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1 DscreteTme Approxmatons of the HolmstromMlgrom BrownanMoton 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 multperod dscretetme and contnuoustme prncpalagent models. We explctly derve the contnuoustme model as a lmt of dscretetme 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 dscretetme 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: Prncpalagent 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: Department of Economcs, Unversty of Munch, Ludwgstr. 8 (Rgb.), D8539 Munch, Germany, emal: 1
2 1 Introducton Ths paper studes the relaton between multperod dscretetme and contnuoustme prncpalagent 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 contnuoustme models. We share the vew of Holmstrom and Mlgrom that the nonlneartes or even dscontnutes of optmal ncentve schemes that are typcal for statc prncpalagent 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 HolmstromMlgrom 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 multperod 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 oneperod model, wth no attenton pad to the order n whch the derent outcomes arse. 3. A contnuoustme model n whch the agent controls the drft rate vector (but not the varance) of a multdmensonal Brownan moton. It s suggested that ths model can be obtaned as a lmt of a sequence of multperod dscretetme models speced n such a way that the overall duraton of the prncpalagent relaton s xed and, along the sequence of models, perods become shorter as well as more numerous. The derent dmensons of the multdmensonal 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 multperod model. As n the multperod dscretetme models, the assumptons of constant absolute rsk averson and statonarty n the technology ensure that the optmal ncentve scheme n the contnuoustme 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 HolmstromMlgrom 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 hghprot 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 contnuoustme 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 contnuoustme models. In ether case t s not clear what analogues these results have n the multperod dscretetme 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 dscretetme and contnuoustme models. To see that ths s a nontrval matter, note that n a dscretetme 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 oneperod, the multperod, and the contnuoustme models of Holmstrom and Mlgrom n a uned framework and explctly derve the contnuoustme model as a lmt of dscretetme 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 multperod dscretetme and the contnuoustme models of Holmstrom and Mlgrom, we look for dscretetme analogues of ther lnearty results for the contnuoustme model. Two sets of results are obtaned. Frst we show that there s no dscretetme 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 dscretetme 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 dscretetme 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 dscretetme analogue for the HolmstromMlgrom 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 dscretetme, 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 dscretetme 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 dscretetme 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 Brownanmoton process; t plays no role n ther multperod dscretetme analyss or n the other results 5
6 they have for the Brownan moton model. Our results show that for ther speccaton of multperod dscretetme and contnuoustme 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 dscretetme 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 HolmstromMlgrom 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 methodorented 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 contnuoustme 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 contnuoustme agency problems concernng jump processes. Bolton and Harrs (1997) also consder contnuoustme problems concernng jump processes as well as dusons. However they are concerned wth rstbest rather than secondbest problems, consderng optmal rsk sharng and optmal actons wthout concern for ncentve compatblty and showng that for arbtrary preference speccatons rsksharng consderatons wll typcally call for nonlnear contracts n a rstbest settng. The abovementoned papers all work drectly n contnuous tme and do not dscuss the relaton between statc or multperod dscretetme and contnuoustme agency models. Multperod dscretetme agency models and ther relaton to contnuoustme 6
7 models are studed by Schattler and Sung (1997) and by Muller (1997). Unlke Holmstrom and Mlgrom (1987) these papers consder multperod 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 rstbest 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 multperod 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 rstbest 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 dscretetme and n a contnuoustme Brownanmoton model. Nether paper asks how a contnuoustme Brownanmoton model would be approxmated by dscretetme 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 oneperod and multperod 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 MultPerod 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 Ndmensonal 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 rstorder 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 amultperod 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 oneperod 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 rstorder 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 multperod dscretetme and contnuoustme 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 lowprot outcome to some hghprot 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 zeroexpectedprots 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 zeroexpectedprots 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 dscretetme 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 tradeo 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 ndvdualratonalty 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 perperod 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 dscretetme model n contnuous tme, we use a lnear nterpolaton to represent the process ~ () by a contnuoustme 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 contnuoustme representaton () where n the dscretetme 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 Ndmensonal 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 Ndmensonal 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 dscretetme models.. Dscretetme approxmatons of the contnuoustme 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 contnuoustme 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 contnuoustme 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 cumulatveoutput process n the dscretetme 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 cumulatvereturns 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 Ndmensonal 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 dscretetme 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 Ndmensonal 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 varancecovarance 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 twodmensonal 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 threedmensonal model we can approxmate any varancecovarance matrx n the twodmensonal 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 hstoryndependent 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 rstorder condtons, one nds that the ncentvecompatblty 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 KuhnTucker 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 KuhnTucker 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 contnuoustme 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 KuhnTucker 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 KuhnTucker 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 KuhnTucker multplers n (35). Usng the twooutcome example one easly veres that not all these KuhnTucker 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 twooutcome example one also sees that the ncentve scheme whch s cheapest for the prncpal, namely s 1 = s = ^c(m 1), nvolves KuhnTucker 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 KuhnTucker multplers u, v. It s convenent to wrte the KuhnTucker 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 dscretetme 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 rstorder 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 rstorder 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 rstorder 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 KuhnTucker 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 rstorder 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 rstorder 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 KuhnTucker 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 multperod 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 multperod 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 contnuoustme model that actually do yeld optmal ncentve schemes that are lnear n aggregates, e.g., total prots. One of these nvolves 6
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