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1、Chapter17Self-Insurance17.1.IntroductionThischapterdescribesaversionofwhatissometimescalledasavingsproblem(e.g.,ChamberlainandWilson,2000).Aconsumerwantstomaximizetheexpecteddiscountedsumofaconcavefunctionofone-periodconsumptionrates,asinchapter8.However,theconsumeriscuto?fromallinsurancemark
2、etsandalmostallassetmarkets.Theconsumercanpurchaseonlynonnegativeamountsofasinglerisk-freeasset.Theabsenceofinsuranceopportunitiesinducestheconsumertousevariationsovertimeinhisassetholdingstoacquire“self-insurance.”Thismodelisinterestingtouspartlyasabenchmarktocomparewiththecompletemarketsmod
3、elofchapter8andsomeoftherecursivecontractsmodelsofchapters20and21,whereinformationandenforcementproblemsrestrictallocationsrelativetochapter8,butneverthelesspermitmoreinsurancethanisallowedinthischapter.Aversionofthesingle-agentmodelofthischapterwillalsobeanimportantcomponentoftheincompletema
4、rketsmodelsofchapter18.Finally,thechapterprovidesour?rstencounterwiththepowerfulsupermartingaleconvergencetheorem.Tohighlightthee?ectsofuncertaintyandborrowingconstraints,weshallstudyversionsofthesavingsproblemunderalternativeassumptionsaboutthestringencyoftheborrowingconstraintandaboutwhethe
5、rthehousehold’sen-dowmentstreamisknownoruncertain.–699–700Self-Insurance17.2.Theconsumer’senvironmentAnagentordersconsumptionstreamsaccordingto∞Eβtu(c),(17.2.1)0tt=0whereβ∈(0,1),andu(c)isastrictlyincreasing,strictlyconcave,twicecon-tinuouslydi?erentiablefunctionoftheconsumptionofasinglegoodc
6、.Theagentisendowedwithanin?niterandomsequence{y}∞ofthegood.Eachtt=0period,theendowmenttakesoneofa?nitenumberofvalues,indexedbys∈S.Inparticular,thesetofpossibleendowmentsisy17、Therearenoinsurancemarkets.Theagentcanholdnonnegativeamountsofasinglerisk-freeassetthathasanetrateofreturnr,where(1+r)β=1.Letat≥0betheagent’sassetsatthebeginningofperiodt,includingthecurrentrealizationoftheincomeprocess.(Laterweshalluseanalte
