Optimal Long-term Contracting with Learning

Transcription

1 Opimal Long-em Conacing wih Leaning Zhiguo He y Bin Wei z Jianfeng Yu x Fis daf: Januay This daf: Mach 3 Absac We inoduce unceainy ino he Holmsom and Milgom (987) in nie-hoizon model o sudy opimal dynamic conacing wih endogenous leaning. The agen s poenial belief manipulaion leads o a hidden infomaion poblem. We chaaceize he opimal conac by solving a dynamic pogamming poblem wih one sae vaiable. We nd ha in he opimal conac, he opimal e o deceases ove ime, exhibiing a fon-loaded paen. Fuhemoe, he opimal conac exhibis an opion-like feaue in ha incenives incease afe good pefomance. Keywods: Execuive Compensaion, Moal Hazad, Bayesian Leaning, Hidden Infomaion, Belief Manipulaion, Pivae Savings, Coninuous-Time Opimal Conacing. We hank Philip Dybvig, Johannes Hone, Navin Kaik, Mak Loewensein, Julien Pa, Yuliy Sannikov, and paicipans a Cheung Kong Gaduae School of Business, he 3 Economeic Sociey a San Diego, Sociey of Economic Dynamics Meeing, and China Inenaional Confeence in Finance fo helpful commens. This wok does no necessaily e ec he views of he Fedeal Reseve Sysem o is sa. All eos ae ou own. Jianfeng Yu gaefully acknowledges nancial suppo fom he Dean s Small Reseach Gan fom he Calson School of Managemen a he Univesiy of Minnesoa. y Univesiy of Chicago, Booh School of Business, and NBER. 587 Souh Woodlawn Ave., Chicago, IL Phone: () zhiguo.he@chicagobooh.edu. z Boad of Govenos of he Fedeal Reseve Sysem, Washingon, DC 55. Phone: () bin.wei@fb.gov. x Univesiy of Minnesoa, Calson School of Managemen, CSOM 3-, 3 9h Avenue Souh, Minneapolis, MN Phone: () jianfeng@umn.edu.

2 Inoducion Many long-em conacual elaionships feaue leaning, as unceainy aises if eihe he pojec s qualiy o he agen s abiliy is unknown when a conac is signed. We inoduce saionay leaning ino a coninuous-ime-in nie-hoizon vaiaion of he classic Holmsom and Milgom (987) model wih a Consan-Absolue-Risk-Avesion (CARA) agen o sudy opimal long-em conacing wih endogenous leaning. Moe speci cally, he pincipal signs a long-em conac wih he agen, whee he obsevable oupu dy each peiod is he sum of he agen s unobsevable e o and he pojec s unknown po abiliy, plus some noise db : dy = ( + ) d + db : () To focus on leaning only, we assume ha boh he pincipal and agen shae he same pio on he iniial pojec po abiliy, and hey lean he pojec po abiliy dynamically fom he obsevable oupu. Dynamic conacing wih unceainy and leaning is heoeically challenging. Wihou leaning, ou seing is simila o he sandad hidden acion dynamic agency models whee he agen s unobsevable shiking has only a sho-lived e ec. In hese pue hidden acion models he hisoy dependence of opimal conacing is fully summaized by he agen s coninuaion payo, and, as a esul, incenive povisions ae essenially independen acoss peiods. Once we allow he oupu o also be a eced by an unceain po abiliy componen (i.e., in equaion ), dynamic leaning endes an iniguing link among incenive povisions acoss di een imes in long-em conacing. The ineempoally linked incenive povisions ae ooed in he following pesisen hidden infomaion poblem. Along he equilibium pah he pincipal knows as much as he agen since boh sa wih a common pio. Howeve, along o -equilibium pahs he agen knows moe, because only he agen knows his acual e o ha is no obsevable o he pincipal and may deviae fom he ecommended level (i.e., hidden acion ). Imagine ha he agen has followed he ecommended e o policy in he pas, and hus boh paies shae he same coec belief oday. Suppose ha he agen shiks oday, i.e., exeing some e o below he ecommended level. The lowe e o deceases oday s oupu on aveage. Fuhemoe, wih Bayesian leaning, he pincipal who anicipaes a highe e o would misakenly aibue oday s weak pefomance o wose po abiliy. Thus, by shiking oday he agen can diso downwad he pincipal s infeence abou po abiliy fom oday on, which is long-lasing (i.e., pesisen hidden infomaion). This belief manipulaion e ec is bene cial o he agen, as he pincipal will misakenly ewad he agen lae wheneve See, fo example, Holmsom and Milgom (987), Spea and Sivasava (987), Sannikov (8), ec.

3 he fuue pefomance beas he pincipal s downwad disoed expecaions. We call his poenial bene due o o -equilibium pivae infomaion he agen s infomaion en, which inceases wih fuue pay-pefomance sensiiviies. Because he infomaion en anishes he agen s cuen woking incenives, incenive povisions a all daes ae inelinked hough his belief-manipulaion e ec. We chaaceize he opimal conac in such a dynamic agency model wih boh hidden acion (i.e., unobsevable e o) and hidden infomaion (i.e., poenially diveging beliefs abou po abiliy). In Secion 3 we s solve he agen s poblem by chaaceizing he s-ode opimaliy condiion fo any given compensaion conac: he agen s opimal e o is simply he insananeous incenive (i.e., pay-pefomance sensiiviy) minus he infomaion en due o he belief manipulaion e ec. As menioned, he infomaion en capues he maginal bene of he agen disoing he pincipal belief downwad, and can be convenienly expessed as he sum of popely discouned fuue incenives. The highe he fuue incenives, he geae he infomaion en, and he lowe he agen s cuen woking moivaions. Fuhe, Poposiion shows ha, given he CARA pefeence, he local s-ode opimaliy condiion is su cien o guaanee he global opimaliy of he agen s poblem. We hen solve he pincipal s poblem of designing he opimal conac in Secion 4. Given he saionay naue of leaning, hee is no mechanical ime e ec. We efomulae he opimal conacing poblem as a dynamic pogamming poblem wih wo sae vaiables: he agen s coninuaion value and he infomaion en. The CARA pefeence feaues no wealh e ec. Alhough he agen is pomised wih di een levels of coninuaion values afe ceain hisoy of pefomance shocks, hese pomised coninuaion values hemselves can be viewed as di een pomised wealh levels o he agen and hus do no a ec he conacing poblem looking fowad. As a esul, he opimal conac can be fully chaaceized by an odinay di eenial equaion (ODE) wih he infomaion en as he only sae vaiable, as we show in Secion 5. Relaive o Pa and Jovanovic () and DeMazo and Sannikov () who focus on he conac ha implemens a consan s-bes level of e o, we solve fo he opimal e o pah endogenously joinly wih he opimal incenives in he long-em conac. We nd ha he opimal e o policy exhibis wo ineesing feaues. Fis, we nd ha in ou model he opimal e o policy, which is always disoed downwad elaive o he s-bes benchmak, has a negaive dif, hus exhibiing a fon-loaded o ime-deceasing paen. In fac, in Secion 5.. we solve in closed fom he opimal deeminisic conac (i.e., he opimal one among he conacs in he subspace ha implemen deeminisic policies only), and shows analyically ha he opimal deeminisic e o policy deceases ove

4 ime. This fon-loaded paen comes fom he belief manipulaion e ec. As menioned, lae incenives incease he agen s cuen infomaion en fo shiking. This implies ha fuue pay-pefomance sensiiviies hu he agen s moive fo woking in ealie peiods, bu no he ohe way aound. conac implemens lowe e o in lae peiods. Given ha lae incenives ae moe cosly, he opimal Ineesingly, he paen of ime-deceasing e o policy in ou pape wih pos-conacing infomaion asymmey is opposie o he dynamic conacing seing wih pe-conacing asymmeic infomaion in Gae and Pavan (): in ha pape, he agen pivaely obseves his poduciviies when signing he conac; and unde he assumpion ha he e ec of he iniial poduciviy on his fuue poduciviy is declining ove ime, he opimal e o policy is ime-inceasing. Inuiively, in Gae and Pavan (), downwad disoion equied fo en exacion is moe sevee in ealie peiods when he majo ficion is pe-conacing pivae infomaion. I is iniguing ha pe-conacing pivae infomaion and pos-conacing infomaion have opposie pedicions fo he ime-seies paen of e o disoion, bu he di eence also lies on he agen being isk-neual wihou wealh consain in Gae and Pavan (). 3 Relaedly, Sannikov () allows fo he possibiliy of he agen s cuen e o o a ec fuue fundamenals, and nds ha he opimal e o policy is inceasing ove ime as well. Second, we nd ha he opimal e o policy is sochasic wih highe incenives afe good pefomance, exhibiing an opion-like feaue. 4 The inuiion comes fom educing he agen s belief manipulaion in a long-em elaionship. Fo a isk-avese agen, he amoun of infomaion en no only depends on fuue pay-pefomance sensiiviies, bu also he agen s maginal uiliies a hese fuue saes when eceiving hose manipulaion bene s. Raising incenives afe good pefomance inoduces a negaive coelaion beween payfo-pefomance and maginal uiliy, i.e., allocaing geae belief manipulaion bene s in saes whee he agen caes less. Hence, he opion-like compensaion conac lowes he agen s infomaion en sanding oday. This esul of e o policy being hisoy-dependen is moe supising given ou seing. To be moe pecise, in ou pape he pos-conacing infomaion asymmey is caused by unobsevable deviaions in ecommended e o levels, and hus exiss only along o -equilibium pahs. 3 In Gae and Pavan (), since he agen is isk neual and has su cien wealh (o, no limied liabiliy), wihou pe-conacing pivae infomaion he pincipal can sell he pojec o he isk-neual agen o achieve he s-bes oucome, and hee is no downwad disoion a all. In ou model, i is e cien fo he pincipal o o e insuance o he isk-avese agen. We expec he fon-loaded paen of opimal e o policy o be aenuaed if we allow fo he agen s pe-conacing pivae infomaion, because of he e ec in Gae and Pavan (). 4 Dimann and Maug (7) show ha he opimal conac implied by sandad pincipal-agen models almos neve conains any opions, a pedicion conay o pacice. Fom his pespecive, he endogenous opion-like esul is paiculaly ineesing. 3

5 Wih a sandad CARA-Nomal seing and leaning, as he poseio vaiance only changes ove ime deeminisically (in ou saionay seing, i is a consan), he esuling equilibium e o po le is usually deeminisic (e.g., Gibbons and Muphy, 99; Holmsom, 999). In conas, in ou pape wih po abiliy unceainy and leaning he opimal long-em conac has an opion-like feaue in ha pay-fo-pefomance ises following good pefomance. We emphasize ha he combinaion of long-em conacing and leaning dives boh esuls. In Secion 5., we s show ha wih long-em conacing bu no leaning, he model is a simple exension of Holmsom and Milgom (987) and a consan e o policy is opimal. On he ohe hand, wih sho-em elaionships and leaning, he belief manipulaion e ec is sill pesen so ha he agen woks less oday if fuue paypefomance sensiiviies ae highe. Howeve, he absence of commimen in he sho-em elaionships implies ha pincipals a di een imes will no ake his belief manipulaion e ec ino accoun. This again implies a consan e o pocess in equilibium, hanks o he Gaussian seing wih saionay Bayesian leaning (as in Holmsom, 999). Though we ely on he speci c seing (e.g., CARA pefeence, Gaussian pocesses, ec.) o chaaceize fully he opimal long-em conac wih leaning, he above wo esuls ae likely o hold qualiaively in a moe geneal seing. Robusness exiss because he economic foces diving hese esuls do no depend on he CARA pefeence o Gaussian pocesses. In any long-em conacing envionmen wih leaning, he agen s infomaion en due o belief manipulaion ha is, he agen would like o shik o diso he pincipal s fuue belief downwad is geneal. The ime-deceasing e o policy comes fom he fac ha lae incenives ene he agen s fowad-looking infomaion en in ealie peiods (bu no he ohe way aound); and he opion-like feaue only elies on he agen s isk-avesion, so ha he maginal value of eaning fuue (poenial) belief manipulaion bene is lowe afe he agen wih good pefomance has eceived highe compensaion. Ou pape is closes o DeMazo and Sannikov () and Pa and Jovanovic (). Boh esic aenion o he opimal conac ha implemens a consan s-bes level of e o. In conas, we solve fo he opimal e o policy endogenously join wih he opimal long-em compensaion conac, and emphasize he geneal economic mechanisms ha shape he opimal e o policy in long-em opimal conacing. Leaning-wise, DeMazo and Sannikov (), like ou pape, conside a famewok wih saionay leaning, while Pa and Jovanovic () sudy paamee unceainy, which implies ha he pojec po abiliy is an unknown consan ahe han a moving age. 5 Theoeically speaking, 5 On he assumpion of he agen s pefeence, he agen in DeMazo and Sannikov () is isk neual. Pa and Jovanovic () cove geneal uiliy funcions fo he agen s poblem, bu hen specialize exponenial uiliy in deiving he opimal conac. 4

6 he exac leaning echnology is less impoan, as boh of he above leaning echnologies lead o a long-lasing belief manipulaion e ec, which is he majo challenge in solving fo he opimal conac. Relaedly, he long-lasing belief manipulaion also exiss in Begemann and Hege (998, 5) and Hone and Samulson (). Thee ae ohe papes ha ae elaed o leaning bu do no deal wih he belief manipulaion e ec. Adian and Wese eld (9) focus on he disageemen beween he pincipal and he agen abou he agen s abiliy and he agen is dogmaic abou his belief (i.e., he agen neve updaes his poseio belief abou po abiliy fom pas pefomance), which eliminaes he belief manipulaion e ec. In ha pape, alhough he agen could diso he pincipal s belief by shiking, he dogmaic agen (who does no ealize ha he m s po abiliy is, in fac, above he one peceived by he pincipal) will no gain anyhing fom his channel, and as a esul hee is no belief disoion e ec. 6 The opic of opimal conacing wih endogenous leaning also elaes o he ecen lieaue sudying opimal long-em conacs wih advese selecion and moal hazad (e.g., Baon and Besanko, 984; Sung, 5; Sannikov, 7; and Cvianic, Wan, and Yang, ). Gae and Pavan () fuhe allow agen s abiliy o be ime-vaying, and sudy he opimal eenion policy. In Geshakov and Pey (), he agen whose skills ae his pivae infomaion faces a nie sequence of asks and leans hei level of di culy. Halac, Kaik, and Liu () inoduce ineacion beween he agen s abiliy and pojec s po abiliy o sudy opimal expeimenaion. In geneal, when he agen has pe-conacing pivae infomaion ha is pesisen, a mechanism design appoach naually aises; see Pavan, Segal and Toikka () and Golosov, Toshkin, and Tsyvinski (), who use he sode-appoach o solve he agen s poblem. 7 Howeve, because ou pape focuses on he poblem wihou pe-conacing pivae infomaion, we do no need o solve fo he opimal menu fo agen s uhful epoing when signing he conac. The es of he pape is oganized as follows. Secion lays ou he model, and Secion 3 solves he agen s poblem. Secion 4 efomulaes he pincipal s poblem which allows fo he use of dynamic pogamming echniques. Secion 5 solves fo he opimal conac and discusses is implicaions. Secion 6 concludes. 6 Moe ecenly, Cosimano, Speigh, and Yun () sudy he long-em conacing poblem wih binay unobsevable poduciviy saes, and show ha he opimal conac ends o be sicky. They assume ha he agen s e o is obsevable bu no conacible, and hence boh he pincipal and he agen always have he same infomaion se, boh on- and o -equilibium pahs. 7 This is he same appoach used in Williams (9, ) and Zhang (9), who sudy pesisen infomaion in a coninuous-ime pincipal-agen seing. We also use he s-ode-appoach o solve he agen s poblem, and veify he global opimaliy of he agen s s-ode-condiion. 5

7 The Model Conside a coninuous-ime in nie-hoizon pincipal-agen model wih a common consan discoun ae >. The pojec geneaes a cumulaive oupu Y up o ime, which evolves accoding o dy = ( + ) d + db ; () whee fb g is a sandad Bownian moion on a complee pobabiliy space (; F; P), is he agen s unobsevable e o level, and he consan > is he volailiy of cash ows. Relaive o Holmsom and Milgom (987), we inoduce pojec po abiliy ino he oupu pocess in equaion (). Equivalenly, we can also inepe as he agen s unknown abiliy. Pojec po abiliy is unobsevable and hus calls fo leaning. If pojec po abiliy is pefecly obsevable (o absen), hee is no need fo leaning; In Secion 5.. we show ha his case educes o Holmsom and Milgom (987). Fo acabiliy, we assume ha unknown po abiliy a ecs oupu dy addiively. We assume ha po abiliy f g follows a maingale pocess whose innovaions ae diven by anohe Bownian moion B ha is independen o fbg, i.e., d = db and B? B ; whee > is a consan. A ime, he pincipal and he agen shae he common nomal pio ha N ;, whee we have nomalized he pio mean o. We mainly focus on saionay leaning, excep ha in Secion 5.. we bie y discuss non-saionay leaning fo obusness check. Fo leaning o be saionay, he pio unceainy is assumed o saisfy =, so ha he poseio vaiance = fo all and Bayesian updaing is ime-independen. When =, ou model feaues no unceainy, and hus is educed o he benchmak model of Holmsom and Milgom (987). Boh paies commi o he long-em elaionship. The isk-neual pincipal (she lae on) o es he isk-avese agen (he lae on) a conac fc ; g, so ha he agen is ecommended o ake he e o policy = f g ; and is compensaed by he wage pocess c = fc g. Boh elemens ae measuable o Y F fy s : s g, which is he laion geneaed by he oupu hisoy. Moal hazad aises fom he agen s unobsevable e o choice. Fuhemoe, we assume ha he agen can pivaely save (i.e., saving is unobsevable) o smooh his consumpion ineempoally. Thus he agen s acual consumpion can di e fom wage c. Denoe he agen s acual consumpion by bc and acual e o by b. Following Holmsom and Milgom (987), we assume ha he agen has an exponenial o CARA pefeence: u (bc ; b ) = a exp [ a (bc g (b ))] ; 6

8 whee a > is he agen s absolue isk avesion coe cien, and g (b ) b is he insananeous quadaic moneay cos fo e o b. The quadaic fom of g () simpli es ou esuls, bu ou analysis goes hough as long as g () is sicly inceasing and sicly convex. A he beginning when boh paies sign he conac, he agen has no pesonal wealh and has an exogenous esevaion uiliy of v. Wihou loss of genealiy, we assume he pincipal has all he bagaining powe. I is a celebaed esul ha CARA pefeences do no have a wealh e ec, and he issue of pivae savings can be easily deal wih (e.g., Fudenbeg, Holmsom, and Milgom, 99; Williams, 9; He, ). In fac, pivae savings allow us o sepaae he agen s coninuaion payo fom anohe sae vaiable in deiving he opimal conac, which endes he acabiliy of ou poblem.. Bayesian Leaning and E o Recall ha a ime, he pincipal and he agen shae he common nomal pio N ; : Boh paies updae hei beliefs based on hei own infomaion ses especively. Recall ha Y = F fy s : s g is he augmened laion geneaed by oupu pah Y. Given any conac fc ; g, he pincipal s infomaion se a ime is F fy s ; s : s g, as he pincipal knows he ecommended e o policy f g. Howeve, he agen s infomaion se also includes his acual e o policy b fb g, i.e., F fy s ; s ; b s : s g. Inuiively, elaive o he pincipal, he agen knows (weakly) moe because he knows his acual pas e o choices b, which may deviae fom he ecommended policy. disincion is impoan fo ou analysis. This If he agen follows he ecommended e o policy, he pincipal s poseio belief abou is coec and fully summaized by he s wo momens: m E [ jy ; ] and ; E ( m ) jy ; : Sandad leing agumen (e.g., Theoem. in Lipse and Shiyayev, 977) implies ha dm = ; dy ( + m ) d = db ; wih m = ; (3) ; = fo all, (4) whee B is a sandad Bownian moion unde he measue induced by he e o policy : db = dy ( + m ) d : (5) Condiional on he acual e o policy fb g, he agen foms his poseio belief as m b E [ jy ; b] and ;b E m b jy ; b : 7

9 The supescip b is used hee o emphasize he dependence on he agen s acual e o policy b (which he pincipal does no know). A simila agumen yields dm b = dy b ;b + m b d = db b ; wih m = ; (6) ;b = fo all ; whee B b policy b: is a sandad Bownian moion unde he measue induced by he acual e o db b dy b + m b d : (7) As we will show nex, he poenial belief divegence m b m gives ise o he agen s incenive o manipulae he pincipal s belief, which plays an impoan ole in he design of he opimal long-em conac wih leaning.. E o and Belief Disoion The di culy of inoducing leaning ino he dynamic moal hazad poblem is no leaning pe se. Rahe, he challenge is o deal wih he issue of belief manipulaion: he agen, simply by shiking fom he ecommended e o oday, can diso he pincipal s fuue beliefs abou pojec po abiliy downwad. Conside he following hough expeimen. Suppose ha a ime he agen chooses an e o level b below he ecommended e o, and hus oupu is lowe han wha is expeced by he pincipal. Cucially, howeve, he pincipal hinks he agen is exeing an e o of, and, hus, hough leaning, misakenly aibues lowe oupu o a lowe value of po abiliy. In conas, he agen updaes po abiliy based on his ue e o level b, leading o a posiive wedge m b m beween he beliefs of he agen and pincipal. In ohe wods, by shiking, he agen makes he pincipal (misakenly) undeesimae pojec po abiliy. This belief manipulaion is bene cial o he agen when fuue oupus un ou o be high, he agen ges ewaded fo high po abiliy (based on he agen s coec infomaion se) ahe han his e o. We now fomalize his e ec. When he agen deviaes fom he ecommended e o pah by choosing e o policy b, he pincipal s poseio mean esimae abou is disoed downwad, and we denoe his disoion by m b m = agen s belief - pincipal s belief. In he Gaussian famewok wih saionay leaning, he poseio vaiances of boh paies coincide o be a consan. Accoding o equaions (3) and (6), he incemen of is 8

10 given by: d = dm b dm = dy ^ + m^ d = ( b ) d; which allows us o solve fo o be (dy ( + m ) d) = Z e (s ) ( s b s ) ds. (8) Hee, we have used he fac ha boh he pincipal and he agen shae he same belief when signing he conac a =, i.e., = as m = m b = m. Inuiively, equaion (8) says ha he cuen belief disoion is a popely-weighed cumulaive e o deviaion in he pas. When =, he zeo pio unceainy = = eliminaes any belief divegence and he issue of belief manipulaion is absen. Suppose ha he employmen conac elies on he agen s unexpeced pefomance along he equilibium pah dy ( + m ) d; (9) which is a maingale incemen db unde he equilibium measue. Fo he agen who deviaes by exeing b 6=, unde his infomaion se he above pefomance is no longe a maingale incemenal. Imagine ha he agen has deviaed befoe so ha b s 6= s whee s <. Even if he agen exes he same e o a ime so ha = b, equaion (7) implies ha dy b + m b d = dy + m b d () is a maingle unde he agen s infomaion se. Compaing equaion (9) o equaion (), we obseve ha he pefomance dy ( + m ) d, which is unexpeced fo he pincipal along he equilibium pah, is no longe a maginale incemen unde he agen s infomaion se. Rahe, i displays a posiive dif of belief divegence, because dy ( + m ) d = dy + m b d + + m b i = hdy + m b d + d; ( + m ) d whee he s em is a maingale incemen unde he agen s infomaion se. Inuiively, if he agen shiked a bi in he pas b s < s wih s <, hen lae on he pincipal would misakenly hink he pojec is wose han i acually is (unde he agen s coec measue), i.e., >. As a esul, he agen can easily bea he pincipal s expecaion, which explains he posiive dif >. As we emphasize lae, his anslaes o a posiive infomaion en o he agen, as long as he pincipal is poviding incenives a ime o ewad he agen fo he unexpeced pefomance. 9

11 .3 Fomulaing he Opimal Conacing Poblem We s sae he agen s poblem. Denoe by S he balance of he agen s savings accoun, which eans inees a he consan ae. poblem is s.. dy = Z max E b fbc;bg b + m b Given he conac c = fc ; g he agen s e u (bc ; b ) d d + db b ; ds = S d + c d bc d wih S = ; wih he ansvesaliy condiion lim T! E b e T S T =. Hee, E b [] denoes he pobabiliy measue induced by he agen s e o policy fb g, and fbc g is he agen s acual consumpion policy. Denoe he opimal soluion o equaion () by fc? ;? g. We call he conac fc ; g incenive-compaible and no-savings if, given he conac fc ; g, he soluion o he agen s poblem in equaion () is c? = c and? =, which fuhe implies S = fo any (i.e., no pivae savings a any ime). In ohe wods, he agen nds i opimal o consume his wages and wok as ecommended. As ypical in he lieaue, he following lemma shows ha hee is no loss of genealiy when we esic ou aenion o incenive-compaible and no-savings conacs. The basic idea is simila o he evelaion pinciple. The pincipal who can fully commi o he conac can save fo he agen, and, once she knows he agen s acual e o policy, she will pefom coec Bayesian updaing based on ha policy. Lemma I is wihou loss of genealiy o focus on conacs ha ae incenive-compaible and no-savings. Poof. See Appendix A. The opimal conac solves he pincipal s poblem: Z e (dy max fc ; g is incenive-compaible and no-savings E c d) () () s.. dy = ( + m ) d + db ; (3) Z e u (c ; ) d = v : (4) In equaion (3), he pincipal s Bayesian updaing of m and db is based on he coec opimal e o policy f g aken by he agen in he incenive-compaible conac. Equaion (4) is he paicipaion consain, as he conac mus aac he agen wih a esevaion value v a =. As sandad in he opimal conacing lieaue wih CARA agens, his paicipaion consain mus bind. E

12 3 The Agen s Poblem In his secion we analyze he agen s poblem. We illusae heuisically he necessay condiions fo a conac fc ; g o be incenive-compaible and induce no pivae savings. We fuhe esablish he su ciency of he necessay condiions. Fomal poofs ae delegaed o Appendix A. 3. Coninuaion Value and Incenives Given he incenive-compaible and no-savings compensaion conac fc ; g, he agen s coninuaion value is de ned as: v E Z e (s ) u (c s ; s ) ds : (5) Accoding o he sandad maingale epesenaion agumen (e.g., Sannikov, 8), hee exiss some pogessively measuable pocess f g so ha Inuiively, ( dv = v d u (c ; ) d + ( av ) (dy m d) (6) = v d u (c ; ) d + ( av ) db. av ) can be inepeed as he incenive loading (in uils) on he agen s unexpeced pefomance dy m d. As we show sholy, av > is he agen s maginal uiliy fom consumpion a ime ; i.e., u c (c ; ). Thus we can inepe as dolla incenives on he agen s unexpeced pefomance. Lae we efe o pay-pefomance sensiiviies f g simply as incenives. 3. No Savings We s show ha, unde CARA pefeence, no-savings condiions imply v = u (c ; ) = a exp [ a (c g ( ))] : (7) The agumen is simila o He (). To see his, we s pesen he following lemma, whee we use o denoe any compensaion conac. Lemma A any ime, conside a deviaing agen who has some abiay savings S and faces he coninuaion conac. Denoe by v (S; ) his deviaion coninuaion value. We have v (S; ) = v (; ) e as = v e as, (8) whee we have used he fac ha v (; ) is he agen s coninuaion value v no-savings pah de ned in equaion (5). Poof. See Appendix A. along he

13 The diving foce behind his esul is simple. Due o he CARA pefeence, he agen s poblem is anslaion-invaian wih espec o his undelying wealh level, as evidenced by u (c s + S; s ) = e as u (c s ; s ). Thus, fo a CARA agen, given he exa savings S, his new opimal policy is o ake he opimal consumpion-e o-leaning policy wihou savings, and consume an exa S moe fo all fuue daes. As suggesed by equaion (8), he opimal deviaion value wih savings S is jus he oiginal value wihou savings, muliplied by he adjusing faco e as fo exa consumpion. By he opimaliy of he agen s consumpion-savings policy in equaion (), his maginal uiliy fom consumpion mus equal his maginal value of wealh, and equaion (8) implies ha: u c (c ; ) (S; = av : (9) S= Thus, equaion (7) follows immediaely fom equaion (9) by using he fac ha unde CARA pefeence, he agen s uiliy level is linea in his maginal uiliy: au (c ; ) = u c (c ; ) : () Plugging equaion (7) ino equaion (6), we nd ha fo no-saving conacs, v follows an exponenial maingale: dv = ( av ) db () Z s Z, v s = v exp a u dbu s a u du fo s >. We can also undesand his esul ha v is a maingale by combining wo obsevaions: s, he agen can smooh ou his consumpion ineempoally so ha his maginal uiliy has o follow a maingale; second, his coninuaion value v is linea in his maginal uiliy u c because of equaions (9) and (). 3.3 Incenive-Compaibiliy Consain and Inuiion 3.3. Soluion o he agen s poblem Poposiion chaaceizes he agen s incenive compaibiliy consain, along wih he equilibium consumpion and coninuaion value heuisically deived above. In Appendix A, we povide a igoous poof fo Poposiion based on he Ponyagin maximum pinciple. Thee, we show ha he policy (c; ) saisfying he (local) necessay condiions is also globally opimal. In he concuen pape by Pa and Jovanovic (), he global opimaliy is guaaneed wih ceain su cien condiions, which ae dependen on some endogenous vaiables and, hus, migh be had o veify. Wih he CARA pefeence and pivae savings

14 consideed in his pape, we ae able o esablish he su ciency of he incenive compaibiliy consain. Poposiion Fo he conac fc ; g o be incenive-compaible and no-savings, f g mus saisfy = {z} insananeous incenive Z = E e (+)(s Z E e (+)(s ) av s ( av s ) ds {z } ) s exp infomaion en Z s a u db u In addiion, equaion (7) implies ha consumpion (o wage) follows c = g ( ) and he coninuaion payo fom he conac is Z v = v exp a s dbs Poof. See Appendix A. Z s () a u du ds :(3) ln ( av ) ; (4) a Z a s ds : (5) We devoe he es of his secion o explain he inuiion behind he e o opimaliy condiion in equaion (). In a sandad dynamic agency poblem wihou po abiliy unceainy (e.g., = ), he agen s e o a ime should only depend on he ime- incenive o eed by he conac (i.e., = ). Wih leaning and associaed beliefmanipulaion, he agen s e o decisions acoss peiods ae inelinked, as suggesed by he fowad-looking naue of he second downwad adjusmen em in equaion (). The fowad-looking downwad adjusmen em epesens he infomaion en o he agen. Inuiively, his em capues he maginal bene of manipulaing he pincipal s belief downwad fom hen on. 8 Because he pincipal ewads +s o he agen s fuue unexpeced posiive pefomance a + s fo all s >, loosely speaking, he oal bene is he expeced (popely) discouned fuue incenives +s : s >. The discoun ae is + due o he combinaion of saionay leaning discouning and ime discouning. In addiion, as explained lae, we also adjus fo maginal uiliies (which is saes. av s ) a fuue 8 This infomaion en em capues he maginal en ha he agen may enjoy by deviaing fom he ecommended e o slighly, ahe han he en ha he agen acually enjoys in equilibium. I is because in equilibium he pincipal knows he agen s acual e o and hence he agen does no know moe han he pincipal does. Howeve, as any ypical moal hazad models, he maginal deviaion bene (maginal en) is impoan in chaaceizing he agen s incenive-compaibiliy condiion. 3

15 3.3. Inuiion fo incenive-compaibiliy consain Conside he agen who educes his e o o slighly below he ecommended e o level, say, only a he ime ineval [; + d]. In ohe wods, given he ecommended policy f s g, he deviaion e o policy is s fo s = [; + d] = : (6) s ohewise Wha is he impac on he agen s oal payo onwads? Sanding a ime, he agen s oal payo s include his insananeous uiliy u (c ; ) and his coninuaion payo. Moe speci cally, we wie he agen s oal payo fom ime onwads as follows: 9 u (c ; ) d + v + E = u (c ; ) d + v + E = u (c ; ) d saving e o cos insananeously E 8 >< >: Z +d Z e (s ) dv s ( av ) (dy ( ) m R d) + e (s ) +d s ( av s ) [dy s ( s + m s ) ds] + v + ( av ) (dy ( ) m d) + {z } " huing pefomance insananeously e (s ) s ( av s ) dy s s + m d + s ds s maingale unde info se geneaed by fuue belief divegences s {z } ceaing belief divegence pesisenly In he s equaliy, E implies ha he agen foms his expecaion based on his infomaion se induced by. In he second equaliy, we have used he esul in equaion (). Thee should be anohe coecion em in ( s s ) ds in he hid equaliy, bu i is zeo because of (6), i.e., we conside an one-sho deviaion a ime fom he equilibium e o policy. As he second line in equaion (7) shows, hee ae wo channels hough which shiking a ime a ecs fuue changes in he agen s coninuaion payo s. The s channel capues he insananeous pefomance e ec, i.e., he agen s e o a ecs insananeous pefomance dy and, hus, his coninuaion payo. To see his, wie pefomance dy ( ) ove [; + d] as a funcion of ime- e o. Exeing e o sho-em pefomance ove [; + d] because dy ( ) = ( ) d + m d + db = dy ( ) d: will hen hu he 9 To see he s line is he agen s oal payo fom ime onwads given any e o policy and c, de ne G () R e s u s ds + e v and G () is he agen s oal payo. Due o pivae savings, u s = v s, and we have dg () = e dv. Theefoe he oal payo (in aed by e ) is E [e G ()] = e G () + E R e (s ) dv s, which is u (c ; ) d + v + E R e (s ) dv s by ignoing uiliies occuing befoe. Unde equilibium e o, dv s is maingale incemen and hus E R e (s ) dv s =. 4 9 # >= (7) >;

16 Modulaed by incenives, his leads o a dop in he agen s coninuaion payo by ( av ) d, via he channel of huing pefomance insananeously. The second channel is he pesisen e ec due o belief manipulaion: as discussed in Secion., he agen s shiking a ime shifs he belief divegence pah f s g away fom he equilibium pah f s = g fo s >. We have seen fom equaion (8) ha once shiking ceaes a posiive ime- belief divegence beween he pincipal and he agen, his belief divegence will pesis in fuue conacual elaions. Now we show ha he incenive-compaibiliy consain in equaion () is implied by equaion (7). By educing e o cos insananeously in equaion (7), he agen s maginal gain fom shiking a is u (c ; ) d. Since u (c ; ) = u c (c ; ) = av, his maginal gain is ( av ) d. On he ohe hand, shiking hus pefomance insananeously in equaion (7), which gives ise o a maginal cos of ( av ) d. In sandad models wihou belief manipulaion, hese wo foces fully deemine he agen s ade-o in choosing his opimal e o a ime. Now we analyze he novel em ceaing belief divegence pesisenly in equaion (7). Thee, because dy s s + m s d ae maingale incemens, he em is educed o Z e (s ) s ( av s ) s ds : (8) E As equaion (6) suggess, he agen is shiking only a ime [; + d], bu neve befoe o afe. Thus, using equaion (8) we deive he belief divegence in any fuue ime s > o be s = Z s e (s ) ( ) d = e (s ) d: (9) As new infomaion ows in, he belief divegence pesiss bu decays ove ime exponenially. Plugging equaion (9) ino equaion (8), he maginal impac of shiking via he channel of belief manipulaion is Z Z E e (+)(s ) s ( av s ) ds d = E e (+)(s ) s ( av s ) ds d + o (d) whee he second em is because he di eence beween he expecaion ems E [] and E [] is (a mos) in he ode of d. Inuiively, if he pincipal misakenly believes ha he pojec is less po able han i should be, he agen s nomal pefomance will be consideed supeb. The highe-poweed he incenives f s g, he geae he infomaion en ha he agen can enjoy. Finally, fo a isk avese agen, he infomaion en depends on As we ae mainly fo illusaing inuiion, his saemen is heuisic. Fo igoous agumen, see he Appendix fo he poof of Poposiion. 5

17 he agen s fuue maginal uiliy ( esul ha is useful lae in Secion 5.4. av s ) when eceiving hose manipulaion bene s, a Combining hee pieces ogehe and canceling d (and ignoing highe ode ems), he agen s local incenive-compaibiliy consain is Z ( av ) ( av ) + E e (+)(s ) s ( av s ) ds = : Afe dividing boh sides by ime- maginal uiliy ( av ) and using equaion (5), we each he key incenive-compaibiliy condiion in equaion (3). Poposiion fuhe esablishes ha his local incenive-compaibiliy consain guaanees he global opimaliy of he agen s soluion. 4 The Pincipal s Poblem Fom now on we focus on incenive-compaible conacs such ha boh paies will have he same infomaion se. Fo ease of noaion we use db o denoe db, and wie E as E. 4. Rewiing he Pincipal s Poblem In ligh of Poposiion, we s ewie he pincipal s poblem in equaion (). Poposiion esablishes an impoan link beween ecommended e o f g and incenives f g in any incenive-compaible conacs. Moeove, he pincipal can choose he opimal f g o maximize he value, and he coesponding opimal consumpion pocess fc g and he opimal e o policy f g ae deemined by equaion (4) and equaion (), especively. Theefoe, we can ewie he pincipal s poblem in equaion () as Z max E e (dy c d) f g s:: dy = ( + m )d + db and dm = db ; (3) ln ( av ) c = g ( ) ; whee g ( a ) = ; (3) dv = ( av ) db, given v ; (33) Z Z s Z = E s e (+)(s ) s exp a u db u a u du ds(34) : Hee, equaion (3) descibes he dynamics of oupu and poseio belief; equaions (3)-(34) ae deived fom equaions (3)-(5) in Poposiion. We will use he echnique of dynamic pogamming o solve he above poblem. The fomulaion above suggess ha we need o keep ack of wo sae vaiables: one is he 6 (3)

18 agen s coninuaion payo v and he ohe is he he second em infomaion en Z Z s Z E s e (+)(s ) s exp a u db u a u du ds on he igh-hand side of equaion (34). As we discussed befoe, he infomaion en is also he agen s maginal bene of shiking in manipulaing he pincipal s belief. Impoanly, we obseve ha in equaion (34) he infomaion en is independen of he agen s coninuaion value v. Indeed, hanks o he CARA pefeence, he opimal conacing poblem can be ewien wihou v. We sa fom he pincipal s objecive in equaion (3). Fo oupus, using equaion (3) we have (m is a maingale wih an iniial value m = ): Z E Z Z e dy = E e ( + m )d = E e d ; In Appendix A, fo compensaion cos, we use equaion (3) and inegaion by pas o show ha E e c d Z = E e g ( ) = a ln ( av ) d ln ( av ) + E 4 a ceainy equivalen of ouside opion v The oal compensaion cos is he ceainy equivalen Z g ( ) + e o cos a isk compensaion ln( av ) a 3 A d5 : of deliveing he agen s ouside opion v, plus he (moneay) e o cos g ( ) = = and he discouned isk compensaion due o incenive povisions. Thus, he ceainy equivalen ln( av ) a sepaaes fom he poblem, and he opimal soluion f g will be independen of he agen s iniial ouside opion v. This esul comes fom he anslaion invaiance of CARA pefeence. In sum, he pincipal s poblem can be simpli ed o Z max E e f g a d Z Z s s:: = E s e (+)(s ) exp a u db u [ M; M] fo some su cienly lage M. Z s (35) a u du ds ; We impose he las consain as he ansvesaliy condiion on ou in nie hoizon poblem. Essenially, we esic he feasible incenive slopes f g o be bounded, i.e., hee exiss some su cienly lage consan M such ha [ M; M]. Impoanly, as we will pove lae, he deived opimal policy is independen of M when M is su cienly lage. 7

19 4. Recusive Fomulaion of he Poblem 4.. Infomaion en as he unidimensional sae vaiable In ligh of he incenive-compaibiliy consain in equaion (3), de ne he infomaion en p as R p E s e (+)(s ) exp Z s a u db u Z s (a u) du ds : (36) As shown sholy, he infomaion en p seves as he only sae vaiable fo he pincipal o design he opimal conac. De ne M p M + : The following lemma shows ha he infomaion en p is bounded in [ M p ; M p ] if incenives is bounded in [ M; M], and he boundaies fo p ae absobing. This also gives us a simple bounday condiion fo he value funcion V. Lemma 3 Suppose ha [ M; M] whee M is a given consan. Then he sae vaiable p eaches M p if and only if s = M; 8s, which implies ha M p ae absobing saes fo p. As a esul, when p = M p, V (M p ) is quadaic in M: V (p = M p ) = M + ( + ) + a M : (37) Poof. See Appendix A. When he sae vaiable p ( Appendix A we show ha he dynamics of p follows M p ; M p ) is away fom boundaies, using equaion (36) in dp = [( + ) p + (a p )] d + p db ; (38) whee p is some pogessively measuable pocess deived fom he maingale epesenaion of equaion (36). Fom now on, we inepe f p ; g as ou conol because he pai deemines he dif and di usion of p in equaion (38). Noe ha we will deive p and as a funcion of he auxiliay sae p. Combined wih he bounday condiion ha s = M fo s > wheneve p his M p, he conol pai f p ; g, as funcions of he auxiliay sae p, gives he full hisoy of f : g ha we ae afe. Now we ae able o wie he poblem (35) in he sandad dynamic pogamming language. Subsiuing = max f ; p g E p in he pincipal s objecive, we have Z e ( p ) ( p ) a d s:: dp = [( + ) p + (a p )] d + p db fo all >, and p = p s [ M; M], p [ M p ; +M p ], and p = M p ae absobing. 8

20 We ae afe he opimal policy f ; p; g as funcions of he sae vaiable p. 4.. Relaxed poblem We s conside he pincipal s elaxed maximizaion poblem given M (and M p = M Z V (p; M) max f ; p g E e s:: dp = [( + ) p + (a p p is absobing a M p, a d )] d + p db fo all >, and p = p + ): (39) can exceed M (bu emains nie) when p ( M p ; M p ). (4) The poblem (39) is a elaxed vesion of he pincipal s oiginal poblem (35), due o equaion (4). Essenially, given M, in he oiginal poblem (35) we equie [ M; M] fo any ime ; while in poblem (39) we only equie [ M; M] wheneve p his M p in ligh of Lemma 3. Thus, when p = M p, he bounday condiions ae he same beween hese wo poblems. Howeve, his elaxaion helps because he elaxed poblem (39) allows us o use he ineio s-ode condiion of when p ( M p ; M p ). We will show ha fo su cienly lage M he value achieved in he elaxed poblem is he same as ha in he oiginal poblem, which implies ha he soluion o he elaxed poblem is also ha o he oiginal poblem. The Hamilon-Jacobi-Bellman (HJB) equaion fo poblem (39) is saighfowad: V = max ; P ( p) ( p) a + V p ( + ) p + a P Unde he assumpion ha + a + a (V p) in Poposiion 5, he s-ode opimaliy condiions ae given by: V pp + V pp P : (4) > and V pp < ha we will veify lae = + p V p and p V p = a : (4) + a + a (Vp) V pp V pp Plugging hem back ino he HJB equaion (4) and dopping M in V (p; M) wihou isk of confusion, we have V = ( + p V p ) + a + a V p V pp p p + V p ( + ) p: (43) We will solve he poblem in equaion (39) by analyzing he above Odinay Di eenial Equaion (ODE) in equaion (43) wih he bounday condiion in equaion (37). 9

21 5 Opimal Conacing 5. Two Impoan Benchmaks Befoe we move on o solve he opimal long-em conac wih leaning, we pesen esuls fo wo benchmak cases: one is obsevable po abiliy wihou leaning while he ohe is wih leaning bu only sho-em conacual elaions. 5.. When Po abiliy is Obsevable We s conside he benchmak case whee he po abiliy is obsevable. This case is almos he same as he sandad Holmsom and Milgom (987) model, excep ha he opimal conac always benchmaks he agen s pefomance o. consain =, he opimal soluion is HM = + a ; Using he incenive and he pincipal s value is V HM = = ( ( + a )). The opimal conac can be implemened by a consan equiy shae = ( + a ) (wih pope benchmaking). Because he pincipal can ignoe he diec infomaion abou, he value V HM deived in he obsevable po abiliy case seves as an uppe bound fo ou value funcion V (p) when po abiliy is unobsevable: V (p) V HM = 5.. When Conacual Relaion is Sho-em ( + a ) : (44) Conside anohe pola case whee he conacual elaionship is sho-em. Imagine he following seing, whee a long-lived agen wih unknown abiliy is woking fo a coninuum of pincipals. A any ime > hee is one pincipal who signs a sho-em incenive conac wih he agen. The elaion, howeve, only lass fo he ineval [; + d]. The sho-em conac consiss of a xed wage, an incenive, and he ecommended e o, so ha a he beginning of + d he agen eceives a compensaion ow of d + (dy d m d) ; given dae belief, E [ ] = m, and oupu pefomance dy ove [; + d]. A he end of peiod +d, he elaion beaks, and he agen is signing anohe conac +d ; +d wih anohe pincipal indexed by + d. Impoanly, sho-em elaions ule ou ine-peiod commimen, so ha each pincipal akes ohe pincipals equilibium o es as given.

22 Fo simpliciy, o deemine he hisoy of xed wages f g, we assign all he bagaining powe o pincipals. The following poposiion saes ha he equilibium incenives and e o ae consan ove ime. Poposiion When a conacual elaionship is sho-em and pincipals have all he bagaining powe, incenives ae consan ove ime: = ST = + + a ( + ) and he equilibium e o is consan ove ime as well Poof. See Appendix A. = + ST = + a ( + ) fo all ; fo all : Hee is he inuiion. When he pincipals have all he bagaining powe, Poposiion sill applies o he agen s poblem. Thus, given oday s incenive and coninuaion incenives +s : s >, he agen ses = p, whee p is discouned fuue incenives as in equaion (36). The pincipal a ime who akes p as given will maximize he expeced oupu + m = p + m, minus he oal compensaion which is he sum of he e o cos g ( p ) and he isk compensaion a. Ignoing he given pojec qualiy m, he pincipal s maximizaion poblem a ime is only maximizing he ow payo in equaion (35) o equaion (4) max ( p ) ( p ) a (45) This impoan obsevaion implies ha, wih sho-em conacing, each pincipal ignoes he long-em consequence of his sho-em incenives. The opimaliy condiion fo his simple quadaic poblem (45) yields ST = + pst + a, whee p ST is de ned as in equaion (36) wih fuue incenives ST +s. Saionaiy implies boh ST and p ST ae consans; inuiively, each pincipal will solve he same poblem, giving ise o he consan equilibium oucome in Poposiion. When he agen does no have any bagaining powe, he poof in Poposiion shows ha fo he agen s poblem, he sho-em incenives f g hee play he same ole as he incenives f g in long-em conacs analyzed in Poposiion. Fo moe explanaion, see foonoe. Holmsom (999) consides leaning when conacual elaions ae sho-em, and when leaning is saionay he equilibium oucome is consan ove ime. Compaing o ou analysis, he subsanial di eence is ha Holmsom (999) sudies wage conacs wihou incenives, and pincipals ae compeiive

23 Thus, wihou commimen, in sho-em conacing each pincipal a di een poins of ime solves his individual myopic poblem in equaion (45). In conas, wih long-em conacing, a single pincipal no only maximizes he ow payo in equaion (45), bu also akes ino accoun he e ec of +s on he fowad-looking infomaion en p. 3 The nex wo subsecions show ha, he full commimen in long-em conacing wih leaning makes he opimal e o policy ime-deceasing and sochasic. 5. Opimal Deeminisic Conac Befoe we chaaceize he opimal conac fully, in ode o gain bee inuiion we s consain he incenive slopes fg o be deeminisic. conacs, we solve he opimal conac in closed fom. 5.. Time-deceasing opimal e o policy Wihin he class of deeminisic When fg is deeminisic, we can move he condiional expecaion in equaion (36) inside R s he inegal. Because exp a R vdbv s a v dv is a maingale, he infomaion en p is simply p = R e (+)(s ) s ds: In ohe wods, deeminisic incenives imply deeminisic infomaion ens, i.e., he volailiy p is consained o zeo. Denoe by V d (p) he value funcion wih deeminisic policies. Plugging p = ino equaion (4), we have d (p) = + p = ( + a ), wih he HJB equaion as: V d p V d (p) = + p V d (p) + a p p + V d p (p) ( + ) p: (so he agen has all he bagaining powe). Thus, evey peiod compeiive pincipals always pay he agen some xed wage equal o his/he expeced abiliy, and agens wok fo epuaion concens. Even if we inoduce sho-em incenives in Holmsom (999), whee epuaion concen is pesen wih compeiive pincipals (o, he agen has all he bagaining powe), ou saionay seing should again lead o consan equilibium incenives/e o po les. Ou assumpion of pincipals having all he bagaining powe eliminaes his epuaion concen, as he agen always ges zeo coninuaion payo fo fuue elaionships. As a esul, we only have he belief manipulaion e ec. Fuhe, his implies ha he cuen sho-em conac is he only souce of vaiaion in he agen s coninuaion value, and hus he sho-em incenive hee has he same inepeaion of in ha of long-em conacing. This allows us o highligh he ole played by commimen. 3 This esul is in conas o Fudenbeg, Holmsom, and Milgom (99), who show ha wih moal hazad, unde CARA pefeences he opimal long-em conac can be implemened by sho-em ones. In a way, hei esul suggess ha commimen iself is no ha impoan. In conas, ou model shows ha he commimen in long-em conacing is impoan because of he long-lasing belief manipulaion e ec wih endogenous leaning.

24 We solve his ODE in closed-fom: whee A d and B d ae consans so ha B d = =, and A d = V d (p) = Ad p + B d p; (46) ( + ) a + + q( + ) a 4 + a ( + ) + + > : We have he following poposiion. Poposiion 3 Wihin he class of deeminisic policies, he value funcion fo he opimal conac is given by equaion (46), and i is opimal o se ime- infomaion en o be p d = p d Bd A d > ; whee p d sais es V d p p d =. The evoluion of infomaion en p is given by p d = p d exp ( ) ; wih + + Ad + a > : In he deeminisic opimal conac, he incenive is given by and he opimal e o policy is Poof. See Appendix A. d (p) = + Ad + a pd = + Ad + a pd exp ( ) ; d = d p d = Ad a + a p d exp ( ) : (47) The above poposiion shows ha in he opimal deeminisic conac, he infomaion en p d, he incenive d, and he opimal e o d all follow ceain exponenially decaying pahs (owad zeo), wih he same ae of. Moeove, a =, fom equaion (47) we have d = Ad a p d + a = a p d < + a + a = HM : Thus, he iniial opimal e o level is below he obsevable po abiliy benchmak, and he opimal e o pah is deceasing ove ime. The opimaliy of he fon-loaded e o policies comes fom he fowad looking naue of infomaion ens. Fom he agen s incenive-compaibiliy condiion in equaion (), he belief manipulaion e ec implies ha giving incenives lae ends o make he agen shik ealie, bu no he ohe way aound. 3 This implies ha lae incenives ae moe

25 cosly han ealy ones, and, consequenly, he opimal conac implemens highe e o in ealie peiods. Clealy, his esul elies on he commimen abiliy in long-em conacing: Secion 5.. shows ha when elaionships ae sho-em so ha each pincipal does no ake he fowad-looking infomaion en ino accoun, he esuling incenives and e o policies ae consan ove ime. 5.. Discussion Infomaion ens and fon-loaded e o policy Boh Pa and Jovanovic () and ou model nd fon-loaded incenives o be opimal. Because Pa and Jovanovic () implemen a consan e o, 4 he fowad looking naue of infomaion ens implies ha he compensaion conac has o o e fon-loaded incenives. Ou model allows he opimal conac o adjus on he e o magin (no jus incenives), and cheape incenive povisions in ealie peiods naually push he opimal conac o implemen a fon-loaded e o po le. The fon-loaded incenive/e o policies also aise in models wih caee concens (e.g., Gibbons and Muphy (99), Holmsom (999)), bu hough a disinc mechanism. Thee, agens in hei ealy caees face highe unceainy in hei abiliies, and, hus, wok hade o impess he make (bu he make will no be fooled in equilibium, a sandad signal jamming poblem). This foce is no pesen in ou saionay model, as he unceainy of he po abiliy/abiliy (i.e., he poseio vaiance of ) says consan ove ime. Pehaps moe ineesingly, Gae and Pavan () sudy a dynamic conacing poblem wih pe-conacing asymmeic infomaion. Unde he assumpion ha he e ec of an agen s iniial poduciviy on his fuue poduciviy is declining ove ime, hey nd ha he opimal e o policy is ime-inceasing. This esul is opposie o he fon-loaded e o policy deived in ou pape wihou pe-conacing pivae infomaion (bu wih leaning). Inuiively, in hei seing, facing infomaion asymmey, he equilibium e o is downwad disoed fo en exacion. The downwad disoion esuling fom infomaion asymmey is moe sevee in ealie peiods. Compaed o ou pape whee pos-conacing pivae infomaion is always o -equilibium, in Gae and Pavan () he pe-conacing pivae infomaion occus on-equilibium. I is iniguing ha hese wo ypes of pivae infomaion have opposie pedicions on he ime-seies paen of disoion. On he ohe hand, he di eence also lies on he agen being isk-neual wihou wealh consain in Gae and Pavan (). 5 Relaedly, Sannikov () allows fo he possibiliy of he 4 Pa and Jovanovic () assume ha he e o cos is linea ove he feasible ineval [; ] and focus on implemening he highes e o level. A simila sucue is assumed in DeMazo and Sannikov (). 5 In Gae and Pavan (), since he agen is isk neual and has enough wealh (o, no limied liabiliy), wihou pe-conacing pivae infomaion he pincipal can sell he pojec o he isk-neual 4