ieb生產(chǎn)能力約束下兩階段競(jìng)爭(zhēng)

3.11Two-stagecapacity-constrained(生產(chǎn)能力約束下的兩階段競(jìng)爭(zhēng)Rigidcapacityconstraints剛性生產(chǎn)能力約束B(niǎo)ackgroundofKSKrepsandScheinkman(1983)developanextensionoftheCournotandBertrandduopolymodels.Inthestage,twoproducerssetcapacities(likeCournot),theninthesecondstageuseBertrand-likepricecompetitiontodeterminedemandwhichissubjecttocapacityconstraintsgeneratedbythe-stagedecisions.Theequilibriumofthistwo-stagegameisthatinthestagefirmsselectcapacitieswhicharejustsufficienttoproducetheCournotequilibriumoutputs,andinthesecondstagefirmsproducethoseKrepsandScheinkmanmodelisafundamentalexampleofysisofcompetitioninanimperfectlycompetitivemarket,inwhich dependcriticallyonthesequenceofdecisiontakingandonthewayinwhichearlierdecisionsconditionthepayoffsassociatedwithlaterTherationingruleofthismodelisthatifthecapacitychosenbyalower-pricefirmcannotsupplytheentiretydemandedatthelowerprice,thehigher-pricefirmswoulddeterminesthetyKrepsandScheinkmanassumethattheproductisKrepsandScheinkmanproductsdifferentiatedmodelInthissection,wediscussaversionofKSmodelwithdifferentiatedFirmssetcapacitiesinthestageofthegame.Inthesecondstageofthegame,firmsnoncooperativelysetpricesto izeownprofits,subjecttocapacityconstraintsandsupplyingthetiesdemandedatthenameprices.WiththeBowleymodelofproductdifferentiation,ifthereisnoexcesscapacityinthesecondstage,thetiesdemandedinthesecondstagesatisfy.??=(1?θ)α???1+θ??2=

=(1?θ)α???2+θ??1= Writingcapacitiesinplaceof??1=?????(??1+ ??2=?????(??2+ periodfirmsselectcapacitiesnoncooperativelyto profit:??istheunitcostofcapacity??1=[???????????(??1+ ??2=[???????????(??2+ Equilibriumcapacities(k*,k*)mustbejustsufficienttoallowfirmstoproducetheequilibriumoutputs(q Ifthereisnotexcesscapacity(q*<k*,q*<k*)inthesecondstagethentheobjectivefunctionsinthestageofthetwo-stagegameandtheobjectivefunctionsintheCournotgamearethesame.Onethereforeexpectsthattheequilibrium esinthetwo Iffirm1hasexcesscapacityinthesecondperiod,itselectsitspriceto

???)(?????)???????+????????(???ThedemonstrationsequenceofthismodelisasInthe yzingpricebest-responseInthesecond yzingequilibriumThen,workingbackfromsecond-stagepayoffsto-stagepayoffsintermsofForthefirms’capacitychoices,firmswillhaveexcesscapacityintheequilibriumoftheprice-settinggame.Suchcapacitychoiceswillnotbemadeinequilibrium,butitisnecessarytoshowthattheseconsequenceswillnotbechoseninequilibrium.Afirm’spricebest-responsecurvemayhaveuptofoursegments,dependingonthe-periodcapacitychoice.Branchone:capacityconstrainisnotBranchtwo:capacityconstrainisBranchthree:doesnotappealinKSPDBranchfour:p2risessufficiently,q2goesto0.Atthatpoint,q1<min(k1,qm(c)),whereqm(c)istheoutputofasingle-varietymonopolistwithmarginalcostcperunitandthenonnegetivityconstraintq2≥0esbinding.Branchfive:p2risessufficiently,q1=min(k1,qm(c)),atwhichpointfirm1’spricereactionfunction esvertical:firm2’spriceissohighthatfirm1cansellasclosetomonopolyoutputasitscapacitylevelpermits,withoutcreatingapositivedemandforvariety2.Best-responsefunctionofcapacityspacearefunctionallyidenticaltotheCournotbest-responsefunctionsandthereforeintersectatthecapacitylevelsthatjustallowfirmstoproducetheCournotequilibriumoutputs.Permeablecapacityconstraints(具有滲透性的生產(chǎn)能力約束Güth(1995)andMaggi(1996)extendtheKSPDmodeltoallowafirmtoexpandcapacityinthesecondperiod,ifnecessarytomeetdemand,atadifferentiallyhighercostcomparedto-periodinstallation.Theunitcostofaddingcapacityinthesecondperiodisρ+δ>ρ,firm1’spayoffwhenitaddssuchcapacityis

????

(?????)?????+ ????????? ??(??? ? Branchthree:firms’pricebest-responsefunctionwhenitchoosestoexpandcapacitytomeetdemand.IntheGüth-Maggimodel,iffirmsselectmoderateorhighcapacitylevelsintheperiod,equilibriumtypesareasintheKSmodel.Ifafirmselectsalowcapacitylevelintheperiod,insecond-stageequilibriumitproducescapacityandisonbranchthreeofitspricebest-responsefunction.ItisneveroptimalforafirmtoholdexcessIf??issufficientlyhigh,theninthestagefirmsareinasituationsimilartothatofKSmodel.If??islow,theninthesecondperiodfirmsareverynearlyinthesituationofBertrandduopolistswithmarginalcost??+??+??;theirequilibriumcapacitychoiceisthecapacitythatjustallowsthemtoproducetheirBertrandequilibriumoutputwiththislevelofmarginalcost.Consideringtheuncertainty,iffirmscancommitinadvancetoschedulesthatrelatethetydtoprice,equilibriumsupplycurvesimplymarketbehaviorthatisintermediatebetweenCournotandBertrand Thesupply-functionoligopolymodelappealsinformallytofirmormarketcharacteristicswhichimplythatfirmscommittosupplyschedulesbeforeanuncertainpartofmarketdemandisrealized.ThemodelsofKSandGMaregoodexampleswhichexaminebehaviorwhenfirmstakesomedecisionsinearlyperiods,conditioningthenatureofstrategicinteractioninlaterperiods.ReallifeElectricityTheelectricitymarketismodeledasatwostagegameunderuncertainty.In thesecondstagefirmscompeteinpricesundercapacityconstraints.Duringtheinvestmentstagethereisuncertaintyaboutthefuturedemand,whichisresolvedrightbeforethepricingstage.Wewanttofindthesubgameperfectsolvethepricingstageandthen,takingthissolutionasgiven,weproceedtothesolutionoftheinvestmentInvestmentStage:attheinvestmentstage,firmschoosesimultaneouslytheamountofcapacitytheywanttobuild.Theirdecisionsareirreversible.DemandRealization:demandcantaketwovalues,eitherthelowdemandvalue????,withprobabilityp>0,orthehighdemandvalue????,with1-p.WeassumethatPricingstage:duringthepricingstage,firmschoosesimultaneouslypriceswhichitholdsr-mc>c.BothfirmshavethesameMC.Thecompetitionheldatthepricingstageinvolvesonlythenewlyinstalledcapa