Domar-Mode課件教學教程

DOMARGROWTHMODELTheFrameworkThebasicpremisesoftheDomarmodelareasfollows:1.AnychangeintherateofinvestmentflowperyearI(t)willproduceadualeffect:itwillaffecttheaggregatedemandaswellastheproductivecapacityoftheeconomy.2.ThedemandeffectofachangeinI(t)operatesthroughthemultiplierprocess,assumedtoworkinstantaneously.ThusanincreaseinI(t)willraisetherateofincomeflowperyearY(t)byamultipleoftheincrementinI(t).Themultiplierisk=1/s,wheresstandsforthegiven(constant)marginalpropensitytosave.OntheassumptionthatI(t)istheonly(parametric)expenditureflowthatinfluencestherateofincomeflow,wecanthenstatethat(1)3.Thecapacityeffectofinvestmentistobemeasuredbythechangeintherateofpotentialoutputtheeconomyiscapableofproducing.Assumingaconstantcapacity-capitalratio,wecanwrite(=aconstant)where(theGreekletterkappa)standsforcapacityorpotentialoutputflowperyear,and(theGreekletterrho)denotesthegivencapacity-capitalratio.Thisimplies,ofcourse,thatwithacapitalstockK(t)theeconomyispotentiallycapableofproducinganannualproduct,orincome,amountingtodollars.Notethat,from(theproductionfunction),itfollowsthat,and(2)InDomar'smodel,equilibriumisdefinedtobeasituationinwhichproductivecapacityisfullyutilized.Tohaveequilibriumis,therefore,torequiretheaggregatedemandtobeexactlyequaltothepotentialoutputproducibleinayear;thatis,.Ifwestartinitiallyfromanequilibriumsituation,however,therequirementwillreducetothebalancingoftherespectivechangesincapacityandinaggregatedemand;thatis(3)WhatkindoftimepathofinvestmentI(t)cansatisfythisequilibriumconditionatalltimes?FindingtheSolutionToanswerthisquestion,wefirstsubstitute(1)and(2)intotheequilibriumcondition(3).Theresultisthefollowingdifferentialequation:or(4)Since(4)specifiesadefinitepatternofchangeforI,weshouldbeabletofindtheequilibrium(orrequired)investmentpathfromit.Inthissimplecase,thesolutionisobtainablebydirectlyintegratingbothsidesofthesecondequationin(4)withrespecttot.Thefactthatthetwosidesareidenticalinequilibriumassurestheequalityoftheirintegrals.Thus,Bythesubstitutionruleandthelogrule,theleftsidegivesusWhereastherightsideyields(beingaconstant)Equatingthetwosidesandcombiningthetwoconstants,wehave(5)Toobtain|I|fromln|I|,weperformanoperationknownas“takingtheantilogofln|I|,”whichutilizesthefactthat .Thus,lettingeachsideof(5)becometheexponentoftheconstant,e,weobtainorwhereIfwetakeinvestmenttobepositive,then|I|=I,sothattheaboveresultbecomes,whereAisarbitrary.Togetridofthisarbitraryconstant,wesett=0intheequation,togetThisdefinitizestheconstantA,andenablesustoexpressthesolution--therequiredinvestmentpath--aswhereI(0)denotestheinitialrateofinvestment.(6)Thisresulthasadisquietingeconomicmeaning.Inordertomaintainthebalancebetweencapacityanddemandovertime,therateofinvestmentflowmustgrowpreciselyattheexponentialrateof,alongapathsuchasillustratedinFig1.Obviously,thelargerwillbetherequiredrateofgrowthofinvestment,thelargerthecapacity-capitalratioandthemarginalpropensitytosavehappentobe.Butatanyrate,oncethevaluesofandsareknown,therequiredgrowthpathofinvestmentbecomesveryrigidlyset.TheRazor'sEdgeItnowbecomesrelevanttoaskwhatwillhappeniftheactualrateofgrowthofinvestment--callthatrater--differsfromtherequiredrate.Domar'sapproachistodefineacoefficientofutilization[u=1meansfullutilizationofcapacity.]andshowthatsothatasInotherwords,ifthereisadiscrepancybetweentheactualandrequiredrates(),wewillfindintheendeitherashortageofcapacityorasurplusofcapacity,dependingonwhetherrisgreateroflessthanWecanshow,however,thattheconclusionaboutcapacityshortageandsurplusreallyappliesatanytimet,notonlyas.Foragrowthrateofrimpliesthatandtherefore,by(1)and(2),wehaveTheratiobetweenthesetwoderivatives,shouldtellustherelativemagnitudesofthedemand-creatingeffectandthecapacity-generatingeffectofinvestmentatanytimet,undertheactualgrowthrateofr.Ifr(theactualrate)exceeds(therequiredrate),then,andthedemandeffectoutstripthecapacityeffect,causingashortageofcapacity.Conversely,if,therewillbeadeficiencyinaggregatedemandand,hence,asurplusofcapacity.Thecuriousthingaboutthisconclusionisthatifinvestmentactuallygrowsatafasterratethanrequiredtheendresultwillbeashortageratherthanasurplusofcapacity.Itisequallycuriousthatiftheactualgrowthofinvestmentlagsbehindtherequiredrate,wewillencounteracapacitysurplusratherthana