relational incentive contracts - stanford universityjdlevin/papers/ric.pdf · 2003-08-13 ·...

Relational Incentive Contracts

By JONATHAN LEVIN*

Standard incentive theory models provide a rich framework for studying informa-tional problems but assume that contracts can be perfectly enforced. This paperstudies the design of self-enforced relational contracts. I show that optimal con-tracts often can take a simple stationary form, but that self-enforcement restrictspromised compensation and affects incentive provision. With hidden information, itmay be optimal for an agent to supply the same inefficient effort regardless of costconditions. With moral hazard, optimal contracts involve just two levels of com-pensation. This is true even if performance measures are subjective, in which caseoptimal contracts terminate following poor performance. (JEL C73, D82, L14)

Incentive problems arise in many economicrelationships. Contracts that tie compensation toperformance can mitigate incentive problems, butwriting completely effective contracts is oftenimpractical.1 As a result, real-world incentives fre-quently are informal. Within firms, compensa-tion and promotion often are based on difficultto verify aspects of performance such as team-work, leadership, or initiative. Employees un-derstand this without the precise details beingcodified in a formal contract and firms live up totheir promises because they care about theirlabor market reputation.2 Similarly, firms often

expect a level of flexibility and adaptation fromsuppliers that goes well beyond contractual re-quirements. The give and take of the relation-ship allows prices or details of delivery to adjustin response to specific circumstances.

The need for a relational contract is a matterof degree. Sovereign nations comply with tradeagreements and repay foreign debt because theydesire the continued goodwill of trading part-ners. Politicians have an incentive to assist largedonors because they will need to raise money infuture campaigns. On the other hand, when afirm contracts with its employees or other firms,or when a government agency regulates an in-dustry, a formal contract may provide a reason-able starting point. In these cases, good faithallows for more flexibility and nuance in incor-porating information.

Information plays the same role in relationalcontracting as in standard incentive theory. Bet-ter performance measures generate more effec-tive incentives. But a relational contract canincorporate a much broader range of subjectiveinformation. For instance, the best gauge ofemployee performance is often the subjectiveevaluations of peers or supervisors. Firms reg-ularly use such measures for compensation andpromotion decisions. In a survey of law firmcompensation by the consulting firm AltmanWeil, Inc., 50 percent of law firms report using

* Department of Economics, Stanford University, Stanford,CA 94305 (e-mail: [email protected]). This paper is arevised chapter of my M.I.T. Ph.D. thesis. I am indebted toGlenn Ellison, Robert Gibbons, Jerry Hausman, and BengtHolmstrom for their generous advice and encouragement. Thepaper has benefited from the suggestions of Susan Athey,Drew Fudenberg, Eric Maskin, Richard Levin, Paul Oyer,Antonio Rangel, Lars Stole, Steve Tadelis, Jean Tirole, and thereferees. I thank the NSF for financial support.

1 For example, it may not be practical to precisely cap-ture what is desired in a written contract. It may be difficulteven to explain the details of performance to an outsidersuch as the court. In some cases, the best measure ofperformance may be a subjective judgment. In other cases,shortcomings in the court system itself may prevent effec-tive contractual enforcement.

2 As an example of failed relational contracting, considerthe case of United Airlines in the summer of 2001. Duringcontract negotiations, the pilots’ union urged its members towork “to the letter of our agreement.” What followed was asummer of delays and cancelled flights that cost United$700 million. United’s management contemplated going tocourt, but determined that it could not afford to further

antagonize the pilots. Ultimately, negotiations resulted in alarge salary increase (Roger Lowenstein, 2002).

835

subjective measures of performance to deter-mine partner compensation (Altman Weil,2000). Subjective measures are also commonlyused by Wall Street firms to help determine theyear-end discretionary bonuses that constitute asignificant fraction of compensation.

The combination of subjectivity and discre-tion can give rise to costly differences of opin-ion. James B. Stewart (1993) describes theturmoil at First Boston in the early 1990’s afteryear-end bonuses were lower than expected.Senior managers were incensed at what theyperceived as a breach of trust. MeanwhileCredit Suisse, First Boston’s parent company,argued that the bonuses merely reflected disap-pointing financial results. The dispute endedwith many top managers leaving the firm. LisaEndlich (1999) details a related wave of depar-tures that occurred when Goldman Sachs dra-matically cut year-end bonuses in 1994.Employees at all levels, who felt they had beenpromised more, quit despite the fact that seniormanagement was predicting rapid growth.

In this paper, I provide a simple analysis ofoptimal relational contracting in a broad rangeof settings. This includes settings where thereare problems of moral hazard or hidden infor-mation as well as environments where perfor-mance evaluation is purely subjective. I firstshow that to characterize optimal contracts itoften suffices to focus on a simple class ofstationary contracts in which the same compen-sation scheme is used at every date. Using thisresult, I explain how self-enforcement limits thescope of incentive provision and characterizeoptimal incentive schemes. I also show how theuse of subjective performance measures necessar-ily gives rises to costly disputes, although optimalincentive contracts are still relatively simple.

The results are established in a general andbroadly applicable agency setting. There are aprincipal and agent who can trade repeatedly.They can contractually specify some fixed pay-ment and can incorporate further performancemeasures in a relational contract that specifieshow payments and future trade will adjust toreflect performance. In the case of employment,the fixed compensation can be interpreted as asalary, while the relational contract describeshow discretionary bonuses, raises, and promo-tions will be awarded based on performance.

In principle a relational contract can be quitecomplicated because performance in any givenperiod can affect the whole future course ofthe relationship. Nevertheless, I show that tocharacterize Pareto-optimal contracts in set-tings of moral hazard or hidden information itsuffices to consider only stationary contractsin which the principal promises the samecontingent compensation in each period. Thiscan be understood as follows. Even with acomplex agreement, the parties have only twoinstruments to provide incentives—presentcompensation and continuation payoffs. Theseinstruments are substitutable given risk neutral-ity. Thus an incentive scheme that uses contin-uation payoffs to reward or punish an agent canbe replicated by a scheme that uses only imme-diate compensation. The one caveat is that di-rect compensation cannot replicate an outcomethat punishes both parties. However, I show thatwhen the principal’s behavior is perfectly ob-served, optimal contracts never involve jointpunishments in equilibrium. It follows that sta-tionary contracts are optimal.

In economic terms, this result says that rela-tionships should optimally respond to variableoutcomes through adjustments in price ratherthan through changes in the underlying incentives.In practice, a pricing system that grants discountsto customers whose last delivery was flawed, ora compensation system that bases raises or bo-nuses on a review of the previous year’s perfor-mance, can serve precisely this purpose.

The simple structure of stationary contractsmakes it easy to characterize optimal relationalincentive schemes. These schemes differ fromstandard incentive theory because contracts areself-enforced. Under a relational contract, eachparty has an incentive to pay promised compen-sation because reneging would bias future tradeterms against the deviator or even end the rela-tionship. But because each party has the optionto walk away, the gap between the highest andlowest payments promised as a function of per-formance cannot exceed the present value of therelationship. This constraint limits what can beachieved in terms of incentives.

As an example, suppose that the agent hasprivate information each period about his costconditions. A well-known result is that optimalscreening contracts will perfectly separate cost

836 THE AMERICAN ECONOMIC REVIEW JUNE 2003

types under certain regularity conditions. More-over, such contracts never distort the choices ofthe most efficient agents. In contrast, when self-enforcement is a binding constraint, an optimalrelational contract never fully distinguishes be-tween different cost conditions and never asks forefficient effort. Indeed, the agent may be asked toprovide the same (inefficient) performance with-out any regard for his cost conditions. These dis-tortions arise because self-enforcement creates ashadow cost to providing effective incentives bylimiting the variability of compensation.

The model predicts that relationships will bemore flexible in adapting to changing cost con-ditions when trading is more frequent or whenparties rely more on each other. The literatureon vertical supply contracting suggests thatadaptability is a key feature of successful long-term relationships (Oliver E. Williamson,1985). Here, if a relationship does not generatea great deal of surplus, the terms of the rela-tional contract will be quite rigid. In contrast, ifthe relationship is more productive, there willbe scope to fine-tune performance to currentconditions.

The use of subjective performance measurespresents problems that do not arise when thereis asymmetric information about costs or whenperformance measures are simply noisy. Thereason is that a successful contract must simul-taneously give the agent an incentive to performand the principal an incentive to assess perfor-mance honestly. This makes stationary con-tracts ineffective. Nevertheless, if the principalreviews performance each period, an optimalcontract still has a straightforward structure.The principal pays the agent a base salary eachperiod, adding a fixed bonus if she judges per-formance to be above some threshold. If theprincipal claims nonperformance and withholdsthe bonus, a dispute results and the agent walksaway. The optimal contract displays both in-formation compression—the reduction of apotentially nuanced signal into just two paylevels—and conflict. Interestingly, conflict ofthis sort does not arise in earlier relationalcontracting models where parties can agree onperformance measures. In contrast, the combi-nation of subjectivity and discretion provides anexplanation for the real-world compensationdisputes described above.

Related Literature.—Many authors, and notjust economists, have emphasized the role ofongoing relationships in contracting [early ref-erences include Stewart Macaulay (1963) andIan R. Macneil (1974)]. In contrast to thepresent paper, previous relational contractingmodels have generally focused on environmentswhere the parties have symmetric information[for instance, Benjamin Klein and Keith Leffler(1981); Carl Shapiro and Joseph E. Stiglitz(1984); Clive Bull (1987); and David M. Kreps(1990)]. W. Bentley MacLeod and James M.Malcomson (1989) provide a definitive treat-ment of this model. They show that optimalcontracts can take a variety of forms: fromefficiency wages where the principal pays ahigh fixed salary and threatens to terminate anagent for poor performance, to a system ofperformance bonuses paid by the principal in eachperiod (see also MacLeod and Malcomson, 1998).

The assumption of symmetric informationcontrasts with the traditional incentive theoryview that asymmetric information, rather thanenforcement, is the central impediment to effec-tive contracting (e.g., Bengt Holmstrom, 1979;Jean-Jacques Laffont and Jean Tirole, 1993).This paper brings these views together. I showthat the connection is in fact quite close. Inessence, with risk neutrality, optimal relationalcontracts are distinguished from standard opti-mal incentive contracts by the presence of asingle enforcement constraint that limits therange of contingent payments.3 This is true evenif there is hidden information, moral hazard, orsubjective performance evaluation.

In practice, real-world contracting often in-volves both formal and relational incentives. Forinstance, a procurement contract may include ex-plicit penalties for late delivery, while the prospectof getting repeat business supplies the incentive tobe flexible in making (nonverifiable) adaptations.George Baker et al. (1994) develop an agencymodel in which the principal has available both a

3 A dynamic contracting literature in macroeconomicsincludes some papers that model contractual self-enforcement.Early examples include Jonathan Thomas and Tim Worrall(1988) and Andrew Atkeson (1991). This line of researchfocuses on the insurance and consumption-smoothing as-pects of long-term relationships, problems that are absent inthe present risk-neutral setup.

837VOL. 93 NO. 3 LEVIN: RELATIONAL INCENTIVE CONTRACTS

verifiable and a nonverifiable performance mea-sure. They show that the principal can benefitfrom using both measures, but do not characterizefully optimal agreements.4 Here I focus only onnonverifiable performance measures, but many ofthe results on optimal contracting extend to envi-ronments where richer formal contracts are possi-ble (see Section II for discussion).

B. Douglas Bernheim and Michael D. Whin-ston (1998) develop the theme of formal andinformal incentives further, arguing that whensome performance measures are not verifiable,parties may deliberately leave formal contractsincomplete. This “strategic ambiguity” allows acontract to be adapted in the face of observablebut nonverifiable information. Effective con-tracts in this paper have precisely the strategicambiguity described by Bernheim and Whin-ston—the parties specify a fixed payment butleave open the possibility of adjusting this pay-ment ex post. This flexibility is crucial becauseit allows good performance to be rewarded.

The basic problem I address—how to struc-ture contractual relationships when the partieshave useful information that cannot be effec-tively translated into an enforceable formalcontract—also bears some relationship to anal-yses of the hold-up problem. Hold-ups arisewhen one party to a relationship makes a spe-cific investment that is not contractible. Withoutcontractual protection, the investing party maybe unable to appropriate the returns to his in-vestment. There are many potential responses tothis problem. Sanford Grossman and OliverHart (1986) argue that vertical integration cansubstitute for contractual protection, whileAaron S. Edlin and Stefan Reichelstein (1996)show that standard legal remedies such as spe-cific performance and expectation damages maysuffice to encourage investment.

These remedies to noncontractibility do notdepend on repeated interaction, but rather on theability of the parties to renegotiate the terms oftrade after performance is observed but beforetrade occurs. The parties structure their initialagreement to frame this renegotiation. In con-

trast, I focus on problems where the benefit ofthe agent’s action accrues directly to the prin-cipal. By the time performance can be assessed,there is nothing left to negotiate over. Thepresent model naturally describes situationswhere an agent performs an ongoing service forthe principal (such as an employment relation-ship), or provides a product whose quality canonly be observed in the process of using it (suchas the advice provided by a consulting firm). Onthe other hand, hold-ups often are associatedwith large discrete investments of the sort thatmight take place at the start of a relationship.5

Finally, my analysis of the dynamic structure ofrelational contracts makes use of the repeated gametechniques developed by Dilip Abreu et al. (1990)and Drew Fudenberg et al. (1994). The crucialdistinction between relational contracting and themodels studied in those papers is that here theparties can use monetary transfers as well as changesin equilibrium behavior to provide incentives. Idiscuss this point in more detail in Section II.

I. The Model

A. The Agency Problem

I consider two risk-neutral parties, a principaland an agent, who have the opportunity to tradeat dates t � 0, 1, 2, ... . At the beginning ofdate t, the principal proposes a compensationpackage to the agent. Compensation consists ofa fixed salary wt and a contingent paymentbt : � 3 �, where � is the set of observedperformance outcomes, to be described momen-tarily. The agent chooses whether or not toaccept the principal’s offer. Let dt � {0, 1}denote the agent’s decision.6

If the agent accepts, he observes a costparameter � t, representing variable aspects ofthe environment such as task difficulty, thecost of materials, or the opportunity cost oftime. The cost parameter is an independentdraw from a distribution P� with support� � [�, �� ]. The agent then chooses an action

4 See also Klaus Schmidt and Monika Schnitzer (1995),David G. Pearce and Ennio Stacchetti (1998), and LuisRayo (2002).

5 Baker et al. (2002) integrate relational contracting withthe hold-up literature by developing a model of repeatedspecific investments.

6 Allowing the agent to propose compensation, and theprincipal to accept or reject, leads to an equivalent analysis.


et � E � [0, e� ], incurring a cost c(et, � t).The agent’s cost is increasing in e with c(e �0, � ) � 0 for all �. The agent’s actiongenerates a stochastic benefit yt for the prin-cipal, where yt has distribution F( � �e) andsupport Y � [ y, y� ]. Then S(e, � ) � �y[ y�e]� c(e, � ) denotes the expected joint surplusas a function of � and e.

During the course of trade, the agent ob-serves all of the relevant information: his costparameter � t, his action et and the principal’sbenefit yt. The principal observes her benefityt, but may or may not observe � t and et. Inthis way, the model allows for problems ofhidden information (� is private) and moralhazard (e is private) as well as symmetricinformation. The performance outcome is thesubset � t � {� t, et, yt} observed by bothparties. Let � be the set of possible realiza-tions of � t.

7

At the end of date t, the principal is obligatedto pay the fixed salary wt. The parties thenchoose whether to adjust this by the contingentpayment bt(�t). This decision belongs to theprincipal if bt � 0 and to the agent if bt � 0.Let Wt denote the total payment actually madefrom principal to agent—either wt � bt(�t) ifthe agreed payment is made, or wt if not. Thusthe agent’s payoff is Wt � c(et, �t) while theprincipal’s is yt � Wt.

If the agent rejects the principal’s originaloffer, both parties receive fixed payoffs: u� forthe agent and �� for the principal. I assume theseoutside opportunities are less desirable than ef-ficient trade, but sufficiently attractive that theparties weakly prefer not to trade if the agentcannot be given incentives to perform. Definings� � u� � �� , I assume that for all �, maxeS(e,� ) � s� � S(0, � ).

Over the course of repeated interaction, theparties care about their discounted payoffstream. Starting from date t, the respective pay-offs for principal and agent, discounted by acommon discount factor � � [0, 1), are:

� t � 1 � �� t

�

�� t�d� y� � W�

1 � d� �� ,

ut � 1 � �� t

�

�� t�d� W�

� ce� , �� 1 � d� u� .

Here, I multiply through by 1 � � to expresspayoffs as a per-period average.

B. Relational Contracts

Because contingent compensation is merelypromised, there is a temptation to renege on pay-ments once production occurs. In a one-time in-teraction this means that the principal can crediblypromise only the fixed salary w. As this providesno performance incentive, mutually beneficialtrade cannot occur in static equilibrium and bothparties receive their outside options. In contrast,ongoing interaction allows the parties to base fu-ture terms of trade on the success of presenttrade—potentially allowing for a good faith agree-ment that provides effective incentives.

A relational contract is a complete plan forthe relationship. Let ht � (w0, d0, �0, W0, ... ,wt � 1, dt � 1, �t � 1, Wt � 1) denote the historyup to date t, and H t the set of possible date thistories. Then for each date t and every historyht � H t, a relational contract describes: (i) thecompensation the principal should offer (andwhich should be paid); (ii) whether the agentshould accept or reject the offer; and in theevent of acceptance, (iii) the action the agentshould take as a function of his realized costs �t.Such a contract is self-enforcing if it describes aperfect public equilibrium of the repeatedgame.8 Note that a relational contract describesbehavior both on the equilibrium path and off—for instance after a party reneges on a payment.

7 If both � t and et are private to the agent, the par-ties can generally benefit from having the agent an-nounce � t prior to choosing et and using thisannouncement to tailor incentives. I ignore this possibil-ity as it does not benefit the parties in the main cases ofinterest.

8 Perfect public equilibrium requires that play followingeach history be a Nash equilibrium. Here, it essentiallyimposes the same sequential rationality requirement thatsubgame perfection would impose in a complete informa-tion model (Fudenberg et al., 1994).


In general, there may be many self-enforcingcontracts, leading to a question of which toselect. In an important sense, however, theproblem of efficient contracting can be sepa-rated from the problem of distribution: if thereis a self-enforcing contract that generates somesurplus beyond the outside surplus s� , it can bedivided in any way that respects the parties’participation constraints.

THEOREM 1: If there is a self-enforcingcontract that generates expected surpluss � s�, then there are self-enforcing contracts thatgive as expected payoffs any pair (u, �) satis-fying u � u� , � � �� and u � � s.

The reasoning behind Theorem 1 is that bychanging the fixed compensation in the initialperiod of the contract, the parties can redistrib-ute surplus without changing the incentives thatare provided after the initial compensation isproposed and accepted.

Given this result, it is natural to focus oncontracts that maximize the parties’ joint sur-plus (subject to the constraint of being self-enforcing). I say that a self-enforced contract isoptimal if no other self-enforcing contract gen-erates higher expected surplus.

II. Stationary Contracts

This section shows that the problem of de-scribing optimal relational contracts can be sig-nificantly simplified by focusing on contractsthat take a stationary form.

Definition 1: A contract is stationary if on theequilibrium path Wt � w � b(�t) and et �e(�t) at every date t, for some w � �, b : �3�, and e : � 3 E.

Under a stationary contract, the principalalways offers the same compensation planw � � and b : � 3 �, and the agent alwaysacts according to the same rule e : � 3 E.The contract must also specify what happensif a party fails to make a specified payment orif the principal fails to offer the expectedcompensation. Since these events never occurin equilibrium, there is no loss in assumingthat the parties respond by breaking off trade,

as this is the worst equilibrium outcome(Abreu, 1988).9

THEOREM 2: If an optimal contract exists,there are stationary contracts that are optimal.

Theorem 2 says that instead of conditioningfuture trade on the agent’s performance, theparties can “settle up” each period using discre-tionary payments and then proceed to the sameoptimal agreement at the next date.

To understand the argument behind Theorem2, consider a simple example where the agenthas constant cost conditions and chooses aneffort that results in either Low or High output.To induce the agent to exert effort, an agree-ment must reward the agent following Highoutput. This reward could come in one of twoforms: through an immediate payment or frommoving to a continuation equilibrium that givesa high expected payoff. Given risk neutrality,the agent views these as equivalent. Thus, forany contract that relies on changes in the equi-librium behavior to provide incentives, it shouldbe possible to provide the same incentives withpayments alone. The potential catch is that dis-cretionary payments can redistribute surplus butcannot replicate the creation or destruction ofsurplus that might occur if, for instance, thecontract called for the agent to be fired follow-ing Low output. Because the principal’s behav-ior is observed perfectly, however, an optimalcontract would never call for surplus to be de-stroyed on the equilibrium path. Thus stationarycontracts suffice for optimality.10

One small point pertains to the issue of rene-gotiation. The proof of Theorem 2 shows thatoptimal stationary contracts can be constructedto split the surplus arbitrarily provided the par-ticipation constraints are satisfied. This meansthat even in the event of a deviation there is no

9 More specifically, one can assume the parties revert tostatically optimizing behavior. This means that if the prin-cipal deviates by offering (w�, b�) � (w, b) with w� � u� ,the agent will accept the offer but exert zero effort.

10 In contrast a contract that generates less than the optimalsurplus might call for variation in the joint surplus over time.By placing some additional structure on the model, however,Theorem 2 can be strengthened to say that a stationary contractcan achieve any surplus between s� and the optimal surplus.


need to terminate the relationship. Instead, theterms of trade can be altered to hold the deviatingparty to his or her outside option. Say that aself-enforcing contract is strongly optimal if givenany history ht � H t, the continuation contractfrom ht is optimal. A strongly optimal contract hasthe desirable property that even off the equilib-rium path the parties cannot jointly benefit fromrenegotiating to a new self-enforcing contract.

COROLLARY 1: For any optimal stationarycontract, there is a strongly optimal stationarycontract with the same equilibrium behavior.

The results in this section can be applied to avariety of other contracting settings. To see whythis is the case, it is useful to connect with thegeneral theory of repeated games. Recall that ina repeated game with perfect information, opti-mal equilibria can always be stationary becausedeviations are never observed in equilibrium(Abreu, 1988). This observation applies toMacLeod and Malcomson’s (1989) symmetricinformation model of relational contracting.Here, however, the agent’s behavior is not per-fectly observed. Performance might appear goodor bad in equilibrium and must be rewarded andpunished accordingly. In the models of EdwardJ. Green and Robert H. Porter (1984), Abreu etal. (1990), Fudenberg et al. (1994), and SusanAthey and Kyle Bagwell (2001), it is preciselythis sort of imperfect monitoring that cause op-timal equilibria to be nonstationary.

Two features make stationary relational con-tracts optimal despite monitoring imperfections.First, the combination of quasi-linear utility andmonetary transfers means discretionary trans-fers can substitute for variation in continuationpayoffs.11 Second, the fact that the principal’sactions are perfectly observed means that thesetransfers can be balanced.12 Because Theorem 2

continues to hold in any repeated game settingwith these features, it can be used to character-ize optimal relational contracts in environmentswell beyond the ones here. One example is themodel considered by Baker et al. (2002) in theirstudy of relational contracts as a solution torepeated hold-up problems.

III. Optimal Incentive Provision

The previous section showed that in a broadrange of environments it is possible to study op-timal relational contracts by considering only asmaller class of stationary agreements. This sec-tion uses this observation to study optimal incen-tive provision in response to problems of hiddeninformation and moral hazard. These two infor-mational problems have been the central focus ofresearch in incentive theory and its applications tolabor and credit markets, procurement and regu-lation, macroeconomics and international trade.

I start by observing that relational contractshave a particular limitation. Because partieshave the option to renege on payments, there arebounds on the compensation that can be credi-bly promised. I show how these bounds affectthe structure of optimal incentives and how theresulting distortions differ from those in theclassic incentive theory models where contractenforcement is not a problem.

A stationary contract is described by an effortschedule e(� ) and a payment plan W(�) �w � b(�) that the principal offers the agentin each period. Such a contract provides per-period payoffs:

� � ��,y �y � W��e � e��,

u � ��,y �W� � c�e � e��.

The expected joint surplus depends only on theeffort schedule:

s � ��,y �y � c�e��.

For the compensation schedule to be

11 The underlying connection between continuation payoffsin repeated games and monetary payments in contracting prob-lems is already highlighted, and plays a fundamental role, inFudenberg et al.’s (1994) derivation of the Folk Theorem.

12 This follows because optimal continuation payoffs arenecessarily on the (linear) frontier of the equilibrium payoffset. Note that this type of sequential optimality is also afeature of the complete contracting agency models ofFudenberg et al. (1990) and Malcomson and Frans

Spinnewyn (1988). In those models, risk aversion can createan intertemporal smoothing reason to use future utility toreward good performance.


self-enforcing, neither the principal nor the agentmust have an incentive to renege. Assuming thatfailure to pay results in the worst possible futureoutcome, a self-enforcing contract must satisfy

�

1 � �� sup

�

b�

so that the principal will not walk away, and

�

1 � �u � u� � �inf

�

b�

so that the agent will not walk away.A key point here is that by adjusting the fixed

component of compensation, w, up or down,slack can be transferred from one payment con-straint to the other. Thus what matters is thedifference between the highest payment speci-fied under the contract and the lowest payment.This observation allows a characterization ofthe set of self-enforcing stationary contracts.

THEOREM 3: An effort schedule e(�) that gen-erates expected surplus s can be implemented witha stationary contract if and only if there is apayment schedule W : �3 � satisfying:

(IC) e� � arg maxe

�y�W��e� � ce, �

for all � � �, and

(DE)�

1 � �s � s� � sup

�

W� � inf�

W�.

The dynamic enforcement (DE) constraintstates that the variation in contingent paymentsis limited by the future gains from the relation-ship. The tightness of this restriction depends onthe discount factor �, as well as on the produc-tivity of the relationship relative to the outsideoption s� . If � is near one, the range of paymentsis essentially unbounded.13 On the other hand, if

� is near zero, the range of payments may be solimited that incentives cannot be provided. Theinteresting question that I now address is whathappens when self-enforcement is a bindingconstraint, but trade is nevertheless possible.

A. Hidden Information

Problems of hidden information arise when theprincipal cannot observe the precise details of theagent’s environment. A classic example is a reg-ulatory agency that can observe the quality andquantity of the delivered product, but not the mar-ginal costs of the regulated firm. Efficiency dic-tates lowering price and increasing productionwhen costs are low, but regulated firms alwayshave an incentive to claim high costs. Similarproblems arise in procurement if production costsare unknown to the procuring firm. I capture thisby assuming the principal can observe the agent’sbehavior et but not his cost parameter �t.

An optimal relational contract displays twostriking features when enforcement is a bindingconstraint. First, it distorts production away fromthe efficient level regardless of realized cost con-ditions. Second, even under the regularity condi-tions that ensure full separation of cost types instandard incentive theory models, an optimal re-lational contract does not fully distinguish be-tween different cost conditions. Indeed, ifenforcement is very limited, it requires the sameproduction level regardless of cost conditions.

To proceed, I place some further structure onthe model. I take both total and marginal coststo be increasing in � and make standard assump-tions about the production process. I assumethat effort choice is continuous, E � [0, e� ], andthat the cost function c is differentiable with ce,cee � 0, c� � 0, ce� � 0, and c�ee, c�e� � 0.I also assume that for each �, S(e, � ) is differ-entiable and concave in e with an interior max-imizer eFB(� ). Finally I assume the distributionof costs P is concave and admits a density p.These conditions ensure that the optimal con-tracting problem is concave and would alsoensure full separation of cost types (i.e., nopooling) in a standard self-selection setting.14

13 As a consequence, the principal can induce first-besteffort. As the agent is risk neutral, there may be many waysto do this, but one way is to pay the agent the full benefit yminus some constant.

14 The usual regularity assumption is that log P is con-cave (i.e., that P/p is nondecreasing), which is implied by


The first step to characterizing optimal incen-tives is to specialize Theorem 3 to the hiddeninformation context.

THEOREM 4: With hidden information, an ef-fort schedule e(�) that generates expected surpluss can be implemented by a stationary contract ifand only if (i) e(�) is nonincreasing and (ii)

(IC–DE)�

1 � �s � s�

� ce�, �

��

��

c� e�, � d�.

Theorem 4 uses self-selection arguments tocombine the (IC) and (DE) constraints fromTheorem 3. The basic idea is that the paymentschedule W : [0, e� ] 3 � must perform twotasks. First, it must induce each type � to selectthe designated effort e(� ) rather than mimick-ing some other type � � � and choosing e(�)(similar to standard incentive compatibility).Second, it must deter all types from choosing aneffort e that is not on the schedule (similar tostandard interim participation).

Of these latter deviations, none is more at-tractive than choosing zero effort. Thus takeW(0) to be the lowest payment. The compen-sation schedule is then pinned down by self-selection: for all �,

(1) We� � W0 ce�, �

��

��

c� e�, � d�.

The final term is the information rent to in-duce self-selection. The highest transfer is W(e(�))and dynamic enforcement implies that W(e(�)) �W(0) must be less than the future gains from therelationship. This gives Theorem 4.

The optimal contract solves:

maxe�

s � ��

��

��y�e�� ce�, � dP�

subject to�

1 � �s � s�

� ce�, � ��

��

c� e�, � d�,

and e � is nonincreasing.

The standard approach to solving self-selectionproblems is to ignore the monotonicity con-straint [that e(� ) is nonincreasing] and solve therelaxed problem by pointwise optimization. Itturns out, however, that the monotonicity con-straint always binds so that approach cannot beused.

Instead I adopt a control theory approach.Define �(� ) � e(� ) as the control variable, andlet � and (�) denote the multipliers on, respec-tively, the (IC–DE) and monotonicity con-straints. The Appendix shows that a necessaryand sufficient condition for e(� ) [along with�(�), �, (�)] to solve the optimal contractproblem is that:

(2) Se e�, �p�

��

1 ��

1 � �

� ce� e�, � �1

� �� .

At the optimum, the marginal benefit of in-creasing the agent’s effort in the event of eachcost realization � is equated with a shadow costarising from the (IC–DE) and monotonicityconstraints. The solution must also satisfy thecomplementary slackness conditions:

(3) � � 0, �� 0 and �� 0,

and two boundary conditions:

(4) � � �ce e�, � and �� 0.

the assumption that P is concave. This assumption and theassumption that eFB(� ) is interior are easily relaxed. SeeAppendix B for a discussion.


The form of the optimal schedule depends onthe degree to which self-enforcement is a bind-ing constraint. At one extreme, if � is small oroutside opportunities appealing, no schedulemay satisfy the constraints making productionunder a relational contract impossible. At theother extreme, the (IC–DE) constraint is slack.Then � � 0 and the first-best schedule eFB(� )is optimally chosen.

Interesting outcomes arise when productionis possible but self-enforcement binds so that � �0. In this case, the boundary conditions imply that (�) � 0, so by complementary slackness e(�) �0. That is, the most efficient cost types are pooledat a single effort level. This then gives rise to twopossibilities. When the enforcement constraint isvery restrictive, all types are pooled at some con-stant effort level. When the enforcement con-straint is somewhat less restrictive, the mostefficient types are pooled at a relatively high effortlevel, while less efficient types are screened toprogressively lower effort levels.

THEOREM 5: If production is possible undera relational contract with hidden information,the optimal effort schedule e(� ) takes one ofthree forms:

1. Pooling: e(� ) is the same for all cost types.2. Partial Pooling: e(� ) is constant on [�, �],

and strictly decreasing on (�, ��], for some� � (�, ��).

3. First-Best: e(� ) � eFB(� ) for all cost types.

In either second-best scenario, e(� ) �eFB(� ) for all �.

Surprisingly, second-best contracts call forinefficient effort regardless of cost conditions.There is a simple intuition for this. Asking formore effort from a given cost type requires anincrease in the slope of the transfer schedule.Because the total variation in payments is lim-ited, this will mean decreasing the incentives ofsome other cost type. Consequently, requiringan efficient level of effort for a given cost typecannot be optimal: at the margin, it would havezero benefit and a positive shadow cost.

The limit on compensation is also responsiblefor the pooling or partial pooling in a second-bestcontract. Starting from zero effort, compensation

need not be increased much to raise the effort leveluniformly, so it is optimal to do so. But as theeffort level rises, it becomes expensive to raise theeffort of less efficient types and hence optimal toscreen them to lower levels of effort while theeffort level of more efficient types continues to rise.

One implication is that optimal contracts aremore flexible when the surplus from a relation-ship is greater or the interaction more frequent.Relationships that create a small amount of sur-plus cannot adapt to changing cost conditions.As an illustration, imagine a supply relationshipin manufacturing. Even with simple fixed-pricecontracts each period, a functioning relationshipmay allow a base level of noncontractible qual-ity above the bare minimum. A close supplyrelationship can make it possible to tailor detailsof delivery or specific quality requirementsmore closely to current cost conditions, even ifthese conditions are known only to one party.

In the present setting, an optimal contractbalances efficiency with what is effectively alimit on total incentives. In contrast, in the clas-sical screening problem where the principal of-fers a profit-maximizing contract to the agentsubject to an interim participation constraint,the fundamental trade-off is between efficiencyand rent extraction. Increased incentives boostefficiency but mean that the principal mustleave information rents to the agent. For a giventype �, this leads to a shadow cost of incentivesthat is proportional to P(� )/p(�). Since P(�) �0, the most efficient type produces efficiently,but not the others. Here, Theorem 1 implies thatrent extraction is not an issue; rather, the prob-lem is self-enforcement. The different trade-offsgenerate quite different predictions.

Although the motivation is different, the limiton contingent compensation imposed by self-enforcement also can be interpreted as repre-senting a form of limited liability—i.e., arequirement that W� � W(e) � W� for some(here endogenous) liability limits W� and W� .Models of contracting under limited liabilitydate back to David Sappington (1983), whostudied hidden information problems under theone-sided constraint that W(e) � 0.15 Sapping-

15 Many papers study moral hazard problems with aone-sided limited liability constraint on the agent. The idea


ton’s work showed that a one-sided constrainthas the same qualitative effect as the standardinterim participation constraint. An implicationof Theorem 5 is that hidden information incen-tives differ dramatically when both parties havelimited resources.

B. Moral Hazard

With hidden information, optimal contractscan reflect a rich trade-off between screening,incentives, and enforcement. In moral hazardenvironments, where the principal observes only anoisy measure of performance, self-enforce-ment leads to a very stark form of optimalcontract. Although there may be many levels ofmeasured performance, an optimal contractcompresses this information into just two levelsof compensation. The principal sets a base pay-ment that is adjusted up if yt exceeds somethreshold or down if yt misses the threshold.

To focus on moral hazard, I now assume thatthe agent’s cost parameter �t is observable butthat his action et is not. Thus a stationary con-tract specifies compensation W(�, y) � w �b(�, y) and an associated effort schedule e(� ).As in the previous section, I assume that effortis a continuous choice and that ce, cee � 0. Inaddition, I assume the Mirrlees-Rogerson con-ditions are satisfied: that is, the density ofF( y�e) has the monotone likelihood ratio prop-erty and F( y�e � c�1( x; � )) is convex in x forany �.16

THEOREM 6: An optimal contract implementssome effort schedule e(� ) eFB(� ). For each�, either e(� ) � eFB(� ) or e(� ) � eFB(� ) andthe payments W(�, y) are “one-step”: W(�, y)equals W� for all y � y(� ) and W� for all y �

y(� ), where W� � W� �

1 � �s � s� and

y(� ) is the point at which the likelihood ratio( fe/f )( y�e(� )) switches from negative to posi-tive as a function of y.

The intuition for Theorem 6 is that the par-ties’ risk neutrality makes it desirable to use thestrongest possible incentives. Self-enforcementlimits the maximal reward and punishment. The“one-step” scheme that results is reminiscent ofthe static model of Robert D. Innes (1990).Innes shows that under a one-sided limited lia-bility requirement that W( y) � 0 and the fur-ther requirement that W( y) y, the optimalcontract pays 0 for low outcomes and y for highoutcomes. Here self-enforcement imposes asimilar lower bound and also a fixed upper bound.These bounds define the two payment levels.

IV. Subjective Performance Measures

The previous sections assumed that althoughperformance measures might be imperfect andnoncontractible, the parties at least could agree onthem. In practice, relational contracting often in-volves performance measures that are inherentlysubjective—for instance, the opinions of supervi-sors or peers. Firms that rely heavily on theirincentive systems, such as the Lincoln ElectricCompany (Norman Fast and Norman Berg, 1975),often make extensive use of subjective perfor-mance reviews. Incorporating subjective assess-ments allows for a more nuanced form ofcompensation, but also creates the potential fordisputes over how to interpret past performance.

I model subjective performance measurementby assuming that the agent’s action et is pri-vately chosen, while the principal privately ob-serves the output yt. That is, the principal has anoisy signal of the agent’s performance notbased on an objective measure the agent canobserve. Having learned yt, the principal candeliver a report mt � M (where M is somelarge set of possible messages). Compensationat time t is composed of a base payment wt �� and an adjustment bt : M3 �. For simplic-ity, I suppose that the agent’s environment isnot stochastic, i.e., c : [0, e� ] 3 � is the samein each period, and I maintain the assumptionson costs and output from the previous section.

With subjective performance evaluation,stationary contracts are no longer effective. To

is that it becomes costly (in terms of rents) for the principalto provide incentives even if the agent is risk neutral.Rosella Levaggi (1999) sets up, but does not solve out, ahidden information problem with two-sided limited liabil-ity.

16 These conditions ensure that the agent’s incentivecompatibility constraint can be replaced with a first-ordercondition for his action choice (William P. Rogerson,1985).


see why, observe that the principal will only bewilling to make distinct reports m and m� inresponse to distinct outcomes y and y� if the tworeports yield the same future profits [under astationary contract, this would require thatb(m) � b(m�)]. Thus if a relational contract isto provide both productive and monitoring in-centives, the agent’s payoff going forward mustvary with performance but the principal’s mustnot. It follows that production cannot be optimalafter every history. Joint punishments or dis-putes must occur in equilibrium.

The question then is how best to provideincentives for the agent while inducing the prin-cipal to make honest evaluations. This problemis complicated by the great flexibility in struc-turing performance reviews over time.17 Tomake progress, I focus on contracts where theprincipal provides a full performance evaluationafter each period. That is, I consider contractswith the following full review property: givenany history up to t and compensation offer at t,then any two outputs yt � y�t must generatedistinct reports mt � m�t.

18 Given this restric-tion, it is natural to suppose that these contractssimply call for the principal to report her trueassessment at each date—i.e., call for her toreport mt � yt.

To characterize optimal full review contracts,I first define a simple form of termination con-tract. Like a stationary contract, a terminationcontract requires the same compensation in eachperiod that trade occurs, but it also allows forthe possibility that the parties might dissolve therelationship.

Definition 2: A contract is a termination con-tract if in every period t that trade occurs, Wt �w � b( yt), et � e, mt � yt and trade continuesbeyond t with probability �t � �(yt) and other-wise ceases forever, for some w � �, b : Y3 �,e � [0, e�], and � : Y3 [0, 1].

In environments with subjective performancemeasures, termination contracts are optimalamong all contracts with the full review property.The logic is by now familiar. Rather than usingvaried continuation behavior to provide incen-tives, the parties can use a combination of imme-diate compensation and termination. As above,this means that rather than having to study a rangeof potentially complex relational contracts, it ispossible to focus on a simple subset.

THEOREM 7: If an optimal full review con-tract exists, a termination contract can achievethis optimum. An optimal termination contracttakes a one-step form. It specifies some e eFB

and a threshold y: if yt � y, Wt � w and therelationship terminates; if yt � y, Wt � w �b and the relationship continues. Compensationis given by w � u� � c(e) � k and b

��

1 � �s � s� � k for some k � [0, s �

s�], where the expected surplus is

s � s� 1 � �� y � c � s� �e�

1 � ��1 � F y�e�.

The reason for the one-step form of optimalcompensation is again that the strongest incen-tives come from maximal and minimal rewards.The difference here is that the principal mustnot have a preference for reporting a bad out-come. Thus an output y � y results in no bonus,but also in separation.19

The fact that even optimal contracts involvedisputes creates a role for mediation or disputeresolution. Many firms use mediation systemsto resolve internal disputes or disputes withsuppliers. Proponents of these systems oftenargue that they help preserve important relation-

17 Specifically, this involves looking for optimal equilib-ria in a repeated game with private monitoring, a problemthat has seen little theoretical progress.

18 The full review property means that in equilibrium, theprincipal does not maintain any private information from pe-riod to period. This restriction can also be stated in terms of theequilibrium concept. I focus on perfect public equilibria of therepeated game rather than sequential equilibria.

19 An additional difference here is that an optimal con-tract cannot involve discretionary downward adjustments tothe base compensation. The reason is that a poor perfor-mance review is always followed by termination so theagent would never volunteer to take reduced compensation.An optimal contract might involve discretionary bonuses,b � 0, or simply a flat efficiency wage where effort ismotivated by the threat of being laid off. One consequenceis that explicit communication (i.e., the announcement m) isunnecessary, since the principal can “communicate” simplyby unilaterally raising compensation or by firing the agent.


ships (Todd Carver and Albert A. Vondra,1994). Here, because inefficiency results fromthe parties having different beliefs about perfor-mance, a mediator can create value by provid-ing objective information that narrows the spaceof disagreement, hence allowing a transfer thatprevents the relationship from breaking up.20

Even apart from instituting formal mecha-nisms such as mediation, one might ask whetherthere are alternative systems for subjective per-formance evaluation that improve on the onestudied here. It turns out in this regard that a fullperformance review each period is limiting in asubtle way. Some evidence of this is providedby the fact that even as � 3 1 the optimalsurplus under a full review contract remainsbounded away from the first-best as a conse-quence of endogenous separations.21 Workby Abreu et al. (1991), Olivier Compte (1998),and Michihiro Kandori and Hitoshi Matsus-hima (1998) on repeated games suggests aclever improvement. They suggest that the prin-cipal could review performance only every Tperiods. With a full review every T periods, theoptimal contract takes a similar form—a bonusfor a favorable review and termination follow-ing a poor review—but the threshold for termi-nation depends on performance in all Tpreceding periods. For sufficiently large �, thedelay in reviewing performance can improveincentives by allowing the principal to moreaccurately assess performance at each reviewdate.22

This leaves open the question of whetherthere are still more effective systems of ongoingperformance review. The previous paragraphsuggests a benefit to infrequent reviews. But inany realistic setting, there are likely to be ben-efits to providing frequent feedback that go be-yond adjusting compensation. To take anobvious example, agents cannot learn from mis-takes without being informed of them. Furtherresearch is needed to understand how perfor-mance review systems should resolve thesetrade-offs.

V. Conclusion

Good faith is an essential ingredient to manycontracting relationships. This paper has takenthe view that even in relatively complex envi-ronments where parties may not be able to pre-cisely observe each other’s costs or efforts, itmay be possible to construct good faith agree-ments that parties will live up to. These con-tracts are limited relative to what could beachieved if an infallible court system couldmonitor and enforce all agreements. Neverthe-less, I have shown that this limitation can becharacterized succinctly in the form of boundson promised compensation. As a result, optimalrelational incentive schemes can be character-ized and compared with predictions from stan-dard incentive theory. For instance, I showedthat optimal contracts will forgo screening forprivate cost information unless parties are suf-ficiently patient, and even then will involvesystematic distortions from efficiency. I alsoconsidered the difference between objective andsubjective nonverifiable measures of perfor-mance. More generally, the framework devel-oped here may perhaps find application in manyareas where incentive theory has provided fruit-ful insights—regulation, employment contract-ing, vertical supply contracting—but where theassumption that parties can perfectly committhemselves via detailed written contracts isstrained.

APPENDIX A: STATIONARY CONTRACTS

This Appendix proves Theorems 1–3 and Corollary 1. To begin, consider a contract that in itsinitial period calls for payments w, b(�), and effort e(� ). If the offer is made and accepted and thediscretionary payment made, suppose the continuation contract gives payoffs u(�), �(�) as a

20 MacLeod (2003) extends the model in this section toallow the agent to observe a signal that is correlated with theprincipal’s assessment. He shows that higher correlationincreases efficiency.

21 See the secondary Appendix, available from the au-thor, for a proof of this result.

22 In fact, a T-review contract can allow an approximatelyfirst-best outcome as �3 1 if the parties also let T3 �. Theidea is that with T-review, the threshold for termination takesthe form of a likelihood ratio test. As T3 �, the power of thistest becomes very high, allowing for very small terminationprobabilities (see the cited papers for details).


function of the observed outcome �. A deviation—an unexpected offer or rejection or a refusal tomake the discretionary payment—implies reversion to the static equilibrium. There is no loss in assum-ing this worst punishment off the equilibrium path of play (Abreu, 1988). Define W(�) � w � b(�).

Let u, � be the expected payoffs under this contract:

u � 1 � ��,y �W� � c�e�� ,y �u��e��,

� � 1 � ��,y �y � W��e�� ,y ��e��.

Let s � u � � denote the expected contract surplus:

(A1) s � 1 � ��,y �y � c�e�� ,y �s��e��.

where s(�) � u(�) � �(�) is the continuation surplus following outcome �.This contract is self-enforcing if and only if the following conditions hold: (i) the parties are

willing to initiate the contract:

u � u� and � � �� ;

(ii) for all �, the agent is willing to choose e(� ),

e� � arg maxe

�y�W� �

1 � �u��e � ce, �;

(iii) for all �, both parties are willing to make the discretionary payment,

b� �

1 � �u� �

�

1 � �u� ,

�b� �

1 � ��

�

1 � �� ;

and (iv) each continuation contract is self-enforcing. In particular, for each �, the pair u(�), �(�)correspond to a self-enforcing contract that will be initiated in the next period.

PROOF OF THEOREM 1:Consider changing the initial compensation w in the above contract. This changes the expected payoffs

u, � but not the joint surplus s. If the original contract was self-enforcing, the altered contract will stillsatisfy (ii)–(iv) as these do not depend on w, and will satisfy (i) so long as the resulting payoffs u�, ��satisfy u� � u� and �� . In this event, the new contract also will be self-enforcing.

Let s* be the maximum surplus generated by any self-enforcing contract. By Theorem 1, the set ofpayoffs possible under a self-enforcing contract is then {(u, �) : u � u�, � � �� and u � � s*}. Thusfor a given contract described as above to be self-enforcing, its continuation payoffs for each � must satisfy:

u�, �� u, � : u � u� , � � �� , and u � s* .

In particular, for all �, s� s(�) s*.

LEMMA 1: Any optimal contract is sequentially optimal. Following any history that occurs withpositive probability in equilibrium, it specifies an optimal continuation contract.


PROOF OF LEMMA 1:Consider increasing �(�) for some � in the above contract. This does not affect the effort

constraint (ii), relaxes the participation and payment constraints (i) and (iii), and is consistent witha self-enforced continuation contract, (iv), so long as u(�) � �(�) s*. Moreover, this changeincreases the expected surplus s. Thus for a contract to be optimal, it must have u(�) � �(�) �s* for all � that occur with positive probability. As this argument also applies to each continuationcontract, it follows that any optimal contract must be sequentially optimal.

PROOF OF THEOREM 2:Suppose that the contract specified above is optimal and generates surplus s � s*. By Lemma 1,

s(�) � s* for all � that occur with positive probability in the initial period. Hence if e(� ) is theinitial effort schedule, (A1) implies that

��,y �y � c�e�� s*.

I now construct a stationary contract that implements e(� ) in every period and thus is optimal. Theidea is to take the original contract and transfer any variation in the continuation payoffs u(�), �(�)into the discretionary payments.

Let u* � [u� , s* � �� ] be given, and define stationary discretionary payments:

b*� � b� �

1 � �u� �

�

1 � �u*.

From here, define the fixed payment w* so that the agent’s per-period payoff will be u*:

w* � u* � ��,y �b� � c�e��.

Let W*(�) � w* � b*(�).The stationary contract has the principal propose w*, b*(�) in each period and the agent

respond with effort e(� ). Any deviation causes reversion to static equilibrium behavior. Thiscontract gives the agent an expected payoff: ��,y[W*(�) � c�e(� )] � u* and the principal �*� s* � u*.

To see that this stationary contract is self-enforcing, observe that (i) by definition u* � u� and also�* � �� . Moreover, the discretionary payments are defined so that for all �,

(A2) b*� �

1 � �u* � b�

�

1 � �u�.

Consequently, the agent’s expected future payoff at the time of choosing his effort and both parties’expected payoffs at the time of making discretionary payments are precisely the same as in the initialperiod of the original contract.

Substituting (A2) into (ii) and (iii) above implies that the stationary contract defined by w*,b*(�), e(� ), u*, and �* will satisfy (ii) and (iii). Thus the agent will choose e(� ) and there is noincentive to renege on a discretionary payment. Finally, because this stationary contract repeats ineach following period, the continuation contract is self-enforcing, (iv). Thus the constructed contractis self-enforcing and gives a per-period surplus s*.

PROOF OF COROLLARY 1:The proof of Theorem 2 constructs a family of optimal stationary contracts giving the agent any

payoff u* � [u� , s* � �� ] and the principal a corresponding payoff �* � s* � u*. Each of these


contracts remains optimal on the equilibrium path, but reverts to no trade following an observeddeviation. Suppose that in a period following an observed deviation, the parties instead initiate eithera stationary optimal contract that gives the agent u� (if the agent deviated), or one that gives theprincipal �� (if the principal deviated). The resulting contract is still stationary, is optimal after anyhistory ht, and is self-enforcing because it provides a payoff-equivalent punishment for an observeddeviation.

PROOF OF THEOREM 3:(f) Consider a self-enforcing stationary contract with effort e(� ), payments W(�) � w � b(�),

and per-period payoffs u, �. Since for any �, the agent can choose e � e(� ) and continue with thecontract, (IC) is a necessary condition for self-enforcement. Similarly, as either party can renege ona discretionary payment and exit the relationship, then for all �:

(A3)�

1 � �u � u� � �b� and

�

1 � �� b�.

The agent’s constraint must hold for inf� b(�) and the principal’s for sup� b(�). Summing theseconstraints implies that

�

1 � �s � s� � sup

�

b� � inf�

b�.

Since b(�) � W(�) � w, this is equivalent to (DE).(d) Conversely, suppose there is a payment schedule W(�) and an effort profile e(� ) that satisfy

(IC) and (DE). Define:

b� � W� � inf�

W�,

and a corresponding fixed payment:

w � u� � ��,y �b� � c�e��.

Now consider a stationary contract with payments w, b(�), and effort e(�) and deviations punishedwith reversion to the static equilibrium. This contract gives per-period payoffs u� to the agent and � �s � u� to the principal. By (DE), s � s�, so � � �� and both parties are willing to initiate the contract.Moreover, (IC) implies that for any �, the agent prefers e(�) to any e � e(�), while from (DE) it is easyto check that (A3) both hold, so there is no incentive to renege on payments.

APPENDIX B: OPTIMAL INCENTIVE PROVISION

PROOF OF THEOREM 4:By Theorem 3, there is a stationary self-enforcing contract with effort e(�) if and only if there exist

payments W(e) satisfying (IC) and (DE). I first derive necessary and sufficient conditions for e(�), W(e)to satisfy (IC). I then extend this to conditions under which both (IC) and (DE) can be satisfied.

Let W(e) be a stationary payment schedule and define U(� ) as the resulting single-period payofffor an agent with cost type �:

U� � maxe

We � ce, �.


Regardless of the payment schedule, U(� ) will be decreasing in � as a consequence of the EnvelopeTheorem.

Given that the agent selects his effort to maximize W(e) � c(e, � ), the schedule W(e) will induceeffort e(� ) [i.e., will satisfy (IC)] if and only if (i) e(� ) is nonincreasing; (ii) for all �,

(B1) U� � We� � ce�, � � U��

��

c� e�, � d�;

and finally (iii) for all � and e � {e(� )}��,

(B2) U� � We� � ce�, � � We � ce, �.

Conditions (i) and (ii) follow from standard incentive compatibility arguments. They imply thateach type � prefers to choose e(� ) rather than mimic type �� by choosing e(��). Condition (iii)implies that each type � prefers e(� ) to some e that is not a specified choice for any cost type. It isanalogous to an interim participation constraint.

Now, assume e(� ) and W(e) satisfy (IC). Then e(� ) is nonincreasing, and rewriting (B1),

(B3) We� � ce�, � U��

��

c� e�, � d�.

Moreover, since any type can deviate to e � 0, and c(0, � ) � 0, (iii) implies that

(B4) U�� We�� ce�� , �� W0.

Combining (B4) and (B3) implies that

We� � W0 � ce�, � ��

��

c� e�, � d�.

Thus a necessary condition for e(� ) and W(e) to satisfy both (IC) and (DE) is that

(IC–DE)�

1 � �s � s� � ce�, � �

�

��

c� e�, � d�.

Conversely, suppose that e(� ) is nonincreasing and satisfies (IC–DE). I construct a payment planW(e) that satisfies (IC) and (DE). Let W(0) be given arbitrarily. Define W(e(�� )) to satisfy (B4) withequality. Then U(��) � W(e(��)) � c(e(��), ��) � W(0). For each � � [�, ��), define W(e(�)) to satisfy (B3),and for each e � {e(� )}��, define W(e) � W(0).

I now verify that e(� ), W(e) satisfy both (IC) and (DE). By construction, e(� ) and W(e)automatically satisfy (i) and (ii). And because U(� ) � maxeW(e) � c(e, � ) is nonincreasing in �and c(e, � ) � 0, the fact that U(�� ) � W(0) by definition implies that (iii) is also satisfied.Consequently, e(� ), W(e) satisfy (IC). Now, because e(� ) is nonincreasing, the construction of Wusing (B3) implies that W assumes its largest value at W(e(�)) and smallest at W(0). Since (IC–DE)

implies precisely that�

1 � �s � s� � We� � W0, it follows that (DE) is satisfied,

completing the proof.


Optimal Hidden Information Contracting.—I now derive the solution to the optimal contractingproblem with hidden information. To formulate this as a problem in optimal control, take �(�) � e(�) tobe the control and e(�) to be a state variable. It is useful to define a second state variable,

K� � ��

� � �

1 � �Se�, �p� � c� e�, �� d�.

The control problem is given by the Hamiltonian:

H � Se�, �p� ��

1 � �Se�, �p� � c� e�, � ,

with the monotonicity constraint:

�� 0

and the (IC–DE) constraint (now rewritten as a boundary constraint):

(B5) K� � 0 and K��

1 � �s� � ce�, � � 0.

Here �(�) and �(�) are co-state variables assigned to e(� ) and K(� ).This problem is concave, so the Pontryagin conditions are both necessary and sufficient for

a solution (e.g., Daniel Leonard and Ngo van Long, 1992, pp. 250 –51). Letting (�) be themultiplier on the monotonicity constraint and � the multiplier on the (IC–DE) inequality, thesolution satisfies:

� � H� � ��

�� He � Se e�,�p�1 ��

1 � �� ce� e�,�

�� Hk � 0

e� � H� � ��

K� � H� ��

1 � �Se�,�p� � c� e�,�;

as well as the boundary conditions:

K� � 0, �� , �� ce e�, �, and �� 0,

the complementary slackness conditions on the monotonicity constraint:

� � 0, �� 0 and �� 0,

and the self-enforcement constraint:


� � 0, K��

1 � �s� � ce�, � � 0

and ��K��

1 � �s� � ce�,� � 0.

To obtain the conditions in the text, I make two observations. First, because �(�) � 0 for all �,and �(��) � �, it must be that �(�) � � for all �. In addition, because �(�) � (�) for all �, then�(�) � (�) as well. Thus, it is possible to substitute for �(�) and �(�) in the above equations, leadingto the conditions in the text.

Beyond the conditions in the text, the solution must also satisfy the complementary slacknessconditions on the self-enforcement constraint. This leads to two cases, depending on whetherself-enforcement is binding.

Case 1: Suppose that � � 0. Then (IC–DE) is slack at the solution, so it suffices to consider the problemof maximizing the joint surplus subject to the monotonicity constraint. Because eFB(�) is decreasing, itboth maximizes the joint surplus and satisfies monotonicity. Hence e(�) � eFB(�) at the optimum.

Case 2: Suppose that � � 0. Then (IC–DE) binds at the solution. I proceed in a series of steps. Ifirst define the schedule eR(� ) as the unique solution to:

(B6) Se eR� , � p� ��

1 ��

1 � �

�ce� eR� , � .

Under the assumptions in the text, eR(� ) is decreasing in �, and eR(� ) � eFB(� ) for all �.

LEMMA 2: On any interval where the solution e(� ) is decreasing, e(� ) � eR(� ).

PROOF:On any interval where e(� ) � 0, it must be that (�) � 0. Hence (�) � 0 as well. Eliminating

in the Pontryagin condition (2) yields the result.

LEMMA 3: If for some �, e(�) � 0, then e(� ) is decreasing on [�, ��].

PROOF:I argue by contradiction. Suppose that for some �0, e(� ) is decreasing below �0 and constant

above it. By complementary slackness, (�0�) � 0 and (�0

�) � 0. Hence (�0) � 0. Moreover, itfollows from the Pontryagin condition (2) that over any interval where e(� ) is constant, (�) mustbe increasing (using the assumption that Se and p are decreasing in � and that ce� is increasing in�). Thus, (�) is both positive and increasing above �0. Consequently (�) � 0 for all � � �0 ,contradicting the boundary condition that (��) � 0.

PROOF OF THEOREM 5:I use the two lemmas above and the optimality conditions in the text. First, observe that

at the optimum (�) � �ce(e(�), �). Thus, if self-enforcement is binding, (�) � 0, whichimplies by complementary slackness that e(�) � 0. So there must be pooling of the mostefficient types. Combined with the two lemmas, this yields two possibilities: either e(� ) isconstant below some � � (�, �� ) and decreasing above it, or e(� ) is constant on the entireinterval [�, �� ].


If there is partial pooling, then for all � � �, e(� ) � eR(� ), while for all � �, all types arepooled at some e. By continuity, this means that e � eR(�). To identify the “cut-off” type �, observethat (�) � �ce(e(�), �) and (�) � 0. Integrating the Pontryagin condition (2) from � up to �, andsubstituting these boundary conditions yields:

(B7) ��

�

Se eR�, �p� d� ��

1 ��

1 � �

�ce eR�, � .

From the assumptions on primitives, it is easy to check that there is at most one value � � (�, ��) thatsatisfies (B7). If this � � �� , the solution has e(� ) � eR(�) for � � and e(� ) � eR(� ) for � ��. Furthermore, because the solution satisfies e(� ) eR(� ) for all � and eR(� ) � eFB(� ) for all�, it follows that e(� ) � eFB(� ) for all �.

In the event that there is no cut-off type � � (�, ��) that solves (B7), there is complete pooling atsome e. To identify e, the Pontryagin condition can again be integrated from � up to �� andcombined with the boundary conditions on (�) and (��) to yield:

��

��

Se e, �p� d� ��

1 ��

1 � �

�ce e, �� .

In this case e � eFB(�� ), so e(� ) � e � eFB(� ) for all �.

Remark: The strong assumption that P is concave is used to guarantee that the separat-ing schedule eR(� ) defined by (B6) is decreasing. If the assumption is relaxed, a partialpooling optimum still involves pooling of the least efficient cost types but may also in-volve additional pooling regions. To derive the optimal contract, the boundary of the lowerpooling interval, �, is identified as before but above � the candidate schedule eR(� ) mustbe “ironed” if it fails to satisfy monotonicity. It is still the case that all types choose less thanefficient effort.

The assumption that eFB(� ) is interior can also be relaxed. The only change is that even in asecond-best environment e(� ) may equal eFB(� ) for some cost types. Specifically, it may be thate(� ) � eFB(� ) � 0 for very high costs or that e(� ) � eFB(� ) � e� for very low costs.

PROOF OF THEOREM 6:The optimal stationary contract e(� ), W(�, y) � w � b(�, y) solves:

maxe�,W�,�

s � ��,y �y � c�e��

subject tod

de��y �W�, y � ce, ��e � e�� 0 for all �,

�

1 � �s � s� � sup

�,y

W�, y � inf�,y

W�, y.

In addition, the total compensation W(�, y) must be structured so that neither party will want to renegeon payments—this can be done as in Theorem 4. Here I have substituted the agent’s first-order conditionfor the (IC) constraint. This is valid under the Mirrlees-Rogerson conditions. The result now follows fromthe fact that the optimization problem is linear in W(�, y). I omit the details.


APPENDIX C: SUBJECTIVE PERFORMANCE MEASURES

I sketch the analysis with subjective performance measures. Details are contained in a secondaryAppendix available from the author.

PROOF OF THEOREM 7:Assume that an optimal full-review contract exists and generates surplus s*. The argument is first

to show that for any optimal full-review contract there is a termination contract that achieves thesame expected surplus. The next step is to argue that the optimal termination contract takes a cut-offform.

1. The first observation is that, similar to Theorem 1, the set of payoffs that can be achieved witha self-enforcing full-review contract is equal to {(u, �) : u � u� , � � �� and u � � s*}.

2. The second step is to argue constructively, along the lines of Theorem 2, that a terminationcontract can achieve the optimal surplus s*. To do this, one considers an optimal full-reviewcontract that in its initial period specifies payments w, b( y), effort e, and equilibrium pathcontinuation payoffs u( y), �( y), and continuation surplus s( y) � u( y) � �( y). A terminationcontract is then constructed with payments w*, b*(y), effort e, continuation probabilities �*(y), and(in the event of continuation) optimal continuation payoffs u* � �* � s*. The idea is to choose thediscretionary payments and continuation payoffs so as to exactly duplicate the incentive struc-

ture under the initial contract—in particular, so that b*y �

1 � ��*yu* � u � by

�

1 � �uy and also �b*y

�

1 � ��*y�* � �� by

�

1 � ��y. The fixed

compensation w* is chosen to ensure the per-period payoffs are u* and �*. Given this constructionthe termination contract will generate the same surplus as the original contract and be self-enforcingas a result of the original being self-enforcing.

3. The final step is to argue that the optimal continuation contract will take a one-step form. Similar toTheorem 6, this result follows from risk neutrality. To see that bad outcomes are followed bytermination consider the following argument. To provide effort incentives, good outcomes mustgenerate a high future payoff for the agent, and consequently a low future payoff for the principal. Toprovide reporting incentives, the principal must be indifferent between outcomes. This means that badoutcomes must result in a low future payoff for both parties. Optimization implies this low payoffshould be as low as possible—hence termination.

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