MacroModel(中级宏观经济学,香港中文大学)

Chapter5AClosed-EconomyOne-PeriodMacroModelTheModel•Theeconomythatweconsiderisaclosedeconomy,i.e.onethatdoesnottradewithothereconomies.•Besidesrepresentativeconsumerandrepresentativefirm,thereisanadditionalagent:thegovernment.•Government•Theonlyactionofthegovernmentistoimplementfiscalpolicy.Fiscalpolicyreferstothegovernment’schoicesoveritsexpenditures,taxes,transfersandborrowing.TheGovernment•Supposethegovernmentwishestopurchaseagivenquantityofconsumptiongood,G.•Sincethereisonlyoneperiod,thegovernmentcannotborrowtofinanceG.ThusGispaidbytaxingtherepresentativeconsumer.•Thegovernmentmustobservethebalancedbudgetconstraint,G=T.CompetitiveEquilibrium•Competitivereferstothefactthatallconsumersandfirmsareprice-takers.•Equilibriumreferstothestatewhentheactionsofallconsumersandfirmsareconsistent.FormalDefinition:•AcompetitiveequilibriumisasetofendogenousquantitiesC,Ns,Nd,TandY,andanendogenousrealwagew,suchthat,giventheexogenousvariablesG,z,andK,thefollowingaresatisfied:1)CandNssolvestheconsumer’sproblem,maxU(C,h-Ns)C,NssubjecttoC=wNs+-TandC0,Ns0.Inotherwords,CandNsmustsatisfytheFOC,U2(C,h-Ns)-wU1(C,h-Ns)=0.2)Ndsolvesthefirm’sproblemgivenz,Kandw.ThustheFOCissatisfied.wNKzFd,23)Market-clearingconditions:–LabormarketclearingNs=Nd=N4)BalancedBudgetconstraint:G=T.Wehave:GoodsmarketclearingY=C+G•CombiningthetwoFOCs,wehaveU2(C,l)-zF2(K,h-l)U1(C,l)=0.•Since=zF(K,h-l)-wNdandG=T,fromthebudgetconstraintC=zF(K,h-l)-G.•Fromthese,wecansolvefortheequilibriumCandl.GraphicalIllustrationsStep1:Derivetheproductionpossibilitiesfrontier(PPF),whichdescribesthetechnologicalpossibilitiesfortheentireeconomy,intermsoftheproductionofCandl.Step2:PutthePPFtogetherwiththeconsumer’sindifferencecurves,sothatwecananalyzeacompetitiveequilibriuminasinglediagram.ProductionPossibilitiesFrontier•Inequilibrium,wehaveNd=Ns=N=h-l.•OutputisgivenbyY=zF(K,h-l).whichisarelationshipbetweenoutputandleisure.•Y*isthelevelwhenl=0.NMPdNdYdldYslopeProductionPossibilitiesFrontier•Inequilibrium,C=Y-G=zF(K,h-l)-GarelationshipbetweenCandl,giventheexogenousvariablesz,KandG.•ThisisthePPFwhichcapturesthetrade-offbetweenleisureandconsumptiongiventheproductiontechnology.•OnlythepointsonDBareavailable(whereC0).ProductionPossibilitiesFrontier•NegativeoftheslopeofthePPFiscalledthemarginalrateoftransformation,MRTl,Cwhichistherateatwhichonegoodcanbeconvertedintoanother.MRTl,C=MPN=-(SlopeofPPF)•Inequilibrium,MPN=w.Thus,wemusthaveMRTl,C=w.•ThishelpsusdeterminetheequilibriumvaluesC*andl*.CompetitiveEquilibrium•Inequilibrium,C*andl*arechosenbytherepresentativeconsumer.•NotethatADBisthebudgetconstraint.BytheFOCoftheconsumer’sproblemMRSl,C=w.•Hence,inequilibriumMRTl,C=MRSl,C=w.Optimality•Questions:Isthecompetitiveequilibriumefficient?Arethereanyotherwaystoobtainabetteroutcome?•AcompetitiveequilibriumisParetooptimalifthereisnowaytorearrangeproductionortoreallocategoodssothatsomeoneismadebetteroffandnooneismadeworseoff.•Sincethereisonlyoneconsumer,wecanignorehowconsumptiongoodsareallocatedamongconsumers.•Rather,wefocusonhowproductionisarranged.SocialPlanner’sProblem•ConsiderasocialplannerwhorunstherepresentativefirmandchoosesthequantitiesCandlsoastomaximizeconsumer’sutility.•Graphically,thesocialplannerchoosesaconsumptionbundlethatisonthePPFandisonthehighestpossibleindifferencecurvefortheconsumer.SocialPlanner’sProblem•Comparison:–Representativeconsumerfacesalinearorkinkedbudgetconstraint.–SocialplannerfacesaconcavePPF.•TheParetooptimumisatBwheretheequalityholdsMRTl,C=MRSl,C=MPN.SocialPlanner’sProblem•Thesocialplannersolvesthefollowingconstrainedmaximizationproblem:MaxU(C,l)C,lsubjecttoC=zF(K,h-l)-G•LagrangianL=U(C,l)+[zF(K,h-l)-G-C].•First-order(Necessary)conditions:•Combiningthefirsttwoconditions,whichstatesthattheParetooptimumisthepointwheretheindifferencecurveistangenttothePPF.•Notethatwehavealsoderivedthesameconditionforacompetitiveequilibrium.ClClMRTzFUUMRS,212,.0,,0,,,0,221CGlhKzFlhKzFlCUlCUTwoWelfareTheorems•Thefirstfundamentaltheoremofwelfareeconomics–Undercertainconditions,acompetitiveequilibriumisParetooptimal.–“Indivisiblehand”:anunrestricted,freemarketeconomycanproducesociallyoptimaloutcome.•Thesecondfundamentaltheoremofwelfareeconomics–Undercertainconditions,aParetooptimumisacompetitiveequilibrium.•Remark:Paretooptimalityignoresthedistributionissueamongindividualsandisthusanarrowconceptofsocialoptimality.SourcesofSocialInefficienciesAcompetitiveequilibriummaynotbeParetooptimaldueto:•Externalities–Anexternalityisanyactivityforwhichanindividualfirmorconsumerdoesnottakeaccountofallassociatedcostsandbenefits.–Externalitiescanbenegative(e.g.pollution)orpositive(e.g.knowledgespillover).–Therootcauseofanexternalityisthatitistoocostly,ifnotimpossible,tosetupamarkettotradeforthebenefitsandcostsassociatedwiththeexternalities(marketfailure).•DistortingTaxes–SupposeGisfinancedbyaproportionalwageincometaxatratet,ratherthanbyalump-sumtaxT.–Thewageincomeisnoww(1-t)(h-l).Sotheeffectivewageisw(1-t).–TheconsumeroptimizesbysettingMRSl,C=w(1-t),whilethefir