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1、MIT14.04IntermediateMicroeconomicTheoryFall2003Answerkeysforproblemset7Exercise15.4Expectedpayo?whenvalueisv:(value-price)*probofwinning=(v?v)v=22v.2Exercise15.6ComputethebestresponseBRtosolvethisgame.?IfColumnismixingwithprobabilitypi.e.playspL+(1?p)RbyplayingTRowgets2?5pbyplayingBhegets1
2、?pBRrow(p)=Tifp<.25,=Bifp>.25,and={T,B}ifp=.25.?IfRowismixingwithprobabilityqi.e.playsqT+(1?q)BbyplayingLColumngets2?5qbyplayingRhegets1?qBRcol(q)=Lifq<.25,=Rifq>.25,and={L,R}ifq=.25.Thetwopurestrategyequilibriaare(B,L)and(T,R).Themixedstrategyequilibriumis(.25T+.75B,.25L+.75R)Probofbothte
3、enagerssurvive=1?.252=1516Exercise16.4SolvefortheBertrandandCournotquantities:b2aibj+cajyi=bi24b1b2?cc2αiβj?γαjyi=24β1β2?γThen?ndtherelationbetweentheGreeklettersandtheRomanonesaibj+cajαi=b1b2?c2bjβi=b1b2?c2cγ=b1b2?c21ExpresstheCournotquantitieswiththeRomanletters2abb+cba?ca2ca2yc=iijiji=y
4、b?ii4bb?c2i4bb?c21212As4bb?c2>0isthestabilitycondition,theCournotquantitiesarealways12lowerthantheBertrandones.Thenusethedemandpi=αi?βiyi?γyj,toconcludethattheCournotpricesarealwayshigherthantheBertrandprices.MoreintuitiveapproachTheintuitionhereisthatinaBertrandequilib-riumifa?rmcutsitspr
5、icetheother?rmwillalsocutitsprice,whereasinaCournotequilibriumifa?rmdecreasesitsquantityinordertoincreasethepricethenitscompetitorwillincreaseitsownquantityandwillenjoypartofthepriceincrease.Onewaytoshowtheresultinthecaseoflineardemandistoprove?rstthatifa?rmchoosesitsCournotpriceintheBertr
6、andcompetitionthentheother?rmwillchooseapricelowerthanitsCournotonei.e.p(pc)7、c+bpc=iiiusingthedemandfunctiony=a?bp+cpiiiij2bi1+bi=pcβii2bi≤pcequalityifandonlyifγ=0i.e.independentmarketiTherestoftheproofiseasytodowithagraph.Firstshowthatasthereactioncurvesareupwardslopping,bothCournotpricesareeitheraboveorbelowtheBertrandprices.Thenshow
