平狄克《微觀經(jīng)濟(jì)學(xué)》課后答案 13-14

千里之行，始于足下。第2頁/共2頁精品文檔推薦平狄克《微觀經(jīng)濟(jì)學(xué)》課后答案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