内生增长模型AK模型endogrowth

LectureNotesonEndogenousGrowthEcon5140MacroeconomicAnalysisFall2014JennyXuHKUSTInattemptingtobuildamodelwhichcanaccountfortheprincipalfactsconcern-inggrowthexperienceacrosscountries,itwouldseemnecessarytoincorporateanendogenousmechanism,topermiteconomicfactorstodeterminelong-rungrowthrate.InSolowmodel,weknowthatthegrowthrateofoutputpercapitaisdeterminedbytheexogenoustechnologygrowth.However,inthelong-run,thetechnologylevelistheoutcomeofthedecisionsoffirmsandindividuals.Itwewishtounderstandgrowth,wemustmodelthedevelopmentofnewtechnology.1TheSimpleAKmodelInthismodel,weassumethatthereisnoexogenoustechnologygrowthandseeifitispossibletogenerategrowthinthelong-run.1.1TheEconomyProductionFunctionThetotaloutputintheeconomyinAKmodelisgivenbyYt=F(Kt;Lt)=AKt(1.1)1whereA0isanexogenousconstantparameter.Forsimplicity,weassumethatthereisnopopulationgrowthinthiseconomyeither,thatis1Lt=¯L;8t(1.2)Soinintensiveform,wecouldrewritetheproductionfunctionas:yt=f(kt)=Akt(1.3)whereyt=YtLtandkt=KtLt.HouseholdsInthiseconomy,itisassumedthatallthehouseholds(consumers/workers)havethesameutilityfunction,thesamebudgetconstraintandthesamehorizonofliving.Inotherwords,theyarehomogenous.2Sinceallconsumersarealikewecanthinkoftheoneconsumerasrepresenting‘thewholeeconomy’andonlyneedtofocusontherepresentativeconsumer’sproblem.Eachconsumerenterstheeconomywithagivencapitalstockk0.Startingfromperiod0,ineachperiod,therepresentativehouseholdwilldecidehisconsumptionandhisinvestmentfornextperiod.3Meanwhile,theconsumerearnswagesfromworkingandrentsfromrentinghiscapitaltothefirm.Notethat,thecapitalownedbyhouseholdsisdepreciatingattherate.Theconsumersderiveutilityfromconsumptionineachperiod.Therefore,therepresentativehousehold’problemisgivenby:(PH)max{ct;kt+1}1t=0∞∑t=0tu(ct)(1.4)1ThepropertyoftheAKmodelwillnotchangeevenifthereexistspositivepopulationgrowthrate.2Recallthatintheoverlappinggenerationmodel,eachgenerationofagentsarebornindifferentperiodandtheyonlylivetwoperiods.SotheagentsinOLGmodelareheterogenous.3Therefore,ktisdeterminedatperiodt1,soitispredeterminedatperiodt.2s.t.ct+kt+1=wt+(1+rt)ktkt(1.5)k00givenwherectistheconsumptionofeachrepresentativeconsumerattimet,u(
)istheinstantaneousutilityfunction,whichgivestheutilityateachperiod.wtisthewagerateandrtisthecapitalrentalrate.Finally,01isthediscountfactor,theloweris,thelessthehouseholdvaluesfutureconsumptionrelativetocurrentconsumption.Solveforthehousehold’sproblem,wecouldgetthefollowingLagrangian:L=∞∑t=0tfu(ct)+t[wt+(1+rt)ktktkt+1ct]g(1.6)wheretisthelagrangianmultiplierassociatedwiththeperiodtbudgetconstraint.Thefirstorderconditionassociatedwithctandkt+1arethengivenby4:@L@ct=0)u′(ct)t=0(1.7)@L@kt+1=0)t+t+1(1+rt+1)=0(1.8)Substituting1.7into1.8,wecangetu′(ct)=u′(ct+1)(1+rt+1)(1.9)ThisiscalledEulerequationorintertemporaloptimalityconditionasitdeter-minedtheoptimalallocationofconsumptionovertime.TheLHSof1.9represents4Noticethatwhensolvingforthefirstorderconditionw.r.tkt+1,youshouldconsiderbothperiodtandperiodt+1terms.3themarginalcostofsavingintermsofutilityatperiodt.Whentheconsumersaveonemoreunitofgoodsatperiodt,hehastogiveuponemoreunitsofgoodsinconsumption,whichwilldecreasehisutilitybytheamountu′(ct).TheRHSof1.9givesthemarginalbenefitsofsaving,alsointermofutilityatperiodt.Foreachunitofgoodsavedatperiodt,theconsumerwillget1+rt+1unitsofgoodsatperiodt+1,whichwillincreasetheconsumer’sutilityatperiodt+1byu′(ct+1)(1+rt).Discountedperiodt+1’sutilitybythediscountfactor,wegettheRHSof1.9.IftheutilityfunctionisaCRRA(ConstantRelativeRiskAversionCoefficient)utilityfunction,i.e.,iftheutilityatperiodtisgivenby:u(ct)=c1−t1(1.10)WecouldrewritetheEulerequationasct+1ct=[(1+rt+1)]1(1.11)FirmsEachfirminthiseconomytriestochoosehowmanyworkerstohireandhowmuchcapitaltorenttomaximizeitsprofits.Therefore,firm’sproblemcanbegivenby:(PF)maxKt;LtΠt=YtrtKtwtLt(1.12)s.t.Yt=F(Kt;Lt)=AKt(1.13)Solvingforfirm’sproblem,wegetrt=@F(Kt;Lt)@Kt=A(1.14)wt=@F(Kt;Lt)@Lt=0(1.15)4Also,weassumethatthegoodsmarketinthiseconomyareperfectlycompetitive,thusΠt=0)Yt=rtKt+wtLt(1.16)1.2CompetitiveequilibriumThecompetitiveequilibriumforthiseconomyisacollectionofsequencesofpricesfrtgandallocationsfct;kt;ytgsuchthat(i)Givenfrtg,thehousehold’sdecisionssolvethehousehold’sintertemporalmaximizationproblem,(PH);(ii)Givenfrtg,thefirm’scapitaldemandsolvesthefirm’sprofitmaximizationproblem,(PF);(iii)Goodsmarketclear(i.e.,demandequalssupplyinthegoodsmarkets):56ct+kt+1=f(kt)+(1)kt=(1+A)kt(1.17)Therefore,thedecentralizedequilibriumofthiseconomyisdecidedbythefol-lowingequations:ct+1ct=[(1+rt+1)]1rt=A(1.18)ct+kt+1=(1+A)ktOrct+1ct=[(1+A)]15Inthiseconomy,sincelabourisnotrequiredinproduction,thereisnolabourmarketclearingcondition.6Anotherwaytogetthegoodsmarketclearingconditionisusingthehousehold’sbudgetcon-straint1.5andsubstitutetheexpressionforrtandwtintoit.5ct+kt+1=(1+A)kt(1.19)Tosolveforthesolution,weuseamethodcalled“guessandverify”.Sincetotalresource(theRHSof1.17)arelinearink,aneducatedguessisthatoptimalconsumptionandinvestmentareals