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

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1、NeoclassicalMonetaryModel_WeijieChenDepartmentofPoliticalandEconomicStudiesUniversityofHelsinkiUpdated3Jan,20120、-100、200、300、-400、-50030200101015202530AbstractThismodelisthesiblingversionofneoclassicalgrowthmocleLthesemodelsformthefoimclationofNewNeoclassicalSynthesis1.ThisnotehighlyextractfromGali

2、nstextbook:MonetaryPolicy.Inflation,andtheBusinessCycle.Thepurposeofthisnoteistoreproduceallomittedderivationstepsandhighlightthekeyeconomicideasscatteringamonghispresentation.AndprovideDynarecodeforthesimulationofthemodel1 IntroductionWhatweareabouttoseeisthebaselinemodelforNew-Keynesianschoolandit

3、sextensionofmonej-in-utility(MIU).2 HouseholdsTherepresentativehouseholdseekstomaximiseherlifetimeutilityfunction1+?whereaisinverseelasticityofintertemporalsubstitution,卩isinverseelasticityoflaboursupplytorealwageSubjecttodynamicbudgetconstraint,PG+QtBt0(3)Ttoo2.1OptimalityConditionsTheoptimalconsum

4、ptionpathcanbefoundbydynamicprogrammingorLagrangian,howeverbotharehighlyunnecessary.TheeasiestwayispresentedbyGalisbookbytakingtotaldifferentialtooptimisedutilityfunction,UcdCt+UNdNt=0whereUcispartialderivativewithrespecttoCt,orequivalentlyUc,tWeassumetheutilityhasalreadybeenoptimisedthenanydepartur

5、eofCtandNttogetherwillremainontheoptimalpathTheeconomicmeaningisthatincreaseofconsumptioninducestheincreaseofworkingliouriQRearrange砂，UndCt$、呢FThenrearrangethebudgetconstraint,Ct=君(-Q旧+民一1+WtM-Tt)TakederivativewithrespecttoNt,(6)dCt_Wt丙=耳Combine阿and砂，wegetintratemporaloptimalitycondition,Tofigureout

6、theexplicitformofUnandUc,dudeduonSubstitutebacktoQ,wegetAnotheroptimalityconditionisEulerequation,whichisalsonamedintertemporaloptimalitycondition*.Eulerequationservesasascaletobalanceeachsubsequentconsumptionpairtoengineertheoptimalconsumptionpathitfunctionsasifallotherperiodsareheldstillexceptfort

7、andt+10Uc、tdCt+t3EtUc.t+idCt+i=0BecauseweknowUn0.thenequationcanhold.Youcanchooseanytwosubsequentperiod,suchasZ98andt99.whichisthefirsthalfofEulerequation,itmeanstodecreasetheconsumptionattinducesanincreaseofconsumptiont+1onanoptimaltimepath.Rearrange,yields(9)-?EtThesecondhalfrequiressomeslightmani

8、pulationofdynamicbudgetconstraint,moreoneperiodforwardsanclrearrange,R+iG+i+Qt+it+i=Bt+Wt+lNt+i-Tt+iEt=R+G+i+Qt+iD+iWt+iM+i+Tt+i(10)Substitute何fintobudgetconstraint越，PG+QtPt+CM+QmBm一Wt+1Nt+i+Tt+i)=D_i+WtNt一TtSeparateCtononeside,Ct=一计(R+G+i+Qt+iBt+iWt+iNt+i+7t+i)+Bt_+WtNtTtTakepartialderivativewithre

9、specttoallresttermsvanish(11)(12)OS_QtPt+idCPTOrequivalently,dCt+i_PtOCt=QtPi+iInordertofullyspecify曾，weneedtoknowUct+i%t+i+i,M+i)=c灵Substitute也Jand(|12|backtothe越，QtPt+iJ,3Et=110Rearrange,thefinalformofEulerequation,(13)Toproceed,welog-linearisebothoptimalitycondition.wt-pt=ct+pnt(14)ct=EtS+i-(it-p

10、)(15)(TwhereiQInQandp=In/?IfwedefineQt=(1+In(1-I-0_1=-In(1+i)=InQtidenotesnominalinterestrate3 FirmsTherepresentativefirmemploysproductionfunctionX=AtN严(16)orinlog-linearterms./=仇+(1a)ntFirmsseekstomaximisetheprofitsateveryperiod,PtVt-WtNt(17)Substituteinto倉，thentakeF.O.C.withrespecttoNt,(1-QAtNp=兽w

11、hichmeansthemarginalproduct(left-handside)equalstherealwage(right-handside).Log-linearisedform.wtpt=atant+In(1a)(18)Notethatwearepresentinganeoclassicalmodel,soperfectcompetitionmakesallfirmsprice-takers.Besides,weneedtocharacterisethestochasticsoftechnology,wedefinelog(At)=at,then（wherepa6(0.1)and?

12、iid0,aa).4 InterestruleHerewesimplyusetheTaylorrule,i=p+。开兀+(t)yyInterestrateisadjustedbyinflationandoutputgap5 Equilibrium(20)WeonlyassumegoodmarketclearinthismocleLthusyt=ctSupplyalwaysequalsdemancLgoodmarketalwaysclearswhichleavesnoroomformonetarypolicy.Thusinsummery,wecharacterisetheequilibriumo

13、fthemodelbyfollowingequation:w；一pt=act+甲ntyt=Etyt+i-(it-Et7rt+i-p)wtpt=atant+In(1ci)Vt=血+(1-Q)弘(it=padt-l+；Thesecondequationusestheidentityyt=ct.6 DynarecodeandexpositionThedynarecodeisasfollowing:7.7. VARIABLEDECLERATIONSvarynipiac;varexoepsilon.a;7.7. PARAMETERDECLARATIONSparameterssigmaphiphi_yph

14、i_pirhoalpharho_a;INITIALPARAMETERCALIBRATIONsigma=5;phi=2;rho=0.9;alpha=05;rho_a=07;phi_pi=1.5;phi_y=l.l;Modelmodel;/*1*/y=y(+1)-(1/sigma)*(i-pi(+l)-rho);/*Eulerequation,dynamicIScurve*/%/*2*/%w-p=sigma*c+phi*c;/*Intratemporaloptimalitycondition*/%/*3*/%w-p=a-alpha*n+log(1-alpha);/Labourmarketclear

15、ingcondition*/*35*/sigma*c+phi*c=a-alpha*n+log(1-alpha);/Combineequation2and3toreducerank*/*4*/y=c;/Goodsmarketequilibrium*/*5*/y=a+(1-alpha)*n;/*Loglinearisedproductionfunction*/*6*/a=rho_a*a(-l)+epsilon_a;/*AR(1)processoftechnology*/*7*/i=rho+phipi*pi+phi_y*y;/Evolutionofinterestrate*/%Initialvalu

16、esettinginitval;%w=0;/Allinitialvaluesare0,log-deviationsteady-state%p=0；y=o；n=0;i=0;pi=0;a=0;end;steady;/Checksteady-stateandsimulatebeginfromsteady-statecheck;/CheckBKconditionshocks;varepsilon.a=0.009J;/Specifytechshocksend;stoch.simul(order=l,periods=500,irf=20,aim.solver);/order=lmeanslinearmod

17、el,irf=20means40periodsforIRFs,aim.solverisAnderson-MoorealgorithmNextwewillgothroughtheresultsofDynareindetailandgivefurtherexplanationThissectionshallbestudiedthroughcarefullyandtrytomodifythemodelandparameterstogainexperienceConfiguringDynaremexGeneralizedQZmexSylvesterequationsolution.mexKroneck

18、erproductsmexSparsekroneckerproductsmexBytecodeevaluationmexk-orderperturbationsolvermexk-ordersolutionsimulation.TheseareautomaticconfigurationwhichMatlabwillperformwhenloadingDynarefunctions.StartingDynare(version4.2.4)StartingpreprocessingofthemodelfileFound6equation(s)EvaluatingexpressionsdoneCo

19、mputingstaticmodelderivatives:-order1Computingdynamicmodelderivatives:-order1ProcessingoutputsdonePreprocessingcompletedStartingMATLAB/OctavecomputingThesecondtothefourthlinetellthatDvnareistranslatingmodellanguageintomachinelanguageStaticmodelmeansmodelinsteady-state,derivativesareJocobianmatrices.

20、STEADY-STATERESULTS:y-0.0866434n-0.173287i1.09062pi0.190615a0c-0.0866434Thesearesteady-statevaluesofallendogenousvariables,whichisevaluatedbyrecursiveblockpartitionmethod(defaultoption).Asyoucanseey=c=0.0866434,sincethesteady-statevalueisnotfarawayfromourinitialvaluessetting,thusnodifficultyforDynat

21、etocalculatenumerically.EIGENVALUES:ModulusRealImaginary0.70.701.51.5InfInf0Thereare2eigenvalue(s)largerthan1inmodulusfor2forward-lookingvariable(s)TherankconditionisverifiedThisisBlanchard-Kahnconditionchecking,thesufficient,andnecessaryconditionforlinearrationalexpectationmodeltobesolvedonsaddle-p

22、athInfmeansinfinite,dynarewilltreatitaslargerthanoneThereareonly3eigenvalues,butwehave6equations,soquestioniswheredorestofequationsgo?Tounderstandthisquestionpi7V+yyTherearenodynamicsinthereidentities,whichmeanstheyarenotdifferenceequations,thusinordersolvethewholesystemweneedtosubstitutethemintores

23、tofthreeequationstoreducerankThisiswhywehaveonlythreeeigenvalueshereMODELSUMMARYNumberofvariables:6Numberofstochasticshocks:1Numberofstatevariables:15Numberofjumpers:Numberofstaticvariables:Summarisationofthemodel,twojumpvariablesareytand矶，staticvariblesarect,ntandandonestatevariablewhichisanexogeno

24、usprocessandmeanwhileanendogenousvariableat.Oneshockissimplethetechnologyshock%MATRIXOFCOVARIANCEOFEXOGENOUSSHOCKSVariablesepsilon.aepsilon.a0.000081Sinceweonlyhaveonlyexogenousshocks,nocomparisonwithothershocks,thecovariancematrixismerelyascalarPOLICYANDTRANSITIONFUNCTIONSPiConstant-0.086643-0.1732

25、871.0906150.190615-0.086643a(-l)0.175000-1.050000-0.477292-0.4465280.7000000.175000epsilon.a0.250000-1.500000-0.821528-0.7310191.0000000.250000HereisthenumericalsolutionofDSGEmodel-policyfunction,whichtellshowsixendogenousvariablesmovealongthesaddle-pathtogether.Mathematically,yt-0.0870.1750.250nt-0

26、.173-1.050-1.500it1.090+t-i-0.478+Cfl-0.821矶0.190-0.446-0.731at00.7001.0000.-0.0870.175.0.250_ThepurposeofsolvinglinearisedDSGEmodelistofindsuchamatrixsystemwhichdescribeshowallendogenousvariablesmovetogether.Andofmostimportancetoremember,theyaredescribedbythepastexogenousprocessaandexogenousshocks%

27、Sincethereisonlyoneshock,thusthesolutionsystemisunivariate,ifwehavetwoshocksweshallseethatsolutionisaVAR(l)modelThisiskeyideaofimpulseresponseandDSGE-VARThesamplemomentsarepresentedinrestofreport,veryeasytoread.NotnecessaryformetoreproducethemhereThefilialresultisimpulseresponsefunctionsflredhorizon

28、tallinemeanssteady-state,inthelongrunendogenousvariableswillreturntosteadystateafteronestandarddeviationshockup.Andverticalaxismeansthepercentagedeviationfromsteady-state.Aswecanseethatasuddenincreaseintechnologywillboostoutputaboveitssteady-stateby0.002percentAnditwillalsoshinklaboursupplyby0.01per

29、centButinflationdropswhichdoesnotmatchtherealworldobservations.7 SummaryInthisnote,wehavewalkedthroughallprocessofsimulatingaclassicalmonetarymodel,whichisaepitomeofpracticalDSGEmodel.MysuggestionisthatyoustudythedynamicsofthemodelbymodifyingallpartsxW3ynFigure1:ImpulseresponsefunctionsxW3ofthemodel

30、toseehowtheresultwillchange,thisisexactlytheideaofcalibration.hereweonlyhaveaninitialcalibration,whichusuallydoesnotapplyinrealresearch.Calibrationneedstobecloneinatrial-and-errormanner,butwithsomesystematicinsightyoucoulddoitquickly,butthisexperienceisbestgainedbyyourownoperation8 AppendixILog-line

31、arisationToshowhowtolog-linearise爭=Cf、wefirstuseUhligsmethodtoreplaceallvariables.Removethestationarycondition.First-orderTaylorexpansion,Wt-pt=(TCt+Tolog-linearisetheEulerequation.Removestationarycondition,Collectingterms.1=1+几一a(EtCt+1-ct)+pt-Etpt+iUseEtpt+ipt=7rt,rearrangewehavect=Etc中-(it-Em+ip)