mit-science-lectures-ans7

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1、MIT14.04IntermediateMicroeconomicTheoryFall2003Answerkeysforproblemset7Exercise15.4Expectedpayoffwhenvalueisv:(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−γThenfindtherelationbetweentheGreeklettersandtheRomanonesaibj+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-riumifafirmcutsitspr

5、icetheotherfirmwillalsocutitsprice,whereasinaCournotequilibriumifafirmdecreasesitsquantityinordertoincreasethepricethenitscompetitorwillincreaseitsownquantityandwillenjoypartofthepriceincrease.OnewaytoshowtheresultinthecaseoflineardemandistoprovefirstthatifafirmchoosesitsCournotpriceintheBertr

6、andcompetitionthentheotherfirmwillchooseapricelowerthanitsCournotonei.e.p(pc)

7、c+bpc=iiiusingthedemandfunctiony=a−bp+cpiiiij2bi1+bi=pcβii2bi≤pcequalityifandonlyifγ=0i.e.independentmarketiTherestoftheproofiseasytodowithagraph.Firstshowthatasthereactioncurvesareupwardslopping,bothCournotpricesareeitheraboveorbelowtheBertrandprices.Thenshow