DSGE模型讨论之六——新古典增长模型(入门级DSGE)的推导和Dynare模拟

NeoclassicalMonetaryModelWeijieChenDepartmentofPoliticalandEconomicStudiesUniversityofHelsinkiUpdated3Jan,20120102030051015202530−500−400−300−200−1000AbstractThismodelisthesiblingversionofneoclassicalgrowthmodel,thesemod-elsformthefoundationof`NewNeoclassicalSynthesis'.ThisnotehighlyextractfromGal'stextbook:MonetaryPolicy,Ination,andtheBusinessCycle.Thepurposeofthisnoteistoreproduceallomittedderivationstepsandhighlightthekeyeconomicideasscatteringamonghispresentation.AndprovideDynarecodeforthesimulationofthemodel.1IntroductionWhatweareabouttoseeisthebaselinemodelforNew-Keynesianschoolanditsextensionofmoney-in-utility(MIU).2HouseholdsTherepresentativehouseholdseekstomaximiseherlifetimeutilityfunctionE01Xt=0tC1t1N1+'t1+'(1)whereisinverseelasticityofintertemporalsubstitution,'isinverseelas-ticityoflaboursupplytorealwage.Subjecttodynamicbudgetconstraint,PtCt+QtBtBt1+WtNtTt(2)Bt1denotesone-periodrisklessbonds,purchasedinperiodt1,outstand-inginperiodt.Theright-handsideofbudgetconstraintmeans,labourincomeWtNtplusoutstandingbondpurchasedatt1,thensubtractlump-sumtransfer(canbepositiveornegative).Theleft-handsideshowstheconsumptionshouldnotexceedthedisposalincome,householdwillconsumegoodsPtCt,thenbuybondsatpriceQt.Besidse,weassumehouseholdneverbealoaner,thusno-Ponzi-gamecondition,limT!1EtBT0(3)2.1OptimalityConditionsTheoptimalconsumptionpathcanbefoundbydynamicprogrammingorLagrangian,howeverbotharehighlyunnecessary.Theeasiestwayispre-sentedbyGal'sbook,bytakingtotaldierentialtooptimisedutilityfunc-tion,UCdCt+UNdNt=0(4)1whereUCispartialderivativewithrespecttoCt,orequivalentlyUC;t.Weassumetheutilityhasalreadybeenoptimised,thenanydepartureofCtandNttogetherwillremainontheoptimalpath.Theeconomicmeaningisthatincreaseofconsumptioninducestheincreaseofworkinghours1.Rearrange(4),UNUC=dCtdNt(5)Thenrearrangethebudgetconstraint,Ct=1Pt(QtBt+Bt1+WtNtTt)TakederivativewithrespecttoNt,dCtdNt=WtPt(6)Combine(5)and(6),wegetintratemporaloptimalitycondition,UNUC=WtPt(7)TogureouttheexplicitformofUNandUC,@U@C=Ct@U@N=N'Substitutebackto(7),wegetWtPt=CtN't(8)AnotheroptimalityconditionisEulerequation,whichisalsonamed`in-tertemporaloptimalitycondition'.Eulerequationservesasascaletobal-anceeachsubsequentconsumptionpairtoengineertheoptimalconsumptionpath,itfunctionsasifallotherperiodsareheldstillexceptfortandt+12,UC;tdCt+Et[UC;t+1dCt+1]=01BecauseweknowUN0,thenequationcanhold.2Youcanchooseanytwosubsequentperiod,suchast98andt99.2whichisthersthalfofEulerequation,itmeanstodecreasetheconsump-tionattinducesanincreaseofconsumptiont+1onanoptimaltimepath.Rearrange,yieldsEtUC;t+1UC;tdCt+1dCt=1(9)Thesecondhalfrequiressomeslightmanipulationofdynamicbudgetconstraint,moreoneperiodforwardsandrearrange,Pt+1Ct+1+Qt+1Bt+1=Bt+Wt+1Nt+1Tt+1Bt=Pt+1Ct+1+Qt+1Bt+1Wt+1Nt+1+Tt+1(10)Substitute(10)intobudgetconstraint(2),PtCt+Qt(Pt+1Ct+1+Qt+1Bt+1Wt+1Nt+1+Tt+1)=Bt1+WtNtTtSeparateCtononeside,Ct=QtPt(Pt+1Ct+1+Qt+1Bt+1Wt+1Nt+1+Tt+1)+Bt1+WtNtTtTakepartialderivativewithrespecttoCt+1,allresttermsvanish,@Ct@Ct+1=QtPt+1Pt(11)Orequivalently,@Ct+1@Ct=PtQtPt+1Inordertofullyspecify(9),weneedtoknowUC;t+1UC;t+1(Ct+1;Nt+1)=Ct+1(12)Substitute(11)and(12)backtothe(9),EtCt+1CtPtQtPt+1=13Rearrange,thenalformofEulerequation,EtCt+1CtPtPt+1=Qt(13)Toproceed,welog-linearisebothoptimalitycondition,wtpt=ct+'nt(14)ct=Etct+11(itEtt+1)(15)whereilnQand=ln.IfwedeneQt=(1+i)1,ln(1+i)1=ln(1+i)=lnQtidenotesnominalinterestrate.3FirmsTherepresentativermemploysproductionfunctionYt=AtN1t(16)orinlog-linearterms,yt=at+(1)ntFirmsseekstomaximisetheprotsateveryperiod,PtYtWtNt(17)Substitute(16)into(2),thentakeF.O.C.withrespecttoNt,(1)AtNt=WtPtwhichmeansthemarginalproduct(left-handside)equalstherealwage(right-handside).Log-linearisedform,wtpt=atnt+ln(1)(18)4Notethatwearepresentinganeoclassicalmodel,soperfectcompetitionmakesallrmsprice-takers.Besides,weneedtocharacterisethestochasticsoftechnology,wedenelog(At)=at,thenat=aat1+at(19)wherea2(0;1)andatiid(0;a).4InterestruleHerewesimplyusetheTaylorrule,i=++yyInterestrateisadjustedbyinationandoutputgap.5EquilibriumWeonlyassumegoodmarketclearinthismodel,thusyt=ct(20)Supplyalwaysequalsdemand,goodmarketalwaysclearswhichleavesnoroomformonetarypolicy.Thusinsummery,wecharacterisetheequilibriumofthemodelbyfol-lowingequation:wtpt=ct+'ntyt=Etyt+11(itEtt+1)wtpt=atnt+ln(1)yt=at+(1)ntat=aat1+ati=++yy5Thesecondequationusestheidentityyt=ct.6DynarecodeandexpositionThedynarecodeisasfollowing:%%VARIABLEDECLERATIONSvarynipiac;varexoepsilon_a;%%PARAMETERDECLARATIONSparameterssigmaphiphi_yphi_pirhoalpharho_a;%%INITIALPARAMETERCALIBRATIONsigma=5;phi=2;rho=0.9;alpha=0.5;rho_a=0.7;phi_pi=1.5;phi_y=1.1;%%Modelmodel;/*1*/y=y(+1)-(1/sigma)*(i-pi(+1)-rho);/*Eulerequation,dynamicIScurve*/%/*2*/%w-p=sigma*c+phi*c;/*Intratemporaloptimalitycondition*/6%/*3*/%w-p=a-alpha*n+log(1-alpha);/*Labourmarketclearingcondition*//*3.5*/sigma*c+phi*c=a-alpha*n+log(1-alpha);/*Combineequatio