lecture-9-Stackelberg-Competition

1Lecture9:StackelbergCompetition2IntroductionInawidevarietyofmarketsfirmscompetesequentiallyThesearedynamicgames.3StackelbergCompetitionFirmschooseoutputssequentially（有序的）Leadersetsoutputfirst,andthechoiceisobservedbythefollower.Followerthensetsoutputuponobservingtheleader’schoice.ThefirmmovingfirsthasaleadershipadvantageItcananticipate（预测）thefollower’sactions4StackelbergCompetitionTimePeriodt=1firm1choosingitsoptimalquantity(q*1)tomaximizeitsprofit.t=2firm2observestheoptimalquantitychoiceofoffirm1(q*1)andsetsitsoptimalquantity(q*2(q1))5StackelbergCompetitionAssumethattherearetwofirmswithidenticalproductsAsinourearlierCournotexample,letdemandbe:P=A–B.Q=A–B(q1+q2)MarginalcostforforeachfirmiscFirm1isthemarketleaderandchoosesq1Indoingsoitcananticipatefirm2’sactions.Soconsiderfirm2.Demandforfirm2is:P=(A–Bq1)–Bq2Marginalrevenuethereforeis:MR2=(A-Bq1)–2Bq26StackelbergCompetitionMR2=(A-Bq1)–2Bq2MC=cq*2=(A-c)/2B-q1/2q2q1R2(A–c)/2B(A–c)/BDemandforfirm1is:P=(A-Bq2)–Bq1P=(A-Bq*2)–Bq1P=(A-(A-c)/2)–Bq1/2P=(A+c)/2–Bq1/2Marginalrevenueforfirm1is:MR1=(A+c)/2-Bq1(A+c)/2–Bq1=cq*1=(A–c)/2B(A–c)/2Bq*2=(A–c)4B(A–c)/4BSEquatemarginalrevenuewithmarginalcostThisisfirm2’sbestresponsefunctionFirm1knowsthatthisishowfirm2willreacttofirm1’soutputchoiceButfirm1knowswhatq2isgoingtobeSolvethisequationforoutputq1Fromearlierexampleweknowthatthisisthemonopolyoutput.Thisisanimportantresult.TheStackelbergleaderchoosesthesameoutputasamonopolistwould.Butfirm2isnotexcludedfromthemarketSofirm1cananticipatefirm2’sreactionEquatemarginalrevenuewithmarginalcost8Firm1’sbestresponsefunctionis“like”firm2’sStackelbergCompetitionAggregateoutputis3(A-c)/4BSotheequilibriumpriceis(A+3c)/4q2q1R2(A-c)/2B(A-c)/B(A-c)/2BFirm1’sprofitis(A-c)2/8BFirm2’sprofitis(A-c)2/16B(A-c)/BR1SCWeknowthattheCournotequilibriumis:qC1=qC2=(A-c)/3B(A-c)/3B(A-c)/3BTheCournotpriceis(A+c)/3Profittoeachfirmis(A-c)2/9B(A-c)/4BComparethiswiththeCournotequilibriumLeadershipbenefitstheleaderfirm1butharmsthefollowerfirm2Leadershipbenefitsconsumersbutreducesaggregateprofits10StackelbergandPriceCompetitionWithpricecompetitionmattersaredifferentfirst-moverdoesnothaveanadvantagesupposeproductsareidenticalsupposefirst-movercommitstoapricegreaterthanmarginalcostthesecond-moverwillundercutthispriceandtakethemarketsotheonlyequilibriumisP=MCidenticaltosimultaneousgame11Application:Advertising&CompetitionThegame(firm1and2)ThemarketdemandfacedbythetwofirmsFirmsproduceatzerocosts,butfirm1incursadvertisingcostsofTimePeriodt=1firm1choosesadvertisinglevel(a)inordertoenhancedemandt=2firm1and2competeinaCournotfashion(choosingquantityLevel)Firm2observesthechoiceofaoffirm121qqap8123a12(Startfromt=2):SolvetheCournotbestresponsefunctionofthetwofirmsattheendofthegame(t=2),takingtheadvertisingleveldeterminedint=1asgiven.Derivethef.o.c.w.r.t.q1andsolveforq1,wegetthebestresponsefunction.Similarlyderivethebestresponsefu.Forfirm2.2112*2212qaqfqqqqa2221*qaqfq81231211aqqqaApplication:Advertising&Competition13TheCournotNashequilibriumcanbeobtained.Theequilibriumpriceisthen,Hence,firm1’sprofitfunctionasafunctionofais,(Nowatt=1):Firm1choosesitsadvertisementlevel(a)tomaximizeitsprofit.TheSPEstrategyprofileis32*1*aqq333aaaap812381232131211aaaqqqa3081692...*21aaaacof333*21aaqaaqaApplication:Advertising&Competition