Solution-for-Assignment-9

SolutionforAssignment91JackandJillbothobeythetwo-periodFishermodelofconsumption.Jackearns$100inthefirstperiodand$100inthesecondperiod.Jillearnsnothinginthefirstperiodand$210inthesecondperiod.Bothofthemcanborroworlendattheinterestrater.a)YouobservebothJackandJillconsuming$100inthefirstperiodand$100inthesecondperiod.Whatistheinterestrater?WecanuseJill'sintertemporalbudgetconstrainttosolvefortheinterestrate:Jillborrowed$100forconsumptioninthefirstperiodandinthesecondperiodusedher$210incometopay$110ontheloan(principalplusinterest)and$100forconsumptionb)Supposetheinterestrateincreases.WhatwillhappentoJack’sconsumptioninthefirstperiod?IsJackbetterofforworseoffthanbeforetheinterestraterise?TheriseininterestratesleadsJacktoconsumelesstodayandmoretomorrow.Thisisbecauseofthesubstitutioneffect:itcostshimmoretoconsumetodaythantomorrow,becauseofthehigheropportunitycostintermsofforgoneinterest.ThisisshowninthefigurebelowByrevealedpreferenceweknowJackisbetteroff:atthenewinterestratehecouldstillconsume$100ineachperiod,sotheonlyreasonhewouldchangehisconsumptionpatternisifthechangemakeshimbetteroff.c)WhatwillhappentoJill’sconsumptioninthefirstperiodwhentheinterestrateincreases?IsJillbetterofforworseoffthanbeforetheinterestrateincrease?Jillconsumeslesstoday,whileherconsumptiontomorrowcaneitherriseorfall.Shefacesbothasubstitutioneffectandincomeeffect.Becauseconsumptiontodayismoreexpensive,shesubstitutesoutofit.Also,sinceallherincomeisinthesecondperiod,thehigherinterestrateraiseshercostofborrowingand,thus,lowersherincome.Assumingconsumptioninperiodoneisanormalgood,thisprovidesanadditionalincentiveforloweringit.HernewconsumptionchoiceisatpointBinthefigurebelowWeknowJillisworseoffwiththehigherinterestratesbecauseshecouldhaveconsumedatpointBbefore(bynotspendingallofhersecond-periodmoney)butchosenottobecausepointAhadhigherutility.2.Thisproblemrequiresuseofcalculustosolvesomeconsumeroptimizationproblems.a)Ninahasthefollowingutilityfunction:𝑈�=�ln(𝐶�1)�+�ln(𝐶�2)�+�ln(𝐶�3)Shestartswithwealthof$120,000,earnsnoadditionalincome,andfacesazerointerestrate.Howmuchdoessheconsumeineachofthethreeperiods?(Hint:Themarginalrateofsubstitutionbetweenconsumptioninanytwoperiodsistheratioofmarginalutilities.)AccordingtoFisher'smodel,consumersallocatetheirincomeacrosstimeperiodssothatthemarginalrateofsubstitutionbetweenconsumptioninanytwoperiodsisequalto1+r,whereristherealinterestrate.Inthisproblem,therealinterestrateiszero.Themarginalrateofsubstitutionistheratioofthemarginalutilitiesinanytwoperiods.Tofindthemarginalutility,differentiatetheutilityfunctionwithrespecttoCitofindMUi=1/Ci.Fortimeperiods1and2,wefindFortimeperiods2and3,wefindTherefore,C1=C2=C3=$40,000.b)DavidisjustlikeNina,excepthealwaysgetsextrautilityfrompresentconsumption.Fromtheperspectiveofperiodone,hisutilityfunctionis𝑈�=�2ln(𝐶�1)�+�ln(𝐶�2)�+�ln(𝐶�3)Inperiodone,howmuchdoesDaviddecidetoconsumeineachofthethreeperiods?Howmuchwealthdoeshehaveleftafterperiodone?Davidalsosetshismarginalrateofsubstitutionbetweenanytwoperiodsequalto1.Fortimeperiods1and2,wefindFortimeperiods2and3,wefindWealsoknowC1+C2+C3=$120,000.SubstituteinforC1andC3fromtheprecedingequationstofind2C2+C2+C2=$120,000,suchthatC1=$60,000andC2=C3=$30,000.Afterperiod1,Davidhas$60,000inwealth.c)WhenDavidentersperiodtwo,hisutilityfunctionis:𝑈�=�ln(𝐶�1)�+�2ln(𝐶�2)�+�ln(𝐶�3)Howmuchdoesheconsumeinperiodstwoandthree?HowdoesyouranswerherecomparetoDavid’sdecisioninpart(b)?Inperiod2,Davidnowgetstwiceasmuchutilityasinperiod3.Followingthesameprocessasinthepreceding,wefindC2=2C3,suchthatDavidwillconsume$40,000inperiod2and$20,000inperiod3.Davidhasrevisedhisdecisionfromperiod1becausehevaluespresentconsumptiontwiceashighasfutureconsumption.d)If,inperiodone,Davidwereabletoconstrainthechoiceshecanmakeinperiodtwo,whatwouldhedo?Relatethisexampletooneofthetheoriesofconsumptiondiscussedinthechapter.IfDavidcouldconstrainhischoicesinperiod2,hewouldprefertoconsume$30,000inperiod2and$30,000inperiod3.Givenhisutilityfunction,hepreferstoconsume$60,000inyear1and$30,000ineachofthetwonextyears.David'spreferencesareanexampleofLaibson'spullofinstantgratificationmodel.Davidmayknowheisanimperfectdecisionmaker,sohemayprefertoconstrainhisfuturedecisions.