委托代理模型的连续时间版本(Sannikov)

AContinuous-TimeVersionofthePrincipal-AgentProblem.YuliySannikovMarch14,2007AbstractThispaperdescribesanewcontinuous-timeprincipal-agentmodel,inwhichtheoutputisadiusionprocesswithdriftdeterminedbytheagent'sunobservedeort.Therisk-averseagentreceivesconsumptioncontinuously.Theoptimalcontract,basedontheagent'scontinuationvalueasastatevariable,iscomputedbyanewmethodusingadierentialequation.Duringemploymenttheoutputpathstochasticallydrivestheagent'scontinuationvalueuntilitreachesapointthattriggersretirement,quitting,replacementorpromotion.Thepaperexploreshowthedynamicsoftheagent'swagesandeort,aswellastheoptimalmixofshort-termandlong-termincentives,dependonthecontractualenvironment.1Keywords:Principal-agentmodel,continuoustime,optimalcontract,careerpath,retirement,promotionJELNumbers:C63,D82,E2CorrespondenceshouldbesenttoYuliySannikov,UniversityofCaliforniaatBerkeley,DepartmentofEconomics,593EvansHall,Berkeley,CA94720-3880;e-mail:[e-mail];cellphone(650)-303-7419.1IammostthankfultoMichaelHarrisonforhisguidanceandencouragementtodeveloparigorouscontinuoustimemodelfrommyearlyideaandforhelpingmegetvaluablefeedback,andtoAndySkrzypaczforadetailedreviewofthepaperandformanyhelpfulsuggestions.IamalsogratefultoDarrellDue,YossiFeinberg,BengtHolmstrom,ChadJones,GustavoManso,PaulMilgrom,JohnRoberts,ThomasSargent,SergioTurnerandRobertWilsonforvaluablefeedback,andtoSusanAtheyforcommentsonanearlierversion.11Introduction.Theunderstandingofdynamicincentivesiscentralineconomics.Howdocompaniesmoti-vatetheirworkersthroughpiecerates,bonuses,andpromotions?Howisincomeinequalityconnectedwithproductivity,investmentandeconomicgrowth?Howdonancialcontractsandcapitalstructuregiveincentivestothemanagersofacorporation?Themethodsandresultsofthispaperprovideimportantinsightstomanysuchquestions.Thispaperintroducesacontinuous-timeprincipal-agentmodelthatfocusesonthedynamicpropertiesofoptimalincentiveprovision.Weidentifyfactorsthatmaketheagent'swagesincreaseordecreaseovertime.Weexaminethedegreetowhichcurrentandfutureoutcomesmotivatetheagent.Weprovideconditionsunderwhichtheagenteventuallyreachesretirementintheoptimalcontract.Wealsoinvestigatehowthecostsofcreatingincentivesandthedynamicpropertiesoftheoptimalcontractdependonthecontractualenvironment:theagent'soutsideoptions,thedicultyofreplacingtheagent,andtheopportunitiesforpromotion.Ournewdynamicinsightsarepossibleduetotechnicaladvantagesofthecontinuous-timemethodsoverthetraditionaldiscrete-timeones.Continuoustimeleadstoamuchsimplercomputationalproceduretondtheoptimalcontractbysolvinganordinarydier-entialequation.Thisequationhighlightsthefactorsthatdetermineoptimalconsumptionandeort.Thedynamicsoftheagent'scareerpathisnaturallydescribedbythedriftandvolatilityoftheagent'spayos.Thegeometryofsolutionstothedierentialequationallowsforeasycomparisonstoseehowtheagent'swages,eortandincentivesdependonthecontractualenvironment.Finally,continuoustimehighlightsmanyessentialfeaturesoftheoptimalcontract,includingtheagent'seventualretirement.Inourbenchmarkmodelarisk-averseagentistiedtoarisk-neutralprincipalforeverafteremploymentstarts.Theagentinuencesoutputbyhiscontinuousunobservableeortinput.Theprincipalseesonlytheoutput:aBrownianmotionwithadriftthatdependsontheagent'seort.Theagentdislikeseortandenjoysconsumption.Weassumethattheagent'sutilityfunctionhastheincomeeect,thatis,astheagent'sincomeincreasesitbecomescostliertocompensatehimforeort.Also,weassumethattheagent'sutilityofconsumptionisboundedfrombelow.Attime0theprincipalcancommittoanyhistory-dependentcontract.Suchacontractspeciestheagent'sconsumptionateverymomentoftimecontingentontheentirepastoutputpath.Theagentdemandsaninitialreservationutilityfromtheentirecontractin2ordertobegin,andtheprincipaloersacontractonlyifhecanderiveapositiveprotfromit.Afterwesolveourbenchmarkmodel,weexaminehowtheoptimalcontractchangesiftheagentmayquit,bereplacedorpromoted.Asinrelateddiscrete-timemodels,theoptimalcontractcanbedescribedintermsoftheagent'scontinuationvalueasasinglestatevariable,whichcompletelydeterminestheagent'seortandconsumption.Afteranyhistoryofoutputtheagent'scontinuationvalueisthetotalfutureexpectedutility.Theagent'svaluedependsonhisfuturewagesandeort.Whileindiscretetimetheoptimalcontractisdescribedbycumbersomefunctionsthatmapcurrentcontinuationvaluesandoutputrealizationsintofuturecontinuationvaluesandconsumption,continuoustimeoersmorenaturaldescriptorsofemploymentdynamics:thedriftandvolatilityoftheagent'scontinuationvalue.Thevolatilityoftheagent'scontinuationvalueisrelatedtoeort.Theagenthasincentivestoputhighereortwhenhisvaluedependsmorestronglyonoutput.Thus,highereortrequiresahighervolatilityoftheagent'svalue.Theagent'soptimaleortvarieswithhiscontinuationvalue.Todetermineoptimaleort,theprincipalmaximizesexpectedoutputminusthecostsofcompensatingtheagentforeortandtheriskrequiredbyincentives.Iftheagentisverypatient,sothatincentiveprovisioniscostless,theoptimaleortdecreaseswiththeagent'scontinuationvalueduetotheincomeee