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千里之行,始于足下。第2页/共2页精品文档推荐平狄克《微观经济学》课后答案13-14CHAPTER13
GAMETHEORYANDCOMPETITIVESTRATEGY
Chapter13continuesthediscussionofcompetitivefirmsinthecontextoftwo-playergames,withthefirstthreesectionscoveringalltopicsintroducedinChapter12.IfyoudidnotpresentSection12.5,youshoulddosoafterdiscussingSections13.1and13.2.Sections13.4through13.8introduceadvancedtopics.Thepresentationthroughoutthechapterfocusesontheintuitionbehindeachmodelorstrategy.TheexercisesfocusonrelatingChapter13toChapter12andonbehaviorinrepeatedgames.
Twoconceptspervadethischapter:rationalityandequilibrium.Assumingtheplayersarerationalmeansthateachplayermaximizeshisorherownpayoffwhetherithurtsorhelpsotherplayers.Rationalityunderliesmanyoftheequilibriainthechapter.UnderlyingallthesemodelsisthedefinitionofaNashequilibrium,whichthestudentswillfindesoteric.Whenpresentingeachmodel,askwhetherauniqueNashequilibriumexists.Ifthereismorethanone,discusstheconditionsthatwillfavoreachequilibrium.
Theanalysisinthelastfivesectionsofthechapterismoredemanding,buttheexamplesaremoredetailed.Section13.4examinesrepeatedgames,anditwillbeimportanttodiscusstheroleofrationalityintheachievementofanequilibriuminbothfinite-andinfinite-horizongames.Example13.2pointsoutconditionsthatleadtostabilityinrepeatedgames,whileExample13.3presentsanunstablecase.Sections13.5,13.6,and13.7introducestrategyinthecontextofsequentialgames.Tocapturethestudents’attention,discussthephenomenalsuccessofWal-Martinitsattempttopreempttheentryofotherdiscountstoresinruralareas(seeExample13.4).First,defineastrategicmove;second,discusstheadvantageofmovingfirst;third,presentExample13.4;andfourth,continuewithotherformsofstrategicbehavior,includingtheuseofnewcapacityandR&Dtodeterentry(seeExamples13.5and13.6).Youmaywishtoreintroducethecaseofbilateralmonopolyduringthediscussionofstrategicbehaviorincooperativegames,whichconcludesthischapter.
1.Whatisthedifferencebetweenacooperativeandanoncooperativegame?Giveanexampleofeach.
Inanoncooperativegametheplayersdonotformallycommunicateinaneffortto
coordinatetheiractions.Theyareawareofoneanother’sexistence,butact
independently.Theprimarydifferencebetweenacooperativeandanoncooperative
gameisthatabindingcontract,i.e.,anagreementbetweenthepartiestowhichboth
partiesmustadhere,ispossibleintheformer,butnotinthelatter.Anexampleofa
cooperativegamewouldbeaformalcartelagreement,suchasOPEC,orajointventure.
Anexampleofanoncooperativegamewouldbearaceinresearchanddevelopmentto
obtainapatent.
2.Whatisadominantstrategy?Whyisanequilibriumstableindominantstrategies?
Adominantstrategyisonethatisbestnomatterwhatactionistakenbytheother
partytothegame.Whenbothplayershavedominantstrategies,theoutcomeisstable
becauseneitherpartyhasanincentivetochange.
3.ExplainthemeaningofaNashequilibrium.Howdoesitdifferfromanequilibriumindominantstrategies?
ANashequilibriumisanoutcomewherebothplayerscorrectlybelievethattheyare
doingthebesttheycan,giventheactionoftheotherplayer.Agameisinequilibriumif
neitherplayerhasanincentivetochangehisorherchoice,unlessthereisachangeby
theotherplayer.ThekeyfeaturethatdistinguishesaNashequilibriumfroman
equilibriumindominantstrategiesisthedependenceontheopponent’sbehavior.An
equilibriumindominantstrategiesresultsifeachplayerhasabestchoice,regardlessof
theotherplayer’schoice.EverydominantstrategyequilibriumisaNashequilibrium
butthereversedoesnothold.
4.HowdoesaNashequilibriumdifferfromagame’smaximinsolution?InwhatsituationsisamaximinsolutionamorelikelyoutcomethanaNashequilibrium?
Amaximinstrategyisoneinwhicheachplayerdeterminestheworstoutcomeforeach
oftheopponent’sactionsandchoosestheoptionthatmaximizestheminimumgainthat
canbeearned.UnliketheNashequilibrium,themaximinsolutiondoesnotrequire
playerstoreacttoanopponent’schoice.Ifnodominantstrategyexists(inwhichcase
outcomesdependontheopponent’sbehavior),playerscanreducetheuncertainty
inherentinrelyingontheopponent’srationalitybyconservativelyfollowingamaximin
strategy.ThemaximinsolutionismorelikelythantheNashsolutionincaseswhere
thereisahigherprobabilityofirrational(non-optimizing)behavior.
5.Whatisa“tit-for-tat”strategy?WhyisitarationalstrategyfortheinfinitelyrepeatedPrisoners’Dilemma?
Aplayerfollowinga“tit-for-tat”strategywillcooperateaslongashisorheropponentis
cooperatingandwillswitchtoanoncooperativestrategyiftheiropponentswitches
strategies.Whenthecompetitorsassumethattheywillberepeatingtheirinteraction
ineveryfutureperiod,thelong-termgainsfromcooperatingwilloutweighany
short-termgainsfromnotcooperating.Becausethe“tit-for-tat”strategyencourages
cooperationininfinitelyrepeatedgames,itisrational.
6.ConsideragameinwhichthePrisoners’Dilemmaisrepeated10times,andbothplayersarerationalandfullyinformed.Isatit-for-tatstrategyoptimalinthiscase?Underwhatconditionswouldsuchastrategybeoptimal?
Sincecooperationwillunravelfromthelastperiodbacktothefirstperiod,the
“tit-for-tat”strategyisnotoptimalwhenthereisafinitenumberofperiodsandboth
playersanticipatethecompetitor’sresponseineveryperiod.Giventhatthereisno
responsepossibleintheeleventhperiodforactioninthetenth(andlast)period,
cooperationbreaksdowninthelastperiod.Then,knowingthatthereisno
cooperationinthelastperiod,playersshouldmaximizetheirself-interestbynot
cooperatinginthesecond-to-lastperiod.Thisunravelingoccursbecausebothplayers
assumethattheotherplayerhasconsideredallconsequencesinallperiods.However,
ifthereissomedoubtaboutwhethertheopponenthasfullyanticipatedthe
consequencesofthe“tit-for-tat”strategyinthefinalperiod,thegamewillnotunravel
andthe“tit-for-tat”strategycanbeoptimal.
7.SupposeyouandyourcompetitorareplayingthepricinggameshowninTable13.8.Bothofyoumustannounceyourpricesatthesametime.Mightyouimproveyouroutcomebypromisingyourcompetitorthatyouwillannounceahighprice?
Ifthegameistobeplayedonlyafewtimes,thereislittletogain.IfyouareFirm1
andpromisetoannounceahighprice,Firm2willundercutyouandyouwillendup
withapayoffof-50.However,nextperiodyouwillundercuttoo,andbothfirmswill
earn10.Ifthegameisplayedmanytimes,thereisabetterchancethatFirm2will
realizethatifitmatchesyourhighprice,thelong-termpayoffof50eachperiodisbetter
than100atfirstand10thereafter.
8.Whatismeantby“first-moveradvantage”?Giveanexampleofagamingsituationwith
afirst-moveradvantage.
A“first-mover”advantagecanoccurinagamewherethefirstplayertoactreceivesthe
highestpayoff.Thefirst-moversignalshisorherchoicetotheopponent,andthe
opponentmustchoosearesponse,giventhissignal.Thefirst-movergoesonthe
offensiveandthesecond-moverrespondsdefensively.Inmanyrecreationalgames,
fromchesstofootball,thefirst-moverhasanadvantage.Inmanymarkets,thefirst
firmtointroduceaproductcansetthestandardforcompetitorstofollow.Insome
cases,thestandard-settingpowerofthefirstmoverbecomessopervasiveinthemarket
thatthebrandnameoftheproductbecomessynonymouswiththeproduct,e.g.,
“Kleenex,”thenameofKleenex-brandfacialtissue,isusedbymanyconsumerstorefer
tofacialtissueofanybrand.
9.Whatisa“strategicmove”?Howcanthedevelopmentofacertainkindofreputationbe
astrategicmove?
Astrategicmoveinvolvesacommitmenttoreduceone’soptions.Thestrategicmove
mightnotseemrationaloutsidethecontextofthegameinwhichitisplayed,butitis
rationalgiventheanticipatedresponseoftheotherplayer.Randomresponsestoan
opponent’sactionmaynotappeartoberational,butdevelopingareputationofbeing
unpredictablecouldleadtohigherpayoffsinthelongrun.Anotherexamplewouldbe
makingapromisetogiveadiscounttoallpreviousconsumersifyougiveadiscountto
one.Suchamovemakesthefirmvulnerable,butthegoalofsuchastrategicmoveis
tosignaltorivalsthatyouwon’tbediscountingpriceandhopethatyourrivalsfollow
suit.
10.Canthethreatofapricewardeterentrybypotentialcompetitors?Whatactionsmightafirmtaketomakethisthreatcredible?
Boththeincumbentandthepotentialentrantknowthatapricewarwillleavetheir
firmsworseoff.Normally,suchathreatisnotcredible.Thus,theincumbentmust
makehisorherthreatofapricewarbelievablebysignalingtothepotentialentrant
thatapricewarwillresultifentryoccurs.Onestrategicmoveistoincreasecapacity,
signalingalowerfutureprice,andanotheristoengageinapparentlyirrational
behavior.Bothtypesofstrategicbehaviormightdeterentry,butfordifferentreasons.
Whileanincreaseincapacityreducesexpectedprofitsbyreducingprices,irrational
behaviorreducesexpectedprofitsbyincreasinguncertainty,henceincreasingtherate
atwhichfutureprofitsmustbediscountedintothepresent.
11.Astrategicmovelimitsone’sflexibilityandyetgivesoneanadvantage.Why?Howmightastrategicmovegiveoneanadvantageinbargaining?
Astrategicmoveinfluencesconditionalbehaviorbytheopponent.Ifthegameiswell
understoodandtheopponent’sreactioncanbepredicted,astrategicmoveleavesthe
playerbetteroff.Economictransactionsinvolveabargain,whetherimplicitor
explicit.Ineverybargain,weassumethatbothpartiesattempttomaximizetheir
self-interest.Strategicmovesbyoneplayerprovidesignalstowhichanotherplayer
reacts.Ifabargaininggameisplayedonlyonce(sonoreputationsareinvolved),the
playersmightactstrategicallytomaximizetheirpayoffs.Ifbargainingisrepeated,
playersmightactstrategicallytoestablishreputationsforexpectednegotiations.
1.Inmanyoligopolisticindustries,thesamefirmscompeteoveralongperiodoftime,settingpricesandobservingeachother’sbehaviorrepeatedly.Giventhatthenumberofrepetitionsislarge,whydon’tcollusiveoutcomestypicallyresult?
Ifgamesarerepeatedindefinitelyandallplayersknowallpayoffs,rationalbehavior
willleadtoapparentlycollusiveoutcomes,i.e.,thesameoutcomesthatwouldresultif
firmswereactivelycolluding.Allpayoffs,however,mightnotbeknownbyallplayers.
Sometimesthepayoffsofotherfirmscanonlybeknownbyengaginginextensive(and
costly)informationexchangesorbymakingamoveandobservingrivals’responses.
Also,successfulcollusionencouragesentry.Perhapsthegreatestproblemin
maintainingacollusiveoutcomeisthatchangesinmarketconditionschangethe
collusivepriceandquantity.Thefirmsthenhavetorepeatedlychangetheir
agreementonpriceandquantity,whichiscostly,andthisincreasestheabilityofone
firmtocheatwithoutbeingdiscovered.
2.Manyindustriesareoftenplaguedbyovercapacity--firmssimultaneouslymakemajorinvestmentsincapacityexpansion,sototalcapacityfarexceedsdemand.Thishappensinindustriesinwhichdemandishighlyvolatileandunpredictable,butalsoinindustriesinwhichdemandisfairlystable.Whatfactorsleadtoovercapacity?Explaineachbriefly.
InChapter12,wefoundthatexcesscapacitymayariseinindustrieswitheasyentry
anddifferentiatedproducts.Inthemonopolisticcompetitionmodel,downward-sloping
demandcurvesforeachfirmleadtooutputwithaveragecostaboveminimumaverage
cost.Thedifferencebetweentheresultingoutputandtheoutputatminimum
long-runaveragecostisdefinedasexcesscapacity.Inthischapter,wesawthat
overcapacitycouldbeusedtodeternewentry;thatis,investmentsincapacity
expansioncouldconvincepotentialcompetitorsthatentrywouldbeunprofitable.
(Notethatalthoughthreatsofcapacityexpansionmaydeterentry,thesethreatsmust
becredible.)
3.Twocomputerfirms,AandB,areplanningtomarketnetworksystemsforofficeinformationmanagement.Eachfirmcandevelopeitherafast,high-qualitysystem(H),oraslower,low-qualitysystem(L).Marketresearchindicatesthattheresultingprofitstoeachfirmforthealternativestrategiesaregivenbythefollowingpayoffmatrix:
FirmB
HL
H
FirmA
L
a.Ifbothfirmsmaketheirdecisionsatthesametimeandfollowmaximin(low-risk)
strategies,whatwilltheoutcomebe?
Withamaximinstrategy,afirmdeterminestheworstoutcomeforeachoption,then
choosestheoptionthatmaximizesthepayoffamongtheworstoutcomes.IfFirmA
choosesH,theworstpayoffwouldoccurifFirmBchoosesH:A’spayoffwouldbe30.If
FirmAchoosesL,theworstpayoffwouldoccurifFirmBchoosesL:A’spayoffwouldbe
20.Withamaximinstrategy,AthereforechoosesH.IfFirmBchoosesL,theworst
payoffwouldoccurifFirmAchoosesL:thepayoffwouldbe20.IfFirmBchoosesH,the
worstpayoff,30,wouldoccurifFirmAchoosesL.Withamaximinstrategy,B
thereforechoosesH.Soundermaximin,bothAandBproduceahigh-qualitysystem.
b.Supposebothfirmstrytomaximizeprofits,butFirmAhasaheadstartinplanning,
andcancommitfirst.Nowwhatwilltheoutcomebe?WhatwilltheoutcomebeifFirmBhasaheadstartinplanningandcancommitfirst?
IfFirmAcancommitfirst,itwillchooseH,becauseitknowsthatFirmBwillrationally
chooseL,sinceLgivesahigherpayofftoB(35vs.30).ThisgivesFirmAapayoffof
50.IfFirmBcancommitfirst,itwillchooseH,becauseitknowsthatFirmAwill
rationallychooseL,sinceLgivesahigherpayofftoA(40vs.30).ThisgivesFirmBa
payoffof60.
c.Gettingaheadstartcostsmoney(youhavetogearupalargeengineeringteam).
Nowconsiderthetwo-stagegameinwhichfirst,eachfirmdecideshowmuchmoneytospendtospeedupitsplanning,andsecond,itannounceswhichproduct(HorL)itwillproduce.Whichfirmwillspendmoretospeedupitsplanning?Howmuchwillitspend?Shouldtheotherfirmspendanythingtospeedupitsplanning?Explain.
Inthisgame,thereisanadvantagetobeingthefirstmover.IfAmovesfirst,itsprofit
is50.Ifitmovessecond,itsprofitis40,adifferenceof10.Thus,itwouldbewillingto
spendupto10fortheoptionofannouncingfirst.Ontheotherhand,ifBmovesfirst,
itsprofitis60.Ifitmovessecond,itsprofitis35,adifferenceof25,andthuswouldbe
willingtospendupto25fortheoptionofannouncingfirst.OnceFirmArealizesthat
FirmBiswillingtospendmoreontheoptionofannouncingfirst,thenthevalueofthe
optiondecreasesforFirmA,becauseifbothfirmsweretoinvestbothfirmswould
choosetoproducethehigh-qualitysystem.Therefore,FirmAshouldnotspendmoney
tospeeduptheintroductionofitsproductifitbelievesthatFirmBisspendingthe
money.However,ifFirmBrealizesthatFirmAwillwait,FirmBshouldonlyspend
enoughmoneytodiscourageFirmAfromengaginginresearchanddevelopment,which
wouldbeanamountslightlymorethan10(themaximumamountAiswillingto
spend).
4.Twofirmsareinthechocolatemarket.Eachcanchoosetogoforthehighendofthemarket(highquality)orthelowend(lowquality).Resultingprofitsaregivenbythefollowingpayoffmatrix:
Firm2
LowHigh
Low
Firm1
High
a.Whatoutcomes,ifany,areNashequilibria?
IfFirm2choosesLowandFirm1choosesHigh,neitherwillhaveanincentiveto
change(100>-20forFirm1and800>50forFirm2).IfFirm2choosesHighand
Firm1choosesLow,neitherwillhaveanincentivetochange(900>50forFirm1and
600>-30forFirm2).BothoutcomesareNashequilibria.
b.Ifthemanagerofeachfirmisconservativeandeachfollowsamaximinstrategy,
whatwillbetheoutcome?
IfFirm1choosesLow,itsworstpayoff,-20,wouldoccurifFirm2choosesLow.IfFirm
1choosesHigh,itsworstpayoff,50,wouldoccurifFirm
2choosesHigh.Therefore,with
aconservativemaximinstrategy,Firm1choosesHigh.Similarly,ifFirm2chooses
Low,itsworstpayoff,-30,wouldoccurifFirm1choosesLow.IfFirm2choosesHigh,its
worstpayoff,50,wouldoccurifFirm1choosesHigh.Therefore,withamaximin
strategy,Firm2choosesHigh.Thus,bothfirmschooseHigh,yieldingapayoffof50for
both.
c.Whatisthecooperativeoutcome?
Thecooperativeoutcomewouldmaximizejointpayoffs.ThiswouldoccurifFirm1
goesforthelowendofthemarketandFirm2goesforthehighendofthemarket.The
jointpayoffis1,500(Firm1gets900andFirm2gets600).
d.Whichfirmbenefitsmostfromthecooperativeoutcome?Howmuchwouldthat
firmneedtooffertheothertopersuadeittocollude?
Firm1benefitsmostfromcooperation.Thedifferencebetweenitsbestpayoffunder
cooperationandthenextbestpayoffis900-100=800.TopersuadeFirm2tochoose
Firm1’sbestoption,Firm1mustofferatleastthedifferencebetweenFirm2’spayoff
undercooperation,600,anditsbestpayoff,800,i.e.,200.However,Firm2realizes
thatFirm1benefitsmuchmorefromcooperationandshouldtrytoextractasmuchas
itcanfromFirm1(upto800).
5.Twomajornetworksarecompetingforviewerratingsinthe8:00-9:00P.M.and9:00-10:00P.M.slotsonagivenweeknight.Eachhastwoshowstofillthistimeperiodandisjugglingitslineup.Eachcanchoosetoputits“bigger”showfirstortoplaceitsecondinthe9:00-10:00P.M.slot.Thecombinationofdecisionsleadstothefollowing“ratingspoints”results:
Network2
First
Network1
Second
a.FindtheNashequilibriaforthisgame,assumingthatbothnetworksmaketheir
decisionsatthesametime.
ANashequilibriumexistswhenneitherpartyhasanincentivetoalteritsstrategy,
takingtheother’sstrategyasgiven.Byinspectingeachofthefourcombinations,we
findthat(First,Second)istheonlyNashequilibrium,yieldingapayoffof(23,20).
Thereisnoincentiveforeitherpartytochangefromthisoutcome.
b.Ifeachnetworkisriskaverseandusesamaximinstrategy,whatwillbetheresulting
equilibrium?
Thisconservativestrategyofminimizingthemaximumlossfocusesonlimitingthe
extentoftheworstpossibleoutcome,totheexclusionofpossiblegoodoutcomes.If
Network1playsFirst,theworstpayoffis18.IfNetwork1playsSecond,theworst
payoffis4.Undermaximin,Network1playsFirst.(Here,playingFirstisa
dominantstrategy.)IfNetwork2playsFirst,theworstpayoffis18.IfNetwork2
playsSecond,theworstpayoffis16.Undermaximin,Network2playsFirst.The
maximinequilibriumis(First,First)withapayoffof(18,18).
c.WhatwillbetheequilibriumifNetwork1canmakesitsselectionfirst?IfNetwork2
goesfirst?
IfNetwork1playsFirst,Network2willplaySecond,yielding23forNetwork1.If
Network1playsSecond,Network2willplayFirst,yielding4forNetwork1.
Therefore,ifithasthefirstmove,Network1willplayFirst,andtheresulting
equilibriumwillbe(First,Second).IfNetwork2playsFirst,Network1willplayFirst,
yielding18forNetwork2.IfNetwork2playsSecond,Network1willplayFirst,
yielding20forNetwork2.Ifithasthefirstmove,Network2willplaySecond,andthe
equilibriumwillagainbe(First,Second).
d.Supposethenetworkmanagersmeettocoordinateschedules,andNetwork1
promisestoscheduleitsbigshowfirst.Isthispromisecredible,andwhatwouldbethelikelyoutcome?
Amoveiscredibleif,oncedeclared,thereisnoincentivetochange.Network1hasa
dominantstrategy:playthebiggershowFirst.Inthiscase,thepromisetoschedule
thebiggershowfirstiscredible.Knowingthis,Network2willscheduleitsbigger
showSecond.Thecoordinatedoutcomeislikelytobe(First,Second).
6.Twocompetingfirmsareeachplanningtointroduceanewproduct.EachfirmwilldecidewhethertoproduceProductA,ProductB,orProductC.Theywillmaketheirchoicesatthesametime.Theresultingpayoffsareshownbelow.
Wearegiventhefollowingpayoffmatrix,whichdescribesaproductintroductiongame:
Firm2
ABC
A
Firm1B
C
a.ArethereanyNashequilibriainpurestrategies?Ifso,whatarethey?
TherearetwoNashequilibriainpurestrategies.EachoneinvolvesonefirmintroducingProductAandtheotherfirmintroducingProductC.Wecanwritethesetwostrategypairsas(A,C)and(C,A),wherethefirststrategyisforplayer1.Thepayoffforthesetwostrategiesis,respectively,(10,20)and(20,10).
b.Ifbothfirmsusemaximinstrategies,whatoutcomewillresult?
Recallthatmaximinstrategiesmaximizetheminimumpayoffforbothplayers.ForeachoftheplayersthestrategythatmaximizestheirminimumpayoffisA.Thus(A,A)willresult,andpayoffswillbe(-10,-10).EachplayerismuchworseoffthanateitherofthepurestrategyNashequilibrium.
c.IfFirm1usesamaximinstrategy,andFirm2knows,whatwillFirm2do?
IfFirm1playsitsmaximinstrategyofA,andFirm2knowsthisthenFirm2wouldgetthehighestpayoffbyplayingC.NoticethatwhenFirm1playsconservatively,theNashequilibriumthatresultsgivesFirm2thehighestpayoffofthetwoNashequilibria.
7.WecanthinkoftheU.S.andJapanesetradepoliciesasaPrisoners’Dilemma.Thetwocountriesareconsideringpoliciestoopenorclosetheirimportmarkets.Supposethepayoffmatrixis:
Japan
Open
U.S.
Close
a.Assumethateachcountryknowsthepayoffmatrixandbelievesthattheother
countrywillactinitsowninterest.Doeseithercountryhaveadominantstrategy?
Whatwillbetheequilibriumpoliciesifeachcountryactsrationallytomaximizeitswelfare?
ChoosingOpenisadominantstrategyforbothcountries.IfJapanchoosesOpen,the
U.S.doesbestbychoosingOpen.IfJapanchoosesClose,theU.S.doesbestby
choosingOpen.Therefore,theU.S.shouldchooseOpen,nomatterwhatJapandoes.
IftheU.S.choosesOpen,JapandoesbestbychoosingOpen.IftheU.S.choosesClose,
JapandoesbestbychoosingOpen.Therefore,bothcountrieswillchoosetohaveOpen
policiesinequilibrium.
b.NowassumethatJapanisnotcertainthattheU.S.willbehaverationally.In
particular,JapanisconcernedthatU.S.politiciansmaywanttopenalizeJapaneven
ifthatdoesnotmaximizeU.S.welfare.HowmightthisaffectJapan’schoiceofstrategy?Howmightthischangetheequilibrium?
TheirrationalityofU.S.politicianscouldchangetheequilibriumfrom(Close,Open).
IftheU.S.wantstopenalizeJapantheywillchooseClose,butJapan’sstrategywillnot
beaffectedsincechoosingOpenisstillJapan’sdominantstrategy.
8.Youareaduopolistproducerofahomogeneousgood.Bothyouandyourcompetitorhavezeromarginalcosts.Themarketdemandcurveis
P=30-Q
whereQ=Q1+Q2.Q1isyouroutputandQ2isyourcompetitor’soutput.Yourcompetitorhasalsoreadthisbook.
a.Supposeyouaretoplaythisgameonlyonce.Ifyouandyourcompetitormust
announceyouroutputatthesametime,howmuchwillyouchoosetoproduce?Whatdoyouexpectyourprofittobe?Explain.
Thesearesomeofthecellsinthepayoffmatrix:
Firm2’sOutput
Firm1’s
Output0
5
10
15
20
25
30Ifbothfirmsmustannounceoutputatthesametime,bothfirmsbelievethattheother
firmisbehavingrationally,andeachfirmtreatstheoutputoftheotherfirmasafixed
number,aCournotequilibriumwillresult.
ForFirm1,totalrevenuewillbe
TR1=(30-(Q1+Q2))Q1,orTRQQQQ1112
1230=--.
MarginalrevenueforFirm1willbethederivativeoftotalrevenuewithrespecttoQ1,??TRQQQ1
12302=--.Becausethefirmsshareidenticaldemandcurves,thesolutionforFirm2willbe
symmetrictothatofFirm1:
??TRQQQ2
21302=--.Tofindtheprofit-maximizinglevelofoutputforbothfirms,setmarginalrevenueequal
tomarginalcost,whichiszero:
QQ12152
=-andQQ21152
=-.Withtwoequationsandtwounknowns,wemaysolveforQ1andQ2:
()????
?--=2155.01511QQ,orQ1=10.Bysymmetry,Q2=10.
SubstituteQ1andQ2intothedemandequationtodetermineprice:
P=30-(10+10),orP=$10.
Sincenocostsaregiven,profitsforeachfirmwillbeequaltototalrevenue:
π1=TR1=(10)(10)=$100and
π2=TR2=(10)(10)=$100.
Thus,theequilibriumoccurswhenbothfirmsproduce10unitsofoutputandbothfirms
earn$100.Lookingbackatthepayoffmatrix,notethattheoutcome(100,100)is
indeedaNashequilibrium:neitherfirmwillhaveanincentivetodeviate,giventhe
otherfirm’schoice.
b.Supposeyouar
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