multi-market-single-price-monopolist.pdf

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1、1SolvingtheTwoMarket,OneProductionFa-cilityProblemInclassonThursday,Iaskedyoutothinkaboutthedi¤erencebetweentheproblemofamonopolistwithTWOproductionfacilitiessellinginonemarket,versusamonopolistwithONEproductionfacilitysellinginTWOmarkets...butwhohadtosellatthesamepriceinbothmarkets.ThelatterisaM

2、UCHharderproblemtoanalyze...butto‘sharpenyourchops’itisworthwhilethinkingabouthowtosolvethisproblem.Formally,wecanexpresstheproblemofthe’twomarket’monopolistinthefollowingway:MaxQ1;Q2(Q1;Q2)=P(Q1)Q1+Pe(Q2)Q2C(Q1+Q2)subjecttoP(Q1)=Pe(Q2)whereP(Q1)denotestheinversedemandcurveforthemonopolist’sout

3、putinmarket1andPe(Q2)denotestheinversedemandcurveforthemonopolist’soutputinmarket2.Observethattheinversedemandcurvesmaybequitedi¤erent,whichiswhywewanttoputthe’squiggle’(tilde)onthesecondP,tounderscorethefactthatthesearedi¤erentfunctions.TosolvetheproblemwithoutusingLagrangians(whichmostofyouhave

4、n’tseen....),theeasiestwayistostartwiththeconstraint.Youequatethetwoinversedemandfunctions,andthensolveforoneofthequantitiesasafunctionoftheotherquantity(andtheparameters),i.e.,youobtainanexpressionforQ1asafunctionofQ2(or,ifyouprefer,youcanobtainanexpressionforQ2asafunctionofQ1)-wecanwritethisas:

5、Qb1=Qb1(Q2):Nextyousubstitutethisexpressionbackintoyourobjectivefunction(whichisnowanunconstrainedmaximisationproblem),andmaximisewithrespecttoQ2:MaxQ2(Q2)=P(Qb1(Q2))Qb1(Q2)+Pe(Q2)Q2C(Qb1(Q2)+Q2)!d(Q2)=dPdQb1Qb+P+dPeQ+PedCdQb1+1=0(1)12dQ2dQ1dQ2dQ2dQdQ2TheQwhichsolves(1)istheoptimaloutputleve

6、lformarket2,andyou…nd2theoptimaloutputformarket1bysubstitutingQintoQb(Q):2121Tohelpyouinterpretyourresult,youcanrewrite(1)as:!dPdQb1Qb+P+dPeQ+Pe=dCdQb1+112dQ1dQ2dQ2dQdQ2!!!dPdQb1Qb1dPeQ2Pe=dCdQb1+1P++1+1dQ1dQ2PdQ2PedQdQ201!!dQb1@dQ2A+Pe1dCdQb1P+1+1=+1Q1;P"Q2;PedQdQ2Theexpressionsontheleft-hands

7、idecapturethemarginalrevenueassociatedwithanin…nitesimalincreaseinoutput,whichissplitautomaticallybetweenthetwomarketstoensurethatthepricestaysthesameonbothmarkets.Therighthandsidemeasuresthemarginalcostofthatin…nitesi