lecture-8-Cournot-Competition

Lecture8:CournotCompetition2OligopolyModelsTherearethreedominantoligopolymodelsCournotBertrandStackelbergTheyaredistinguishedbythedecisionvariablethatfirmschoosethetimingoftheunderlyinggameWewillstartfirstwithCournotModel.3TheCournotModelConsiderthecaseofduopoly(2competingfirms)andtherearenoentry..Firmsproducehomogenous(identical)productwiththemarketdemandfortheproduct:Marginalcostforeachfirmisconstantatcperunitofoutput.AssumethatAcTogetthedemandcurveforoneofthefirmswetreattheoutputoftheotherfirmasconstant.Soforfirm2,demandisItcanbedepictedgraphicallyasfollows.1212quantityoffirm1quantityoffirm2PABQABqqqq12PABqBq4TheCournotModelP=(A-Bq1)-Bq2$QuantityA-Bq1如果企业1的产量增加则企业2的需求曲线左移A-Bq’1Theprofit-maximizingchoiceofoutputbyfirm2dependsupontheoutputoffirm1DemandMarginalrevenueforfirm2isMR2==(A-Bq1)-2Bq2MR2MR2=MCA-Bq1-2Bq2=c求解q2q*2=(A-c)/2B-q1/2cMCq*222TRq5TheCournotModelWehavethisisthebestresponsefunctionforfirm2(reactionfunctionforfirm2).Itgivesfirm2’sprofit-maximizingchoiceofoutputforanychoiceofoutputbyfirm1.Inasimilarfashion,wecanalsoderivethereactionfunctionforfirm1.Cournot-Nashequilibriumrequiresthatbothfirmsbeontheirreactionfunctions.2*122AcqqB1*222AcqqB6q2q1Thereactionfunctionforfirm1isq*1=(A-c)/2B-q2/2(A-c)/B(A-c)/2BFirm1’sreactionfunctionThereactionfunctionforfirm2isq*2=(A-c)/2B-q1/2(A-c)/2B(A-c)/BIffirm2producesnothingthenfirm1willproducethemonopolyoutput(A-c)/2BIffirm2produces(A-c)/Bthenfirm1willchoosetoproducenooutputFirm2’sreactionfunctionTheCournot-NashequilibriumisattheintersectionofthereactionfunctionsCqC1qC2TheCournotModel7q2q1(A-c)/B(A-c)/2BFirm1’sreactionfunction(A-c)/2B(A-c)/BFirm2’sreactionfunctionCq*1=(A-c)/2B-q*2/2q*2=(A-c)/2B-q*1/2q*2=(A-c)/2B-(A-c)/4B+q*2/43q*2/4=(A-c)/4Bq*2=(A-c)/3B(A-c)/3Bq*1=(A-c)/3B(A-c)/3BTheCournotModel8TheCournotModelInequilibriumeachfirmproducesTotaloutputisthereforeDemandisP=A-BQ,thuspriceequalstoProfitsoffirms1and2arerespectivelyAmonopolywillproduce1**23ccAcqqB*23AcQB*2233AcAcPA******12122**129ccPcqPcqAcB11111maxMqPcqABqcq12MAcqB214MAcB9TheCournotModelCompetitionbetweenfirmsleadsthemtooverproduce.Thetotaloutputproducedishigherthaninthemonopolycase.Theduopolypriceislowerthanthemonopolyprice.Itcanbeverifiedthat,theduopolyoutputisstilllowerthanthecompetitiveoutputwhereP=MC.Theoverproductionisessentiallyduetotheinabilityoffirmstocrediblycommittocooperatetheyareinaprisoner’sdilemmakindofsituationmoreonthiswhenwediscusscollusion.1*232MAcAcQqBB*12because32mAcAcPPABqAc22MRAcPMCccABQQBTheCournotModel•Question1：为什么古诺市场结构下，企业的产量之和低于垄断企业的产量？•Quesition2：随着企业数目的增加，市场的总产量会增加还是减少？为什么？11TheCournotModel(ManyFirms)SupposethereareNidenticalfirmsproducingidenticalproducts.Totaloutput:Demandis:Considerfirm1,itsdemandcanbeexpressedas:Useasimplifyingnotation:Sodemandforfirm1is:123...NQqqqq123...NPABQABqqqq231...NPABQABqqqBq123...NQqqqThisdenotesoutputofeveryfirmotherthanfirm111PABQBq12P=(A-BQ-1)-Bq1$QuantityA-BQ-1Iftheoutputoftheotherfirmsisincreasedthedemandcurveforfirm1movestotheleftA-BQ’-1Theprofit-maximizingchoiceofoutputbyfirm1dependsupontheoutputoftheotherfirmsDemandMarginalrevenueforfirm1isMR1=(A-BQ-1)-2Bq1MR1MR1=MCA-BQ-1-2Bq1=cSolvethisforoutputq1q*1=(A-c)/2B-Q-1/2cMCq*1TheCournotModel(ManyFirms)13q*1=(A-c)/2B-Q-1/2Howdowesolvethisforq*1?Thefirmsareidentical.SoinequilibriumtheywillhaveidenticaloutputsQ*-1=(N-1)q*1q*1=(A-c)/2B-(N-1)q*1/2(1+(N-1)/2)q*1=(A-c)/2Bq*1(N+1)/2=(A-c)/2Bq*1=(A-c)/(N+1)BQ*=N(A-c)/(N+1)BP*=A-BQ*=(A+Nc)/(N+1)AsthenumberoffirmsincreasesoutputofeachfirmfallsAsthenumberoffirmsincreasesaggregateoutputincreasesAsthenumberoffirmsincreasespricetendstomarginalcostProfitoffirm1isΠ*1=(P*-c)q*1=(A-c)2/(N+1)2BAsthenumberoffirmsincreasesprofitofeachfirmfallsTheCournotModel(ManyFirms)lim1NANccN*211AcQNBN14Cournot-NashEquilibrium:DifferentCostsMarginalcostsoffirm1arec1andoffirm2arec2.DemandisP=A-BQ=A-B(q1+q2)Wehavemarginalrevenueforfirm1asbefore.MR1=(A-Bq2)-2Bq1Equatetomarginalcost:(A-Bq2)-2Bq1=c1Solvethisforoutputq1q*1=(A-c1)/2B-q2/2Asymmetricresultholdsforoutputoffirm2q*2=(A-c2)/2B-q1/215Cournot-NashEquilibrium:DifferentCostsq2q1(A-c1)/B(A-c1)/2BR1(A-c2)/2B(A-c2)/BR2Cq*1=(A-c1)/2B-q*2/2q*2=(A-c2)/2B-q*1/2q*2=(A-c2)/2B-(A-c1)/4B+q*2/43q*2/4=(A-2c2+c1)/4Bq*2=(A-2c2+c1)/3Bq*1=(A-2c1+c2)/3BWhathappenstothisequilibriumwhencostschange?Ifthemarginalcostoffirm2fallsitsreactioncurveshiftstotherightTheequilibriumoutputoffirm2increasesandoffirm1falls16Cournot-NashEquilibrium:DifferentCostsInequilibriumthefirmsproduce:ThedemandisP=A-BQ,thustheeq.priceis:Profitsare:Equilibriumoutputislessthanthecompetitivelevel.Outputisproducedinefficientlythelowcostfirmshouldproducealltheoutput.122112*121222and3323CCCCAccAccqqBBAccQqqB*1212233AccAccPA22**12211222and99AccAccBB17ConcentrationandProfitabilityConsiderthecaseofNfirmswithdifferen