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AContinuous-TimeVersionofthePrincipal-AgentProblem.YuliySannikovMarch14,2007AbstractThispaperdescribesanewcontinuous-timeprincipal-agentmodel,inwhichtheoutputisadi usionprocesswithdriftdeterminedbytheagent'sunobservede ort.Therisk-averseagentreceivesconsumptioncontinuously.Theoptimalcontract,basedontheagent'scontinuationvalueasastatevariable,iscomputedbyanewmethodusingadi erentialequation.Duringemploymenttheoutputpathstochasticallydrivestheagent'scontinuationvalueuntilitreachesapointthattriggersretirement,quitting,replacementorpromotion.Thepaperexploreshowthedynamicsoftheagent'swagesande ort,aswellastheoptimalmixofshort-termandlong-termincentives,dependonthecontractualenvironment.1Keywords:Principal-agentmodel,continuoustime,optimalcontract,careerpath,retirement,promotionJELNumbers:C63,D82,E2CorrespondenceshouldbesenttoYuliySannikov,UniversityofCaliforniaatBerkeley,DepartmentofEconomics,593EvansHall,Berkeley,CA94720-3880;e-mail:sannikov@econ.berkeley.edu;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 ndtheoptimalcontractbysolvinganordinarydi er-entialequation.Thisequationhighlightsthefactorsthatdetermineoptimalconsumptionande ort.Thedynamicsoftheagent'scareerpathisnaturallydescribedbythedriftandvolatilityoftheagent'spayo s.Thegeometryofsolutionstothedi erentialequationallowsforeasycomparisonstoseehowtheagent'swages,e ortandincentivesdependonthecontractualenvironment.Finally,continuoustimehighlightsmanyessentialfeaturesoftheoptimalcontract,includingtheagent'seventualretirement.Inourbenchmarkmodelarisk-averseagentistiedtoarisk-neutralprincipalforeverafteremploymentstarts.Theagentinuencesoutputbyhiscontinuousunobservablee ortinput.Theprincipalseesonlytheoutput:aBrownianmotionwithadriftthatdependsontheagent'se ort.Theagentdislikese ortandenjoysconsumption.Weassumethattheagent'sutilityfunctionhastheincomee ect,thatis,astheagent'sincomeincreasesitbecomescostliertocompensatehimfore ort.Also,weassumethattheagent'sutilityofconsumptionisboundedfrombelow.Attime0theprincipalcancommittoanyhistory-dependentcontract.Suchacontractspeci estheagent'sconsumptionateverymomentoftimecontingentontheentirepastoutputpath.Theagentdemandsaninitialreservationutilityfromtheentirecontractin2ordertobegin,andtheprincipalo ersacontractonlyifhecanderiveapositivepro tfromit.Afterwesolveourbenchmarkmodel,weexaminehowtheoptimalcontractchangesiftheagentmayquit,bereplacedorpromoted.Asinrelateddiscrete-timemodels,theoptimalcontractcanbedescribedintermsoftheagent'scontinuationvalueasasinglestatevariable,whichcompletelydeterminestheagent'se ortandconsumption.Afteranyhistoryofoutputtheagent'scontinuationvalueisthetotalfutureexpectedutility.Theagent'svaluedependsonhisfuturewagesande ort.Whileindiscretetimetheoptimalcontractisdescribedbycumbersomefunctionsthatmapcurrentcontinuationvaluesandoutputrealizationsintofuturecontinuationvaluesandconsumption,continuoustimeo ersmorenaturaldescriptorsofemploymentdynamics:thedriftandvolatilityoftheagent'scontinuationvalue.Thevolatilityoftheagent'scontinuationvalueisrelatedtoe ort.Theagenthasincentivestoputhighere ortwhenhisvaluedependsmorestronglyonoutput.Thus,highere ortrequiresahighervolatilityoftheagent'svalue.Theagent'soptimale ortvarieswithhiscontinuationvalue.Todetermineoptimale ort,theprincipalmaximizesexpectedoutputminusthecostsofcompensatingtheagentfore ortandtheriskrequiredbyincentives.Iftheagentisverypatient,sothatincentiveprovisioniscostless,theoptimale ortdecreaseswiththeagent'scontinuationvalueduetotheincomee e