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.
76
PEARCE AND STACCHETTI
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
IMPLICIT AND EXPLICIT CONTRACTS
77
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..
78
PEARCE AND STACCHETTI
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
IMPLICIT AND EXPLICIT CONTRACTS
79
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
80
PEARCE AND STACCHETTI
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
IMPLICIT AND EXPLICIT CONTRACTS
81
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.
82
PEARCE AND STACCHETTI
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.
88
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.
IMPLICIT AND EXPLICIT CONTRACTS
89
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-
90
PEARCE AND STACCHETTI
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.
IMPLICIT AND EXPLICIT CONTRACTS
91
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 .
92
PEARCE AND STACCHETTI
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
94
PEARCE AND STACCHETTI
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
IMPLICIT AND EXPLICIT CONTRACTS
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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