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1 Dscrete-Tme Approxmatons of the Holmstrom-Mlgrom Brownan-Moton Model of Intertemporal Incentve Provson 1 Martn Hellwg Unversty of Mannhem Klaus M. Schmdt Unversty of Munch and CEPR Ths verson: May 5, 1998 Abstract: Ths paper studes the relaton between mult-perod dscrete-tme and contnuous-tme prncpal-agent models. We explctly derve the contnuous-tme model as a lmt of dscrete-tme models wth ever shorter perods and show that the optmal ncentve scheme n the contnuous model, whch s lnear n accounts, can be approxmated by a sequence of optmal ncentve schemes n the dscrete models. For a varant of the dscrete-tme model n whch the prncpal observes only total prots at the end of the last perod and where the agent can destroy prots unnotced we show, that f the length of each perod s sucently small, then an ncentve scheme that s lnear n total prots s approxmately optmal. Keywords: Prncpal-agent problems, lnear ncentve schemes, ntertemporal ncentve provson, Brownan moton. JEL classfcaton numbers: C61, D8, J33. 1 For helpful comments and dscussons we are grateful to Darell Due, Olver Hart, Bengt Holmstrom, Nobuhro Kyotak, John Moore, Holger Muller, Sven Rady, and Jae Sung. We are also grateful for research support from the Schwezerscher Natonalfonds, the Deutsche Forschungsgemenschaft, and the Taussg Char at Harvard Unversty. Department of Economcs, Unversty ofmannhem, D Mannhem, emal: [email protected] Department of Economcs, Unversty of Munch, Ludwgstr. 8 (Rgb.), D-8539 Munch, Germany, emal: [email protected] 1

2 1 Introducton Ths paper studes the relaton between mult-perod dscrete-tme and contnuous-tme prncpal-agent models n the semnal paper by Holmstrom and Mlgrom (1987). The purpose s to obtan a better understandng of the structural elements underlyng the lnearty of optmal ncentve schemes n some contnuous-tme models. We share the vew of Holmstrom and Mlgrom that the nonlneartes or even dscontnutes of optmal ncentve schemes that are typcal for statc prncpal-agent models are unlkely to be robust to changes n the models, n partcular, to changes that allow for manpulaton of the nformaton requred to mplement them. However, the argument presented by Holmstrom and Mlgrom does not make the underlyng structure entrely clear, at least to us. The Holmstrom-Mlgrom paper nvolves three derent models: 1. A statc model, n whch the agent chooses a (nte) vector of probabltes over possble states of the world. Ths model s used to show that f the acton space of the agent s sucently rch (of \full dmensonalty" n the underlyng probablty space), then for any acton n the nteror of the acton set, there exsts at most one ncentve scheme mplementng ths acton.. A mult-perod model, whch s a T -fold repetton of the statc model. It s shown that f the technology controlled by the agent s statonary and f the prncpal and the agent have utlty functons wth constant absolute rsk averson, then the optmal ncentve scheme n the T -perod model s a smple T -fold repetton of the optmal ncentve scheme n the one-perod model, wth no attenton pad to the order n whch the derent outcomes arse. 3. A contnuous-tme model n whch the agent controls the drft rate vector (but not the varance) of a mult-dmensonal Brownan moton. It s suggested that ths model can be obtaned as a lmt of a sequence of mult-perod dscrete-tme models speced n such a way that the overall duraton of the prncpal-agent relaton s xed and, along the sequence of models, perods become shorter as well as more numerous. The derent dmensons of the mult-dmensonal Brownan moton

3 correspond to the derent states of the world or outcomes n the statc model and the derent \accounts" showng how often a gven outcome arses n a mult-perod model. As n the mult-perod dscrete-tme models, the assumptons of constant absolute rsk averson and statonarty n the technology ensure that the optmal ncentve scheme n the contnuous-tme model depends only on the cumulatve change n ths vector of \accounts", wth no attenton pad to the detals of the underlyng tme paths. As reported so far, the results of Holmstom and Mlgrom provde for \lnearty" of optmal ncentves n \accounts" lstng the frequences of ncdence of derent outcomes, but not necessarly for lnearty n outcome varables such as total revenues and costs, or prots. Ths reects the fact that \outcomes" n the Holmstrom-Mlgrom analyss correspond to \states of the world" n the statc model whch are dened wthout any reference to varables such as revenues, costs, or prots that would permt aggregaton across states or - n the ntertemporal models - across accounts. Even f such outcome varables are ntroduced, t s not clear that ncentves should rely on lnear aggregates that treat, e.g., two prot realzatons of one as equvalent to one prot realzaton of two and one prot realzaton of zero. Ths s obvously not an ssue f the statc model nvolves only two possble outcomes. In ths case, aggregate prots can be wrtten as a constant plus the number of tmes the hgh-prot outcome occurred tmes the derence between prot levels across outcomes, so trvally lnearty n accounts and lnearty n prots are equvalent. However wth more than two outcomes, lnearty n accounts and lneartyn prots are no longer equvalent, and one needs addtonal assumptons to justfy the use of ncentve schemes that are lnear n prots or n some other outcome aggregate. Wthn ther contnuous-tme model, Holmstrom and Mlgrom gve twoassumptons whch yeld the lnearty wth respect to outcome varables that they are really after. One of these s the assumpton that the agent's eort cost depends only on a lnear aggregate of the drft rate vector that he controls. The other s the assumpton that the prncpal observes only a lnear aggregate of the derent \accounts",.e., the derent dmensons of the Brownan moton whose drft rate vector the agent controls. Ether assumpton ensures that an optmal ncentve scheme nvolves only the correspondng lnear aggregate of the vector of cumulatve changes n \accounts". If the weghtng used n these lnear 3

4 aggregates reects some underlyng outcome varable so that e.g. an outcome nvolvng the prot realzaton \two" twce as much weght as an outcome nvolvng the prot realzaton \one", the ncentve scheme s actually lnear n ths outcome varable. Unfortunately, both these routes to obtanng lnearty ofoptmal ncentve schemes n outcome varables are gven only for the contnuous-tme models. In ether case t s not clear what analogues these results have n the mult-perod dscrete-tme models that serve to approxmate the Brownan moton model. In consequence t s dcult to dsentangle the respectve roles of ntertemporal aggregaton n the Brownan moton model and the addtonal assumptons n provdng for the lnearty of optmal ncentve schemes. The dculty s partly due to the fact that Holmstrom and Mlgrom are very sketchy about the relaton between dscrete-tme and contnuous-tme models. To see that ths s a nontrval matter, note that n a dscrete-tme model the process ndcatng how many tmes each outcome has been observed up to tme t s nondecreasng n t, negatve nstances of observaton beng out of the queston. To get such a process to converge to a Brownan moton, one must be lookng at the \accounts" process relatve to some norm so that a negatve change can be nterpreted as shortfalls of the actual frequency of a gven outcome from the norm. The queston then s where ths norm comes from and how t s speced. Holmstrom and Mlgrom do not say anythng about ths. Nor do they say anythng about the speccaton of outcome varables and ther dependence upon the perodzaton. Our paper lls ths gap. We lnk the one-perod, the mult-perod, and the contnuoustme models of Holmstrom and Mlgrom n a uned framework and explctly derve the contnuous-tme model as a lmt of dscrete-tme models wth ever shorter perods. We wll ndcate the class of Brownan models that can be approxmated ths way. Gven our account of the precse relaton between the mult-perod dscrete-tme and the contnuous-tme models of Holmstrom and Mlgrom, we look for dscrete-tme analogues of ther lnearty results for the contnuous-tme model. Two sets of results are obtaned. Frst we show that there s no dscrete-tme analogue for the lnearty result that s based on eort cost beng a functon of a lnear aggregate of the drft vector that the agent controls. For ths speccaton of eort costs, the correspondng dscrete- 4

5 tme models nvolve soquants that are straght lnes n the space of vectors assgnng probabltes to the derent possble outcomes. Therefore, these models typcally have boundary solutons assgnng probablty zero to all but very few outcomes. Indeed, n the absence of any further restrctons the dscrete-tme models wll have solutons concentratng all probablty mass on the two outcomes closest to the target value for mean returns. Thus, except for a gap left by the dscretzaton of outcomes, all rsk and hence all agency costs of an necent allocaton of rsk wll dsappear. If one looks at a sequence of such dscrete-tme models wth ever shorter perods these boundary solutons do not converge to anythng resemblng the soluton that Holmstrom and Mlgrom derve for the Brownan moton model wth ths speccaton of eort costs. Our second major result shows that there s a dscrete-tme analogue for the Holmstrom-Mlgrom result that n the Brownan moton model a lnear ncentve scheme s optmal f the prncpal observes only a lnear aggregate of the derent dmensons of the Brownan moton whose drft vector the agent controls. To obtan ths analogue, we assume that n dscrete-tme, the prncpal does not observe the tme path of the outcome process at all. He only observes a nal aggregate. Moreover, by the tme he observes the aggregate, the agent may have manpulated t by destroyng some returns that had actually been realzed. Wth ths assumpton about asymmetrc nformaton concernng outcomes, we prove that the lnear ncentve scheme that s optmal n the contnuous model s approxmately optmal n the dscrete-tme models f the perod length s sucently short. The argument reles on the contnuty of certan features of the model n the transton between dscrete and contnuous tme. Statonarty of solutons to dscrete-tme models, whch s central for Holmstrom and Mlgrom, plays no role here because under the gven nformaton assumpton such statonarty typcally s not obtaned. Our results lnk up wth the basc ntuton that Holmstrom and Mlgrom had provded for ther analyss. In motvatng ther paper they had argued that nonlneartes - and even more so, dscontnutes - n ncentve schemes are vulnerable to manpulaton by the agent. However much of ther actual analyss does not nvolve ths noton of manpulaton at all. Manpulaton enters ther analyss only n the case where the prncpal observes only a lnear aggregate of the derent dmensons of the Brownan-moton process; t plays no role n ther mult-perod dscrete-tme analyss or n the other results 5

6 they have for the Brownan moton model. Our results show that for ther speccaton of mult-perod dscrete-tme and contnuous-tme models, the lnearty result that s based on asymmetrc output observaton and the scope for manpulaton that ths ntroduces s the only one that has a dscrete-tme analogue. Wthn the context of ther analyss, concerns about manpulaton must be the underlyng force behnd lnearty. Ths beng sad, we must however pont to the companon paper by Hellwg (1998a), whch does develop a lnearty result based on the speccaton of the cost functon, wthout any concern about manpulaton. However, ths result nvolves a speccaton n whch varances and duson parameters are endogenous and eort cost depends on both, drft rates and duson parameters. Ths speccaton cannot be accommodated wthn the framework of Holmstrom and Mlgrom whch presumes exogenously gven duson terms. The Holmstrom-Mlgrom paper has gven rse to a large lterature, but only a few papers are concerned wth the methodologcal and mathematcal underpnnngs of the analyss. Most papers n the lterature just appeal to ther results to justfy the use of lnear ncentve schemes n applcatons; for an example see Holmstrom and Mlgrom (1991). Among the more method-orented papers, Schattler and Sung (1993) develop a general mathematcal framework for the study of agency problems when the agent controls the drft of a Brownan moton n contnuous tme; ther results strengthen and greatly extend the contnuous-tme results of Holmstrom and Mlgrom (1987). Sung (1995) further extends the analyss by allowng for moral hazard wth respect to rsk choces, more precsely, choces concernng the duson parameters of a Brownan moton. Sung (1997) develops the correspondng analyss for contnuous-tme agency problems concernng jump processes. Bolton and Harrs (1997) also consder contnuous-tme problems concernng jump processes as well as dusons. However they are concerned wth rst-best rather than second-best problems, consderng optmal rsk sharng and optmal actons wthout concern for ncentve compatblty and showng that for arbtrary preference speccatons rsk-sharng consderatons wll typcally call for nonlnear contracts n a rst-best settng. The above-mentoned papers all work drectly n contnuous tme and do not dscuss the relaton between statc or mult-perod dscrete-tme and contnuous-tme agency models. Mult-perod dscrete-tme agency models and ther relaton to contnuous-tme 6

7 models are studed by Schattler and Sung (1997) and by Muller (1997). Unlke Holmstrom and Mlgrom (1987) these papers consder mult-perod models as T -fold repettons of a statc model n whch the agent 's eort choce determnes the mean of a normally dstrbuted random varable. Muller (1997) shows that, as n Mrrlees (1974), n a model of ths type, the prncpal's problem typcally does not have a soluton because a rst-best allocaton can be approxmated (but not reached) by a sequence of ncentve schemes usng penaltes for low outcome realzatons to dscourage shrkng, the ncdence of penaltes becomng ever rarer and the penaltes themselves becomng ever more severe as one goes along the sequence. 1 The concluson s ndependent of the number of perods T or the \length of one perod" T n the mult-perod model, but when T becomes large and T goes to zero, the requste penaltes become large and ther ncdence becomes ever rarer even f the shortfall of payo expectatons from rst-best s kept xed. In Schattler and Sung (1997), exstence of a soluton to the prncpal's problem n dscrete tme s also a problem. To get around t the authors mpose a pror restrcton on the class of ncentve schemes they admt; wth ths restrcton, they nd that condtons for mplementng a gven strategy of the agent are smlar n a T -perod dscrete-tme and n a contnuous-tme Brownan-moton model. Nether paper asks how a contnuous-tme Brownan-moton model would be approxmated by dscrete-tme models wth ntely many possble outcomes n any one perod or what s the relaton between lnearty ofncentve payments n \accounts" and lnearty of ncentve payments n outcome aggregates, e.g. n prots. The plan of the paper s as follows. The next secton develops the framework for the analyss by ntroducng one-perod and mult-perod models and specfyng the role of the \length of the perod" as a parameter of these models. Secton 3 consders a sequence of control paths n the dscrete models and shows that f ths sequence converges to a well dened control path n the contnuous model, then the ncentve schemes that mplement the control paths n the dscrete models converge to an ncentve scheme that mplements the lmt path n the contnuous model. Whereas ths result takes the convergence behavor of control paths as gven, Secton 4 shows that ths convergence behavour s actually obtaned for sutable subsequences of optmal control paths and ncentve schemes. Secton 5 deals wth the case where the agent's cost functon depends on expected prots only. 1 In Holmstrom and Mlgrom (1987), ths dculty was avoded by the assumpton that the set of states of the world n the statc model s nte. 7

8 Secton 6 consders the case where the prncpal observes an accountng aggregate, such as total prots, but not the tme paths of ndvdual accounts. There we show that Holmstrom and Mlgrom's man result on lnearty naggregates can ndeed be approxmated by a seres of approprately desgned dscrete models. All formal proofs are relegated to the appendx. In partcular, Appendx A gves the formal proofs of all results n the text. Certan supplementary are gven n Appendces B and C. A Dscrete Mult-Perod Model We start wth the statc model. Suppose that there s one perod of length 1. At the begnnng of the perod the agent chooses an acton whch gves rse to a stochastc outcome ~ f ;:::; N g. The outcome IR s nterpreted as a prot level. Followng Holmstrom and Mlgrom (1987) we assume that the agent chooses the probablty dstrbuton p over possble prot levels drectly at personal cost c(p). Thus, the agent's acton s p =(p ;:::;p 1 )P where P s the N-dmensonal smplex. Throughout the paper we assume the followng: Assumpton 1 The eort cost c(p) of an acton p P s gven by a functon c() on IR N+1 whch s strctly convex as well as contnuously derentable on some open set that contans the nteror of P. The agent s assumed to have a constant coecent of absolute rsk averson r >. Gven an ncentve scheme assocatng the payment s to the outcome,hechooses acton p P so as to maxmze hs expected utlty, = p e,r(s,c(p)) : (1) From the rst-order condtons for ths maxmzaton problem, one easly nds that to mplement an acton p P such that the certanty equvalent of the agent's utlty s, an ncentve scheme s =(s ;:::;s N )must satsfy: s c(p), 1 r 1, rc + r 8 j= p j c j 1 A ()

9 for =;:::;N, wth equalty fp >, where, as usual, c refers to the partal dervatve of the eort cost functon wth respect to p. The argument s a straghtforward adaptaton of Theorem 3 n Holmstrom and Mlgrom to allow for the possblty of boundary solutons,.e., of actons p nvolvng p = for some. As ndcated by (), f p = for some, the correspondng ncentve payment s s not unquely determned by ncentve compatblty consderatons. However, we assume that for actons p nvolvng p = for some, we may set the correspondng ncentve payment s so as to satsfy () wth equalty. The prncpal s assumed to be rsk neutral. Hs payo from mplementng an acton p by an ncentve scheme s = fs ;:::;s N g s gven by: = p (, s ) : (3) Consder now amult-perod verson of ths model n whch the agent can change hs acton at dscrete ponts n tme. We want to keep the total length of the tme nterval (whch s normalzed to 1) xed, however, and ncrease the number of perods wthn ths nterval. Suppose that there are 1 perods, each of length f1; 1 ; 1 ;:::g, whch are 3 ndexed by = f1;:::; 1 g. In order to make the one-perod problem and the 1 -perod problem comparable we have to reformulate the model: In each perod there are N + 1 prot levels, f;:::;ng, whch are gven by3 = 1 8 f;:::;ng : (4) The agent's acton p n each perod and the cost of hs acton wll be evaluated as a functon of the devaton of p from some standard ^p, where ^p P, ^p, s an acton nvolvng zero expected prots,.e. = ^p = = ^p = (5) If the rst-order condtons for the agent's maxmzaton problem are sucent aswell as necessary, ths s wthout loss of generalty. Otherwse, havng a payment s strctly below the rght hand sde of () may serve to dscourage the agent from some \far away" acton p nvolvng p >. 3 In ther dscusson of the relaton between mult-perod dscrete-tme and contnuous-tme models, Holmstrom and Mlgrom do not ndcate that the possble outcome values n any one perod must depend on the length of the perod (1987, p. 318). Such a normalzaton s, however, mplct n ther analyss as t underles ther subsequent appeal to the central lmt theorem. For a more systematc dscusson of the role played by the dependence of the potental values of perod prots on perod length, see the companon paper by Hellwg (1998a). 9

10 for all. The eort cost n a perod of length f the agent chooses p s denoted by c (p ) whch s dened by c (p ) c ^p + p, ^p ;:::;^p 1 N + p N, ^p N 1! : (6) What s the pont of ths speccaton? There are 1 perods each of length. In order to keep total prots over the 1 perods comparable to expected prots n the statc model we have to keep expected prots per perod to an order of magntude. Consder an acton p that shfts probablty mass from some low-prot outcome to some hgh-prot outcome as compared to the acton ^p. Wth prot levels proportonal to 1 ths rases expected prots by an order of magntude 1. However, the agent's cost of ths shft n probablty mass s made to depend on n such away that the agent wll keep the order of magntude of such shfts ordnarly to 1. That s, f the soquants of the cost functon c() exhbt nonzero curvature, then the devaton of the acton p that s actually chosen from the zero-expected-prots acton ^p wll be on the order of 1, and expected prots per perod wll be on the order of. To make these deas more precse, dene = k p, ^p 1 for =1;:::;N, wth k =,. Note that f the agent chooses p n each perod, then total expected prots n the perod of length 1 are gven by 1 = p = 1 = For any, the quantty (p, ^p )(, ) 1 = = (, ) p, ^p 1 = (7) : (8) n (7) and (8) can be nterpreted as the contrbuton to expected prots per unt of \real tme" that stems from the agent shftng probablty mass from state of the world zero to state of the world, relatve to the standard set be the zeroexpected-prots acton ^p. It wll be useful to thnk of the agent choosng the vector =( 1 ;:::; N)neach perod, whch then determnes an assocated acton p ( ) where for =1;:::;N, p ( ) = ^p + p ( ) = 1, 1 k 8 f1;:::;ng ; (9) p = ^p, 1 1 k : (1)

11 Snce p s a probablty vector we have to restrct the agent's choce of by, k ^p 1 k (1, ^p ) 1 (11) for all f1;:::;ng and, 1, ^p 1 k ^p 1 Note that the set of satsfyng (11) and (1) ncreases and goes to the entre set IR N as!. The agent's cost from the choce n a tme perod wth length s gven by: (1) ^c( ) = c (p ( )) ; (13) where by (6) c (p ( )) = c! ^p + p ( ), ^p =c ^p 1,! ; ^p ;:::;^p N + N : (14) k k 1 k N Note that for any gven vector that s ndependent of the argument of c() n (14) s also ndependent of. The tradeo between the vector = ( 1 ;:::; N ) of contrbutons to expected prots per unt of \real tme" that stem from shftng probablty mass to outcomes 1;:::;N and the assocated eort cost per unt of \real tme" ^c( ) s thus ndependent of. We are now ready to prove our rst result, whch shows that gven our speccaton of prots, costs, actons and probabltes t s ndeed possble to compare the dscrete-tme models wth derent perod lengths. Proposton 1 Consder the dscrete problem wth 1 subntervals where pro- t levels are gven by (4) and where the agent's eort cost as a functon of s gven by (14) each perod. If for =1;:::; 1 the agent chooses ; = ( ; 1 ;:::; ; N ), then expected gross prots are equal to P P N =1 ;, and eort costs are equal to P =1 ^c( ; ), where ^c() s gven by (13). In partcular, f ; = regardless of and of the perod length, then expected gross prots and total eort costs of the agent are ndependent of. 11

12 The proposton shows that f the agent chooses a constant, then expected prots and eort costs of the agent are ndependent of. In partcular, the trade-o between expected prots and eort costs s not aected by the length of the perod. To be sure, the mplementaton problem as seen by the prncpal wll also depend on the varance and the other hgher moments of the dstrbuton of prots, all of whch depend on. However, as wll be shown n the next secton, when s small and the agent's acton s close to the standard ^p, these hgher moments are close to beng ndependent of, and the ncentve payments that are requred to mplement p () admt a smple approxmaton. 3 Approxmaton of the Brownan Model So far we assumed that the agent can change hs acton only at dscrete ponts n tme. Now we are nterested n the case where the agent can change hs acton contnuously at any pont n tme. In the followng we wll derve the contnuous case as the lmt of the dscrete model when goes to. The followng dentons wll be useful. Let ~ A ; f;1gbe a random varable such that A ~ ; = 1, ~ ; =, f;:::ng, f 1 g m m=1;:::, f1;:::; 1 g. Clearly, Prob( ~ A ; =1)=p ;, and A ~ ; a Bernoull dstrbuton, but ~ A ; Let s = 1 mples ~ A ; j and ~ A ; j = for all j 6=. Thus, each ~ A ; are not stochastcally ndependent. be the ncentve payment for outcome correspondng to () when the perod length s, the eort cost functon s gven by (6), and the certanty equvalent of the agent's utlty s. Usng (13) and (14) one can wrte s than the acton p ( ) that corresponds to t. Ths yelds: s = ^c( ), 1 r 1, r^c k 1 + r N j= p j ^c j k j 1 has n terms of the vector rather 1 A ; (15) where ^c = c,c k made of the fact that by (6) c s the partal dervatve of^cwth respect to,^c =, and use has been = 1 c for all. (Clearly the convexty and derentablty assumpton on c() mply that ^c() s strctly convex and contnuously derentable on the set of all vectors = ( 1 ;:::; N ) for whch ^p =1;:::;N.) 1 > P N =k and ^p >, =k for

13 Usng a Taylor seres expanson of the logarthmc term n (15), the requste ncentve payment s s now approxmated by: s = ^c( )+ 4^c k, j= p j ( )^c j k j r 4^c k, j= 3 p j ( )^c j k j 5 +O 3 Suppose that the prncpal wants to mplement the tme path of actons f ; g =1;:::; 1. Then, stll assumng that the certanty equvalent of the agent's utlty has to be, the total renumeraton that has to be oered s gven by ~s = =1 + r = =1 + r ^c( ; )+ =1 = ^c( ; )+ =1 = =1 = ~A ; ~A ; 1 4^c k, =1 = ~A ; j= 4^c k, j= p j ( )^c j k j p j ( )^c j k j 5 + O 3 =1 ^c ( ; ) ~ A ;, p (; ) k 1 4^c ( ; )k, where we made use of the fact that P N ~ = A ; j= (16) 3 p j ( ; )^c j ( ; )k j 5 +O( 1 ) (17) = 1 for all f1;:::;g. If we substtute p j ( ; ) n the squared term by (9) and (1), use ^c =,and rearrange by puttng the approprate terms nto O( 1 )weget ~s = =1 + r ^c( ; )+ =1 = =1 = ~A ; ^c ( ; ) ~ A ;, p (; ) k 1 4^c ( ; )k, Equaton (18) admts a smple nterpretaton: 4 j= 3 ^p j^c j ( ; )k j 5 +O( 1 ) (18) The rst term reects the total eort cost of the agent from choosng f ; g, =1;:::; 1. The second term gves the approprate ncentves to the agent to actually choose ths tme path of actons. If outcome s realzed n perod, ths rases the agent's overall ncentve payment by an amount ^c ( ; )k 1, reectng the margnal cost of shftng probablty mass towards outcome. The expected value of ths payment, p (; )c (; )k 1, s subtracted agan because n expectedvalue terms the prncpal's payments to the agent depend only on ndvdual-ratonalty 4 See also Schattler and Sung (1993, p. 337). 13

14 consderatons. Thus the second term n (18) s proportonal to the derence between the actual realzaton of ths prot level and the expected realzaton gven that the agent chooses p ( ; ). It s useful to smplfy ths term by denng ~ ; = k 1 ~A ;, p (; ) (19) The thrd term s the rsk premum that has to be pad to the agent to compensate hm for the randomness of the second term. The last term reects the approxmaton we are usng; t vanshes as goes to zero. Gven the way the per-perod devatons ~ ; of realzed from expected prots depend on the length of the perod, we nd t more convenent to work wth the cumulatve devatons ~ () = k 1 =1 [ A ~ ;, p ( ; )] () from perod one to perod. For =1;:::;N, ~ () can be thought of as the cumulatve devaton (up to ) of realzed prots under outcome from the expected value of these prots under the gven polcy of the agent. of cumulatve devatons from the mean". ~ () wll be called the \stochastc process To embed the dscrete-tme model n contnuous tme, we use a lnear nterpolaton to represent the process ~ () by a contnuous-tme process (t) = 1, t t + ~ t + t, t (t) such that for t [; 1]: ~ where [ t ] denotes the greatest nteger less than or equal to t. t +1 Note that (1) (t) s a random functon takng values n C = C[; 1], the space of contnuous functons on [; 1]. For any determnstc tme path of actons 5 f ; g =1;:::; 1 we use a contnuous-tme representaton () where n the dscrete-tme model (t) = ;[t=] : () We can now state our rst man result: 5 We restrct attenton to determnstc tme paths of actons. Snce Holmstrom and Mlgrom have shown that the agent wll be nduced by the optmal contract to take a constant determnstc acton there s no need to consder stochastc controls explctly at ths stage. 14

15 Theorem 1 Consder a sequence of dscrete models wth perod length, = 1; 1 ; 1 3 ;:::. Suppose that, as!, the tme path of actons (t) converges unformly to some contnuous functon (t), t [; 1] such that ^p > P (t)=k and ^p >, (t)=k for all t and = 1;:::;N. Then, as!, (a) the stochastc process of cumulatve devatons from the mean (t) = ( 1 (t);:::; N(t)) converges n dstrbuton to a process () whch s a drftless N-dmensonal Brownan moton wth covarance matrx = k 1 ^p 1(1, ^p 1 ),k 1 k ^p 1^p,k 1 k N^p 1^p N,k k 1^p ^p 1 k^p (1, ^p ),k k N^p ^p N......,k N k 1^p N^p 1,k N k ^p N^p k N^p N(1, ^p N ) and startng pont () = ; (b) the total cost to the agent converges to R 1 ^c((t))dt; 1 C A (3) (c) the ncentve payments that serve to mplement (t) wth certanty equvalent w converge n dstrbuton to ~s = w+ ^c((t))dt+ where ^c () =(^c 1 ();:::;^c N ()). ^c ((t))d + r ^c ((t))[^c ((t))] T dt (4) Remarks: 1. Theorem 1 s closely related to Theorem 6 of Holmstrom and Mlgrom (1987) and Corollary 4.1 of Schattler and Sung (1993), who show that n the contnuous model a tme path of actons () s mplemented by an ncentve scheme satsfyng s = w + + r ^c((t))dt + ^c ((t))dz, ^c ((t))(t)dt ^c ((t))[^c ((t))] T dt (5) where Z s a process gven by the stochastc derental equaton dz = dt + db (6) 15

16 wth ntal condton Z() = ; here B s an N-dmensonal Brownan moton wth covarance matrx that s equvalent to the process n Theorem 1. 6 In contrast, our Theorem 1 deals wth the dscrete model and shows that the ncentve schemes that serve to mplement the exogenously gven sequence of control paths (t), = 1; 1 ;:::, converge to a contnuous functon (4). It does not show that n the contnuous model the lmt control path (t) s ndeed mplemented by the lmt ncentve scheme (4). However, under the assumptons mposed by Schattler and Sung (1993, Corollary 4.1), whch we wll mpose and dscuss n more detal n Secton 5, the lmt of the ncentve schemes s does ndeed mplement the lmt of the control path (t). Theorem 1 thus shows that the mplementaton condton (5) on ncentve schemes and polces that Holmstrom and Mlgrom obtaned n ther Brownan model can be nterpreted as the lmt of the correspondng condtons on ncentve schemes and polces n approxmatng dscrete-tme models.. Dscrete-tme approxmatons of the contnuous-tme process Z() n (5) can be gven by specfyng ~Z ; = k 1 [ ~ A ;, ^p ] (7) as the contrbuton to total prots stemmng from outcome n excess of some \standard" that s gven by the probablty ^p, and ~Z () = k 1 " =1 ~A ;, ^p as the cumulatve contrbuton. Usng the lnear nterpolaton as before to obtan a contnuous-tme representaton Z (t) = 1, t t + ~Z t + # t, t ~Z t +1 of the process (8), we nd that, by a smple corollary to Theorem 1(a), the processes (8) (9) Z (t) converge n dstrbuton to the process Z() asgoes to zero.7 6 Holmstrom and Mlgrom show that (5) s necessary to mplement (). Schattler and Sung consder a more general model and gve condtons under whch under whch (5) s sucent. 7 Holmstrom and Mlgrom provde a sketch of the relaton between dscrete and contnuous models n terms of the processes Z () and Z() (wthout ndcatng the dependence of perod prot levels on ). We have nstead focussed on the processes () and () because these processes are of central mportance n the applcaton of Donsker's Theorem. 16

17 To better understand the role of the \standard" ^p n the speccaton (7) and (8) note that f s small, and f the agent chooses a constant n some small tme nterval [t; t ], we have E Z (t ), Z (t) k 1 (p (), ^p ) t, t = (t, t ) : (3) Thus, f the agent chooses ^ =(;:::;) for all [t; t ], then p () =^p n each perod and the agent allocates hs probablty mass accordng to the standard. If > ( < ) he puts more (less) weght on achevng state of the world rather than state of the world (as compared to the standard set by ^p). Ths s reected n the account process Z (t) whch n expectaton measures how much weght the agent puts on achevng state. Furthermore, we know from the proof of Proposton 1 that f the agent chooses a constant, expected prots n tme nterval [t; t ] are gven by [t =] =[t=] = p = ([t =], [t=]) (t, t) : (31) Recall that P N = ^p =. Hence, s smply the expected contrbuton to total prots that stems from the realzatons of state of the world n excess of the standard set by ^p, and the account Z (t) measures the actual contrbuton over tme. 3. In the contnuous-tme lmt the agent controls the drft rate but not the hgher moments of the cumulatve output process. Note that he does control the entre dstrbuton of the cumulatve-output process n the dscrete-tme models. However, when s small, mplementaton of a gven drft rate process nvolves actons p (()) close to the constant acton ^p, the derence n any perod beng of order 1. Because of ths convergence of the path of actons to the constant path wth value ^p, the second moments of the cumulatve-returns process n the lmt are fully determned by ^p, and the hgher moments vansh. The crtcal acton vector ^p thus serves a dual functon n our model: Frst, as an acton vector wth a zero prot expectaton t provdes a base devatons from whch measure the mpact of behavour on expected prots (see equaton (3)). Second, t determnes the \nose" of the agency problem n contnuous tme. 17

18 Theorem 1 takes the dscrete model as gven and shows how to obtan a Brownan model as the lmt of a sequence of these dscrete models. We could have proceeded the other way round, askng whether t s possble to approxmate a gven Brownan model wth some sequence of dscrete models. The followng result, whch s an mmedate corollary to Theorem 1, gves an answer to ths queston. Corollary 1 Let B be a gven N-dmensonal Brownan moton wth zero drft and covarance matrx. Suppose that f the agent chooses a control process (), then the agent's cumulatve costs are gven by the derental equaton dc =^c((t))dt (3) and the dsturbance process s B. If there exst real numbers k 1 ;:::;k N postve real numbers ^p 1 ;:::;^p N wth P N ^p 1 such that can be wrtten as n (3), then to ths contnuous tme model there corresponds a dscrete-tme model wth perod length n whch - there are N +1 possble prot levels n each perod whch are gven by and =, 1 ^p k (33)! = 1 k, ^p k (34) - the probablty p ( ) of prot level s gven by (9) and (1), wth ;[t=] = ([t=]), - and the cost to the agent n each perod s gven by c (p ( )) = ^c( ), such that the contnuous model s the lmt of ths dscrete model (n the sense of Theorem 1) as approaches. Corollary requres that the covarance matrx of the N-dmensonal Brownan moton can be wrtten as n (3). If N = 1, ths s always possble. If N > 1, ths condton mples a restrcton on the set of Brownan models that can be approxmated by a sequence of dscrete models. Ths restrcton stems from the fact that n the dscrete model only one state of the world can materalze n each perod. Ths mples that the 18

19 accounts ~ A ; and thus also the accounts ~ ; example, t s mpossble that the ~ ; restrcton s very natural. have a specal covarance structure. For are stochastcally ndependent. Therefore ths In the entre analyss here, the dmenson N of the Brownan moton has referred to the number of derent prot levels that can be dstngushed. Holmstrom and Mlgrom (1987, p. 3) oer a second nterpretaton accordng to whch N refers to derent actvtes of the agent. For example, f N =, account 1 could be a measure of revenues, whle could be a measure of costs. Thus, 1 reects the agent's eort to ncrease revenues, whle reects hs eort to reduce costs. Wth ths nterpretaton the Brownan model could have any varance-covarance structure. It s possble to show that any N- dmensonal Brownan moton model (wth an arbtrary covarance matrx ) can be approxmated by a dscrete model, f we extend the dmenson of the dscrete model to N, 1. However, we do not want to go nto the detals of ths approxmaton here. 8 4 The Convergence of Optmal Control Paths and Incentve Schemes Theorem 1 s not qute satsfactory n that t takes the sequence of control paths () and ther convergence behavour as gven. However () s chosen endogenously by the agent n response to the ncentve scheme s. Moreover s and () together are chosen endogenously, subject to ncentve compatblty, by the prncpal and agent when they ntally agree on a contract. One may therefore wonder how relevant Theorem 1 stll s once the endogenty of s and () staken nto account. 8 Tosketch the basc dea suppose that the Brownan model s two-dmensonal and that 1 s the drft rate of revenues whle s the drft rate of costs. To approxmate ths model by a sequence of dscrete models we need at least two derent levels of revenues R fr 1 ;R gand two levels of costs C fc 1 ;C g. Thus, n each perod there are four derent possble outcomes f(r 1 ;C 1 );(R 1 ;C );(R ;C 1 );(R ;C )g, whch means that there are, 1 =3derent accounts. Wth ths three-dmensonal model we can approxmate any varance-covarance matrx n the two-dmensonal contnuous model by choosng the correlaton between the Bernoull dstrbutons over fr 1 ;R g and fc 1 ;C g approprately. If we want to allow for, say, M r levels of revenues and M c levels of costs, we need a model wth M r M c possble outcomes and hence M r M c, 1 derent accounts. The problem of gettng from these M r M c, 1 derent accounts to ncentve schemes that are dened n terms of just revenue and cost aggregates s then the same as the problem of aggregaton accross accounts that s studed n Sectons 5 and 6. 19

20 Two ssues arse. Frst, for some ncentve schemes t wll be optmal for the agent to choose a control path that s not smply a contnuous functon of t, but that may also depend on the realzatons of the prot process up to tme t. Second, even f () s a hstory-ndependent constant, t s not clear that the sequence () should have a convergent subsequence. Of these two ssues, the rst one s unproblematc. Hstory dependence of the control strateges causes techncal dcultes because the varables ~ ; n (19) are no longer ndependent, but n spte of these dcultes Theorem 1 can be extended to control paths () that are predctable functons of t and the hstory up to t, for detals see Appendx B. More mportantly, Holmstrom and Mlgrom (1987, Theorem 5) have shown that n the repeated dscrete model there always s an optmal soluton to the prncpal's problem n whch he nduces the agent to take the same acton n each perod, regardless of pror hstory. If s such a constant optmal control path n the dscrete model wth perod length, and f the sequence f g has a convergent subsequence, Theorem 1 can automatcally be appled to ths subsequence. However wthout addtonal assumptons, a sequence f g of constant optmal control paths n the dscrete models wll not necessarly have a convergent subsequence. Indeed n the followng secton we shall come across a farly natural example n whch goes out of bounds, wth some of ts components gong to +1 and others gong to,1 as goes to zero. Such a possblty can be ruled out by addtonal assumptons, but then one must worry about the compatblty of such assumptons wth the speccaton underlyng Theorem 1. To see the ssue, suppose for example that the vector must always be chosen from a product K = Q N [m ;M ] of compact ntervals. Wth ths addtonal assumpton a sequence f g of constant optmal control paths n the dscrete models wll obvously have a subsequence that converges to a lmt. Moreover the rst two statements of Theorem 1 wll be satsed for ths subsequence,.e. the process () of cumulatve devatons from the mean converges n dstrbutons to the drftless Brownan moton () wth covarance matrx and the agent's total cost converges to ^c( ). However the requrement that K wll modfy the agent's ncentve compatblty condtons and hence the relaton between desred actons and requred ncentve payments; ths throws

21 doubt on statement (c) of Theorem 1. In choosng hs acton p P, the agent now faces the addtonal constrants p p P, where p = ^p + 1 mnfm =k ;M =k g and P = ^p + 1 maxfm =k ;M =k g, = 1;:::;N. Takng account of these constrants n the agent's rst-order condtons, one nds that the ncentve-compatblty condton (19) must be rewrtten as # s = ^c( ), 1 "1 r ln, (r^c k 1, u N + v )+ (r^c k 1, u + v ) (35) where u and v are the Kuhn-Tucker multplers of the constrants p p and p P n the agent's problem, wth complementary slackness requrng u (p, maxf;p g) = and v (P, p ) = for = 1;:::;N. Gven the appearance of the Kuhn-Tucker multplers u and v n (35), t s not clear that the approxmaton procedure of the precedng secton can be used; for nstance, a Taylor approxmaton of the logarthmc term n (35) would make sense only f u and v were known to go to zero as the contnuous-tme model s approached. Holmstrom and Mlgrom try to avod ths dculty by assumng that optmal actons always le n the nteror of the admssble set. Presumably such an assumpton s justed f the eort cost functon satses sutable Inada condtons. Unfortunately such Inada condtons are not compatble wth the unform boundedness of the dervatves of the eort cost functon ^c() whch has been used extensvely n the proof of Theorem 1(c). Therefore we prefer to tackle the problem posed by the Kuhn-Tucker multplers n (35) drectly, wthout tryng to rule boundary actons out. As t turns out, ths can be done wthout any substantve change n the model. v From the prncpal's perspectve, the appearance of the Kuhn-Tucker multplers u, n (35) reects the fact that when the acton p = p ( ) s at the boundary of the admssble acton set, there s more than one ncentve scheme that wll mplement ths acton, so he must consder whch of these ncentve schemes s cheapest for hm. For example suppose that N =1, 1 >, and consder the acton p whch assgns the smallest possble mass to the outcome 1,.e., let p = p (m 1 ). Ths acton s mplemented by any ncentve scheme (s ;s 1) that satses s 1,s 1 r [ln(1+rp 1^c 1 k 1 1 ),ln(1,rp ^c 1 k 1 1 )] and that s ndvdually ratonal for the agent, the pont beng that the prncpal does not have to provde ncentves for outcome 1 because the agent s unable to lower p 1 below the 1

22 stpulated level anyway. Among the derent schemes that mplement the acton p (m 1 ), the scheme s 1 = s =^c(m 1 )scheapest for the prncpal; t just compensates the agent for hs cost and nvolves no rsk premum. Derent ncentve schemes that mplement the same acton wll nvolve derent values of the Kuhn-Tucker multplers n (35). Usng the two-outcome example one easly veres that not all these Kuhn-Tucker multplers wll be small f s small, so for some of the ncentve schemes mplementng the acton p (m 1 ), no analogue of Theorem 1(c) can be gven. However n the two-outcome example one also sees that the ncentve scheme whch s cheapest for the prncpal, namely s 1 = s = ^c(m 1), nvolves Kuhn-Tucker multplers u 1 = r^c 1k 1 1 and v 1 =,whch do become small as goes to zero. Ths last observaton reects a general prncple. Qute generally the prncpal's concern for mnmzng mplementaton costs leads to the use of ncentve schemes nvolvng payments wth a mean on the order of and devatons from the mean on the order of 1, so as goes to zero an argument analogous to the one underlyng Theorem 1 can be gven. Ths provdes the key to the convergence behavor of optmal control paths and ncentve schemes, whch we now dscuss. Consder the prncpal's optmzaton problem n the dscrete model wth perod length. By Theorem 5 of Holmstrom and Mlgrom, we can restrct attenton to constant controls. A constant control K n each perod, wth assocated ncentve scheme ~s, wll be optmal for the prncpal f t maxmzes hs expected payo U P =, E(~s ) (36) under the gven ncentve compatblty and partcpaton constrants. agent's rst order condton (35), ncentve schemes may be assumed to satsfy ~s = =1 = In vew of the ~A ; (w + s ) (37) where w s the agent's certanty equvalent and, for each, s ( ) satses (35) for a sutable set of Kuhn-Tucker multplers u, v. It s convenent to wrte the Kuhn-Tucker multplers n (35) n the form r 1 k ^u, r 1 k ^v, where for =1;:::;N,^u, ^v, and ^u (, m ) = ^v (M, ) = (38)

23 Wth ths change of notaton (35) can be rewrtten as: s ( ) = ^c( ) (39) 3, 1 N r ln 4 1, r(^c, ^u +^v )k 1 +r p j ( )(^c j, ^u j +^v j)k j 1 5 j=1 Upon substtutng from (37) and (39) and usng the fact that E ~ A ; wrte the prncpal's objectve as: U P ( ; ^u 1 ; ^v 1 ;:::;^u N ;^v N) = + 1 r p ( )ln = p ( ) we can, w, ^c( ) (4) 3 N 4 1,r(^c, ^u +^v )k 1 +r p j ( )(^c j, ^u j +^v j)k j 1 5 j=1 In the dscrete-tme model wth perod length, the prncpal's problem reduces to maxmzng (4) wth respect to K and u V, v V, =1;:::;N, subject to the complementary slackness condton (38) and the ncentve compatblty condton:, p ( )e,r(s,^c( )), whch for ncentve schemes satsfyng (39) s equvalent to p ( )e,r(s,^c( )) (41),1, p ( )e,r(^c( ),^c( )) 1 1, r(^c, ^u +^v )k 1 +r p j ( )(^c j, ^u j +^v j)k j 1 A j=1 (4) for all K. If the rst-order condton (39) was sucent as well as necessary for the mplementaton of through the ncentve scheme ~s, the ncentve constrant (4) would be redundant. However because of the convexty of the exponental functon, (41) s typcally stronger than the agent's rst-order condton (39) so (4) s not redundant. Even so t s nstructve to consder what happens when the prncpal replaces the global ncentve compatblty constrant (41) by the rst-order condton (39),.e., when he maxmzes (4) subject to the complementary slackness condtons (38) wthout payng attenton to (41). For ths \relaxed problem of the prncpal" we obtan: 3

24 Proposton Let K be gven and suppose that, ^v, =1;:::;N, maxmze (4) subject to the complementary slackness condton (38). Then ^u ( )=k and ^v ( )=k, =1;:::;N, where ( ):= max j (k ^c ( ), k j^c j ( )). Proposton suggests that the prncpal wll optmally choose the ncentve scheme ~s so that the Kuhn-Tucker multplers for the agent's maxmzaton problem wll be commensurate to margnal costs and wll have the order of magntude 1. Unfortunately we cannot mmedately use ths result because the global ncentve constrant (41) s not generally mpled by the agent's rst-order condton, and (4) s not n general redundant. However, as goes to zero, ths concern becomes less and less mportant, the derence between the ncentve compatblty condton (41) and the rst-order condton (38) dsappears, and the ncentve constrant (4) ceases to mpose any addtonal constrants. Formally we have: Proposton 3 For =1; 1 ;:::;let IC() be the set of vectors ( ; ^u ; ^v K IR N + IR N + that satsfy the ncentve constrant (4) as well as the complementary slackness condtons (4). As converges to zero, IC() converges (n the Hausdor topology) to the set of vectors (; ^u; ^v) K IR N + IR N that + satsfy ^u (, m ) = ^v (M, ) = (43) for =1;:::;N. Gven Propostons and 3, another applcaton of the maxmum theorem yelds: Theorem Let f g be a sequence of constant optmal control paths that the prncpal wants to mplement n the dscrete model wth perod length = 1; 1 ::: when s constraned to the set K = Q N [m ;M ]. Let ^u, ^v, = 1; :::; N, be the normalzed Kuhn-Tucker multplers nduced by the correspondng optmal ncentve schemes. The sequence f ; ^u 1 ; ^v 1 ;:::;^u N ;^v N g has a subsequence whch converges to a lmt ( ; ^u 1 ; ^v 1 ;:::;^u N ;^v N). Moreover, saconstant optmal control path and the ncentve scheme s s gven 4

25 by s () = w+^c( )+ ( (^c ),^u +^v ) (1)+ r j=1 (^c,^u +^v ) j(^c,^u +v ) ; (44) wth ( j ) = gven by (3), whch s an optmal ncentve scheme n the contnuous model. Turnng to the optmal ncentve schemes, we note that (4) can be rewrtten n the lnear form: N ~s = + (1) ; (45) where = w + s ( ), and, for =1;:::;N, =[s ( ),s ( )]=k 1 and s the process of cumulatve devatons from the mean that was dened n () and (1). From Theorem one mmedately obtans: Corollary For =1; 1 ;:::; let ( ; 1 ;:::; N) be the parameters of an optmal ncentve scheme mplementng a constant optmal control when controls are restrcted to the compact set K = Q N [m ;M ]. The sequence f ; 1 ;:::; nghas a subsequence whch converges to a lmt ( ; 1 ;:::; N ). The lmt ( ; 1 ;:::; N ) denes an optmal ncentve scheme ~s = + (1) (46) n the contnuous model. The coecents, = 1;:::;N, satsfy = ^c ( ), u + v where and u, v, = 1;:::;N, are the correspondng lmts of and u, v along the convergent subsequence. 9 9 In Corollary 1, as n Theorem 1, one can replace the processes () and () by the processes Z () and Z () that were used by Holmstrom and Mlgrom and that are dened n (6) - (9). In terms of these processes, (45) and (46) become ~s =^ +P N ^ Z (1), P P and ~s =^ N + P Z (1), where, for =1;:::;N, and are the same as before and ^ =, N N, ^ =, 5.

26 5 Lnearty n Aggregates So far the optmal ncentve schemes n the dscrete mult-perod and n the correspondng Brownan model are lnear only n accounts, not n total prots. Speccally, total ncentve payments n (45) are equal to a constant plus P N ^c ( ) (1); unless ^c ( ) s the same for all = 1;:::;N, ths s not representable as a functon of P N (1), the total contrbuton to prots from havng outcomes other than. In partcular, f N s large,.e., f there are many derent possble prot levels n each perod, the correspondng Brownan moton s of hgh dmenson and the optmal ncentve scheme may be very complex, much more complex than real world contracts that are often lnear n aggregates, such as total prots. As noted by Holmstrom and Mlgrom, the precedng remarks are moot f N = 1, and the statc model nvolves just two possble outcomes. In ths case, the aggregate bonus for outcomes other than reduces to ^c 1 1 (1) and the problem of aggregaton across outcomes other than does not arse. Snce any Brownan moton can be represented as a lmt of bnomal processes, t s sometmes beleved that ths observaton s enough to support the lnearty of ncentve schemes n Brownan models. For a undmensonal Brownan moton, ths s of course correct. However, Theorem 1 and Corollary show that the undmensonal Brownan moton does not actually yeld the approprate lmt for mult-perod problems wth multnomal rather than bnomal processes. The underlyng structure of the ntertemporal agency problem - n contnuous as well as dscrete tme - s gven by the functon ^c() whch ndcates the tradeo between the vector of contrbutons of the derent accounts to prots per unt of \real tme" and the agent's eort costs per unt of \real tme". The dmenson N of the doman of ths functon s an essental feature of the economc stuaton. Ths dmenson corresponds to the cardnalty of the outcome set n the statc model as well as the dmenson of the Brownan moton n the contnuoustme model. For N > 1, the observaton that undmensonal Brownan motons can be represented as lmts of bnomal processes s therefore rrelevant. As mentoned n the ntroducton, for the case N > 1 Holmstrom and Mlgrom (1987) present two varants of the contnuous-tme model that actually do yeld optmal ncentve schemes that are lnear n aggregates, e.g., total prots. One of these nvolves 6

27 an assumpton that the prncpal observes the tme path of total prots rather than the tme paths of all the accounts (). We wll dscuss ths approach extensvely n the next secton. The other one nvolves an assumpton that the agent's eort cost depends only on expected prots rather than the vector of drft rates for the derent accounts. In the remander of ths paper we ask what are the dscrete-tme analogues of these contnuoustme results. In ths secton we begn wth the second varant. The assumpton that eort cost depends only on expected prots requres that the cost functon n the contnuous-tme model takes the form ^c() = g! ; (47).e, the agent's cost depends only on the expected prots he s gong to produce: the hgher expected prots the more costly t s for the agent. Ths cost functon seems to be very natural, t s easy to deal wth, and t has often been used n applcatons of the Brownan model (see e.g. Holmstrom and Mlgrom (1991) or Itoh (199)). 1 also close to the speccaton of Mrrlees (1974), who consders a one-perod model where prots are lognormally dstrbuted and the agent chooses the mean of ths dstrbuton. Mrrlees assumes that the agent's costs are an ncreasng and convex functon of the mean,.e., of expected prots. 11 It s Usng (13) and (7) t s straghtforward to show that n the dscrete one-perod model the cost functon correspondng to (47) s gven by 1 c(p) = g(e(p)) = g In the dscrete mult-perod model we get c (p )=c ^p+ p,^p 1! =g = ^p + p, ^p 1 = p!!! =g (48)! E (p ) =g( ) (49) 1 Furthermore, t admts the nterpretaton that the \tasks" are perfect substtutes n the agent's cost functon. 11 Schattler and Sung (1997) use the same speccaton n a dscrete-tme model that s used to approxmate a contnuous-tme Brownan-moton model. They run nto Mrrlees' problem that the dscrete prncpal-agent problem has no soluton. 1 Ths speccaton of the cost functon volates Assumpton 1 whch requres that c(p) s strctly convex. So far, however, strct convexty ofc() has been used n Proposton 3 only. All of the results of ths secton hold f g() s strctly convex. 7

28 where and Fnally, let and dene V (p) = E (p ) = = = = p = (5) : (51) p, E (p) = N = p (, E(p)) (5) V 1 (^p) (53) The followng result, whch s an mmedate corollary of Theorem 1(c), shows that n the lmt, as!, the ncentve scheme that mplements a gven acton n the nteror of the admssble acton space s ndeed a lnear functon of total prots. Corollary 3 Suppose that n the dscrete mult-perod model the agent's cost functon s gven by (49) and that the prncpal wants to mplement a constant acton vector, satsfyng < p ( ) < 1 for any sucently small, that gves rse to expected prots. Let w denote the certanty equvalent of the agent's utlty f he does not work for the prncpal. Assume that as converges to zero, converges to. Then also the ncentve schemes that mplement converge to s = w + g()+g (total prots, )+ r (g ) (54) whch concdes wth the ncentve scheme that mplements n the Brownan model. At rst sght, Corollary 3 seems to provde for the dscrete-tme approxmatons of contnuous-tme models nvolvng the cost speccaton (47) and the lnear ncentve scheme (54). However, ths nterpretaton s awed because n the dscrete mult-perod models the prncpal never wants to mplement a constant acton vector such that <p ( )<1for all. To see ths, go back to the statc model wth cost speccaton (48). If the set P of admssble probablty vectors s equal to the entre smplex, there 8

29 s nothng to prevent a soluton whch elmnates practcally all rsk, and under (48) such a soluton wll n fact be optmal. Snce the eort cost of the agent depends on expected prots only and snce total surplus ncreases when the rskness of the outcome s reduced, the prncpal wll try to mplement any target level of expected prots wth as lttle rsk as possble. For ths purpose he nduces the agent to put all probablty mass on the two prot levels just neghborng the target level. If the set (p 1 ;:::;p N ) of prot levels n the statc model has been speced so that neghborng levels are close to each other, the outcome s \almost" determnstc and the rst best can \almost" be mplemented. The very core of the agency problem seems to have dsappeared. Ths result holds n any of the mult-perod dscrete-tme models as well as n the statc model. 13 In each case, the varance of cumulatve prots s bounded by the square of the maxmum of derences between neghborng prot levels n the statc model. In the Brownan moton model, however, the varance s gven exogenously and the optmal soluton of the prncpal-agent problem s bounded away from the rst-best. Theorem 1 and ts Corollares are not applcable because the sequence of optmal actons n the dscrete-tme model does not have a well dened lmt. For any for whch p =, by (7) the correspondng (p ) s negatve and goes out of bounds as!, whereas for the two ndces on whch p s concentrated, the correspondng (p ) s postve (and also goes out of bounds). Not only does the dscrete-tme verson of (48) elmnate the core of the agency problem, but also t gves rse to a dscontnuty n the transton from mult-perod dscrete-tme models to contnuous tme. The dscontnuty n the transton from dscrete-tme models to contnuous tme would not be present f the controls were restrcted to a compact set K, as n Theorem. In ths case however the lmtng ncentve scheme would be gven by (44) rather than (54). The Kuhn-Tucker multplers ^u and ^v n (44) would depend on and would n fact prevent the ncentve scheme from beng a lnear functon of the prot aggregate In (49) ^p cancels out of g() for any and the agent's socost curves are stll lnear n p. 14 In fact, the speccaton (48) does not t nto the framework P of Holmstrom and Mlgrom's dscrete N tme model. If c(p) depends only on the lnear aggregate = p, then the correspondng socost curves n the probablty smplex are lnear and there s no natural mpedment to the agent's substtutng between, say, p and p j. In consequence the prncpal wants to mplement a probablty vector on the boundary of the set of admssble vectors. Ths s ncompatble wth Assumpton A(v) of Holmstrom and Mlgrom (1987, p. 31) whereby anypon the boundary of the set of admssble probabltyvectors s prohbtvely costly for the agent and hence for the prncpal. 9

30 We conclude that the result that the optmal ncentve scheme n the Brownan model s lnear n total prots f the agent's cost functon depends on expected prots only cannot be approxmated by the mult-perod dscrete-tme models. 6 Approxmatng Sharng Rules that Are Lnear n Aggregates In ths secton we develop a dscrete-tme analogue of the proposton of Holmstrom and Mlgrom that optmal ncentve schemes n the contnuous-tme model are lnear n a gven aggregate f the prncpal observes the tme path of ths aggregate wthout observng ts ndvdual components. In terms of formal modellng, the assumpton s that the prncpal observes the tme path of the accountng aggregate z(t) = P N Z (t) - and hence the tme path of total prots - wthout observng the tme paths of the ndvdual accounts Z (t), =1; :::; N. Wth ths assumpton, Theorem 8 of Holmstrom and Mlgrom (1987) shows that the optmal ncentve scheme s a lnear functon of the value of ths aggregate at t =1(e.g. total prots z(1) z at the end of the perod). Ths result reects the basc ntuton that nonlnear ncentve schemes are vulnerable to manpulaton. Unfortunately, however, Theorem 8 has no mmedate dscrete-tme analogue. To see ths, note that n any perod f1;:::; 1 g the prncpal observes P =1 ;. Snce the prot n perod s smply ; the ndvdual accounts ~ Z ; = P =1 ;, P,1 =1 ;, the prncpal can compute even f he observes the tme path of total prots only. Snce ths nformaton can be used to mprove the ncentve scheme, the prncpal wll use t n the same way as before. Thus, for all > we get agan lnearty n accounts only. In the lmt, however, t s mpossble to derve the ndvdual accounts Z (t) from the observaton of z(t). Thus, f we want to approxmate ths result n a dscrete framework, we need a stronger form of asymmetrc nformaton than s used n Theorem 8 of Holmstrom and Mlgrom. As t turns out, ths has a sgncant mpact on the analyss of the dscrete-tme models themselves. The dscrete models consdered n ths secton have the followng structure. The tme nterval [; 1] s devded n 1 perods each of length. At the begnnng of each perod, f1;:::; 1 g, the agent observes the past hstory of prots and chooses hs acton ;. As n Secton 3, we embed the dscrete model n a contnuous model where 3

31 t [; 1] and (t) = ;[t=]. In contrast to the model consdered by Holmstrom and Mlgrom, however, we assume that the prncpal does not observe the tme path of prots. Furthermore, we assume that the agent has the possblty to destroy output before he reports total prots ^z to the prncpal. Thus, the prncpal observes only the revealed prot ^z P N = Z (1) z at date 1. Both of these assumptons are requred to make sure that the prncpal cannot construct the accounts Z (1) for each prot level f;:::;ng. 15 In ths model the prncpal has consderably less nformaton than the agent. However, the prncpal could ask the agent not only to report nal prots but also to report e.g. the tme path of prots. 16 Somewhat more generally, the prncpal could oer an ncentve scheme S(m) whch requres the agent to send a message m out of some message space M(z). Ths message space wll, n general, depend on total prots avalable at date 1. For example, M(z) could be the set of all possble prot paths wth the property that total prots do not exceed z. Snce the agent can destroy prots unnotced, t must be the case that z <zmples M(z ) M(z). The followng proposton shows that wthout loss of generalty we can restrct attenton to ncentve schemes that are non-decreasng functons of reported total prots. Proposton 4 If the prncpal can mplement a control path (t) wth an ncentve scheme S (m), then (t) can also be mplemented wth a nondecreasng ncentve scheme s (z), whch asks the agent to report total prots z = P N Z (1) truthfully. Furthermore, ths can be done so that the utltes of the prncpal and of the agent are the same under S (m) and under s (z). The proof s a standard applcaton of the revelaton prncple and s relegated to Appendx A. Note that f an ncentve scheme mplements a control path () under the nformaton assumptons of Secton and f ths ncentve scheme can be wrtten as a nondecreasng 15 Note that f the agent cannot destroy output the prncpal may be able to construct these accounts even f he does not observe the tme path of prots. To see ths, suppose that there are three derent prot levels, f,1; 1 ;1g, where e s Euler's number. In ths case the total amount of prots reveals e how often each of the three derent states occured for any number of perods Lke Holmstrom and Mlgrom (1987) we rule out the possblty that the prncpal requres the agent to report prots at any ponts n tme t [; 1]. The dea s that the prncpal cannot montor the agent contnuously but only at exogenously gven dscrete ponts n tme. Wthout ths assumpton the analyss s consderably more complcated. 31

32 functon of cumulatve prots, then t also mplements () under the nformaton assumptons n ths secton: Frst, the ncentve scheme s feasble for the prncpal because t depends only on total prots. Second, by Proposton 4, the agent wll report total prots truthfully snce the ncentve scheme s nondecreasng. Fnally, when the agent chooses the control path (), he faces exactly the same ncentves under the nformaton assumptons of Secton and of ths secton. Hence, f the ncentve scheme ~s () mplements () under the nformaton assumptons of Secton, then the condton that ~s () can be wrtten as a monotonc functon of z s not only necessary, but also sucent for the mplementablty of() under the nformaton assumptons we mpose n ths secton. Wth the condton that nal payments to the agentmust be gven by a nondecreasng functon of cumulatve total prots, the nature of the mult-perod dscrete-tme agency problem changes dramatcally. As a result, the solutons to ths problem wll no longer exhbt the statonarty propertes that were so useful n Corollary 1. To see the ssue, note rst that n the statc agency problem, an ncentve scheme nvolvng a nonlnear nondecreasng functon of total prots, e.g., a sutable step functon, mght enable the prncpal to neutralze at least some of the eects of havng to condton on total prots rather than each outcome separately. In a two-perod agency problem - and even more so n a general mult-perod problem -, hs scope for dong so s reduced as the eects of ntertemporal aggregaton set n and he can, e.g., not dstngush whether a cumulatve total prot of two stems from two realzatons of one or from one realzaton of two and one of zero. Even so, hs desre to use nonlneartes of the ncentve scheme n order to neutralze some of the eects of hs nablty to condton on ndvdual accounts s not lkely to dsappear altogether; after all there are lmts to ntertemporal aggregaton, at least as long as the number of perods as well as the set of possble outcomes n each perod s nte. Gven an ncentve scheme that s a nonlnear functon of cumulatve total prots, the agent's optmzaton problem wll typcally have nonstatonary, tme-dependent and hstory-dependent solutons. To see why, suppose that the ncentve scheme s a step functon. If the horzon s far away, the agent's eort choce wll manly depend on consderatons of \global steepness of the starcase",.e., on the sort of \global" tradeo between eort and ncentve payments that s relevant when he can envsage hmself as 3

33 takng the same acton repeatedly and wonderng what s the mpact of the correspondng movement n cumulatve total prots on hs own ncome. In contrast, f the agent s near to the end of hs horzon, he s more concerned about the local propertes of the ncentve scheme, e.g., how far away he s from the nearest ponts of ncrease or decrease,.e. the \nearest steps of the starcase" - and also how large these steps are. These consderatons suggest that under the nformaton assumptons made here the mult-perod agency problem s rather more dcult to study than n the settng of Secton where the soluton of the problem can be taken to be statonary. At the same tme they suggest that the addtonal dcultes may perhaps be relatvely unmportant when there are many perods and for \most" perods the horzon may betaken to be dstant. In ths case consderatons of ntertemporal aggregaton would seem to vtate any attempt of the prncpal to use nonlneartes of the ncentve scheme n order to neutralze the eects of hs nablty to condton on ndvdual accounts. The followng analyss shows that ths s ndeed the case. In Theorem 3 below, we show, roughly, that when ncentve schemes are restrcted to nondecreasng functons of cumulatve total prots, a sutably chosen lnear ncentve scheme s approxmately optmal f the number of perods T = n the mult-perod model s large. Ths provdes one analogue of the optmalty of lnear ncentve schemes n contnuous tme that s establshed n Theorem 8 of Holmstrom and Mlgrom. The stronger analogue that optmal ncentve schemes n the T -perod problem are approxmately lnear when T s large can also be establshed. However ths requres consderably more mathematcs and s presented elsewhere (see Hellwg (1998b)). In ths secton we assume agan that for all = 1; 1 ;:::; the vector must be chosen from a product K = Q N [m ;M ] of compact ntervals, where m < < M. Combned wth Assumpton 1, ths mples that ^c() s contnuously derentable on the nteror of K wth unformly bounded rst dervatves. Holmstrom and Mlgrom (1987) and Schattler and Sung (1993) assume n addton that the control paths chosen by the agent le n the nteror of a compact set. Note that the followng theorem does not requre ths assumpton. 33

34 Theorem 3 In addton to Assumpton 1, suppose that controls are constraned to the compact set K = Q N [m ;M ]. Consder a sequence of dscrete models wth perod length, = 1; 1 ;:::. There exsts an ncentve scheme s (z) whch s lnear n total prots such that for any >there exsts a > wth the property that for all < the prncpal's utlty loss from usng the lnear contract s (z) rather than the optmal contract s ; (z) s smaller than. Furthermore, s (z) s an optmal ncentve scheme n the contnuous model n whch the prncpal observes only the tme path of aggregate prots z(t). Theorem 3 shows that there exsts a dscrete-tme analoque to Holmstrom and Mlgrom's man result, whch says that the optmal ncentve scheme n the contnuous-tme model s lnear n aggregates f the prncpal observes the tme path of these aggregates only. In our dscrete tme model there s not only aggregaton accross accounts, but also aggregaton over tme, snce the prncpal s assumed to only observe the sum of total pro- ts at the end of the last perod. Note, however, that there s some mplct aggregaton over tme n Holmstrom and Mlgrom as well. The prncpal observes the tme path of a Brownan moton. However, at any pont n tme the ncremental change \db" of ths stochastc process s only dened by the ntegral of the stochastc process between two (arbtrarly close) ponts n tme. But ths means that the Holmstrom-Mlgrom assumpton whereby the prncpal observes only the aggregate process P N db nvolves some mplct aggregaton over tme as well. In the ntroducton of ther paper, Holmstrom and Mlgrom argued that nonlneartes n ncentve schemes are vulnerable to manpulaton by the agent. However, n most of ther actual analyss \manpulaton" by the agent does not play any role. Manpulaton enters ther analyss only n the Brownan model of Theorem 8. If the prncpal observes the tme path of total prots only, then the agent cannot be prevented from choosng a control path that yelds a gven tme path of expected prots at least cost to the agent. Ths cost mnmzaton leads to margnal costs of rasng prots beng the same for all accounts so that the term P N ^c (1) n the expresson for nal ncentve payments can n fact be wrtten as ^c 1 P N (t), a constant tmes the devaton of cumulatve total prots from ther mean. 34

35 Our dscrete-tme analyss shows that there has to be some aggregaton over tme n addton to the aggregaton accross accounts. Furthermore, n our model the agent can destroy prots unnotced. Thus, the agent has a lot of dscreton n how to allocate hs eort over tme and what to report to the prncpal. The ablty to destroy prots s mportant because t restrcts the prncpal to nondecreasng ncentve schemes. Ths monotoncty of ncentve schemes s mportant for the clam n the proof of Theorem 3 that as goes to zero the set of ncentve schemes satsfyng (A.44) and (MS) shrnks to the set of schemes satsfyng (A.45) and (MS). Wthout monotoncty, one mght not be able to rule out sequences of ncentveschemes satsfyng (A.44), but uctuatng ever more wldly as the range of possble outcome becomes denser and denser and tends towards llng the whole space. As dscussed n the ntroducton to ths secton, as long as s postve a lnear contract s unlkely to be optmal n the dscrete-tme model. However our result shows that t s at least approxmately optmal. A lnear contract gves a constant ncentve pressure over tme so that the agent wll not explot nonlneartes of the contract by makng hs eort depend on past prot realzatons. Moreover a lnear contract s not vulnerable to manpulaton arsng from the prncpal's nablty to observe anythng other than the cumulatve total prot that the agent has chosen not to destroy. These two consderatons put a bound on any advantage that a nonlner contract mght have over the optmal lnear one; moreover ths bound s close to zero whenever s close to zero. 35

36 Appendx A Proof of Proposton 1: If the agentchooses ; =( ; 1 ;:::; ; N ) n perods =1;:::; 1, expected gross prots are =1 = p (; ) = = = =1 " =1 = =1 = ; ^p 1, N ; =1 1 k, k 1 N A 1 + ; k + ; ; # k 1 k 1 A 1 ; : (A.1) 3 5 Furthermore, we have =1 c (p ( ; )) = =1 c! ^p + p ( ; ), ^p 1 = =1 ^c( ; ) : (A.) Obvously, f ; = for all =1;:::; 1 and all = 1; 1 ;:::,wehave =1 = p () = 1 = (A.3) and =1 c (p()) = 1 ^c() = ^c() : (A.4) Thus, n ths case total expected prots and total costs to the agent depend on but are ndependent of the length of each subperod. Q.E.D. Proof of Theorem 1: (a) Consder any tme nterval [t; t ], t > t. Suppose that the agent takes acton ; n perod f[t=]+1;[t=] + ;:::;[t =]g. Then, n each of these perods the probablty that ~ A ; = 1 equals p (; ) whch s dened by (9) and (1) above. Furthermore, snce the tme path of actons f ; g s determnstc, any A ~ ; 1 and ~A ; are stochastcally ndependent for any 1 ; f[t=]+1;[t=]+;:::;[t =]g, 36

37 1 6=. The dstrbuton of (t ), (t) s characterzed by: " #! E (t ), (t) = E ~ t, ~ Var Cov = k 1 (t ), (t) = Var = k = k t = =t=+1 t = =t=+1 = k t = =t=+1 ~ t = =t=+1 " t p ( ; )(1, p ( ; + ; 1 k 1 ^p, ^p ; t! p ( ; ), p ( ; ) #!, 1, ^p, ; 1 k + ; 1 k t! 1 1 A k, ^p ; 1 k = (A.5), ( ; ) k (A.6) = k (t, t)^p (1, ^p )+O() (A.7) (t ), (t); j (t ), j (t) " #! = Cov ~ t, ~ = k k j = k k j = k k j t = =t=+1 t = =t=+1 t = =t=+1 t ; ~ j p ( ; )p j ( ; + ; 1 ^p j +^p ; j 1 A 1 k j " + ; j +^p j ; #! 1 k j 1 k, ~ j 1 A + ; ; j t! k k j 1 A 1 A (A.8) =,k k j ^p ^p j (t, t)+o() (A.9) where O() contans all terms that vansh as goes to. Thus, n the lmt as goes to zero, the varances and covarances are ndependent of the tme path of actons f ; g. The stochastc process (t) satses the condtons of Prohorov's generalzaton of Donsker's Theorem. 17 In partcular, note that the maxmum of prots that can be 17 See Bllngsley (1968), Theorem 1.1 n conjuncton wth Problem 1.1 (p. 77) and Problem 16.7 (p. 143). 37

38 obtaned n any perod s gven by N = N 1 and goes to zero as goes to zero whch mples that the Lndeberg condton s satsed. Hence, as!, (t) converges n dstrbuton to a multdmensonal Brownan moton wth drft and covarance matrx. (b) Suppose the agent takes the tme path of actons ;, = 1;:::; 1, where (t) converges to (t) unformly n t as goes to zero. Snce (t) s contnuous, there exsts a compact set K IR N such that for any sucently small one has (t) K for all t. Snce ^c() s contnuous - and hence bounded on K - we nd for the agent's cost n the lmt as goes to : lm! =1 ^c( ; ) = lm ^c( (t))dt =! ^c((t))dt : (A.1) (c) Substtutng (19) n (18) yelds: =1 ~s ; = =1 ^c( ; )+ + r =1 = =1 ~ A ; ^c ( ; ) ~ ; 4^c ( ; )k, j=1 3 ^p j^c j ( ; )k j 5 + O( 1 ) (A.11) From (b), the rst term n (A.11) converges to R ^c((t))dt as converges to zero. As for the second term, the same argument as n (b) mples that for any sucently small, ; belongs to a compact set K for all, and ^c ( ; ) s bounded, unformly n. From part (a) therefore, one nds that for any sucently small, the sums =1 ^c ( ;[ = ] ) ~ ; = =1 h ^c ( ; ) ( ), (, ) (A.1) converge n dstrbuton to =1 h ^c ( ; ) ( ), (, ) = ^c ( (t))d (t) ; (A.13) as goes to zero, unformly n. One also has lm! ^c ( (t)) = ^c ((t)) R unformly n t, and hence plm! ^c ( (t))d (t) = R^c ((t))d (t). Therefore the sums P ^c ( ; ) ; n the second term of (A.11) converge n dstrbuton to the stochastc ntegral R ^c ((t))d (t) as goes to zero. 38

39 Turnng to the thrd term n (A.11), we rewrte ths n the form r =1 = + r ~ A ;, p (; ) 4^c ( ; )k, =1 = p ( ; ) 4^c ( ; )k, j=1 j=1 3 ^p j^c j ( ; )k j 5 3 ^p j^c j ( ; )k j 5 (A.14) By the law of large numbers, the rst of these terms converges to zero almost surely R as goes to zero. The second term converges to the ntegral r 1 ^c ((t))[c ((t))] T dt. Thus, f the prncpal wants to mplement the tme path of actons (t), t [; 1], such that the agent's certanty equvalent s w, then the ncentve scheme that mplements (t) converges to s = w + ^c((t))dt + ^c ((t))d + r ^c ((t))[c ((t))] T dt : (A.15) Q.E.D. Proof of Proposton : To smplfy the notaton wrte a := 1, r(^c, ^u +^v )k 1 + N j=1 p j ( )r(^c j, ^u j + v j )k j 1 (A.16) and note that the rst-order condtons for the maxmzaton of (39) wth respect to ^u and ^v under the constrants (38) can be wrtten as: 1 a j=1 p j ( ) a j ; wth equalty f^u >, and ^v =,f =m ; (A.17) ^u = ^v = ; f m < <M ; (A.18) 1 a j=1 p j ( ) a j ; wth equalty f^v >, and ^u =,f =M ; (A.19) Suppose rst that ^u > and ^v j > for some and j. Then (A.17) and (A.19) mply a = a j as well as ^v = ^u j =, and one mmedately obtans k ^u + k j^v j = k ^c ( ), k j^c j ( ) ( ). Suppose next that ^u = for all and ^v > for some. Let J be the set of ndces j for whch ^v j = and note that J so J 6=. Let M := P jj M = P p j ( ) a j and note that (A.19) mples M = P jj p j ( ) a j + P j6j p j ( )M, hence jj p j ( )=a j and PjJ 1 mn p j ( ) M jj a j. For any 6 J, (A.19) mples a = 1, hence M 39

40 a mn jj a j. Ths n turn yelds k v max jj k j^c j ( ),k ^c ( ) ( ). Fnally, f ^v = for all and ^u > for some, a precsely symmetrc argument shows that for all, k ^u max jj k j^c j ( ), k ^c ( ) ( ), where J now s the set of ndces j for whch ^u j >. Q.E.D. Proof of Proposton 3: Trvally,any lmt (; ^u; ^v) of a sequence f( ; ^u ; ^v )g of vectors n K IR N + IR N + vectors n K IR N + IR N + that satsfy (4) as well as (38) for all wll tself belong to the set of that satsfy (43). To prove the proposton t therefore suces to show that any vector (; ^u; ^v) K IR N + IR N + approxmated by a sequence of vectors ( ; ^u ; ^v ) IC(). that satses (43) can n fact be Gven (; ^u; ^v) K IR N + IR N + satsfyng (43), for any >, let s = ^c(), 1 r ln 4 1, r(^c, ^u +^v )k 1 +r N j=1 p j ()(^c j, ^u j +^v j )k j ; (A.) and let be a soluton to the problem of maxmzng, P N p ( )e,r(s,^c( )) over the set K. Moreover for =1;:::;N, let r 1 k ^u, r 1 k ^v be the Kuhn-Tucker multplers correspondng to the constrants m and M n ths maxmzaton, and wrte ^u = (^u 1 ;:::;^u N ), ^v = (^v 1 ;:::;^v N ). Then clearly ( ; ^u ; ^v ) IC(). We also show that ( ; ^u ; ^v ) converges to (; ^u; ^v) asconverges to zero. The argument s based on Berge's (1959) maxmum theorem. For the gven ncentve scheme, the maxmzaton problem denng s equvalent to the problem of mnmzng e,r(^c(),^c( )) " 1, r over the set K. Gven the speccaton of p n turn s equvalent tothemnmzaton of e,r(^c(),^c( )) " (p ( ), p ()(^c, ^u +^v )k 1 1, r # (A.1) (), mnmzaton of (A.1) wth respect to (, )(^c, ^u +^v ) # (A.) wth respect to. Upon subtractng the constant 1 and dvdng by, one nds that gven the mnmzaton of (A.) s equvalent to the mnmzaton of e,r(^c(),^c( )), 1, r (, )(^c, ^u +^v )e,r(^c(),^c( )) : (A.3) 4

41 As goes to zero and converges to say, the mnmand n (A.3) converges to: r(^c( ), ^c()), r (, )(^c (), ^u +^v ): (A.4) By Berge's maxmum theorem t follows that any lmt of a subsequence of mnmzers of (A.3) must tself be a mnmzer of (A.4). Snce ^c() s strctly concave, (A.4) has a unque mnmum at =. Any convergent subsequence of mnmzers of (A.3) must therefore converge to. Snce K s compact and all convergent subsequences have the same lmt, t follows that the sequence f g converges to. Convergence of the assocated vectors of normalzed Kuhn-Tucker multplers ^u, ^v to ^u, ^v follows by takng lmts n the Kuhn-Tucker condtons for. Ths shows that the gven (; ^u; ^v) K IR N + IRN + satsfyng (43) can n fact be approxmated by a sequence of vectors ( ; ^u ; ^v ) IC(). Q.E.D. Proof of Theorem : By Proposton C.1 and Remark C.1 n Appendx C, a constant control path s mplementable by an ncentve scheme s() n the contnuous-tme model f and only f there exst vectors ^u =(^u 1 ;:::;^u N ), ^v =(^v 1 ;:::;^v N ) such that and ^u, ^v satsfy the complementary slackness condton (38), and moreover s() has the representaton: s() = w +^c()+ (), ^u +^v ) (1)+ (^c r j=1 (^c, ^u +^v ) j (^c, ^u +v ) ; (A.5) The prncpal's expected payo from usng the scheme s() to mplement the constant control path s therefore computed as U P (; ^u; ^v) =, w, ^c(), r j=1 (^c, ^u +^v ) j (^c, ^u + v ) ; (A.6) Therefore a constant control path and ncentve scheme s() mplementng are optmal for the prncpal n the contnuous-tme model f and only f and the assocated vectors ^u and ^v maxmze (A.6) subject to the complementary slackness condtons (43). Now consder the transton from the multperod dscrete-tme models to the contnuoustme model. Expandng the logarthmc term n (4) n a Taylor seres and takng lmts as goes to zero we obtan lm! U P ( ; ^u ; ^v ) (A.7) 41

42 =, w, ^c(), r ^p 4 (^c, ^u +^v )k, j=1 3 ^p j (^c j, ^u j + v j )k j 5 = U P (; ^u; ^v) (A.8) for any sequence f ; ^u ; ^v g that converges to a lmt (; ^u; ^v) K IR N + IR N + goes to zero. In vew of (A.7) and Proposton 3, Berge's maxmum theorem mples that f the sequence f ; ^u 1 ; ^v 1 ;:::;^u N ;^v N g of constant optmal control paths and assocated Kuhn-Tucker multplers n the dscrete-tme problems has a subsequence whch converges to a lmt ( ; ^u 1 ; ^v 1 ;:::;^u N ;^v N), then ( ; ^u 1 ; ^v 1 ;:::;^u N ;^v N) maxmzes U P (; ^u; ^v) subject to the constrant (43) so and the ncentve scheme s () gven by (44) are optmal for the prncpal n the contnuous-tme problem. It remans to be shown that the sequence f ; ^u 1 ; ^v 1 ;:::;^u N ;^v N g actually has a convergent subsequence. Snce K for all, certanly the sequence f g has a subsequence whch converges to a lmt. To economze on notaton, we dentfy the convergent subsequence wth the orgnal sequence. For each, let ^u = (^u 1 ;:::;^u N ), ^v = (^v 1 ;:::;^v N ) be the soluton to the prncpal's \relaxed problem" of maxmzng U P ( ; ^u; ^v) wth respect to ^u and ^v subject only to the complementary slackness condtons (38). By Proposton, we have ^u ( ) k and ^v ( ) k for all and all, hence ^u ^ k and ^v ^ k for all and all. The sequence f^u ; ^v g must therefore have a convergent subsequence. Moreover another applcaton of Berge's maxmum theorem shows that any lmt (^u ; ^v ) of a subsequence of f^u ; ^v g must be maxmzng U P ( ; ^u; ^v) over the set of (^u; ^v) satsfyng (43). Gven that the functon U P ( ; ; ) s strctly concave, the latter maxmzer s unque, so the lmt (^u ; ^v ) must be the same for all convergent subsequences of f^u ; ^v g, and the sequence f^u ; ^v g tself must be convergng to (^u ; ^v ). We clam that the \true" vectors of Kuhn-Tucker multplers ^u and ^v must also be convergng to ^u and ^v, so ( ; ^u ; ^v ) s ndeed a lmt pont of the sequence f ; ^u 1 ; ^v 1 ;:::;^u N ;^v N g. Let U P be the maxmum value of U P (; ; ) subject to the constrant (43). In vew of (A.7) and the denton of (^u and ^v, we have: lm sup U P ( ; ^u ; ^v ) lm U P ( ; ^ ; ^v )!! = U P ( ; ^u ; ^v ) U P : (A.9) as 4

43 From Proposton 3 and (A.7), we also know that there exsts a sequence f ;u ;v g such that for any, ;u ;v ) IC() and moreover lm! U P ( ;u ;v ) = U P. Gven the denton of f ; ^u 1 ; ^v 1 ;:::;^u N ;^v N g t follows that lm nf U P ( ; ^u ; ^v ) U P :! (A.3) Upon combnng (A.9) and (A.3), we conclude that lm U P ( ; ^u ; ^v ) = lm U P ( ; ^u ; ^v ) = U P ( ; ^u ; ^v ) = U P : (A.31)!! For any, let (u ;v ) be an arbtrary convex combnaton of (^u ; ^v ) and (^u ; ^v ). Snce U P ( ; ; ) s concave and, by the denton of (^u ; ^v ), U P ( ; ^u ; ^v ) U P ( ; ^u ; ^v ) for all, wehaveu P ( ;^u ;^v )U P ( ;u ;v ) U P ( ; ^u ; ^v ) for all, so (A.31) mples: lm U P ( ;u ;v ) = U P ( ; ^u ; ^v ) = U P : (A.3)! Gven that (^u ; ^v ) maxmzes the strctly concave functon U P ( ; ; ) under the constrant (43) - and any lmt pont of the sequence fu ;v g wll also satsfy (43) -, t follows that any convergent subsequence of convex combnatons of (^u ; ^v ) and (^u ; ^v ) must actually converge to (^u ; ^v ). Ths n turn mples that the sequence f^u ; ^v g tself converges to (^u ; ^v ), so ( ; ^u ; ^v ) s ndeed a lmt pont of the sequence f ; ^u 1 ; ^v 1 ;:::;^u N ;^v N g. Ths completes the proof of Theorem. Q.E.D. Proof of Proposton 4: Consder an ncentve scheme S (m) that mplements (t). If total prots at t = 1 are gven by z, the agent reports m (z) arg max S (m) mm(z) (A.33) Consder now a mechansm s (z) whch s dened by s (z) =S (m (z)). If m (z) s an optmal announcement gven z under S (m), then z must be an optmal announcement under s (z). Suppose not. If total prots are gven by z, then the agent cannot report ^z >z.if he reports ^z <z,hs payo s s (^z) S (m (^z)) S (m (z)) s (z) (A.34) 43

44 snce M(^z) M(z). Hence, the mechansm s (z) nduces the agent to report total prots truthfully. Furthermore, t follows from (A.34) that s (z) s non-decreasng. We nowhave to show that s (z) nduces the agent tochoose the same path of acton (t) ass (m). Snce S (m) mplements (t) t must be the case that EU A (S (m (z)) j (t)) EU A (S (m (z)) j ^ (t)) (A.35) for all admssble paths of actons ^ (t). However, snce s (z) nduces the agent to report total prots truthfully, we have EU A (s (z) j (t)) = EU A (S (m (z)) j (t)) EU A (S (m (z)) j ^ (t)) = EU A (s (z) j ^ (t)) (A.36) for all admssble paths of actons ^ (t). Hence, s (z) also mplements (t). Fnally, snce s (z) mplements the same tme path of actons and yelds the same payments to the agent as the old contract S (m), the expected utltes of the prncpal and the agent must also be the same. Q.E.D. Proof of Theorem 3: Note rst that Z 1 = 1 + =1 ; (A.37) Hence, we can wrte s () as a functon of x = P N f s() s nondecreasng n x, t s also nondecreasng n z. 1 rather than z. Furthermore, For any, let ~ ;, = 1; :::; 1, and s () be a control strategy and an ncentve scheme that solve the prncpal's problem n the dscrete-tme problem wth perod length. Formally, the control strategy ~ ;, =1; ::: 1, and ncentve scheme s () maxmze the prncpal's expected payo, subject to the constrants E 4 =1 ~ ;, s 1!3 5 (A.38) (IR) If the agent chooses the control path ~ ;, = 1;:::; 1 reservaton utlty,e,rw. he gets at least hs 44

45 (IC) Gven the ncentve scheme s () t s ndeed optmal for the agent to choose the control path ~ ;. (MS) The sharng rule s (x) s nondecreasng n x. By Holmstrom and Mlgrom (1987, Theorem 4) we know that the control strategy ^ ; s mplemented by sharng rule s f and only f cumulatve payments under ths scheme can be wrtten as a sum of payments under ncentve schemes s ; that would mplement ^ ; n the statc problem, = 1;:::; 1. Furthermore, we have shown n Proposton of Secton 4 that the acton ~ ; can be mplemented by ~s ; only f ~s ; satses for some vectors ^u ; ~s ; = ^c( ), 1 r ln h 1, r(^c ( ; ), u ; +r j= p j (^c j( ; ), u ; j + v ; )k 1 (A.39) + v ; j )k j 1 = (^u ; 1 ;:::;^u ; N ), ^v; = (^v ; 1 ;:::;^v ; N ). It follows that the prncpal's maxmal expected payo n problem (A.38) wth the constrants (IR), (IC), and (MS) s no larger than the payo he would obtan f he chose a control strategy ^ ;, an ncentve scheme ^s, and vectors ^u ;, ^v ; to maxmze the expresson: E 4 =1 ln ; =1 ^c( ; )+ 1 1, r(^c ( ; ), u ; =1 ~ ; k 1 + p! + v ; N )k 1 + r p j (^c j( ; ), u ; j + v ; j )k j 1 j= 3 5 (A.4) 1 3 A, w 5 under the constrants that s N ln ~ 1! = w + 1, r(^c ( ; ), u ; ^c( ; ), 1 r =1 ~ ; k 1 + p! (A.41) + v ; N )k 1 + r p j (^c j( ; ), u ; j + v ; j )k j 1 j= 1 A wth probablty one, and (MS) s () sanondecreasng functon. Let ^ ;, ^s ;, ^u ;, ^v ; be a soluton to ths problem. By Proposton, ^u ; and ^v ; belong to the compact set [;G] N where G s the maxmum of max () k over K. Usng a Taylor expanson for the logarthmc term, we therefore nd that for any > 45

46 the value of (A.38) s no larger than E 4 =1 ^ ;, ^c(^ ; ), r where ( ; )=^c ( ; ), u ; =1 ^p (^ ; )k, ^p j j (^ ; )k j! 3 5,w, ; (A.4) + v ;, f s sucently close to zero. Here we have made use of the fact that the expected value of the lnear term of the Taylor expanson s zero and the fact that for each and, condtonal on ^ ;, the coecent ( ~ ; =k 1 + p (^ ; )) = ~ A ; of the logarthmc term has expected value p (^ ; )=^p +O( 1 ) and varance p (^; )(1, p (^; )), so for the quadratc term of the Taylor expanson, whch s proportonal to, ths coecent may be approxmated by ^p f s sucently small. At ths pont t s convenent to rewrte (A.4) and (A.38) usng contnuous-tme notaton. In contnuous-tme notaton, our precedng argument can be summarzed as sayng that for any >and any sucently small, n the 1 -perod problem, the prncpal's maxmal expected payo s no larger than plus the maxmum of the expresson E 6 4 ^ (t)dt, ^c(^ (t))dt, r (^ (t))k, j=1 1 ^p j j (^ (t))k j A wth respect to the control strategy () and an ncentve scheme s () satsfyng 3 7 dt5,w (A.43) (PL) () s pecewse lnear wth (t) = ;[t=], and s N, 1 r, 1 r (1)! = w + 1 k 1 ln p ( (t)) ln ^c( (t))dt 1 1, r ( (t))k 1 + r p j j( (t))k j 1 j= A d (t) 1 1, r ( (t))k 1 + r p j j( (t))k j 1 j= wth probablty one, and (MS) s () s a nondecreasng functon. A dt (A.44) By Proposton B.1 n Appendx Bweknow that as goesto zero, the dsturbance process converges unformly to a Brownan moton wth ntal value B() =, zero drft and covarance matrx. Gven that the range of ths Brownan moton can be dented 46

47 wth the set of all contnuous functons of [; 1] nto IR N, t follows that as goes to zero the set of pars of control strateges () and ncentve schemes s () satsfyng (A.44) as well as (MS) shrnks to the set of pars of control strateges () and ncentve schemes s() satsfyng! N s (1) = w + ^c( (t))dt + ((t))d (t)dt + r ((t))[ ((t))] T dt (A.45) wth probablty one and (MS) s() s a nondecreasng functon. Hence, for all > and any sucently small, the maxmum value of (A.43) under the constrants (PL), (A.44) and (MS) s no less than plus the maxmum value of (t)dt, ^c((t))dt, r under the constrants (A.45) and (MS). ((t))[ ((t))] T dt, w (A.46) Now compare the problem of maxmzng (A.46) under the constrants (A.45) and (MS) to the contnuous-tme problem wth controls (t) as dscussed n Appendx C. In ths appendx, as well as n Holmstom and Mlgrom (1987, Theorem 8), the prncpal maxmzes the same payo functon but the ncentve scheme can be a functon of the entre tme path of the aggregate P N (). Proposton C. shows that ths problem s solved by a constant control path and a lnear sharng rule s (z) = + 1 z that depends only on accumulated total prots at tme 1, z, wth 1 = 1 ( )= ( )=::: = N ( ) > and =^c( ),^c ( ) T + r ( )[ ( )] T. Clearly, a constant tme path of actons satses our restrcton (PL). Furthermore, the ncentve scheme also satses (MS') and (IC). Hence, the optmal soluton to Holmstrom and Mlgrom's optmzaton problem s also the soluton to the problem of maxmzng (A.46) under the constrants (A.45) and (MS). If we substtute the maxmzer on the rght-hand sde of (A.46), we nd that for any > and any sucently small, the maxmum value of the prncpal's expected payo (A.38) n the 1 -perod dscrete-tme problem wth the constrants (IR), (IC) and (MS) s no larger than, ^c( ), r ( )[ ( )] T, w + : 47 (A.47)

48 Usng the entre approxmaton argument n reverse, we also nd that for gven > and any sucently small, (A.47) s no larger than E 4 =1, s ~ 1! ; (A.48) where for any, s the agent's (unque) optmal acton when faced wth the lnear scheme s(). Upon settng = =3, we obtan the concluson of Theorem 3. Appendx B Q.E.D. In ths appendx we dscuss the relatonshp between dscrete-tme and contnous-tme strateges and outcome processes when the agent's choce at any one tme may depend on the hstory of the process up to ths tme. In the mult-perod dscrete-tme model wth perod length, the agent chooses a sequence of possbly hstory-dependent controls f~ ; g =1. Ths choce generates a sequence f~ ; g =1 of random prot levels. We assume that for each, the agent's control choce ~ ; s gven by a - possbly degenerate - functon of prot realzatons pror to,.e., that we can wrte ~ ; = ^ ; (~ ;1 ; :::; ~ ;,1 ) (B.1) for all, and we dentfy the agent's strategy wth the sequence of functons f^ ; g =1. The range of each functon ^ ; s taken to be the compact set K = Q N [m ;M ]. Gven the control strategy f^ ; g =1 the prot sequence f~ ; g =1 condton that for each f1; :::; g, ~ ; gven ;1 ; :::; ;,1 s assumed to satsfy the takes values n f 1 ; :::; N 1 g and, for Prob(f~ ; = 1 g j ~ ;1 = ;1 ; :::; ~ ;,1 = ;,1 ) (B.) = p (^; ( ;1 ; :::; ;,1 )) As n the text, a gven control strategy f^ ; g =1 and assocated prot sequence f~ ; g =1 are used to dene countng varables A ~ ; ; = ;1; :::; N; = 1; :::; 1 ; = 1; 1 ; ::: such that A ~ ; = 1, ~ ; = 1 and A ~ ; =,~ ; 6= 1 : From (B.) one obvously has: E[ ~ A ; j~ ;1 = ;1 ; :::; ~ ;,1 = ;,1 ] p (^; ( ;1 ; :::; ;,1 )) (B.3) 48

49 It follows that f we dene ~ ; = k 1 ( ~ A ;, p (^; ( ;1 ; :::; ;,1 )) (B.4) we have E[ ~ ; j~ ;1 = ;1 ; :::; ~ ;,1 = ;,1 ] (B.5).e., for any the process f( ~ 1 (); :::; ~ N ())g =1 s a martngale. Agan usng a lnear nterpolaton to embed the processes f( ~ 1 (); :::; ~ N ())g =1 n a contnuous-tme formulaton, we wrte for any and any t [; 1]: (t) = 1, t + t ~ t + t, t ~ t +1 (B.6) where agan [ t ] denotes the greatest nteger less than or equal to t. For any ; ( (); :::; 1 N ()) s a random functon takng values n the space CN [; 1] of contnuous functons from [; 1] nto IR N. chosen control strategy of the agent. The dstrbuton of ( (); :::; 1 N ()) depends on the Proposton B.1 For =1; 1 ;1 3 ;:::; let f^; g =1 be a control strategy takng values n the compact set K = Q N [m ;M ], and consder the nduced process ( 1 (); :::; N ()). As converges to zero, the processes ( 1 (); :::; N ()) converge n dstrbuton to the Gaussan process B() wth ntal value B() =, zero drft, and covarance matrx. Proof: For any, let f~ ; g =1 be the random prot sequence that s nduced by the control strategy f^ ; g =1. For any and any t [; 1], let F t be the -algebra on the underlyng probablty space that s generated by the random varables ~ ;1 ; :::; ~ ;[t=] : Fx some vector q IR N ; and consder the stochastc process q for any t [; 1], () on [; 1] such that q (t) = N q (t) (B.7) For any t and, (B.5) and (B.6) yeld: E[ q ([t=] + ), q ([t=])jf t ] = E[ N q ~ ;[t=]+ jf t ] = (B.8) 49

50 For any two vectors q 1 ;q IR N, one also has: E[( q 1([t=] + ), q 1([t=])( q ([t=] + ), q ([t=])jf t ] = E[ = E[ = =,E[ j=1 q 1 ~ ;[t=]+ q ~ j ;[t=]+ j jf t ] q 1 q k (1,p (~;[t=] )) p (~;[t=] )jf t ] j6= q 1 q j k k j p (~ ;[t=] ) p j (~ ;[t=] )jf t ] q 1 q k ^p, j=1 j=1 q 1 q j j +O() q 1 q j k k j ^p ^p j +O() (B.9) where j s the j-th element of the matrx (see (3) n the text) and O() s a term that goes to zero when goes to zero, unformly n strateges and hstores. Now consder the vector process Q = ( q 1; :::; q N ) where q 1 ; :::; q N are egenvectors of the matrx ; normed so that q r q r =,1 r s the egenvalue of that corresponds to q r. where r Gven that s a symmetrc, postve dente matrx, the egenvectors q r ;q s ;r 6=s; are mutually orthogonal. For ths choce of q 1 ; :::; q N, (B.9) yelds: E[( q r([t=] + ), q r([t=])( q s([t=] + ), q s([t=])jf t ] = = j=1 where j =1f=jand j =f6=j. q r qs j j +O() q r sq s +O() Fnally we also have, for any ">;any t and any, = rs +O() (B.1) Probfj q ([t=] + ), q ([t=]j "jf t g = (B.11) 5

51 for any < ["= max jk j j] : From (B.8), (B.1), and (B.11), one easly sees that the processes Q ; =1;1 ;1; ::: satsfy the assumptons of Theorems 6 and 7 of Ghman and Skorohod (1979, p.195). It follows that for any t, as converges to zero, the condtonal dstrbutons of fq (t )g t [t;1] gven Ft converge to the condtonal dstrbuton of a standard N-dmensonal Brownan moton W (:) on[t; 1] gven the \ntal" value W (t). In partcular the overall dstrbutons of the processes Q on [; 1] converge to the dstrbuton of a standard Brownnan moton on [; 1] wth nntal value W ()=: Gven ths convergence result, the proposton follows from Theorem 5.1 of Bllngsley (1968, p.3) n conjuncton wth the observaton that for any;one has ( 1 (:); :::; N (:)) = (Q ),1 Q. Q.E.D. Appendx C In ths appendx we analyse the contnuous-tme agency problem wth controls restrcted to a product of compact ntervals. In the contnuous-tme model, the prncpal's problem s to choose a functonal s() on the space of contnuous functons from [; 1] nto IR N and an admssble control process (), takng values n Q N [m ;M ], so as to maxmze the expectaton of R 1 P N (t)dt,s() subject to the constrant () that gven the ncentve scheme s(), the agent s wllng to choose the control process (), and () that the resultng expected utlty of the agent be at least as large as, exp(,rw), hs expected utlty elsewhere. If an ncentve scheme s() and a control process () satsfy both these constrants, we say that s() mplements (). For a precse denton of admssblty of a control process, as well as other detals of the formulaton of the prncpal's problem and ts analyss, we refer the reader to Schattler and Sung (1993). Under our assumptons, ther results are easly adapted to yeld: Proposton C.1 An admssble control process () s mplementable by some ncentve scheme s () f and only f there exst nonnegatve-valued adapted processes u (), v (), =1;:::;N, such that for (almost) every t [; 1], wth probablty one, the (possbly hstory dependent) control (t) mnmzes the 51

52 expressson ^c((t)), [^c ((t)), u (t)+v (t)] (t) (C.1) under the constrants () (t) Q N [m ;M ], and () the ncentve scheme s () has the representaton: s () = w + + r ^c( (t))dt + j=1 [^c ( (t)), u (t)+v (t)]d(t) (C.) [^c ( (t)), u (t)+v (t)] j [^c j ( (t)), u j (t)+v j (t)]dt where = ( j ) s the covarance matrx of the N-dmensonal Brownan moton that s speced n Theorem 1. Remark C.1: Gven the assumpton that ^c() s convex, the condton that mnmze (C.1) over the set K = Q N [m ;M ] s equvalent to the complementary slackness condton that for all. u (t)[ (t), m ] = v (t)[m, (t)] (C.3) Proof of Proposton C.1: The argument nvolves a straghtforward modcaton of the proof of Theorem 4. of Schattler and Sung (1993, pp. 35f) to take account of the possblty of boundary values of the controls. If s () mplements the control process (), then by Theorem 4.1, p. 348, of Schattler and Sung, there exst adapted processes V () and rv (), takng values n IR, and n IR N + such that for almost every t, wth probablty one, (t) maxmzes P N rv (t) (t) +rv (t)^c((t)) over the set Q N [m ;M ]. 18 The rst-order condton for ths problem mples the exstence of nonnegatve u (t), v (t) such that for any, rv (t) rv (t) =,[^c ( (t)), u (t)+v (t)] ; (C.4) wth u (t)( (t), m ) = v (t)(m, (t)) =. For ths speccaton of u (t) and v (t), t s then easy to see that (t) ndeed mnmzes the expressson (C.1) over the set Q N [m ;M ]. The representaton (C.) for the ncentve scheme s () s obtaned by usng (C.4) to substtute n the ncentve scheme representaton gven n Theorem 4.1 of Schattler and Sung (1993). 18 The symbol r s only natatonal and does not stand for a dervatve. 5

53 Conversely f there exst adapted processes u (), v () such that for (almost) every t, wth probablty one, (t) mnmzes the expressson (C.1) over the set Q N [m ;M ], then formula (C.) denes an ncentve scheme s (), and the argument gven n the proof of Theorem 4. of Schattler and Sung shows that ths ncentve scheme mplements the control process (). Proposton C.1 has the followng mmedate corollary: Q.E.D. Corollary C.1 An admssble control process () s mplementable by an ncentve scheme s () takng the form s (()) = s ( P N ()) f and only f there exst nonnegatve-valued adapted processes u (), v (), = 1;:::;N, such that for (almost) every t [; 1], wth probablty one, the (possbly hstory dependent) control (t) mnmzes (C.1) over the set Q N [m ;M ] and moreover ^c ( (t)), u (t)+v (t) s the same for all. Lemma C.1 For any E [ P N m ; P N M ], let (E) be the mnmum of ^c() over the set Q N [m ;M ] under the constrant that P N = E, and let (E) be the correspondng mnmzer. The control process () satsfyng (t) = (E) for all t s mplementable by an ncentve scheme s () takng the form s (()) = s ( P N ()). Indeed s () has the form s () = w + (E)+ (E) where = P N P N j=1 j. (1)! + r (E) (C.5) Proof: The rst-order condtons for (E) can be wrtten as: ^c ((E)), u + v = ; =1;:::;N; (C.6) where u, v, = 1;:::;N, are nonnegatve, wth u (, m ) = v (M, ) =, and s the Lagrange multpler of the constrant P N = E. By the envelope theorem, one also has (E) = : (C.7) The lemma now follows drectly from Proposton C.1, Remark C.1, and Corollary C.1. Q.E.D. 53

54 Proposton C. Suppose that E maxmzes the expresson E, (E), r (E) 4 N ^p k, j=1 ^p ^p j k k j 3 5 (C.8) over the nterval [ P N m ; P N M ]. If the prncpal observes the aggregate process Y () := P N (), but not ts ndvdual components, then a soluton to the prncpal's problem s gven by the control process () satsfyng (t) =(E )for all t, mplemented by the ncentve scheme s () gven as s () = w + (E )+ (E ) (1)! + r (E ) : (C.9) Proof: If the prncpal observes only the aggregate outcome process, any ncentve scheme s () that he uses must take the form s (()) = s ( P N ()). Wth such an ncentve scheme, by Corollary C.1, he can mplement a control process () f and only f there exst adapted processes u ();v ();;:::;N, and (), such that for all, u () and v () take nonnegatve values, and for almost all t, wth probablty one, (t) mnmzes (C.1) over the set Q N [m ;M ], and moreover ^c ( (t)), u (t)+v (t)=(t)for =1;:::;N. Snce (t) mnmzes (C.1) over the set Q N [m ;M ], one also has u (t)( (t), m ) = v (t)(m, (t)) =. Thus (t) satses the rst-order condtons for the problem of mnmzng ^c() over the set Q N [m ;M ] under the constrant that P N (t) = E(t), where E(t) := P N (t). Gven that ^c() s convex, t follows that for almost all t, wth probablty one, (t) =(E(t)), ^c( (t)) = (E(t)), and ^c ( (t)), u (t)+v (t)= (E(t)). Upon usng these equaton to substtute n (C.), we nd that the functon s () has the representaton s (Y )=w+ (E(t))dt + (E(t))dY (t)+ r (E(t)) dt : (C.1) Condtonal on the process E(), the prncpal's net expected payo s then almost surely equal to E(t)dt, w, (E(t))dt, r (E(t)) dt : The denton of E mples that wth probablty one ths s no larger than (C.11) E, w, (E ), r (E ) : (C.1) 54

55 Upon takng expectatons wth respect to the process E(t), one mmedately sees that the prncpal's net expected payo from any control process () whch s mplementable by an ncentve scheme of the form s (()) = s ( P N ()) cannot exceed hs net expected payo from the control process () satsfyng (t) =(E )for all t. Under the gven nformaton assumpton the control process () s therefore ndeed optmal for the prncpal. The representaton (C.9) of the ncentve scheme that mplements () s obtaned by substtutng for E(t) = E n (C.1) and usng the fact that R 1 dy (t) = Y (1), Y () = P N (1), P N () = P N (1). Q.E.D. 55

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