teoria dos contratos

7/27/2019 Teoria Dos Contratos

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C H A P T E R 3

The theory of contracts

Oliver Hart and Bengt Holmstrom

lntroductionThe past decade has witnessed a growing interest in contract theories ofvarious kinds. This development is partly a reaction to o ur r ather thor-ough understanding of th e s tan dar d theory of perfect competi t ion undercomplete markets , but m ore im porta ntly t o the result ing realizat ion tha tthis paradigm is insufficient to acc om m od ate a num ber o f im porta nt eco-nomic phen om ena. Studying in mo re detai l the process of contracting -particularly i ts hazards a nd imperfections - s a na tura l way t o enrich andamend th e idealized competi t ive model in a n at tem pt to f it th e evidencebetter . A t present i t is the maj or al ternative to m odels of imp erfect com-petit ion; we will co mm ent o n i ts compar ative advanta ge below.In one sense, contracts provide the found ation for a large part of eco-nomic analysis. Any trade - a s a q ui d p r o q u o - must be mediated bysome form of cont ract , whether it b e explici t or implicit . In the case ofspot trades , however, where the two s ides of the transaction occur almostsimultaneously, the contractual element is usually downplayed, presum-ably because it is regarded a s trivial (al tho ug h this need not be th e case;see Secti on 3). In recent years, economists have become much m ore inter-ested in long-term relat ionships where a considerable am ou nt of t ime mayelapse between the quid a n d the qu o. In these circumstances, a con tractbecomes a n essential part of the trad ing relat ionship.

Of course, long-term c ontra cts ar e not new in economics. Contin gentcomm odity trades of th e Arrow-D ebreu type ar e examples par excellenceof such contrac ts. W hat does seem new is th e analysis of co ntr acts writtenWe w o~ lld ike 10 thank Jonathan Feinstein, Paul Joskow, John Moore, Sherwin Rosen,Jean Tirole, and Andy Weiss for comments on an earlier draf t . Financial support f romNSF and the Sloan Foun dation is gratefully acknow edged.


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72 Oliver Hart and Bengt Holmstromby an d covering a small number of people. Th at is, there has been a moveaway f rom the impersona l Arrow-Debreu mark et set ting where peoplemak e trades "with t he ma rket ," to a si tuat ion where f irm A and firm B, o rfirm C an d union D , write a long-term c ontr act. This dep artur e is no t with-o ut econo mic significance. Wil l iamson (1985), in particular, has stressedthe im portance o f si tuat ions where a small numbe r of part ies mak e invest-me nts which ar c to s om e extent relationship-specific; that is, once made,they have a muc h higher value inside the relations hip th an o utside. Giventhis "lock-in" effect, each party will hav e som e mon opo ly p ower ex post,al thoug h there may be plenty o f com peti t ion ex an te before investmentsare sunk . Since the parties can no t rely o n the mark et once their relat ion-ship is underway, the obvious way for them t o regula te (and divide thegains from ) trade is via a long-lerm co ntract . Until the advent o f contracttheory, econom ists d id not have the tools to analyze ex an te competi tive,ex post nonc om peti t ive relationships of this type via form al models.

Research o n con tracts has progressed alon g several different l ines, eachwith its own particular interests. It may be useful to begin by mention-ing so me of these directions before out l ining th e subjects we will concen-trate on in this chapter .

O ne s t ran d o f the l i te ra ture has focused on the in ternal organizat ionof the firm , viewing th e firm itself a s a response t o failures in the price sys-tem. Qu est ions of interest include structuring incentives fo r mem bers ofthe t i rm, al locat ing decision aut ho ri ty, a nd choosing decision rules t o beimplemented by sui table reward structures. Of course, the object ive ispart ly t o gain insight in to organizat ion theory as such. B ut perhaps m oreimportant ly, one is interested in knowing whether organizat ion theorym atter s in the aggregate - tha t is, t o w hat extent the con du ct of f i rms willbe different fr o m the assumed profit-maximizing behavior; a nd , if it dif-fers, what ramificat ions follow for m arket outcom es an d overal l al loca-t ions in the econom y.

An oth er p rom inent l ine of research has explored the workings of thelabor m arket . A plausible hypothesis is that conting ent claims for laborservices ar e l imited for reasons o f o pportun ism. This invi tes innovationof o ther types of contracts that can be used as substi tutes. Th e researchhas centered on the structure of op t imal bi lateral labo r contracts (undervar ious assumpt ions a bo ut enforcement oppor tuni t ies) , on the propert iescontractua l equil ibria will have, and in part icular o n whether these equi-libria will cxhibit the commonly claimed inefficiencies associated with real-world adjustments in employment.

lnspired by the possibili ty that long-term contracts may embody pricean d wage sluggishness, a related body of w ork has explored their m acro -


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Theory of contracts 73economic implications [see, e.g., Fischer (1977) and Taylor (1980)l. Un-like most con tract analysis this l i terature has tak en t he for m o f co ntractsas given, typically with nominal wage and price rigidities. This is not assatisfactory as working from first principles, but it has made policy anal-ysis quite tractable.

Financial markets offer ano ther a rena of su bstantial potential for con -tract theoretic studies that is beginning to be recognized. Th e imp or tan ceof limited co ntra ctin g fo r the eme rgen ce of financial services a n d institu-tions has been suggested by D . D iam on d (1984), G al e an d Hellwig (1985),and Townsend (1980). This line of research also offers prospects for acareful modeling of the role of money an d the conduct of mo netary pol-icy [see Townsend, Chapter 11 this volume, and D. Diamond (1985)l .As the field is progressing, i t becomes harder to place models in spe-cific categories. Initially, m ode ls of org aniza tional design igno red m arke tforces, or at least treated them in a very primitive fashion. In contras t ,the theory of labor contracts s tarted out without considerat ion for or-ganizational incentives. More recent models, however, treat both incen-tive and market issues concurrently. Such crossbreeding is fruitful, buti t makes the task of organizing this chapter much harder. Since we havebeen unable to co m e up with a na tur al classification th at w ould av oid thisproblem, we will employ an outline that follows the historical progressrather closely.

We begin in Section 1 with agency the ory as a representative para digmfor the organizat ion-theo ret ic aspects o f co ntract ing. Fro m there we goon to labor contracting (Section 2). Finally, we turn to incomplete con-tracts and th e afor em entio ned lock-in effects (Section 3) . This work, rep-resenting mo re recent m ethodological trend s in con tract research, has notadvanced very far yet, and our discussion will be correspondingly moretentative in nature.Needless to say, we will not attempt a comprehensive survey of thelarge number of contractual models that have appeared to date. Somesubjects (e.g. , models relat ing contracts to macroeconomic policy) areleft out entirely. So are models of financial contracting. Our intentionhas been to be selective an d critical rath er t ha n com prehensive. Alth oug hwe allow ourselves a rather opinionated tone, we trust this chapter stil lgives a good idea of the general n atur e of the on going research a nd a rea-sonably fair assessment of its main contributions.Despite our selective approach, the chapter has grown very long. Inorder tha t it m ay be m or e readily digestible, we hav e written it so th at th ethree parts can be read essentially independently; each part has a con-cluding section that sums up its major points.


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74 Oliver Hart and Bengt HolmstromA word about methodology

Most cont rac t theor ies a re based o n the assumpt ion tha t the par ti es a tsom e init ial da te (say, zero) design a P are to o pt im al ( for them) long-termcontra ct . O ptimality is not to b e un dersto od in a first-best sense, but ratherin a constrained or second-best sense. Indeed, informat ional and otherrest r ic t ions that force the contract to be second-best are at the hear t ofthe analysis - without them on e wou ld quickly be back in the s tandardArrow-D ebreu para digm w here contractu al fo rm is inessential. Becauseinformat ional constraints wi l l play a par t icular ly important role in theensu ing discussion, let us no te right away tha t t l ~ r o u g h o u twe will restrictat ten t ion to cases in which info rma t iona l asymmetr ies ar ise only subse-quent to contract ing. In th e typical language of the l i terature, we will notconsider adverse selection models.The design of a Pareto opt imal contract proceeds by maximizing oneparty's expected uti li ty s ubjec t to the oth er p arty (o r parties) receiving am in im um (res erv atio n) expe cted utility level. Wh ich party's utility level istaken a s a constraint d oes not usual ly m at ter , because most analyses arepar t ia l equi l ibr ium. W hen there is perfect co mpet i t ion ex ante , this reser-vat ion ut il ity can be interpreted a s that par ty 's date-zero op portu ni ty costde termined in th e da te-zero marke t for cont rac ts . W hen ex an te compet i -t ion is imp erfect , th e parties will presum ably barga in ove r the ex ante sur-plus f ro m the relat ionship, an d s o the reservation expected ut i li ty levelsbecome endogenous .T he l i te ra ture has of ten been cavalier abo ut the de term inants of thereservation uti l i ty, because valua ble insights have emerged f ro m th e gen-era l charac ter i s t i cs of Pare to opt imal i ty a lone . On the o ther hand, thefact tha t ma rket forces reduce t o s imple constraints on expected uti li tiesgreatly facil i tates equil ibriu m analysis. Eq uilibratio n in expected util it iesis usual ly t r ivial . This gives the contractual approach i ts main method-ologica l a dvantag e re lat ive to models o f imper fec t com pet i t ion , for in-s tance. Th e analyt ical core of contract theory is a n opt imizat ion problem,whereas in imperfect com pet i t ion it is a n equi l ibr ium p roblem. Metho dsfo r solving opt imizat ion exercises ar e substant ial ly mor e advanced thanmetho ds for so lv ing equi l ibr ium p roblems.

Of course, subst i tut ing an opt imizat ion analysis for an equi l ibr iumanalysis is not a lways econom ical ly meaning ful ( fo r instance, we are notimplying that imperfect com pet i t ion s hou ld be s tudied in this way) . In-deed, the economic credibi li ty of th e contractual a pp ro ac h may be cal ledin to ques t ion when, as of ten happens , opt imal cont rac t s become mon-st rou s s tate-cont ingent prescript ions. H ow a re such con tracts wri t ten an denforced?


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Theory of contracts 75Three responses to this question ca n be offered. Th e first o ne is to a p-peal to the po wers o f th e judicial system a n d its ability t o enfor ce certainexplicitly agreed-upon contractual terms. The assumption is that suffi-cient penalties, either pecuniary or nonpecuniary, will be imposed for

breach an d hence rationa l parties will not breach. T his assump tion m akesa model internally consistent, but is unsatisfactory o n tw o accoun ts. Itmaintains an artificial dichotomy between those contractual provisionsthat are assumed t o be infinitely costly t o enforce a n d tho se tha t ar e as-sumed to be completely costless to enforce. Also, it often predicts (byassumption) explicit terms that are much more complex than those weobserve an d in th at sense is no answer t o w hat p rodd ed the enforcementquestion ab ove .The second response is a pragm atic one: O ne could argu e that quali ta-tive and aggregate features, rather than contractual detail, are the rele-vant ones for judging the success of a mo del. In sup po rt of this view on ecan allude to the implicit nature of contracts in the real world; in otherwords, suggest that equilibrium outcomes in the real world mimic opti-mal, complex state-contingent contracts despite the relative simplicity ofthe explicit agreements we observe. The difficulty with this response is

that we d o not unde rstand well how implicit con trac ts of this type aresustained as equilibrium phenomena.Ideally, on e would like to know what determ ines the division betweenexplicit an d implicit en forcem ent of a con trac t. This leads to the third ap-proach, w hich is t o co nf ro nt th e enforc em ent issue explicitly by includingrealistic legal penalties fo r breach as well as indirect co sts that affect equ i-librium behavior, for instance through reputalional concerns. Whereas

much of the extant literature rests on a combination of the first two re-sponses to the enforcem ent issue, the present tren d is tow ard the more am -bitious, b ut also m ore sa tisfactory, third app roa ch. Th is will be discussedat some length in Section 3 .1 Agency models

Agency relationships a re ubiq uitous in eco nom ic life. W herever there ar egains t o specialization the re is likely t o aris e a relati ons hip in which agen tsact on behalf of a principal, because of com para tive advan tage . E xam plesabound: w orkers supplying labor t o a firm, man agers acting o n behalf ofowners, d octo rs serving patients, lawyers advising clients. Th e econo micvalue of decision making made on behalf of someone else would easilyseem to m atch th e value of individual co nsu m ptio n decisions. In this light


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76 Oliver Hart and Bengt Holmstromthe attention paid to agency problems has been relatively slight. More-over, there are some less obvious instances of the same formal agencystructure - the government taxing its citizens, the monopolist price-dis-criminating customers, the regulator controlling firms - all of which aresubstantial problems in their own right.If agents cou ld costlessly be indu ced to internalize the principal's ob-jectives, there w ould be little reason to stu dy agency. Th ings becom e inter-esting only when objectives can no t be autom atically aligned. S o what isitthat prevents inexpensive alignment? Th e most plausible and commonlyoffered reason is asymmetric information, which of course ties closely tothe source of agen cy: re turn s t o specialization. The sincerity of a worker'slabor input is often h ard t o verify, leading to pro blem s with shirking. In-form ational expertise permits ma nagers t o pursue goals of their ow n suchas enhance d social statu s or im proved career opportu nities. Private infor-ma tion a bo ut individual characteristics causes problem s fo r the govern-ment in collecting taxes.

Th us , underlying each agency m odel is an incentive problem caused bysome fo rm o f asym metr ic inform ation. It is comm on to dis t inguish mod-els based o n the p articular info rm ation asym me try involved. We will usethe following taxonomy. All models in which the agent has precontrac-tual inform ation we place under the heading of adverse selection; exceptfor an occasion al reference, we will no t deal a t all with this category. Ourmodels will assume symm etr ic inform ation a t th e t ime of contracting.W ithin this category, which w e refer t o as moral hazard, a further distinc-tion is useful: the case where the agent tak es un observa ble actions, andthe case where his actions (but not the contingencies under which theywere taken) may be observed. A rrow (1985) ha s recently suggested the in-formative names Hidde n Act ion M odel and Hidden Informat ion Modelfor these two subcategories. T he w orker supplying unobserv able effort isthe pro totypical hidden ac tion case, while the expert ma nager m aking ob-servable investment decisions leads to a typical hidden info rm ation model.

As will become clear sho rtly, the hidden actio n case formally subsumesthe hidden information case. (This rationalizes our use of moral hazardas a joint label.) Nevertheless, it is meaningful to keep the two distinct,because they differ in their eco nom ic implications a s well as in their solu-tion techniques. In this section we focus on the hidden action case. Sec-tion 2 , o n lab or con tracting, will i l lustrate the hidden inform ation case.T h e general objective o f a n agency an alysis is to characterize the opti-mal org aniz ation response t o the incentive p roblem . Typ ically, the anal y-sis delivers a second-best reward structure for the agent, based on infor-mation that can be included in the contract. Characterizing the optimalincentive scheme is imp orta nt but not the prime econ om ic purpose. What


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Theory of contracts 77is more interesting is the allocational distortions that come with the in-centive solution. A lthough on e could oft en design incentive schemes thatinduce the agent t o behave as if n o info rm ation asym metry were present,that is rarely second-best. Instead, som e of the costs of th e info rm atio nasymmetry are borne by distortions in decision rules, task assignments,and other costly institutional arrange men ts. This is what gives the theoryits main economic content.Th e agency para digm has indeed been qu ite successful in sheddin g lighton institutional p heno men a tha t a re beyond received micro econ om ic the-ory. T he second-best natu re of incentive efficient solutions ad m its a host ofarrangements tha t w ould be inexplicable if in form atio n flows were cost-less. Examples ab ou nd in the li terature an d we could easily use u p ou r al-lotted space by describing som e of th em . How ever, we have chosen not tofollow this line, but rather t o be mo re methodologically oriented. Agencymodels are not w ithout pro blem s, an d this is best bro ugh t h om e by goinginto the details of a generic structure.We will begin with thre e different fo rm ula tio ns of the agency pro blem ,each with its own merits. Next we go on to discuss a simple version ofhidden action th at will suffice to su m up th e main insigh ts of th at typ e ofmodel. An econo mic assessment an d critique follow, w hich in turn leadus to a discussion of recent imp rovem ent efforts. These include the roleof robustness in simplifying incentive schemes and the use of dynamicmodels to arrive at richer predictions. Finally, we provid e a summ ary ofwhat agency theory has to offer as well as what (in our view) its short-comings are.1.2 Threefortnuiat ionsLet A be the set of actions av ailable to the agent a n d de no te a generic ele-ment of A by a . Let 8 represent a state of nature d raw n fro m a distr i-bution G. Th e agent's act ion a n d the s tate of natur e jointly determine averifiable outcome x= x (a , 8) a s well a s a mon etary payoff n = i r(a , 8 ) .The verifiable ou tc om e x can be a vector and may include ir. The mone-tary payoff belongs to the p rincipal. His pro blem is to constr uct a rewardscheme s ( x ) that takes outcomes into paym ents for the agent .The principal values money acco rding to the util ity fu nction v (m ) an dthe agent accord ing to the util ity fu nc tion u ( m ) . Th e agent also incurs acost from taking the a ction a , which we den ote c ( a ) . We assum e initiallythat the agent's cost of action is independent of his wealth, that is, thathis total uti li ty is u ( s ( x )) - c ( a ) . Th e principal's ut il ity is v( ir- s( x) ) .The agent a n d the principal agree on the distr ibut ion G, the technol-ogy x (. , . ) , an d the uti li ty a n d cost funct ions.


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78 Oliver Hart and Bengt HolmstromThis is the state-space formulation of th e agency problem as initiatedby W ilson (1969), Spence an d Ze ckh ause r (1971), an d R oss (1973). Its m ainadvantage is that the technology is presented in what appear to be themost n atur al terms. Ec onom ically, however, i t does not lead to a very in-

formative solut ion.The re is another , equivalent way of looking a t the above problem thatyields more economic insights. By the choice of a, the agent effectivelychooses a distribution over x a n d a, which can be derived from G via thetechnology x ( . , ). Let us denote the derived distribution F ( a , x ;a ) a n dits density (or mass fu nction) f a , ; a ) . This parameterized distributionJbr~nulationwas pioneered by M irrlees (1974, 1976) a n d fur ther exploredin Ho lm str om (1979). For later reference, we sta te the principal's problemmathematically in parameterized distribution terms. His problem is to

M ax \ u ( a- ( x ) ) a , ; a ) d x over aEA , s ( . ) E S , (1.1)subject to

u (s (x ) ) a , x ;a ) dx- c(a)r u (s (x ) )( a , x ; a ' ) d x- c ( a l ) ,v a ' ~ A . (1.3)

In this progr am th e principal is seen as deciding on th e action he wantsthe agent to implement and picking the least cost incentive scheme thatgoes along with that action. It is wo rth no ting that because the principalknows the agent (his preferences), he also knows what action the agentwill take even though he cann ot directly observe it. Co nstr aint (1.3) as-sures that th e incentive scheme is consistent with the action the principalwan ts the agent t o choo se, while constraint (1.2) assures the agent a m ini-m um expected utility level a, presumably determined in the m arketplace.A solution to the principal 's program is not au toma tically assured; infact simple examples can be given in which n o opt im al solution exists. Wewill encou nter a non-existence examp le shortly, but o therw ise we merelyassume that a solution exists.Th e third , most ab strac t, formu lation is the following. Since the agentin effect chooses am on g alternative distributions, on e is naturally led totake the dis t ribut ions themselves as the act ions, dropp ing the reference

' Groj>man and Hart (1983a) offer a set of sutficient conditions for existence. A key condi-lion is that the probabilities controlled by the agent are bounded away from zero.


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Theory of contracts 79to a. Let p den ote a chosen density (or mass) function over a a n d x , a n dlet P be the set of feasible densities fr om which the agent can ch oose. Be-cause the agent can rando mize a m on g actions, P can be assumed convex.In the case ( a , ~ )akes on a f inite numb er of values , P is a simplex. T h ecost function in this formulation is written as C ( p ) ,which also will beconvex because of randomization.Of course, the economic interpretation of the agent 's action and theincurred cost is obscu red in this general distr ibut ion form ulat io n, but inreturn on e gets a very stream lined model o f pa rticular use in u ndersta nd-ing the formal structure of the problem.This way of looking at the principal's problem is also very general. Itcovers s i tuations w here the agent may observe some in form ation a bo utthe cost of his actions, or the expected returns from his actions, beforeactually deciding wh at t o do; in othe r words, cases of hidden in for m a-tion. To see this, simply n ot e that whatev er strategy the agent uses fo rchoosing actions contingent o n in for ma tion h e observes, the strategy willin reduced for m m ap in to a distribution ch oice over ( a , x ) . Th us , ex antestrategic choices ar e equivalent t o distribu tion choices in som e P (p ro p -erly restricted, of course). Note also that the primitive cost function foractions, c ( a ) , could be stochastic without affecting the general formu la-tion. Taking expectations over costs c ( a ) would still translate into a costfunction C ( p ) , because the agent 's utility fun ction is separa ble.1.3 The basic hidden action rnodelMuch of the general insights obtained fro m studying hidden action mo d-els can be conveyed in the simplest setting, wher e th e agent has only tw oactions to cho ose fro m . F or con creteness, let us identify them with work -ing hard , H, an d being lazy, L. Also, assume for the moment that x coin-cides with the m onetar y payoff t o the principal an d th at th e principal isrisk-neutral. If the agent works hard, the distribution over x is f H ( x ) ,while if he is lazy, the distribution is f , ( x ) . In view of this language it isnatural t o assume that f , ominates f, n a first-order stochastic do m i-nance sense; that is, the cum ulative dist ribu tion fu nctions satisfy F H ( x )0 and so(by CDFC) the right-hand side is a concave function in a. Consequently,none of the distributions in the original family will be as appealing to theagent as the action the principal is implementing from the local family(1.9). Hence, s(x) remains optimal in the extended family as well.This argument is illustrated in Figure 1. The triangle represents thesimplex of all distributions in the case where there are only three possibleoutcomes x, , x 2 ,and x,, which we assume for ease of diagramming. Oneaxis measures p , , the other p Z ; he third, p , = 1 p , - p 2 , does not appearin the figure. The curved line CBA is the one-dimensional manifold ofdistributions f( x; a ) [here represented as {( p , (a ) ,p , (a ) ) a E A ] ] ; his setis one-dimensional, because the action a is a scalar. Any straight line inthe simplex represents a family satisfying LDFC. The shaded region isthe set P of all distributions that the agent has access to when random-ized strategies are included (cf. the general distribution formulation in


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86 Oliver Hart and Bengt Holmstrom

Figure 1

Section 1.2). The figure does not show the cost function or the incentivescheme. W ith a third dim ension measuring costs an d rewards, the incen-tive scheme would be a hyperplane an d the cost function a convex mani-fold in R3 .

Assume the principal wants to implement the distr ibution at point B(representing the earlier a * ) .Corresponding t o the argument above, hestarts by designing a cost-minimizing scheme that im plemen ts B when theagent 's hypothetical al ternatives are the distr ibutions along the tangentt o B [the tangent represents th e distributio ns in th e linear family (1.9)].This cost-minimizing scheme is characterized by (1.7). Next, CDFC andM L R P assure [using (1.7) and (1.1 I ) ] that none of the distr ibutions alongthe curved line (or in P for that matter) is as at tractive to the agent aspoint B given the scheme in (1.7). Thus, B is indeed implemented in theactual set of feasible distributions P. W i th ou t C D F C a n d M L R P , t h eagent might want t o jum p across, for instance to C , when B is being im -plemented from the tangent set . Then (1.7) would not be valid.As might be expected, MLRP and CDFC are very restr ict ive condi-t ions and economically rather peculiar . Part icularly, CD F C seems to ruleout a num ber of " natural" families, because few of those we might think


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The ory o f contracts 87of are closed under conv ex com bin atio ns. Fo r instance, the re is no fam ilywe know of tha t sat isfies both cond it ions an d is generated fro m the tech-nology x = a + 0 (or x = at)). This doe s not mean that th e set of familiessatisfying both C D F C an d M L R P is small . There i s an easy way of gen-erating sam ple famil ies with b oth propert ies. Simply start w ith any tw odistributions a n d exten d this family by L D F C as in (1.8). If the two initialdistributions can be ord ere d by ML R P , the extended family will have thisproperty. Note that the role of M L R P here is simply to ob tain an increas-ing schedule and not to assure the val idi ty of the fi rst-order approach,which is already guaranteed by LDFC.'The fact that L DF C app ears to be the main inst rum ent for const ruct -ing families with C D F C an d M L R P leaves op en th e quest ion o f w hetherthere are any interest ing cases that d o not sat isfy LD F C but on ly CD FC .Except for add ed conv enienc e in stud yin g examples , this issue is no t veryinteresting either. We alr ead y saw how the tw o-action case was rat her richin generat ing a variety of o ptim al incentive schemes. Th is richness obv i-ously carries over to the LDFC case.

From the preceding discussion on e shou ld infer that the fi rst-order ap -proach works in the case whe re the family of distribu tions con trol led bythe agent is one -dim ension al in distribution spac e (LD FC ). I t also worksin cases which are effectively one-d ime nsio nal, in th e sense that their so -lution is equivalent to a problem with a one-dimensional family (CDFCplus MLRP). Notice that i t is one-dimensionali ty in distribution spacethat makes things simpler, not one-dimensionality in the underlying eco-nomic variable (effort). Even th ou gh effort is take n to be one -dim ens ion -al, the curve it trace s will gen erally (wh en convexified) ge ner ate a highe r-dimensional P, making mat t e rs complex .What is meant ab ov e by "the first-order ap pro ac h w orks" also needs abi t of elaboration. Its precise me anin g is that the optim al schem e is ch ar -acterized by (1.7), which i s a narrow er s ta teme nt than the claim that o necan describe the agent's choice by first-order conditions. Viewing thingsin distributional terms, we note that the agent in Figure 1 has two deci -sion variables: p , a n d p2. If th e cost fu nction over P were strictly conv exand the optimal distribution to implement were interior to P (because,say, the cost goes to infinity toward the b ou nd ary ), then a fi rst-order a p-proach in the traditio nal se nse wo uld w ork perfectly well. N orm ally asingle first-order condition would not be enough to describe the agent 'sbehavior, but two would alway s d o . O ne would th en ob tain a ch aracter i-' Alternatively, of course, o ne ca n work wi th any o ne-p ararn e~e r amily ( for which a solu-

tion is known l o exist) an d the n inlerprel the c ha ra ct er i~ at io n s rel 'erring not t o (h i\ I'arn-ily nece\sarily, b u ~ 0 he langcnt space of dislributions described by (1.9).


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88 Oliver Hart and Bengt Holm stromzation like (1.7), but with tw o multipliers p 1 a n d p2 rather than one. Thisdilutes the in form ation c onte nt of the characterization; the sufficient sta-tistic results will no t be a s crisp (in particular, op tim al incentive schemesmay aggregate more than what the earlier sufficient statistic result indi-cated; see Section 1.6) an d statements ab ou t mon otonicity will be hard toma ke. Needless to say, when o ne goes to higher-dimensional cases, thevalue of a general c harac terization alo ng these lines quickly disappear^.^^^We conc lude that models with a con tinuo us effort variable allow a sim-ple characterization when they look much l ike the two-action case dis-cussed before. In that case the solution, as far a s the optim al reward struc-ture is concerned, exhibits the same features and the same variety. Onedifference is worth stressing, though. In the two-action model it is diffi-cult to say anything about the agent's choice of action, because it is notdetermined by a cont inuous t rade-off. On e has to com pare the solutionthat implements H with the solution that implements L directly. O n theother hand, if effort is a continuous variable and the first-order approachworks, then i t can be proved (Ho lms trom , 1979) that the optim al level ofeffort to implement is such that the principal would like to see it go evenhigher. In other words, in equilibrium we should see principals desiringmo re effort fro m their workers. Because this enrichment can be obtaine dalready, by m oving fro m the two-action case to the LD FC case, there ap-pears to be l it tle reason ever to g o beyond LD FC in a model tha t wan ts toexploit the characterization in (1.7).1 . 5 A n intermediate assessmentT he m ain predictive content o f th e basic agency m odel is in the sufficientstatistic result , which tells what inform ation should enter in to a contra ctin the first place. Simple as it seems, this result turns ou t to have quite abit of e con om ic scope. O ne trivial implica tion is that agency relationshipscreate a demand for monitoring. This has generated substantial interest

Gro ssm an a nd Ha rt (1983a) study cases in which the first-order ap pro ac h may not be ap-plicable. Even with MLRP, incentive schemes need not be monotone . O n the other han d,the result that sufficient statistics are sufficient for designing optimal incentive schemesdoes not d epend o n the f irs t-order approach. Also, a more informative system ( in theBlackwell rense) is strictly better than a less informative one, assuming that the garblingmatrix that connects the two systems ha5 full rank. However, signals that provide addi-tional information about the agent's strategy may not be valuable when the first-orderapproach h i l s .Hidden lnformation Models, viewed in distribution space, are typically of high dimen-sion, because contingent strategies result in rich distrib ution al choices for the agent (seeSection 1.6). This is why the analyses of Hidden lnform ation M odels proceed along quitedifferent lines than the analysis of Hidden Action Models.


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Theory of contracts 89in the accounting literature an d has led to va rious refinements in predict-ing the usefulness of different mon itorin g schemes [fo r a survey, see Bai-man (1982)l.A m ore significant im plication concerns the use of relative pe rform anc eevaluation [Baiman and Demski (1980), Holmstrom (1982a)l. Agents whowork on tasks tha t ar e related - in the sense that one task provides in for-mation ab ou t the other - should n ot be com pensated solely on individualoutput, but partly o n the output of others. N ote that the reason for this(according to the sufficient statistic result) is not that one would like toinduce com petitio n fo r incentive purpos es, b ecause if the agents' techno l-ogies are not stochastically related then relative performance evaluation

is useless at best. Rather, competition is a consequence of the desire toextract info rm atio n ab ou t the circumstances un der which the agents per-formed. This information is used to filter out as much of the exogenousuncertainty a s possible, allowing m ore weight to be placed on individu alperformance.

A furth er con sequence of the sufficient statistic result is th at som etimesaggregate info rm atio n will d o as well as detailed info rm atio n in relativeperformance schemes. For instance, if technologies have normal noisethen weighted averages of peer performance will suffice as a basis for anoptimal scheme. The weights are proportional to the information con-tent of the signals from peers.Predictions like these accord at least broadly with stylized facts. Rela-tive performance evaluations are commonplace, particularly in the formof prizes (e.g., promotions) awarded to top performers in an organiza-tion. Indeed, the labor m ark et as a whole form s a grand incentive struc-ture in which relative evaluations implicitly or explicitly play a dominantrole. The literature on rank order tournaments, initiated by Lazear andRosen (1981), has studied in more detail the performance and design ofsuch contests [see also Green and Stokey (1983) and Nalebuff and Stig-litz (1983)l. We note that the use of rank order as a basis for payment israrely optim al in th e basic agency model; o ne could usually d o better w ithschemes sensitive to ca rdinal measures. How ever, there may be other ad-vantages to rank-order payments not captured by the standard agencymodel. O ne reason is that rank is easier to m easure in many circum stances.Another arg um ent - suggested by Carm ichael (l984), Malcomson (1984a),and Bhattacharya (1983) - s that tourna me nts provide the principal withincentives to honor promised awards even in cases where legal enforce-ment is difficult, because performance can be observed but not verified.I n tournaments the total am ou nt paid by the principal remains consta ntand payment should therefore be easy to verify.


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90 Oliver Hart and Bengt HolmstromExplicit relative p erf or m an ce schemes have recently emerged in execu-tive com pen satio n package s as well. Typically, they relate man agerial per-formance to companies within the industry, which fits the notion thatstochastically closer technologies have more value as a basis for optimal

rewards. A ntle a nd Sm ith (1986) have studied m ore broadly the degree ofrelative performance evaluation in executive compensation, measuringimplicit (as well as explicit) contractual elements. Their statistical testsshow that th e da ta in fact exhibit a comp one nt of relative com pensation,but n ot to the extent predicted by the basic theory . Th is seems puzzling atfirst, but tw o explan ations ca n be suggested f or th e evidence. F irst, exec-utives may be diversifying their portfolio through personal transactionsin the mark et, which d o not sh ow up in the da ta; in fact, Section 1.6 dis-cusses a model with precisely the property that no relative performancepay me nts are necessary because the executive can m anu facture them him-self. Th e other , m ore plausible, reas on is tha t relative pe rform ance evalu-ations distort eco nom ic values an d thereby decision mak ing (e.g., a n ex-ecutive completely insulated fro m systematic risk will disregard such riskin evalu ating investment decisions). In the one-dimensional agency m od-els normally studied, such decisions are excluded. Including more deci-sion dim ensio ns in th e model seems essential fo r gaining a better fit withthe dat a and a better u nderstanding o t 't h e merits of relative performa nceschemes.Given tha t the basic agency m odel is so general, it is perhaps surprisingthat i t has any predictive value at a ll . To this can be adde d the value ofhaving a paradigm within which one may begin considering in more pre-cise terms such su bjects as the manag erial theory of the firm. Jens en a n dMeckling's (1976) pioneering w ork is an ex am ple of w hat insights can bederived from the mere recognition that m anagers need t o be provided withincentives against shirking; ano ther (mo re explicit) model o n the sam e sub-ject is in Gro ssm an a nd H art (1982). Both pap ers derive the capital struc-ture of the firm from the underlying incentive problems (with oppositehypotheses about the manager's options to dilute the firm's resources).Althoug h these studies beg the qu estion of why capital structure needs tobe used for incentive purposes when direct incentive schemes appear tobe cheaper, they still open the do or for furth er investigations into a sub -ject that surely is of substantial economic importance.Let us next turn to the problems with th e basic agency model. Th e ma inon e is its sensitivity t o distribution al as sum ption s. It manifests itself in a nop tima l sharing rule that is complex, responding to th e slightest changesin the informa tion content of the outcom e x. Such "fine-tuning" appearsunrealistic. In the real wo rld incentive schemes d o show variety, but notto the degree predicted by the basic theory. Linear or piece-wise linear


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Theory of contracts 91schemes, for instance, are used frequently an d across a large range of en-v i r o n m e n t ~ . ~heir pop ular ity is hardly explained by shared properties ofthe information technology, as the basic model would have it. It is clearthat other technological or organizational features, excluded from thesimp le mo del, must be responsible f or w hatever regularities in shapes wed o observe empirical ly.Fine-tuned , com plex incentive schemes also stand in the way of seriousextensions a n d applications. O ne can say lit tle about com parative staticsproperties o f the m odel and it is also hard t o introduc e addition al vari-ables into the analysis. This is a critical dra w bac k, because the unobs erv-able v ariable in the mo del (say, effort) is not of primary interest preciselybecause i t cannot be observed. Instead one would be interested in whatconsequences the agency model has for such observable variables as in-vestment decisions and task assignments, for example. Little has beendo ne in this regard, du e to the complexity of the basic solution. [F or oneattempt that reveals these difficulties, see Lambert (1986).]T hu s casual empiricism, a s well a s the desire to include decision vari-ables of allocational an d aggreg ate significance, strongly point to a needto refine agency models in the direction of predicting simpler incentiveschemes. We turn next to such an effort.

1.6 Robuslness and linear sharing rulesTh e prevalence o f relatively simple incentive schemes could partly be ex-plained by the costs of writing intrica te contracts .' But tha t is hard ly thewhole story. A more fundamental reason is that incentive schemes needto perform well across a wider range of circumstances than specified instandard agency models. In other words, incentive schemes need to berobust.O ne way o f expressing the dem and for robustness is to allow the agenta richer set of ac tion s or strategies. Intuitively, the mo re options th e agenthas, the mo re poorly intricate schemes will per form . T o give a familiar ex-amp le: If there is a secondary mark et for goods , arb itrage will take awayall opportunities for price discrimination. Linear schemes are optimal,because they a re the only ones tha t a re o p e r a t i ~ n a l . ~

Sha recrop ping rules are almost exclusively linear, despite great variations in stochasticenvironments .

' We remind the reader o f ou r discussion of explicit versus implicit incentive schemes in theint ro du c~ ion . o me would argue that real-world schemes are quite complex when viewcda s equilibrium phenom ena.

"his could be o n e reason fo r the prevalence o f linear shari ng rules in share crop ping. 1 rmay also explain why corp orat e tax schemes ar e mo re l inear than income tax schemeb;


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92 Oliver Hart and Bengt HolmstromAno ther e lementary example of how added opt ions contr ibute to s im-plifications can be given in the context of our basic agency model. Wenoted th at an op t imal incent ive scheme need not be mon otone in generalunless M L R P holds. O n the othe r han d, if the agent is allowed free dis-

posal of o utput , then the only operational schemes are monotone n o mat-ter what the stoch astic technology looks like. This i llustrates the kind ofnondistributional con siderations that one is led to look for in understand -ing mo re universal pro perties of incentive schemes.Recently, H olmstrom an d M ilgrom (1985)have proposed a simple agen-cy model in which linear schemes are optim al because the agent is assumedt o have a rather rich action sp ace. Th e main idea can best be grasped bydescribing an exam ple, du e to Mirrlees (1974), in which no o ptim al solu-tion exists. M irrlees's exam ple has a risk-neutral principal, an a gent withunbo unde d m arginal ut il ity fo r consum ption, an d a technology with o ut-put x = a+ 6,where 6 is a normally distributed erro r term with zero meana n d a is the agent 's lab or supply. In other w ords, the agent controls themean of a normally d istributed out pu t. This technology is the most obvi-ou s candidate f or a n agency analysis, an d it is qui te a shock t o learn thatthe problem has n o solution. Th e reason is that first-best can be approx -imated arbitrarily closely by step-function schemes that offer first-bestrisk sharing (a flat re wa rd) fo r almost all outcom es except the extremelybad ones for which a severe punishment is applied. This approximationresult is in fact easy to understand using the statistical intuition that thebasic model offers. The normal technology has a l ikelihood ratio f , / fthat is unb oun ded below (it is l inear in x ) . Therefore, very low x valueswill be very informative ab ou t the agent 's action, a nd o ne can act on thatinfo rm ation almost as if i t revealed com pliance perfectly. T he step func-tions app rox ima te forcing contracts, w hich ar e well know n to be optimalif there a re ou tcomes that revea l devia t ions with ~ e r t a i n t y . ~

Th e examp le is clearly unrealistic a n d there are ways t o patch it (e.g. ,bound utility or bound the likelihood ratio). But this would be mislead-ing, because the example points t o a mo re fundam ental f law. Step func-tions com e close to first-best o nly unde r the unrealistic assum ption tha ton e knows exactly the param eters o f the problem (util i ty functions, tech-nology, etc.) , an d they will generally perf orm poorly a s soon as on e intro-duces slight variations or uncertainty in to the model. In oth er wo rds, theexample represents the extreme case of fine-tuning discussed earlier.

Footnote 8 (cant.)presumably, corporations can circumvent nonlinearities in tax schemes more easily thanindividuals. (Some would ar gue that individuals can d o a lot of arbitra ge as well, makingincome tax a lot less progressive than it appears.)

' Harris and Raviv (1979) study optimal forcing contracts.


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Theory of contracts 93For instan ce, think of a d ynam ic context, where the agent is paid (say)at the end of the week, a nd assum e he can observe his own performanceduring the week so that he can adjus t h is labor input a s a function of therealized path of output. Then step functions will induce a path of effort

that will be both erratic and, on average, low (generally, the agent willbide his time to see if there is any need to work at all). In con tras t, a lin-ear scheme, which applies the same incentive pressure no matter whatthe outcome history, will lead to a more uniform choice of elTort. Thissuggests that the optim ality of step func tions is highly sensitive to the as-sumption that the agent chooses his labor input only once.1This intuition can be made precise by considering a dynamic versionof the normal example. Specifically, let the agent control the drift ratep o f a one-dimensional Brownian m ot ion ( x ( t ) ; E [O , 11) over the unitt ime interval . Formally , th e process x ( f ) is defined as the solut ion to th estochas tic differential equ atio n:

Here B is s tandard Brownian motion (zero drif t and unitary variance).Note th at the instantaneous variance, a2dt, s assumed to be cons tant .Th e agent in the model is assumed to have an e xponential ut il ity func-t ion, a nd th e cost of effort (unlike in o ur earl ier m odel) is assumed to beindepe ndent of t he agent 's income. In other word s, the agent 's payoff is:

as evaluated at the end of the horizon, w here x = x( 1) is the final positionof t he process (the profit level at tim e 1, say), c( p ) is a convex (ins tant an-eous) cost function, a nd r is the coefficient of a bso lute risk ave rsio n. Th eparticular form of the utility function assures that a linear scheme willindeed apply the same incentive pressure over time. In general, incomeeffects would cause distortions.Notice tha t if the agent were unable to observe the path x ( i ) , then i twould be optimal for h im to choose a constant drif t rate p(i) = p [be-cause e ( - ) s convex], an d the end-of-period posit ion x would be norm allydistr ibuted with mean p a nd variance a. In other w ords, we would have amodel identical to ou r previously discussed one-pe riod exam ple tha t ha dl o This can be simply illustrated in the case of a risk-neutral agent, where an intinity of

schem es will be tirst-best. T hey include a linear schem e with un itary slo pe as well as theaforementioned step function. However, if the agent receives some noisy informationabout the technology before choosing his effort, the linear scheme will be uniquely opti-mal. This idea is used in Laffont and Tirole (1986).


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94 Oliver Hart and Bengt Holmstromno o ptim al solution, because step functions appro xim ate first-best. W henthe agent can observe x ( t ) and base his choice p( f) on the history of thepath of x ( t ) (which we denote x' ), the s ituation is significantly cha nged .Instead of being constrained t o a one-parame ter family of outcom e dis-t r ibut ions, the r ich set {p(xt) ; E O , 11) of con tingen t strategies permitsa vastly wider choice. The enormous expansion of the agent 's opportu-nity set limits the principal's op tion s dram atically; in fa ct, fo r each stra t-egy tha t the principal w ants to im plement th ere is essentially a un ique in-centive scheme that he must use, which stand s in sha rp con trast to theusual flexibility in choice that the principal has in one-dimensional staticmodels.The one-to-one mapping between strategies and sharing rules makesthe m odel solvable technically (recall the discussion in Section 1.4). T h erelationship can be written out explicitly, after which it is easy to showthat the op timal rule is linear. Th e interested reader is referred to th e orig-inal paper (Holmstrom and Milgrom, 1985) for details.Intuitively the result can be seen as follows. Consider a discrete ver-sion of the Brownian mo del, o ne in which the agent contro ls a Bernoulliprocess. Because of exponential utility it is easy to see that the optimalcomp ensation scheme, if i t could be mad e contingent on the whole pathof periodic outcomes, would be to pay the agent the same bonus eachtime he has a "success"; the problem is stationa ry, becau se there are no in-com e effects. Viewed as an end-of- period pay me nt schem e, this rule paysthe agent a co nstant plus the numb er of successes t imes the b onu s, whicham oun ts to a l inear scheme in end-of-period profits. Th e Brownian mo d-el - being the limit of a Bernoulli process - should therefore be expectedto have a l inear optimum as well , and indeed i t does.Notice tha t this line of reasoning show s that the principal need not usethe detailed information of the path of the outcome process even if heha s access t o it. Th is is a case whe re an insufficient statistic with respect t othe agent's distributio nal ch oice (th e end-of-pe riod level of profits) is stillenough for constructing an o ptim al rule; in other words, a case where theprincipal uses mo re aggregated inform ation tha n th e sufficient statistic re-sults of one-dimen sional models would suggest. Th e reason is that thereis n o conflict of interest in the timing of effort, on ly in the agg regate levelof effort; hence information about t iming is of no value."" W e venture the guess that ( in mult idimensional agency models) addit ional s ignals are

~ a l u a b l e reci5ely when they give intorm atio n abo ut d imension s of choice in which thereis a conflict of interest. In one-dimensional mo dels there is a conflict of interest alway s(b y assumption). Th e result thal add it ional s ignals have value if they are inform ative isalways true in that case.


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Theory of contracts 95The rema rkable th ing abo ut th is model i s tha t by ma king the incent iveproblem apparen t ly muc h m ore compl ica ted ( the r igorous p roof tha t a

linear schem e is optim al is nontrivia l) , i t delivers in the en d a m uc h sim -pler solut ion. In fac t , once we know that the opt imal incent ive schemeis linear it is trivial t o so lve f o r its coefficients. A linear scheme will in-duce the agent t o cho ose a con stant level of ef for t . There fore w e can t rea tthe problem a s a s ta t ic on e (cf . th e d iscuss ion ab ove) in which th e agentchooses the mean of a normal d is t r ibut ion, but th is t ime with the con-s traint tha t the pr incipal is only a l lowed to use linear rules . Th e dyn am -ics ra tional izes a n a d hoc res tr ict ion to l inear ity in t he s ta t ic model an din the process resolves th e nonexistence prob lem th at Mirrlees originallyposed!

Com putat io nal ease gives the model sub stant ia l methodological va lue.In contras t to genera l agency models it i s easy to con duct c om parat iv es ta tics exerc ises . M ore imp ortant ly , o ne can use the m odel a s a bui ld ingblock in s tudying r icher appl ica t ions of m oral hazar d . Suc h appl ica t ionsare further facil i tated by the fact th at the l ineari ty results extend t o si tua-tions in which t he agent co ntrols the vector of drif t rates of a mu ltidime n-s ional Brown ian process; or , in s ta t ic term s, chooses the m ean vector of amultivariate normal distr ibution.

As a brief i l lustrat ion, let us discuss the effects of agency cos ts on in-vestment decisions, assuming that investments are made jointly by thepr incipal and the agent . (We cannot le t the agent make the choice pr i -vate ly , because that would amount to having him control the var iance ,which wo uld upset th e l inearity results .) S up po se there is a collection ofprojects avai lable for inves tment . E ach project re turns x = p+ 8, where 8is a normally distr ibuted variable with mean m an d va r iance u2 n d p isthe agent's effort . F or a clos ed-form solu tion , assum e the cost of effort tobe quadratic: c(p)= p2/2. T o make the example a b i t r i che r, a s sum e inaddi t ion that there is a market index z , normal ly d is t r ibuted wi th var i -ance an d zero me an, tha t corre la tes wi th x. The n each p ro jec t can becharacterized by the triple (m , a2, ) , where p is the correlation coeffi-cient between z a n d x.

To determine th e best inves tment on e solves f irst fo r the opt im al in-centive scheme an d net re turn to th e pr incipal , g iven a par t icular projec t .The optimal scheme is l inear in x a n d z - tha t i s , of the form s(x,z )=a,x+a2z+P. T h e best coefficients ar e easy to calculate. O n e finds tha tthe principal should set


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96 Oliver Hart and Bengt HolmstromT he c onsta nt coefficient l3 is determined by the agent's participation con-straint. If he has to be assured a z ero certain equivalent, then t he princi-pal will be left with an expected net return equal to

Note that the optimal incentive scheme exhibits relative performanceevaluation. The agent is not merely rewarded based on the project out-come x, but a lso on the marke t ou tcom e z. Th e s ign of a 2 s the oppositeof p as one would expect. This is in accordance with the general resultthat an o ptim al design sho uld filter out a s much ui?controllable risk as pos-sible. Using z as a filter reduces unco ntroll able risk by the fa ctor (1- 2 ) .If x an d z happen to be perfectly correlated, all risk can be filtered outand first-best can be achieved. (I11 first-best, a ,= 1 and a =m+ + . ) ' ITh e best p roject is th e on e that maximizes (1.16). Because of the agen-cy problem, we see that project choice depends o n the degree of idiosyn-cratic r isk as measured by a 2 (1- 2 ) (which is th e conditio nal variance ofx given z ). Th e price of th at risk is a func tion o f the agent's risk aversion(and in general also the cost of effort). There is no price for systematicrisk, because the principal is r isk-neutral. O ne could allow a r isk-averseprincipal (with exponential utility) without altering the linearity result;then systematic r isk w ould also enter th e decision criterion. But the m ainpoint is that, unlike standard portfolio theory, idiosyncratic risk will playa role in investment decisions.Because idiosyncratic risk carries a price, diversification will generallyhave value [see A ro n (1984) for th e sam e point] . Also, a concern for id-iosyncratic risk will give rise to a m arket portfo lio that is m ore concen-tra ted th an un der full inform ation . Firm s will find value in choosing proj-ects that are more heavily correlated with the market, because that willenab le a better incen tive design. (T his assumes all proje cts are positivelycorrelated.) Thus, agency costs could amplify aggregate swings in theeconomy.This discussion is merely suggestive of what one might be able to dowhen linear schemes are opt im al. I t appea rs that linearity has th e poten-tial to t ake us towa rd som e livelier an d m ore serious econom ic analyses.[F or some othe r illustrative examples, see the original paper (Ho lm stromand Milgrom, 1985) .] On the other hand, the Brownian model is quite

I Z It is worth noting that in this example the agent could privately ma nufa cture t he opti-mal degree of relative p erfor ma nce evaluation by trad ing in other firms' assets. In otherwords, the principal could equally well pay the agent based on x alon e and leave it u p tothe agent to filter out uncontrollable risk. (Of course, the agent must no t be allowed totrade stock in his own firm.)


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Theory of contracts 97special . Th e technological opt io ns a re very l imited; fo r instance, the factthat the agent cann ot be allowed t o ma ke private investment decisions is a nunfortunate constraint for app l icat ions. Th e etfectiveness of the Bro wn-ian model is restr ic ted, because i t does not capture th e dem an d fo r robust-ness in th e most intui t ive way. Presum ably, on e will hav e to g o outs idethe Bayesian f ramew ork a nd in t roduce boun ded ra t iona l i ty in order t ocapture the t ru e sense in which incent ive schemes need to b e rob ust in th ereal wo rld.

1.7 Dynamic extensionsDynamic extensions of the basic agency model a re of interest fo r two ra th-er opposi te reasons. O ne has t o d o with the relevance of the incent ive is-sues portrayed in the s ta t ic models , th e other with the add ed predict ionsthat m ight be had fro m introducing dynam ics. In the form er category wehave theoretical studies that suggest that t ime may resolve agency prob-lems cost lessly. T his ha s been argu ed b oth fr om th e perspective of su per-games, in w hich all coop erative gain s can be realized between tw o parties,and in term s of re puta t ion effects created by the ma rket . Wh ile we d o notconcur in ei ther case with the conc lusion tha t incent ive problems disap-pear, i t i s wo rth understanding th e argum ents . Th ey will take us to dy -namic models tha t can expa nd an d sharp en the predic tions f rom the s ta t ictheory.

The first studies of dyn am ic agency were th ose by Rad ner (1981) an d Ru -binstein (1979). Bo th sh ow th at in an infinitely rep eated v ersion of t he ba -sic one-period mo del, the first-best solutio n (complete risk-sharing to geth-er with corre ct incentives) can be atta ine d if uti li ties a re not disco unte d.The analysis does not offer an opt im al solut ion, but r a the r a c lass of con -tracts within which first-best can b e reached. The se cont rac ts op era te l ikecontrol charts , pu nishing th e agent f or a per iod of t im e if his aggregateperformance falls sufficiently below expectations. Over t ime, as uncer-ta inty is f i l tered out by the law of large numbers , the punishments be-come mo re severe an d the con t ro l reg ion t igh ter . Th e assum pt ion of nodiscounting assures that only events in the dis tant fu ture , w here the con -trol is t ight a nd few violat ions occu r , m atter .

These models appear to formal ize the in tu i tion tha t in long- te rm re la -tionships o n e ca n cope m or e effectively w ith incentive prob lem s, becausetime permits sharper inferences abou t t rue performance ." T he fact that

This interpreta tion is dispute d in Fud enb erg et al. (1986). There it is argued that repeti-tion changes the agent 's preferences rather than improves monitoring accuracy.


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98 Oliver Harl and Bengl Holmstromf irst-best can be achieved is more incidental and a consequence of theunrealistic assumption of no discounting paired with infinite repetition.Even though Radner (1981) has subsequently shown that with some dis-cou nti ng on e can still get close to f irst-best, there is little reaso n t o believethat incentives are costless in reality. T he main questio n then is whetherdynamics a l ters the ins ights an d results f rom one-per iod models . In thestudies above, as well as in subse quent w ork by Rog erson (1985a) an dLambert (1983) [see also Roberts (1982) and Townsend (1982)], memoryplays a key role, suggesting tha t a n optim al long-term contrac t might lookrather different fro m a sequence o f short- term c ontracts .

Jum ping to such a conclusion is premature , however. T he models dis-cussed abo ve assume that the agent canno t bo rrow a nd save, in which caselong-term con trac ts substitu te in part for self- insurance that would in factbe available to agents (saving is certainly a real op tio n an d limited bo r-rowing as well) . Co uld it be true that the gains t o long- term co ntractingidentified in the ear ly models a re in fact du e to restr ic t ions o n borrowingand savings?Recent stud ies by Allen (1985), Malco mson an d Spinnew yn (1985), andFude nberg et al. (1986) sho w that this may indeed be the case. M ore spe-cifically, if o ne goes to th e oth er extreme an d assumes tha t the agent canaccess capita l mark ets f reely an d on the sa me interes t terms as t he pr inci-pal , th en long- term contracts will be no better than a sequence of shor t-term c ontrac ts in the ( independently) repeated model.Fo r ins tance, Allen noted that if there is n o discounting, then o ne cansimply appeal t o Yaari's (1976) early wo rk o n consum ption under uncer-tainty t o conclude that a f irst-best solutio n can be achieved by having theagent rent th e production technology from the principal at a f ixed price.Th e agent, by borrowing o r saving, need not be concerned abou t f luctua-t ions in incom e, s ince they can be smo othe d out a t n o cost. In this caseself- insurance is perfect a nd risk carries n o premiu m.Allen also studies the f inite horizon case, but in a pure in surance con-text [specifically, Townsend's (1982) model], which is simpler than theagency mode l we have been discussing. Here a lso he finds that long-termcontrac ts do not domin a te shor t -te rm contrac ts. T he sam e resu lts for theagency model are established by Malcom son a nd Spinnewyn an d by Fu-denberg et a l . These two pap ers dif fer in that th e former assumes that theagent's borrow ing a nd saving decisions ca n be verif ied (an d hence his con-sum ption ca n be controlled contractually) , whereas the la t ter t reats thesedecisions as pr ivate t o th e agent. Th e basic idea of the argum ent is verysimilar , however. T he key observa tion is that long-term co ntra cts can beduplic ated by a sequence of short- term contrac ts by rearrangin g th e pay-


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Theory of contracts 99ment stre am t o the agent withou t altering its net present value along an yrealized path. Roughly speaking, the rearrangem ent wo rks so that theprincipal clears his balance with the agent in utility terms each period.Because there is a capital m arket , the timing of paym ents doe s not m at-ter . The agent re turns to the consumption s tream implied by the long-term contract by borrowing an d saving appropr ia te ly .Of course , the assumption tha t th e agent can b orrow a nd save freelyin the capital market is rather unrealistic. (In addition, the Fudenberget a l . model asumes that the agent can consum e negative am ou nts , whichcertainly is unrealistic.) Nevertheless, th e models d o ma ke clear that on eshould not rush to th e conclusion that lon g-term cont racts, at least in re-peated settings, hav e subs tantial benefits; in som e situatio ns, th e insightsof the one-period m odels remain unaltered w ith the introdu ction of dy-namics. More importantly, however, these findings suggest that - becausewe do observe long- term rela t ionships and long- term contracts - someforces other tha n incom e sm oothing a re l ikely to be behind the benef its.

There are many potentia l forces one could think of . Informationallinkages between periods ar e discussed in Fuden berg et al., a n d som e oth erreasons wili be taken up in Section 3. H ere we want t o stress that whencontingences are hard or impossible to verify, so that explicit contractscannot be easily enforced, long-term relationships are likely to providemajor ad vant ages. They can implicitly (via repu tation effects) support co n-tracts that ma y be infeasible to duplicate in short- te rm relationships. Bull(1985) offers a m odel of t his variety, which we will com e back t o in Sec-tion 3. Radner 's a n d Rubinstein's m odels are also best interpreted in thisfashion; both have self -enforcing equil ibr ia that d o not require ou ts ideenforcement. Lazear 's (1979) model on mandatory retirement is in thesame vein. L azear arg ues that age-earnings profiles slope upw ard [as a nabund ance of em pir ical evidence corroborates ; see , however, Ab raha mand Farber (1985) for contradicting evidence], because that way incen-tives for work a re maintained over the agent's em ploym ent horizon . T heimplication is that termination of employment should be mandatory, be-cause m argina l pro du ct will be below pay at later stages in the career. Al-though t he argum ent needs some refinement, Lazear 's model serves wellas an illustration of how introducing dynamics can yield additional pre-dictions into the basic agency set-up.

As a related example of reputation modeling, let us consider Fama's(1980) argu me nt t ha t incentive problems, particularly m anagerial incen-tive problem s, ar e exaggerated in the agency literature, because in realitytime will help alleviate them . His reaso ning is different f ro m Rad ner's an dRubinstein's in that it focuses o n the power of th e market t o police m an a-


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102 Oliver Hart and Bengt HolmstromAs in the case of R adne r's a nd Rubinstein's mod els, the result th at first-best can sometimes be achieved is of little interest per se. It requires veryspecial and implausible assumptions, in particular that the manager isr isk-neutral and does not discount future payoffs. The main point with

the model is rather t o illustrate that reputation c an indeed enforce an im-plicit con tract of som e form , when learning ab ou t characteristics is a keyfactor as it often would seem to be. In the particular example, the im-plicit c ontra ct pe rform s exactly like an explicit contrac t would (in a w orldwith known competence) if that contract were of the form s (x )= kx+ b,where k is given in (1.19). It is imp ort an t to note , however, th at when re-lying on re puta tion effects (at least as determined in the m arket ) there islittle freedom to design the contract in desirable ways.Wolfson (1985) has con ducted an empirical study of the returns t o rep-utatio n in the marke t fo r general partn ers of oil-drilling ventures. The re-sults con form broadly w ith the implications of the example. In the m ar-ket for oil-drilling ventures myopic behavior would dictate that generalpartn ers com plete fewer wells tha n limited pa rtners desire (because of thetax code). However, because new ventures come up frequently and newpartnerships a re formed , o ne might expect general partners to tak e intoaccount their reputation an d comp lete mo re wells than would be optim alin the short run. Indeed, Wolfson finds statistically significant evidencefor tha t to be the case. Repu tation is priced in the mark et m uch as in themodel described. The results correspond to a case where k < 1, becauseW olfson also finds that residual incentive problems remain a nd th at theseare reflected in the price of the shares of limited partners.These empirical findings give reason t o explore furthe r the w orkings ofreputation and learning. The general idea can be pursued in many direc-tions and some interesting work has already been done. Gibbons (1985)has considered what organizations can d o to align reputation incentivesm ore closely with true productivity. It is evident from the model desc ribedthat there need not be a very close relationship, particularly in the earlyperiods, between the returns t o reputation for a m anager an d his presentmarginal product. Indeed, if we think of young managers in lower posi-tions, their returns from effort may vastly exceed the actual product ofwhat they do, because the future value of being considered competentmultiplies in general throu gh e nha nced responsibility. O ne way of copingwith the problem, suggested by Gibbons, is to control the flow of infor-mation abo ut perform ance potential so that the initial impact of perfor-mance is diminished. Perhaps the phenomenon of young professionalsjoining larger pa rtnersh ips before establishing their own firms can be seenas a way of protecting oneself against overly strong reactions by the ma r-ket if mistakes happen in the early career.


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Theory of contracts 103Another paper that elaborates on this simple learning model is Aron

(1984). She uses the learning effects to derive a number of implicationsconcerning the correlation between the growth rate of firms, the degreeof diversification within firms, and the size of firms.Although ou r example supp orted the comm on intuit ion that incentiveproblems are alleviated by long-term considerations, it is important tostress that this is by no means true universally. In fact, career concernscan themselves be a source of incentive problem s. F or instance, in Ho lm -strom and Ricart-Costa (1986)a mod el is analyzed in which incongruitiesin risk-taking between man agers a nd sh areholders arise purely because ofreputation effects. T h e reason is that m anage rs look up on investments asexperiments that reveal inform ation a bo ut their com petence, while share-holders of co urse view them in terms of financial returns. T he main p oint

is that there is no reaso n f or a project 's hu m an capital retu rn t o be closelyaligned with its financial return; hence the problem requires explicit in-centive alignment. For those who distrust incentive models that rely oneffort aversion, such a m odel provides a new chann el for analyzing ma na-gerial risk-tak ing incentives.14Finally, we want to mention the work of M urphy (1986)as an exampleof how dynamics can help discriminate between competing theories ofcompensation. M urphy compares two hypotheses for why age-earningsprofiles tend to be upward-sloping. One is the earlier mentioned modelof Lazear. Th e other theory suggests tha t the upw ard slope comes fromlearning abo ut productivity an d th e contracting process associated withinsurance against that risk [see e.g. H arris and Ho lms trom (1982)l. Mur-phy argues th at if the incentive hypothesis were true then th e variance inindividual earnings sho uld increase with te nur e due to income smoothing .The reverse should be true if the learning hypothesis held, because thenthe effects of performance information are strongest in the early years.Mu rphy tests these competing posit ions o n panel d ata for executive com-pensation drawn from prospectuses. His results are rather inconclusive,perhaps because both effects ar e really present. But the m ain po int is thatin principle dyn amic models allow discrim ination th at is plainly unavail-able from single-period studies.1.8 Summary and conclusionsDespite the length of this section we have covered only a few dimensionsof the extensive literature on principal-agent models. Before summ ing upl4 A related repu tation model concerning risk-taking, which derives very interesting pre-dictions about the nature of debt contracts and credit rating in capital markets, is in

D. Diamond (1985).


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104 Oliver Hart and Bengt Holmstromwe want to me ntion tw o impo rtan t omissions. On e is the lack of exam plesof Hidden Information Models, which have played a visible role in thel i terature, of ten under the na m e of Mechanism Design [see Harris an dTow nsend (1981) an d M yerson (1979) for seminal contributions an d Green(1985) for a unified look a t the field]. We will partly ma ke u p for th is omis-sion in Section 2, where a model of hidden information is analyzed inconnect ion with labor con tract ing. Th e mechanism design ap proac h hasbeen quite successful in explaining a ran ge of institutions th at a re beyondthe scop e of sta nda rd theory, an d it has also offered insights in to norm a-tive pro blem s such a s taxation [Mirrlees (1971)], auc tion design [H arrisa n d Raviv (1981), M yers on (1981), an d M askin a n d Riley (1984)], an d reg-ulat ion [Baro n an d M yerson (1982), Baron a nd Besank o (1984), an d La f-fon t a nd Tirole (1986)l. In th e sam e way as the models we have discussedhere, these models ar e plagued by an excessive sensitivity t o inform ation alassumptions, which makes it hard to go beyond qualitative conclusions.Th e other ma jor omission is that we have not discussed a t all the gen-eral equilibrium effects from contracting, an area in which Stiglitz hasbeen p articu larly active. As Stiglitz has no ted in a variety of different con-texts [see, e.g. , Arn ott an d Stiglitz (1985)], the im perfections of second-best contr acts will have externa l effects that may be imp orta nt. T he gen-eral idea, fam iliar fro m second-best theory, can be described as follows.In all econom ies, contracting between two parties will have so me equilib-rium effect on th e rest of th e econom y. How ever, in the idealized Arrow -Debreu world, equi l ibr ium occurs at a social opt im um an d so the impactof m arginal chang es in a bilateral c on tract will have zero social costs. Incon tras t, when we ar e in a second-best world (fo r whatever reason), ma r-ginal chan ges in con tract s will have a first-order effect o n the social wel-fare function, which is not accounted for by the contracting parties.I5Perh aps one relevant examp le would be the consequences of nominal con-tracts in o ne part of the econom y o n the use of indexed contracts in otherpar t s .Naturally, such externalit ies could give reason for government inter-vent ion. How ever, on e should be careful in making sure that there is animproving policy that acts solely on info rmatio n tha t the government h asavailable. A s a mod eler i t is easy to spo t improvem ents, because the mod-eler sees all the relevant information. But that does not imply automat-ically that the government can improve things, particularly if the morestringent notions of efficiency that are associated with incom plete infor-l 5 A somewhat different dimension of the same problem appears when a party contractswith many independent agents in a decentralized fashion. This has been recently looked

at by Cremer and Riordan (1986), but it deserves much more attention.


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Th eo ry of contracts 105m ation models are ap plied . Ope ration al welfare schemes in this sense seemto have been little explored in the literature t o date .To summarize:

1. In reduced form all agency models have the agent choose from afamily of distributions over observab le variables, such as out pu t. A keysimplification in Hidden Action Models is to assume that the agent con-trols a one-dimensional family of distributions. This leads to a simpleand intuitive characterization of an optimal scheme. One-dimensionalitydoes not refer here to any economic variable, like effort, but to the setof distributions that the agent can choose from. Understanding this isimp ortan t fo r resolving the confusion s associated with th e validity of thecharacterization of the optimal rule, which is sometimes (but mislead-ingly) referred to as the validity of the first-order approach.2. Th e main insight of the basic H idde n Action M odel is that th e opti-mal incentive scheme looks like on e based o n an inference ab ou t the agent'saction from observable signals. This implies that the optimal scheme ishighly sensitive to th e information content of the technology that the agentcontrols, which has only loose ties with the physical properties of thattechnology. Con sequen tly, fiddling with the infor m ation technology willaccommodate almost any form of incentive schedule and the theory isreally without predictive content in this regard.What does have some predictive content, however, is the result that acontract should use all relevant inform ation th at is available up t o a suf-ficient statistic. Am ong othe r things, it leads to statements a bo ut th e useof relative perfo rm anc e evaluation that seem to m atch em pirical evidenceat least broadly.

3 . However, the extreme sensitivity to informational variables thatcomes across from this type of modeling is at odds with reality. Real-world schemes are simpler than the theory w ould d ictate and surprisinglyuniform across a wide range of circumstances (e.g., linear schemes arequite comm on in a variety of situations). The conclusion is th at som ethingother t ha n info rm ation al issues drives whatever regularities on e might o b-serve. One possibility that has recently been suggested is that the usualagency models are overly simplistic and fail to account for the need tohave schemes that perform well in a variety of circumstances - that is,schemes that are robust. We gave one example of a model in which ro-bustness issues lead to linear schem es. It seems tha t research in this direc-tion could have high payoffs in the future.Another reason why schemes in reality are simpler and less sensitiveto environ men tal differences is th at exotic contrac ts are hard t o evaluatein terms of b oth their implied performanc e a nd their value for the parties


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106 Oliver Hart and Bengt Holrnstrorninvolved. T his is not som ething which we addressed, because it seems tofall outside the com mo n Bayesian parad igm; but that is not t o say it isunimportant. Research along these lines may also have high payoffs.

4. Th e com mo n H idden A ction Models are rather weak predictively.One reason is that complex incentive schemes make it hard to say any-thing ab ou t distribu tiona l choices. Th e othe r reason is tha t the actions inthe model are not observable economic variables. (I n this regard H iddenInfo rma tion M odels are more useful, because actions - e.g ., levels of in-vestment or employment - are usually observable; see Section 2.) Model-ing efforts should be directed mo re toward including interesting economicquantities that focus on allocational consequences of agency. Robustnessargume nts tha t predict simpler schemes should be helpful in this endeav-or, as indicated in Section 1.6.

5. Another useful direction for sharpening predictions from agencymodels is to g o to dy namic form ulations. These bring t o bear time seriesan d panel d at a that allow discriminations tha t are impossible to make instatic models. Dynamic models also bring atten tion to reputation effectsand long-term explicit and implicit contracting that may well be at thecenter of real-world incentive problems.2 Labor contractsOne of the first applications of contract theory was to the case of con-tracts between firms and worke rs [th e seminal papers a re by A zariadis(1975), Baily (1974), an d G ordo n (1974)l. Section 2 is concerned with thiswork and various extensions, including the introduction of asymmetricinfo rma tion and macroec onom ic applications. We begin with the Azari-adis-Baily-Gordon model itself. [Fo r an excellent recent survey of laborcontract theory with a rather different focus from the present one, seeRosen (1985).]

2.1 The Azariadis-Baily-Gordon (ABG) modelTh e AB G m odel was developed to explain non-W alrasian employment de-cisions, particularly layoffs, and to und erstand deviations between wagesand the marginal pro duct of labor. It is based o n the idea that a firm of-fers its risk-averse workers wage and employment insurance via a long-term contract.Th e model can be described as follows. Ima gine a single firm th at hasa long-term relation ship with a grou p of workers.I6 Presum ably a lock-inl6 O n the empir ical importance of such relationships, see Hall (1980).


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Theory of contracts 107effect of so m e so rt explains why the relatio nsh ip sh ould be lon g-term , al-though this is not modeled explicitly. To simplify, assume that the rela-tionship lasts two periods. At da te 0, the firm an d w orkers sign a con tractwhile employm ent an d pro duction occur a t da te 1. AB G stress the idea ofan implicit co ntr ac t; we postp one discussion of th is until Se ction 3.4 andrely on the contract being explicit and legally binding.Let the firm's date-1 revenue be f( s , L ) , where s represents an exoge-nous demand or supply shock, and L is total employment at date 1. As-sume that the date -0 work force consists of m iden tical work ers, w here mis given.17 Each work er h as a n (indirect) von N eum ann-M orgenstern util-~ t yunction U (I , P; p ) where I represents incom e or wages received f ro mthe firm, P is employment in the firm, and p refers to a vector of con-sumption goods prices. We shall suppose that p is constant and there-fore suppress it in what follow^.'^ We assume that UI > 0 , U ,cO , and Uis concave in I an d P with UIIc (i.e., work ers are risk-averse). T h e firm,on the othe r h an d, is sup pos ed t o be risk-n eutral. We shall assu m e th at Pis a continuous variable, in con trast to ABG wh o suppose that i t equalsOorl .In the ABG model, the s ta te s is taken t o be publicly observable at d ate1, although unkno wn to bo th parties at da te 0. In this case, a co ntract can

1 be contingent in the sense of m akin g I and P functions of s: I = I ( s ) , P =' f(s). Since Y is sm oo th a nd U is con cav e in P, it is desirable to hav e work/ sharing at date 1; that is, P(s) = (L (s) /m ) (so this vers ion of ABG doesnot explain layoffs; see, however, Section 2.4B). Therefore, an optimal, date-0 contract solves:11 Max Es[ f ( s , mP (s) )- m I(s ) l sub ject t o E s [U ( I ( s ) . P ( s) )]a i , (2.1)where both exp ectatio ns ar e take n with respect t o the ob jective prob abil-

i ty distribution of s , which is assumed to be com mo n knowledge at d ate0. We are ado ptin g the ass um ption that the firm gets all the surplus fr omthe contract while the workers are held down to their date-0 reservationexpected utility levels ii. Nothing that follows depends on tl+lisex antedivision of the surplus, however.The solution to (2.1) is very simple. Und er th e usual interiority assu mp -tions, it is characterized by

fo r all s , (2.2)" In a more general model, the size of the workforce would be a choice variable.laTwo assumptions are embodied here. First, that p is independent of the shock s hittingthe firm; and , second , that the firm and workers a re sufficiently small that their action s

do not affect prices. We shall maintain both assumptions throughout Sections 2 and 3.


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108 Oliver Hart and Bengt Holmstromfor all s.

where X is a Lagrange multiplier. Equation (2.2) tells us that the margin-al rate of substitution between consumption and employment equals themarginal rate of transformation in each state; (2.3) tells us that a work-er's marginal utility of income is constant across states. I t is the condi-tion for optimal insurance between a risk-averse agent and a risk-neutralagent. [Note that (2.3) implies that if P(s,)= P(s,) then I(s,)= I(s,); thatis, wages vary only i f employment does.]Several observations can be made. First, i t follows from (2.2) that em-ployment decisions will be ex post Pareto efficient in each state. Hence toemphasize what is by now well known, the ABG model does not explaininefficient employment levels. Although there was some initial confusionabout this result, i t is not exactly surprising given that an ex ante optimalcontract should exploit all the gains from trade ex post (under symmetricinformation). Employment levels, however, although efficient, are notgenerally the same as in a standard Walrasian spot market, where thewage w(s) in state s satisfies

The point is that the possibility of income transfers across states permitsa divergence between (I(s)/Y(s))nd w(s)= ( a /a L) (s , mY(s)). In fact, i flabor is a normal good and the Walrasian labor supply is upward-sloping,Kosen (1985) has pointed out that employment will generally vary morein a contractual setting than in a Walrasian spot market."An important special case is where labor causes no disutility for a work-er per se, but simply deprives him of outside earning opportunities at dateI. This can be represented by

U(I, Y ) = U(I+R(?- Y)),where ?is the worker's total endowment of labor and R is the wage in al-ternative date-l employment [in (2.6), labor is neither normal nor infe-rior]. Equations (2.2) and (2.5) both then becomeI y The reason is the following. In a spot ma rke t, a worker's incentive to work hard in a

goo d s tat e where the wag e rate is high (th e substitu tion effect) will be otl'set by his desir eto co nsume a lot of leisure given that his income is high ( th e income effect); an d con-versely in a bad state. In a contractual setting, the income effect is reduced in size be-cause the tirm provides income insurance across differen1 stales of the world.


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110 Oliver Hart an d Bengt Holmstromcess may involve considerable costs relat ive t o the case o f insura nce bythe f irm.

In fac t , to prov ide optimal insurance, i t is not necessary that th e insur-ance compan y obs e rve s , only tha t i t obse rve wages I ( s ) and employmentY(s) .Ho wev er, even if it can obs erve these variables, new p roblem s ariseif som e aspect of a w orker 's perfo rma nce is unob servab le to the insurancecom pany . For example , suppose tha t t o mak e employment p roduc tive itis necessary that a worker exert effort, e. The n the opt im al r isk-shar ingco ntra ct would insure a worker 's wage subject to the worker exert ing ef-for t . If the insurance com pany - which canno t observe s o r e - offers in-suranc e, the worke r m ay exert n o effort an d claim that his low wage wasa result of a bad s. Ag ain this proble m is reduced if the f irm, which doesobserve e, acts a s insurer.

T he reader may wo nder how, if s a n d e are no t obse rvab le to ou t s ide rssuch as insurance companies , a contrac t be tween the f i rm and workersmaking I a n d Y funct ions of s a n d e can be enforced. T his is an impor-tant questio n

teoria dos contratos

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