DSGE模型讨论之五——CES-functions-and-Dixit-Stiglitz-Formu

CESfunctionsandDixit-StiglitzFormulationWeijieChen1DepartmentofPoliticalandEconomicStudiesUniversityofHelsinki12September,201105100510010203040CapitalLabourCapitalLabour02468100123456789101Anysuggestionandcommentpleasesendemailto:weijie.chen@helsinki.AbstractConstantelasticityofsubstitution(CES)functionsarethemostextensivelyusedfunctionalformineconomicssofar,buttextbooksseldomgivegoodandenoughillustrationsonhowtousethem.ThepurposeofthisnoteistoshowyouhowitcanbederivedanditscontributiontoDixit-Stiglitzformulation.1CESfunctionsWestartwiththemostbasicCESfunction,u(x1;x2)=(x1+x2)1where1.LetusrstverifythisisaCESfunction.Takepartialderivativesw.r.t.bothxi's,@u(x1;x2)@x1=1(x1+x2)11x11=(x1+x2)1x11(1)@u(x1;x2)@x2=1(x1+x2)11x12=(x1+x2)1x12(2)Thenrecallhowwedenepriceelasticityofdamand,p=
x=x
p=pMostoftime,weworkwithcontinuouscase,p=dx=xdp=p=dxdppxWeassumeonlyonegoodinthiscase.Thepriceelasticityofsubstitutionlookssimilar,butalittlebitcom-plicated,sinceitisaboutsubstitution,weneedatleasttwogoods,sayx1andx2.Thisisaveryimportantconceptincomparativestatics,denedas:therelativechangeoftheratiooftheconsumptiongoodsx1andx2overtherelativechangeoftheratiooftheaccordingpricep1andp2,sub=d(x1=x2)x1=x2d(p2=p1)p2=p1Weneedtoshowthecaseattheoptimum,thusthepoint-elasticityattheoptimumis,sub=d(x1=x2)x1=x2d(p2=p1)p2=p1=d(x1=x2)x1=x2d(u2=u1)u2=u1=dx1x2
dp2p1

p2p1x1x2(3)1whereu1=@u=@x1,u2=@u=@x2.Attheoptimum,p2p1=u2u1Use(1)and(2),u2u1=(x1+x2)1x12(x1+x2)1x11=x2x11Thusx2x11=p2p1x1x2=p2p111(4)Wecanuse(4)tocalculate,dx1x2
dp2p1
=11p2p11(5)Thenaccordingto(3)weneedp2=p1x1=x2=p2p1p2p1
11=p2p1111=p2p121(6)Multiply(5)and(6),following(3)dx1x2
dp1p2
p1p2x1x2=11p1p221p1p221=11WehaveverieditisCESfunction,theelasticityofsubstitutionis1=(1)whichisaconstant.1.1GeneralCaseofCESThemostgeneralfunctionalformofCESisgivenbyF=A11x11+12x1212whereAisaconstantorastochasticprocess(cf.technologyparameterAinCobb-Douglasproductionfunction),denotesdistributionparameterastheexponentinC-Dfunction,besidesweassume1+2=1.Inthiscase,isexactlytheelasticityofsubstitutionparameteraswehaveseeninrstcaseassub.Thepurposeoftheouterexponent1isdesignedtorenderthefunctionasrst-orderhomogeneous(linearhomogeneous).Wecanshowthisasfollows:F=A11(tx1)1+12(tx2)11=At111x11+12x121=tA11x11+12x121Astherstcase,wedenetheelasticityofsubstitutionatoptimum(point-elasticity)assub=d(x1=x2)x1=x2d(p2=p1)p2=p1=d(x1=x2)x1=x2d(F2=F1)F2=F1=dx1x2
dp2p1

p2p1x1x2(7)Takepartialderivativew.r.t.x1andx2,F1=A111x11+12x1211111x111=A11x1111x11+12x1211=A11x11F1=A11Fx11(8)Similarly,F2=A12Fx21(9)Divide(9)by(8),F2F1=211x1x21=p2p1(10)3Nextweshallisolatex1x2ononesideatpointofoptimum,21x1x2=p2p1x1x2=p2p112(11)Togetp2p1=x1x2,wemodify(11),x1x2=p2p1p2p1112x1x2p2p1=p2p1112p2p1x1x2=p2p1121(12)From(11),wegetdx1x2
dp2p1
=p2p1112(13)Accordingto(7),wemultiply(13)and(12),p2p1112p2p1121=ThuswehaveconrmedthepropertyofCESfunction.Thelargerthethegreatthesubstitutability,if=0,bothproductshavetobeinaxedproportion,suchasrightandleftshoes.1.2Cobb-DouglasFunctionAsTheSpecialCaseWewillseeinthissectionthatCobb-DouglasfunctionissimplyalimitingversionofCESfunction,whichrendersC-Dfunctionasaspecialcase.Let'sgobacktothegeneralCESform,F=A11x11+12x121(14)4when!0,(14)willapproachesC-Dfunction.Howeverwecannotsimplyequalto1here,sincethedenominatorofouterexponentisundened.Thus,themostnaturalwaytoproceedistouseL0H^opital0srule.FirstdivideFbyAandtakenaturallog,lnFA=ln11x11+12x121=m()n()(15)Toseewhetheritis00form,weset=1,thedenominatorobviouslyequals0,andnumeratorcanbeshownln(1x01+2x02)=ln1=0WecanperformL0H^opital0sruleto(15),butwenditwillbecomeex-tremelymessyinnotationtohandlethisproblem,sincewehavetousethreetimesderivativeproductruleinanestedmannerandexponentderivativerule.ActuallyitisunnecessarytoworkonsuchageneralfunctiontoderivetheC-Dfunction,wecansimplify(15)abitandkeepitsCESfeatures.Therewillbesomeslightnotationchangeinthissimpliedversion,F=AK+(1)L1(16)asyoucanseewereplacex1andx2bycapitalKandlabourL.AandhavetheircorrespondingmeaninginC-Dfunction,butnotyet.Besides,wecannoticethatwehaveset11=implicitly,if=1then=1.Thusif!0,(16)approachesC-Dfunction.Divide(16)byAthentakenaturallog,lnFA=ln[K+(1)L]=m()n()Toseethe00,simplyplug0intonumerator,ln[+1]=0.Next,useL0H^opital0srulem0()n0()=1K+(1)L[KlnK(1)LlnL]5Thenset=0,wegetlnK+(1)lnL+1=lnK+(1)lnL=lnK+lnL1=lnKL1SincewehavedividedbyAandtakingnaturallog,weneedtoperformtheinverseoperationtorecovertheCESform,thusAelnKL1=AKL1(17)whichisthestandardformofCobb-Douglasfunction.1.3CESProductionFunctionWehavealreadyseentheCESproductionfunctionwithinputcap