucrative in the profiles E* and E. Therefore, E* is an equilibrium and nAŽ E*. s nAŽ E .. This is a contradiction: n Ž E . is not efficient in V. Q.E.D. 6 In either case, the lack of equality between C9ŽUŽ c .. and yf 9Ž u. permits a small change in the bonus and an exactly offsetting change in the agent’s continuation value, which leaves the principal strictly better off. If either party cheats, it receives the same payoff as it would had it cheated in the original equilibrium. Hence, incentives to conform are the same for the agent and they are enhanced for the principal.

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Theorem 2 describes precisely how the explicitly and implicitly promised rewards are used in combination to create the appropriate incentives for the agent. Suppose that it is desirable to give the agent greater total rewards following some history h Žending in some profit realization. than following another history h9. The current compensation, the salary, and the agent’s expected future payoff are all higher after h than after h9, but the bonus moves in the opposite direction: the more an agent is being rewarded, the smaller is the bonus he receives. Although this initially sounds counterintuitive, note that when an agent is being rewarded, his Žguaranteed. salary is relatively high, and this is only partially offset by a low bonus. The variation in the bonuses has a moderating effect on total compensation when the agent conforms to the implicit agreements; this enhances efficiency without threatening incentives. THEOREM 2. Adopt the assumptions and notation of Proposition 1, and suppose that compensation c k is not constant. Then salaries sk must be positi¨ e and the following statements are equi¨ alent: Ži. Žii. Žiii. Živ. Žv.

ck - cl ; uk - ul ; ¨ k ) ¨ l; bk ) bl ; sk - sl .

Proof. If sk were zero for some k, the agent’s payoff from deviating in period 1 would be y`, and hence none of the agent’s incentive constraints Žnot including the participation constraint . would be binding. Then compensation could be smoothed slightly Žas in Theorem 1., thereby improving the agent’s payoff without violating incentive compatibility. This contradicts the fact that E is efficient relative to V. From Proposition 1, C9ŽUŽ c q .. s yf 9Ž u q ., q s k, l, and since U is strictly concave, C9 is strictly increasing. Therefore Ži. and Žii. are equivalent. Since E is efficient relative to V, Ž u q , ¨ q . is an efficient point of V, q s k, l, so Žii. is equivalent to Žiii.. Theorem 1 implies that bq s ¨ q y ¨ for q s k, l; hence Žiii. is equivalent to Živ.. Also, s q s c q y bq , so Ži. Žand Živ.. implies Žv.. Finally, suppose c k G c l . Then bk F bl and sk s c k y bk G c l y bl s sl . Hence wnot Ži.x implies wnot Žv.x. Q.E.D. Without invoking ŽA4. or differentiability of the efficient frontier of the equilibrium value set, one can obtain somewhat less tidy versions of the results above. The equality yC9ŽUŽ c k .. s f 9Ž u k . in Proposition 1 is replaced by the statement that yC9ŽUŽ c k .. lies in the subdifferential of f at u k . Consequently, it is possible for two different compensations to correspond to the same expected future payoff. Nonetheless, a slight modifica-

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tion of Theorem 2 still applies: if any one of the conditions Ži. ] Žv. Žin the statement of the theorem. holds Žwith strict inequality as before., the other four conditions must be satisfied as weak inequalities. Thus, a comparison of rewards in two equilibrium contingencies will never reveal bonuses moving in the same direction as salaries, compensations, or expected future values.

4. CONCLUSION In a repeated agency model in which the principal has better information than the courts regarding the agent’s actions, optimal cooperation between the players requires the use of both explicit and implicit agreements. This article illustrates the interplay of external and self-sustaining enforcement mechanisms by studying the constrained efficient equilibria of supergames based on a particular, fairly standard agency model. The equilibria conform to a simple pattern. When an equilibrium history calls for the agent to be rewarded generously, his guaranteed salary is high, his total current compensation Žsalary plus bonus. is high, his expected future payoff is high, and the bonus Žthe voluntary component of the payment made by the principal. is low. The presence of bonuses helps to discourage the agent from cheating, and their variation Žacross contingencies . partially smooths the risk-averse agent’s consumption. Through the award of bonuses, the payments specified by an explicit contract are replaced ex post by other payments Žthose that were promised implicitly.. We conjecture that a model with long-term explicit contracts would exhibit a more involved kind of contract substitution: long-term contingent contracts are frequently replaced on the equilibrium path by new long-term contracts, despite the fact that the eventually realized terms of the relationship could have been specified in the first contract that was offered. To an external observer, it would appear that the long-term contracts were renegotiated frequently, although to the contracting parties, all of this renegotiation is anticipated at the beginning of their relationship. Similarly, in the short-term contract model we study here, the payment of bonuses can be interpreted as a degenerate form of a perfectly anticipated form of renegotiation. Thus, while renegotiation of explicit contractual arrangements can be explained by appealing to various problems of complexity and incomplete information, we wish to emphasize that it also emerges naturally as a way of exploiting multiple equilibria to provide incentives as efficiently as possible.

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APPENDIX We show here that the equilibrium value set V of the supergame is nonempty and compact. This Appendix draws extensively on results in Abreu, Pearce, and Stacchetti Ž1986, 1990.. The proofs of Theorems 3 and 4 below are omitted; they are straightforward modifications of the self-generation and factorization theorems found in those papers. The reader is directed to Abreu, Pearce, and Stacchetti Ž1990. for a more formal and detailed account. The definition of admissibility below captures all incentive constraints that an equilibrium of the repeated game must satisfy in the first period. ˆ is factorizable into its first period recommendation An equilibrium E of G mq 1 m Ž s, a i , b . g Rq = A = Rq to the players, and the values Ž u k , ¨ k . g V, k s 0, . . . , m, of the strategies induced by E on the subgames beginning in period 2. Since each subgame beginning in period 2 is identical to G, the strategies induced by E on these subgames must be equilibria of G, and ˆ Initially we do not know the therefore their values must be in V s co V. ˆ so we draw values from an arbitrary set W ; R 2 and use them as if set V, ˆ Thus, the conditions Ži. ] Živ. of admissithey were equilibrium values of G. bility have the following interpretation: Ži. Continuation values implicitly promised in the first period are equilibrium values. Žii. The agent has no incentives to deviate in the first period: a i , b, u. is the supergame payoff he expects in equilibrium and Ž FA s, a j . is his expected payoff when he chooses action a j instead. FAU Ž s,

Žiii. The bonus bk promised in equilibrium is no greater than the wedge between the principal’s expected future value and the worst punishment value. Živ. The principal has no incentive in the first period to offer a contract s9 different from the contract s prescribed by the equilibrium: FPU Ž s, a i , b, ¨ . is the supergame payoff she expects in equilibrium and FP Ž s9. is her expected payoff if she offers contract s9 instead. DEFINITION ŽAdmissibility.. Assume W ; R 2 is bounded and let wA [ inf  wA ¬ Ž wA , wp . g W for some wp 4 wP [ inf  wP ¬ Ž wA , wp . g W for some wA 4 .

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mq 1 m The tuple Ž s, a i , b, u, ¨ . g Rq = A = Rq = R mq1 = R mq1 is admissible w.r.t. W if:

Ži. Ž u k , ¨ k . g co W for each k s 0, 1, . . . , m. Žii. FAU Ž s, a i , b, u. G FAŽ s, a j . for each j / i, where

¡Ý p m

FAU Ž s, a i , b, u . [

ik

~

Ž 1 y d . U Ž sk q bk . q d u k y Ž 1 y d . d i

ks1

if i / 0

¢

Ž 1 y d . U Ž r q s0 . q d u 0 y Ž 1 y d . d 0

and

if i s 0

¡Ý p m

FA Ž s, a j . [

~

jk

Ž 1 y d . U Ž sk . q d wA y Ž 1 y d . d j

ks1

if j / 0 Ž 1 y d . U Ž r q s0 . q d wA y Ž 1 y d . d 0 if j s 0.

¢

Žiii. Ž1 y d . bk F d Ž ¨ k y wp . for each k s 1, . . . , m. mq 1 Živ. FPU Ž s, a i , b, ¨ . G FP Ž s9. for all s9 g Rq , where

¡Ý p m

FPU

Ž s, ai , b, ¨ . [~

ik

Ž 1 y d . Ž p k y sk y bk . q d ¨ k

if i / 0

ks1 m

¢

d ¨0 q Ž1 y d .

žÝ

p 0 k p k y s0

ks1

/

if i s 0

and FP Ž s9. is the optimal value of the following optimization problem: inf

FPU Ž s9, a j , b9, ¨ 9 .

m s.t. a j g A, b9 g Rq , Ž uXk , ¨ Xk . g co W for each k s 1, . . . , m

FAU Ž s9, a j , b9, u9 . G FA Ž s9, a l .

Ž1 y d .

bXk

FdŽ

X ¨k

y wP .

for each l / j

for each k s 1, . . . , m.

mq 1 m The value of a tuple Ž s, a, b, u, ¨ . g Rq = A = Rq = R mq1 = R mq1 U U is F*Ž s, a, b, u, ¨ . s Ž FA Ž s, a, b, u., FP Ž s, a, b, ¨ ...

DEFINITION.

For each W ; R 2 bounded, let

B Ž W . [  F* Ž s, a, b, u, ¨ . ¬ Ž s, a, b, u, ¨ . is admissible w.r.t. W 4 .

IMPLICIT AND EXPLICIT CONTRACTS

PROPOSITION 2.

93

If W ; R 2 is compact, B ŽW . is compact.

Proof. The reader may check that if W is bounded, B ŽW . is bounded. Hence, we show that B ŽW . is closed. Let Ž s q, a q, b q, u q, ¨ q .4 be a sequence of admissible tuples w.r.t. W such that F*Ž s q, a q, b q, u q, ¨ q . \ w q ª w*. We will show that w* g B ŽW .. Since W is bounded,  u q 4 ,  ¨ q 4 , and  b q 4 are bounded Žfor the latter, use condition Žiii. of admissibility., and therefore we can assume w.l.o.g. that u q ª u*, ¨ q ª ¨ *, and b q ª b*. Since A is infinite, we can assume that a q s a i for all q. Finally, if i s 0, condition Živ. of admissibility implies that  s0q 4 is bounded, and w.l.o.g. we assume that skq s 0 for each k s 1, . . . , m and q G 1. Similarly, if i / 0, w.l.o.g. we assume that s0q s 0 for all q G 1, and condition Živ. implies that  s q 4 is bounded. Therefore, in either case we can assume that  s q 4 is bounded and that s q ª s*. Because co W is compact, Ž uUk , ¨ Uk . g co W, and since bkq F ¨ kq y wp for all q, bUk F ¨ Uk y wp for all k s 1, . . . , m. Clearly, FAU Ž s, a, b, u. is continuous in Ž s, b, u. and FAŽ s, a. is continuous in s. Hence, for all j / i, FAU Ž s q, a i , b q, u q . G FAU Ž s q, a j . for all q implies that FAU Ž s*, a i , b*, u*. G FA Ž s*, a j .. Finally FPU Ž s, a, b, ¨ . is continuous in Ž s, b, ¨ . so FPU Ž s*, a i , b*, ¨ *. G FP Ž s9. for all s9. Thus Ž s*, a i , b*, u*, ¨ *. is admissible w.r.t. W, and since by continuity w* s F*Ž s*, a i , b*, u*, ¨ *., w* g B ŽW .. Q.E.D. DEFINITION.

W ; R 2 is self-generating if W ; co B ŽW ..

THEOREM 3 ŽSelf-generation.. ˆ then co B ŽW . ; V.

If W ; R 2 is compact and self-generating,

The following theorem corresponds to the factorization theorem in Abreu, Pearce, and Stacchetti Ž1990.. The converse inclusion is stated below in the corollary to Theorem 5. However, to show the converse inclusion, it is first necessary to establish, for example, that the worst continuation values for the agent and the principal are attained; this is implied by the fact that V is compact. THEOREM 4. V ; co B Ž V .. Let Ž s*, a*, b*. be the path of a Nash equilibrium of the principal]agent component game described in Section 2. ŽClearly we can assume b* s 0 ˆ specifying that in every period, after w.l.o.g.. Let E* be the profile of G any history, the principal offers s* and pays no bonuses, and the agent takes a myopic best response to the current contract in that period Žif there is more than one, choose one that is best for the principal.. Suppose

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a* s a i and let

¡ Ý p UŽ s . y d ~ u* [ ¢U Ž s q r . y d ¡ Ý p wp y s x m

U k

ik

i

if i / 0

ks1

U 0

0

m

¨* [

~

ik

k

ks1 m

¢Ý p

0 kp k

U k

y sU0

if i s 0 if i / 0 if i s 0.

ks1

ˆ with value n Ž E . s Ž u*, ¨ *.. It is easy to see that E* is an equilibrium of G ˆ Therefore V / B. Let p [ max 1 F i F n Ý m and recall that by ŽA3., 0 s ks1 pi k p k m m min 0 F i F n Ý ks1 pi k p k s Ý ks1 p 0 k p k . The principal can guarantee herself a supergame payoff of 0 by offering the contract s ' 0 in every period. On the other hand, the best outcome she can ever expect is that in every period the agent chooses the best action for the principal and receives no payment. This gives the principal an expected supergame payoff of p . Let p [ min pi k <1 F i F n, 1 F k F m4 and d [ min d i 4 . Since the principal knows that her continuation value, after any history, is in the interval w0, p x, we can assume she will never offer a contract s having s0 ) dprŽ1 y d . or sk ) dprw pŽ1 y d .x for some k s 1, . . . , m. The agent can guarantee himself a salary r in every period by taking his alternative job in ˆ every period. Hence, for each Ž u, ¨ . g V, U Ž r . y d0 F u F U

ž

dp pŽ1 y d .

/

y d and 0 F ¨ F p .

ˆ V is nonempty and bounded. Since V s co V, In the next theorem we use the fact that B is monotone in the sense that if W ; W9 ; R 2 , then B ŽW . ; B ŽW9.. THEOREM 5. V is a nonempty compact set. Proof. We need only show that V is closed. We have V ; co B Ž V . ; co B Žcl V ., and since V is bounded, cl V is compact and co B Žcl V . is

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95

compact. Hence, cl V ; co B Žcl V ., and by self-generation, cl V ; V. Therefore, V is closed. Q.E.D. COROLLARY. V s co B Ž V ..

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Maskin, E. Ž1977.. ‘‘Nash Equilibrium and Welfare Optimality,’’ mimeo. Pearce, D. Ž1987.. ‘‘Renegotiation]Proof Equilibria: Collective Rationality and Intertemporal Cooperation,’’ Cowles Foundation Discussion Paper 855, Yale University. Radner, R. Ž1985.. ‘‘Repeated Principal]Agent Games with Discounting,’’ Econometrica 53, 1173]1198. Rogerson, W. Ž1985.. ‘‘Repeated Moral Hazard,’’ Econometrica 53, 69]76. Rubinstein, A. Ž1979.. ‘‘Offenses that May Have Been Committed by Accident}An Optimal Policy of Retribution,’’ in Applied Game Theory ŽS. J. Brams, A. Schotter, and G. Schwodiauer, Eds... Wurzburg: Physica-Verlag. ¨ ¨ Schwartz, A. Ž1990.. ‘‘The Myth that Promisees Prefer Supracompensatory Remedies: An Analysis of Contracting for Damage Measures,’’ Yale Law J. 100, 369]407. Spear, S., and Srivastava, S. Ž1987.. ‘‘On Repeated Moral Hazard with Discounting,’’ Re¨ . Econ. Studies 54, 599]618. Tirole, J. Ž1986.. ‘‘Procurement and Renegotiation,’’ J. Polit. Econ. 94, 235]259.