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AdvancedMicroeconomicsTheory
Chapter8:GameTheoryandImperfectCompetition
Outline
• GameTheoryTools• BertrandModelofPriceCompetition• CournotModelofQuantityCompetition• ProductDifferentiation• DynamicCompetition• CapacityConstraints• EndogenousEntry• RepeatedInteraction
AdvancedMicroeconomicTheory 2
Introduction
• Monopoly:asinglefirm• Oligopoly:alimitednumberoffirms–Whenallowingfor𝑁 firms,theequilibriumpredictionsembodytheresultsinperfectlycompetitiveandmonopolymarketsasspecialcases.
AdvancedMicroeconomicTheory 3
GameTheoryTools
AdvancedMicroeconomicTheory 4
GameTheoryTools• Considerasettingwith𝐼 players(e.g.,firms,individuals,orcountries)eachchoosingastrategy𝑠$fromastrategyset𝑆$,where𝑠$ ∈ 𝑆$ and𝑖 ∈ 𝐼.– Anoutputlevel,aprice,oranadvertisingexpenditure
• Let(𝑠$, 𝑠*$) denoteastrategyprofilewhere𝑠*$representsthestrategiesselectedbyallfirms𝑖 ≠ 𝑗,i.e., 𝑠*$ = (𝑠/, … , 𝑠$*/, 𝑠$1/, … , 𝑠2).
• Dominatedstrategy:Strategy𝑠$∗ strictlydominatesanotherstrategy𝑠$4 ≠ 𝑠$∗ forplayer𝑖 if
𝜋$(𝑠$∗, 𝑠*$) > 𝜋$(𝑠$4, 𝑠*$)forall𝑠*$– Thatis,𝑠$∗ yieldsastrictlyhigherpayoffthan𝑠$4 does,regardlessofthestrategy𝑠*$ selectedbyallofplayer𝑖’srivals.
AdvancedMicroeconomicTheory 5
FirmBLowprices Highprices
FirmALowprices 5,5 9,1Highprices 1,9 7,7
GameTheoryTools• Payoffmatrix(NormalFormGame)
AdvancedMicroeconomicTheory 6
• “Lowprices”yieldsahigherpayoffthan“highprices”bothwhenafirm’srivalchooses“lowprices”andwhenitselects“highprices.”– “Lowprices”isastrictlydominantstrategyforbothfirms(i.e., 𝑠$∗).
– “Highprices”isreferredtoasastrictlydominatedstrategy(i.e.,𝑠$4).
GameTheoryTools
• Astrictlydominatedstrategycanbedeletedfromthesetofstrategiesarationalplayerwoulduse.
• Thishelpstoreducethenumberofstrategiestoconsiderasoptimalforeachplayer.
• Intheabovepayoffmatrix,bothfirmswillselect“lowprices”intheuniqueequilibriumofthegame.
• However,gamesdonotalwayshaveastrictlydominatedstrategy.
AdvancedMicroeconomicTheory 7
GameTheoryTools
• 𝐴𝑑𝑜𝑝𝑡 isbetterthan𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡 forfirm𝐴 ifitsopponentselects𝐴𝑑𝑜𝑝𝑡,butbecomesworseotherwise.
• Similarly argumentappliesforfirm𝐵.• Hence,nostrictlydominatedstrategiesforeitherplayer.• Whatistheequilibriumofthegamethen?
AdvancedMicroeconomicTheory 8
FirmBAdopt Notadopt
FirmAAdopt 3,1 0,0
Not adopt 0,0 1,3
GameTheoryTools
• Astrategyprofile(𝑠$∗, 𝑠*$∗ ) isaNashequilibrium(NE)if,foreveryplayer𝑖,
𝜋$ 𝑠$∗, 𝑠*$∗ ≥ 𝜋$ 𝑠$, 𝑠*$∗ forevery𝑠$ ≠ 𝑠$∗
– Thatis,𝑠$∗ isplayer𝑖’sbestresponsetohisopponentschoosing𝑠*$∗ as𝑠$∗ yieldsabetterpayoffthananyofhisavailablestrategies𝑠$ ≠ 𝑠$∗.
AdvancedMicroeconomicTheory 9
GameTheoryTools
• Inthepreviousgame:– Firm𝐴’sbestresponsetofirm𝐵’splaying𝐴𝑑𝑜𝑝𝑡 is𝐵𝑅A 𝐴𝑑𝑜𝑝𝑡 = 𝐴𝑑𝑜𝑝𝑡,whiletofirm𝐵 playing 𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡is𝐵𝑅A 𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡 = 𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡.
– Similarly,firm𝐵’sbestresponsetofirm𝐴 choosing𝑈 is𝐵𝑅C 𝐴𝑑𝑜𝑝𝑡 = 𝐴𝑑𝑜𝑝𝑡,whereastofirm𝐴 selecting𝐷 is𝐵𝑅C 𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡 = 𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡.
– Hence,strategyprofiles(𝐴𝑑𝑜𝑝𝑡, 𝐴𝑑𝑜𝑝𝑡) and(𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡, 𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡) aremutualbestresponses(i.e.,thetwoNashequilibria).
AdvancedMicroeconomicTheory 10
Mixed-StrategyNashEquilibrium
• Insofarwerestrictedplayerstouseoneoftheiravailablestrategies100%ofthetime(commonlyknownas“purestrategies”)
• Generally,playerscouldrandomize(mix)theirchoices.– Example:Choosestrategy𝐴 withprobability𝑝 andstrategy𝐵 withprobability1 − 𝑝.
AdvancedMicroeconomicTheory 11
Mixed-StrategyNashEquilibrium
• Mixed-strategyNashequilibrium(msNE):Considerastrategyprofile𝜎 = (𝜎/, 𝜎H, … , 𝜎I),where𝜎$ isamixedstrategyforplayer𝑖.Strategyprofile𝜎$ ismsNE ifandonlyif
𝜋$ 𝜎$, 𝜎*$ ≥ 𝜋$ 𝑠$4, 𝜎*$ forall𝑠$4 ∈ 𝑆$– Thatis,𝜎$ isabestresponseofplayer𝑖,i.e.,𝜎$ ∈ 𝐵𝑅$(𝜎$),tothestrategyprofile𝜎*$ oftheother𝑁 − 1 players.
AdvancedMicroeconomicTheory 12
Mixed-StrategyNashEquilibrium
• ThreepointsaboutmsNE:1. Playersmustbeindifferentamongall(oratleastsome)
oftheirpurestrategies.2. Sinceplayersneverusestrictlydominatedstrategies,a
NEassignsazeroprobabilitytodominatedstrategies.3. Ingameswithafinitesetofplayersandafinitesetof
availableactions,thereisgenerallyanoddnumberofequilibria.
AdvancedMicroeconomicTheory 13
Mixed-StrategyNashEquilibrium
• Example (noNEinpurestrategies):
AdvancedMicroeconomicTheory 14
FirmBAdopt Notadopt
FirmAAdopt 3,-3 -4,0
Not adopt -3,1 2,-2
– Thereisnocellofthematrixinwhichplayersselectmutualbestresponses.
– ThuswecannotfindaNEinpurestrategies.– FirmA(B)seekstocoordinate(miscoordinate)itsdecisionwiththatoffirmB(A,respectively).
Mixed-StrategyNashEquilibrium
• Example (continued):– Giventheiropposedincentives,firmAwouldliketomakeitschoicedifficulttoanticipate.
– IffirmAchoosesaspecificactionwithcertainty,firmBwillbedriventoselecttheoppositeaction.
– AnanalogousargumentappliestofirmB.– Asaconsequence,playershaveincentivestorandomizetheiractions.
AdvancedMicroeconomicTheory 15
Mixed-StrategyNashEquilibrium
• Example (continued):– Let𝑝(𝑞) denotetheprobabilitywithwhichfirmA(B,respectively)adoptsthetechnology.
– IffirmAisindifferentbetweenadoptingandnotadoptingthetechnology,itsexpectedutilitymustsatisfy
𝐸𝑈A 𝐴𝑑𝑜𝑝𝑡 = 𝐸𝑈A(𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡)3𝑞 + −4 1 − 𝑞 = −3𝑞 + 2(1 − 𝑞)
6𝑞 = 6 1 − 𝑞 ⇒ 𝑞 = 1/2– HencefirmBadoptsthetechnologywithprobability𝑞 = 1/2.
AdvancedMicroeconomicTheory 16
Mixed-StrategyNashEquilibrium
• Example (continued):– Similarly,firmBmustbeindifferentbetweenadoptingandnotadoptingthetechnology:
𝐸𝑈C 𝐴𝑑𝑜𝑝𝑡 = 𝐸𝑈C(𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡)−3𝑝 + 1 1 − 𝑝 = 0𝑝 + (−2)(1 − 𝑝)
1 − 𝑝 = 3𝑝 ⇒ 𝑝 = 1/2– HencefirmAadoptsthetechnologywithprobability𝑝 = 1/2.
– Combiningourresults,weobtainthemsNE12𝐴𝑑𝑜𝑝𝑡
12𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡,
12𝐴𝑑𝑜𝑝𝑡
12𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡
AdvancedMicroeconomicTheory 17
Mixed-StrategyNashEquilibrium
• Example (Technologyadoptiongame):
AdvancedMicroeconomicTheory 18
FirmBAdopt Notadopt
FirmAAdopt 3,1 0,0
Not adopt 0,0 1,3
– ThegamehastwoNashequilibriainpurestrategies:(𝐴𝑑𝑜𝑝𝑡, 𝐴𝑑𝑜𝑝𝑡)and(𝑁𝑜𝑡𝐴𝑑𝑜𝑝𝑡, 𝑁𝑜𝑡𝐴𝑑𝑜𝑝𝑡).
– Thereis,however,athirdNashequilibriainwhichbothfirmsuseamixedstrategy.
Mixed-StrategyNashEquilibrium
• Example (continued):– Let𝑝(𝑞) denotetheprobabilitywithwhichfirmA(B,respectively)adoptsthetechnology.
– IffirmAisindifferentbetweenadoptingandnotadoptingthetechnology,itsexpectedutilitymustsatisfy
𝐸𝑈A 𝐴𝑑𝑜𝑝𝑡 = 𝐸𝑈A(𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡)3𝑞 + 0 1 − 𝑞 = 0𝑞 + 1(1 − 𝑞)
3𝑞 = 1 − 𝑞 ⇒ 𝑞 = 1/4– HencefirmBadoptsthetechnologywithprobability𝑞 =1/4.
AdvancedMicroeconomicTheory 19
Mixed-StrategyNashEquilibrium
• Example (continued):– Similarly,firmBmustbeindifferentbetweenadoptingandnotadoptingthetechnology:
𝐸𝑈C 𝐴𝑑𝑜𝑝𝑡 = 𝐸𝑈C(𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡)1𝑝 + 0 1 − 𝑝 = 0𝑝 + 3(1 − 𝑝)
𝑝 = 3 − 3𝑝 ⇒ 𝑝 = 3/4– HencefirmAadoptsthetechnologywithprobability𝑝 = 3/4.
– Combiningourresults,weobtainthemsNE34𝐴𝑑𝑜𝑝𝑡
14𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡,
14𝐴𝑑𝑜𝑝𝑡
34𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡
AdvancedMicroeconomicTheory 20
Mixed-StrategyNashEquilibrium
• Example (continued):best-response
AdvancedMicroeconomicTheory 21
Sequential-MoveGames
AdvancedMicroeconomicTheory 22
Sequential-MoveGames• Whenplayerschoosetheirstrategiessequentially,ratherthan
simultaneously,thedefinitionofstrategybecomesmoreinvolved.
• Strategyisnowacompletecontingentplandescribingwhatactionplayer𝑖 choosesateachpointatwhichheiscalledontomove,giventheprevioushistoryofplay.
• Suchhistorycanbeobservableornotobservablebyplayer𝑖.
AdvancedMicroeconomicTheory 23
Sequential-MoveGames
• Sequential-movegamesarerepresentedusinggametreesratherthanwithmatrices.– The“root”ofthetree,wherethegamestarts,isreferredtoastheinitialnode.
– Thelastnodesofthetree,wherenomorebranchesoriginate,aretheterminalnodes.
AdvancedMicroeconomicTheory 24
Sequential-MoveGames
AdvancedMicroeconomicTheory 25
Sequential-MoveGames
• Basicrules:1. Atreemusthaveonlyoneinitialnode.2. Everynodeofthetreehasexactlyoneimmediate
predecessorexcepttheinitialnode,whichhasnopredecessor.
3. Multiplebranchesextendingfromthesamenodemusthavedifferentactionlabels.
4. Everyinformationsetcontainsdecisionnodesforonlyoneoftheplayersinthegame.
5. Allnodesinagiveninformationsethavethesameimmediatesuccessors
AdvancedMicroeconomicTheory 26
Sequential-MoveGames
AdvancedMicroeconomicTheory 27
Sequential-MoveGames
AdvancedMicroeconomicTheory 28
Sequential-MoveGames
• Informationsetsareusedtodenoteagroupofnodesamongwhichaplayercannotdistinguish.
• Acommonfeatureoftreesrepresentinggamesofincompleteinformation.
• Informationsetsarisewhenaplayerdoesnotobservetheactionthathispredecessorchose.
AdvancedMicroeconomicTheory 29
Sequential-MoveGames
AdvancedMicroeconomicTheory 30
Sequential-MoveGames
AdvancedMicroeconomicTheory 31
Sequential-MoveGames
• CanwesimplyusetheNEsolutionconceptinordertofindequilibriumpredictionsinsequential-movegames?
• Wecan,butsomeoftheNEpredictionsarenotverysensible(credible).
AdvancedMicroeconomicTheory 32
Sequential-MoveGames
• Example (Entryandpredationgame):– Consideranentrant’sdecisiononwhethertoenterintoanindustrywhereanincumbentfirmoperatesortostayout.
AdvancedMicroeconomicTheory 33
Sequential-MoveGames
• Example (continued):– InordertofindtheNEofthisgame,itisusefultorepresentthegameinmatrixform.
AdvancedMicroeconomicTheory 34
IncumbentAccommodate Fight
EntrantIn 2,2 -1,-1Out 0,4 0,4
– TwoNEs:(𝐼𝑛, 𝐴𝑐𝑐𝑜𝑚𝑚𝑜𝑑𝑎𝑡𝑒) and(𝑂𝑢𝑡, 𝐹𝑖𝑔ℎ𝑡)– Thefirstequilibriumseemcredible,whilethesecondequilibriumdoesnotlookcredibleatall.
Sequential-MoveGames• Theprecedingexampleindicatestheneedtorequireanotionof
credibilityinsequential-movegamesthatdidnotexistintheNEsolutionconcept– Arequirementcommonlyknownas“sequentialrationality”
• Player𝑖’sstrategyissequentiallyrationalifitspecifiesanoptimalactionforplayer𝑖 atanynode(orinformationset)ofthegame,eventhoseinformationsetsthatplayer𝑖 doesnotbelievewillbereachedintheequilibriumofthegame.– Thatis,player𝑖 behavesoptimallyateverynode(orinformationset),
bothnodesthatbelongtotheequilibriumpathofthegametreeandthosethatlieoff-the-equilibriumpath.
• Howcanweguaranteethatitholdswhenfindingequilibriainsequential-movegames?– Backwardinduction:startingfromeveryterminalnode,eachplayer
usesoptimalactionsateverysubgame ofthegametree.
AdvancedMicroeconomicTheory 35
Sequential-MoveGames• Asubgame canbeidentifiedbydrawingarectanglearound
asectionofthegametreewithout“breaking”anyinformationset
AdvancedMicroeconomicTheory 36
Sequential-MoveGames• Thebackwardinductionrequirestofindthestrategythateveryplayer𝑖 findsoptimalateverysubgamealongthegametree.– Startbyidentifyingtheoptimalbehavioroftheplayerwhoactslast(inthelastsubgameofthetree).
– Takingtheoptimalactionofthisplayerintoaccount,movetotheprevioustothelastplayerandidentifyhisoptimalbehavior.
– Repeatthisprocessuntiltheinitialnode.• SubgameperfectNashequilibrium (SPNE):Astrategyprofile(𝑠/∗, 𝑠H∗, … , 𝑠]∗ ) isaSPNEifitspecifiesaNEforeachsubgame.
AdvancedMicroeconomicTheory 37
Sequential-MoveGames
• Example (Entryandpredationgame):– Identifythesubgamesofthegametree
AdvancedMicroeconomicTheory 38
– TheSPNEis(𝐼𝑛, 𝐴𝑐𝑐𝑜𝑚𝑚𝑜𝑑𝑎𝑡𝑒),whichcoincideswithoneoftheNEofthisgame.
Sequential-MoveGames
• Example (backwardinductioninthreesteps):
AdvancedMicroeconomicTheory 39
Sequential-MoveGames
• Example (backwardinductionininformationsets):
AdvancedMicroeconomicTheory 40
Sequential-MoveGames
• Example (continued):– Thesmallestsubgameisisstrategicallyequivalenttooneinwhichplayer1and2choosetheiractionssimultaneously.
AdvancedMicroeconomicTheory 41
Player2X Y
Player1A 3,4 1,4B 2,1 2,0
– TheNEofthesubgameis(𝐴, 𝑋).
Sequential-MoveGames
• Example (continued):– Oncewehaveareduced-formgametree,wecanmoveonestepbackward(theinitialnode)
AdvancedMicroeconomicTheory 42
– TheSPNEofthisgameis(𝑈𝑝|𝐴, 𝑋).– Player1’sstrategy:play𝑈𝑝 inthefirstnodeand𝐴 afterwards– Player2’sstrategy:play𝑋
Sequential-MoveGames
• Example (continued):– Normal-formrepresentationofthesequentialgame
AdvancedMicroeconomicTheory 43
Player2X Y
Player1
Up/A 3,4 1,4Up/B 2,1 2,0
Down/A 2,6 2,6Down/B 2,6 2,6
– ThreeNEs:(𝑈𝑝|𝐴, 𝑋),(𝐷𝑜𝑤𝑛|𝐴, 𝑌),(𝐷𝑜𝑤𝑛|𝐵, 𝑌).– Onlythefirstequilibriumissequentiallyrational.
Simultaneous-MoveGamesofIncompleteInformation
AdvancedMicroeconomicTheory 44
Simultaneous-MoveGamesofIncompleteInformation
• Thestrategicsettingspreviouslyanalyzedassumethatallplayersareperfectlyinformedaboutallrelevantdetailsofthegame.
• Thereareoftenreal-lifesituationswhereplayersoperatewithoutsuchinformation.
• Playersactunder“incompleteinformation”ifatleastoneplayercannotobserveapieceofinformation.– Example:marginalcostsofrivalfirms
AdvancedMicroeconomicTheory 45
Simultaneous-MoveGamesofIncompleteInformation
• Forcompactness,werefertoprivateinformationasplayer𝑖’s“type”anddenoteitas𝜃$.
• Whileplayer𝑗 mightnotobserveplayer𝑖’stype,heknowstheprobabilitydistributionofeachtype.
• Example:– Marginalcostscanbeeitherhighorlow,wherebyΘ$ = 𝐻, 𝐿 .
– Theprobabilityoffirm𝑖’scostsbeinghighis𝑝 𝜃$ = 𝐻 = 𝑝 andtheprobabilityofitscostsbeinglowis𝑝 𝜃$ = 𝐿 = 1 − 𝑝 ,where𝑝 ∈ (0,1).
AdvancedMicroeconomicTheory 46
Simultaneous-MoveGamesofIncompleteInformation
• Example (technologyadoption):– Afirstmoveofnaturedeterminestheprecisetypeof𝜃$.– FirmAhastwopossibletypes,eitherhighorlowcosts,withassociatedprobabilities2/3and1/3.
– FirmAobservesitsowntype,butfirmBcannotobserveit.– Graphically,firmAknowswhichpayoffmatrixfirmsareplaying,whilefirmBcanonlyassignaprobability2/3(1/3)toplayingtheright-hand(left-hand)matrix.
AdvancedMicroeconomicTheory 47
Simultaneous-MoveGamesofIncompleteInformation
AdvancedMicroeconomicTheory 48
Simultaneous-MoveGamesofIncompleteInformation
• Everyplayer𝑖’sstrategyinanincompleteinformationcontextneedstobeafunctionofitsprivatelyobservedtype𝜃$
𝑠$(𝜃$)• Player𝑖’sstrategyisnotconditionedonotherplayers’types
𝜃*$ = (𝜃/, 𝜃H, … , 𝜃$*/, 𝜃$1/, … , 𝜃I)• Thatis,wedonotwrite𝑠$(𝜃$, 𝜃*$) becauseplayer𝑖 cannot
observethetypesofallotherplayers.– Ifallplayerscouldobservethetypesofalloftheirrivals,we
wouldbedescribingacompleteinformationgame.• Forsimplicity,typesareindependentlydistributed,which
entailsthateveryplayer𝑖 cannotinferhisrivals’types𝜃*$afterobservinghisowntype𝜃$.
AdvancedMicroeconomicTheory 49
Simultaneous-MoveGamesofIncompleteInformation
• BayesianNashequilibrium (BNE):Astrategyprofile(𝑠/∗ 𝜃/ , 𝑠H∗ 𝜃H , … , 𝑠]∗ 𝜃] ) isaBNEofagameofincompleteinformationif
𝐸𝑈$ 𝑠$∗ 𝜃$ , 𝑠*$∗ 𝜃*$ ; 𝜃$, 𝜃*$≥ 𝐸𝑈$ 𝑠$ 𝜃$ , 𝑠*$∗ 𝜃*$ ; 𝜃$, 𝜃*$
foreverystrategy𝑠$∗ 𝜃$ ∈ 𝑆$,everytype𝜃$ ∈ Θ$,andeveryplayer𝑖.
• Whenallotherplayersselectequilibriumstrategies,theexpectedutilitythatplayer𝑖 obtainsfromselecting𝑠$∗ 𝜃$ whenhistypeis𝜃$ islargerthanthatofdeviatingtoanyotherstrategy𝑠$ 𝜃$ .
AdvancedMicroeconomicTheory 50
Simultaneous-MoveGamesofIncompleteInformation
• Approach1:FourstepstofindallBNEsinsimultaneous-movegamesofincompleteinformation.
• Example (technologyadoption):1. Strategysets:Identifythestrategysetforeachplayer,
whichcanbeafunctionofhisprivatelyobservedtype𝑆/ = 𝐼g𝐼h, 𝐼g𝑁𝐼h, 𝑁𝐼g𝐼h, 𝑁𝐼g𝑁𝐼h𝑆H = 𝐼, 𝑁𝐼
AdvancedMicroeconomicTheory 51
Simultaneous-MoveGamesofIncompleteInformation
• Example (continued):2. Bayesiannormal-formrepresentation:Usethe
strategysetsidentifiedinstep1toconstructthe“Bayesiannormal-form”representationoftheincompleteinformationgame.
AdvancedMicroeconomicTheory 52
FirmB𝐼 𝑁𝐼
Firm A
𝐼g𝐼h𝐼g𝑁𝐼h𝑁𝐼g𝐼h𝑁𝐼g𝑁𝐼h
Simultaneous-MoveGamesofIncompleteInformation
• Example (continued):3. Expectedpayoffs:Findtheexpectedpayoffsthat
wouldgoineverycell.
AdvancedMicroeconomicTheory 53
FirmB𝐼 𝑁𝐼
Firm A
𝐼g𝐼h 5,1 2,0𝐼g𝑁𝐼h 4,2/3 21/3,1𝑁𝐼g𝐼h 1,1/3 2/3,2𝑁𝐼g𝑁𝐼h 0,0 1,3
Simultaneous-MoveGamesofIncompleteInformation
• Example (continued):4. Findbestresponsesforeachplayer:Followan
approachsimilartothatinsimultaneous-movegamesofcompleteinformationtofindbest-responsepayoffs.
– TheBNEsare(𝐼g𝐼h, 𝐼) and(𝐼g𝑁𝐼h, 𝑁𝐼).
AdvancedMicroeconomicTheory 54
Simultaneous-MoveGamesofIncompleteInformation
• Approach2:FindthesetofBNEsbyfirstanalyzingbestresponsesfortheprivatelyinformedplayer,andthenusethoseinouridentificationofbestresponsesfortheuninformedplayer.
AdvancedMicroeconomicTheory 55
Simultaneous-MoveGamesofIncompleteInformation
• Example (technologyadoption):– Thetwopossiblegamesthatfirmscouldbeplaying.
AdvancedMicroeconomicTheory 56
Simultaneous-MoveGamesofIncompleteInformation
• Example (continued):– First,welookattheprivatelyinformedfirmA.– IffirmAisofthehightype,𝐼𝑛𝑣𝑒𝑠𝑡 strictlydominates𝑁𝑜𝑡𝑖𝑛𝑣𝑒𝑠𝑡.
– IffirmAisofthelowtype,neitherstrategystrictlydominatestheother.
– Needtocomparetheexpectedutilities𝐸𝑈A 𝐼𝑛𝑣𝑒𝑠𝑡 𝐿𝑜𝑤 = 3 j 𝛽 + 0 j 1 − 𝛽 = 3𝛽𝐸𝑈A 𝑁𝑜𝑡𝑖𝑛𝑣𝑒𝑠𝑡 𝐿𝑜𝑤 = 0 j 𝛽 + 1 j 1 − 𝛽 = 1 − 𝛽
– FirmAinvestsif3𝛽 ≥ 1 − 𝛽 or𝛽 ≥ 1/4.
AdvancedMicroeconomicTheory 57
Simultaneous-MoveGamesofIncompleteInformation
• Example (continued):– Next,welookattheuninformedfirmB.– SincefirmBdoesnotknowfirmA’stype,wehavetomodelintheprobability(𝑝)thatfirmAisofthehightype.𝐸𝑈C 𝐼𝑛𝑣𝑒𝑠𝑡
= 1 j 𝑝l
mnnopqrostoutvwxy,ovoz{ysvs
+ 1 − 𝑝 j 1 j 𝛾}~$��C$I��������I�������
+ 0 j 1 − 𝛾~$��C����I��$I�������I���
����
mnnopqros���vwxy
= 𝑝 + (1 − 𝑝)𝛾
AdvancedMicroeconomicTheory 58
Simultaneous-MoveGamesofIncompleteInformation
• Example (continued):𝐸𝑈C 𝐼𝑛𝑣𝑒𝑠𝑡
= 1 j 𝑝}mnnopqrostout
vwxy,~$��Coz{ysvs
+ 1 − 𝑝 j 1 j 𝛾}~$��C$I��������I�������
+ 0 j 1 − 𝛾~$��C����I��$I�������I���
����
mnnopqros���vwxy
= 𝑝 + 1 − 𝑝 𝛾
𝐸𝑈C 𝑁𝑜𝑡𝑖𝑛𝑣𝑒𝑠𝑡
= 0 j 𝑝}mnnopqrostout
vwxy,~$��Coz{ysvs
+ 1 − 𝑝 j 0 j 𝛾}~$��C$I��������I�������
+ 3 j 1 − 𝛾~$��C����I��$I�������I���
����
mnnopqros���vwxy
= 3(1 − 𝑝)(1 − 𝛾)
AdvancedMicroeconomicTheory 59
Simultaneous-MoveGamesofIncompleteInformation
• Example (continued):– Therefore,firmBinvestsif
𝑝 + (1 − 𝑝)𝛾 ≥ 3(1 − 𝑝)(1 − 𝛾)– Since𝑝 = 2/3,theaboveinequalityreducesto
2 ≥ 3 − 4𝛾𝛾 ≥ 1/4
– TwoBNEs:1. If𝛾 ≥ 1/4, 𝐼g𝐼h, 𝐼 .2. If𝛾 < 1/4,(𝐼g𝑁𝐼h, 𝑁𝐼).
AdvancedMicroeconomicTheory 60
Sequential-MoveGamesunderIncompleteInformation
AdvancedMicroeconomicTheory 61
Sequential-MoveGamesunderIncompleteInformation
• TheBNEsolutionconcepthelpsusfindequilibriumoutcomesinsettingswhereplayersinteractunderincompleteinformation.
• Whiletheapplicationsintheprevioussectionconsideredthatplayersactsimultaneously,wecanalsofindtheBNEsofincompleteinformationgamesinwhichplayersactsequentially.
AdvancedMicroeconomicTheory 62
Sequential-MoveGamesunderIncompleteInformation
AdvancedMicroeconomicTheory 63
• Investmentgame:
Sequential-MoveGamesunderIncompleteInformation
• InordertofindthesetofBNEs,wefirstrepresenttheBayesiannormal-form representationofthegametree.
• Thematrixincludesexpectedpayoffsforeachplayer.
AdvancedMicroeconomicTheory 64
Player2𝐴 𝑅
Player 1
𝑂C𝑂]C 4-p,3.5p -3+p,-3+5p
𝑂C𝑁]C 3p,3.5p -2p,2p
𝑁C𝑂]C 4-4p, 0 -3+3p,-3+3p
𝑁C𝑁]C 0,0 0,0
Sequential-MoveGamesunderIncompleteInformation
• TherearetwoBNEsinthisgame:(𝑂C𝑂]C, 𝐴)(𝑁C𝑁]C, 𝑅)
• ThefirstBNEisrathersensible– Player1makestheofferregardlessofhistype,andthustheuninformedplayer2choosestoaccepttheofferifhereceivesone.
AdvancedMicroeconomicTheory 65
Sequential-MoveGamesunderIncompleteInformation
• ThesecondBNEisdifficulttorationalize– Notypeofsendermakesanofferinequilibrium,andtheresponderrejectsanyofferpresentedtohim.
– Ifanofferwaseverobserved,thereceivershouldcomparetheexpectedutilityofacceptingandrejectingtheoffer,basedontheoff-theequilibriumbelief𝜇.
𝐸𝑈H(𝐴) = 3.5 j 𝜇 + 0 j 1 − 𝜇 = 3.5𝜇𝐸𝑈H 𝑅 = 2 j 𝜇 + −3 j 1 − 𝜇 = −3 + 𝜇
– Player2acceptstheoffer,since3.5𝜇 > −3 + 𝜇 ⇒ 2.5𝜇 >− 3,whichholdsforall𝜇 ∈ (0,1).
– Therefore,theofferrejectionthat(𝑁C𝑁]C, 𝑅) prescribescannotbesequentiallyrational.
AdvancedMicroeconomicTheory 66
Sequential-MoveGamesunderIncompleteInformation
• Inordertoavoididentifyingequilibriumpredictionsthatarenotsequentiallyrational,weapplythePerfectBayesianEquilibrium(PBE)thatcandealwithsequentialmovegameswithincompleteinformation.
• ThePerfectBayesianEquilibrium(PBE):Astrategyprofit(𝑠/, 𝑠H, … , 𝑠]) andbeliefs𝜇 overthenodesatallinformationsetsareaPBEif:1. eachplayer’sstrategiesspecifyoptimalactions,giventhe
strategiesoftheotherplayers,andgivenhisbeliefs,and2. beliefsareconsistentwithBayes’s rule,whenever
possible.
AdvancedMicroeconomicTheory 67
Sequential-MoveGamesunderIncompleteInformation
• ThefirstconditionresemblesthedefinitionofBNE.• Thesecondconditionwasnotpresentinthedefinitionof
BNE.– ItstatesthatbeliefsmustbeconsistentwithBayes’s rule
wheneverpossible• ApplyingBayes’s ruleintheinvestmentgame,player2’s
probabilitythattheinvestmentisbeneficialafterreceivinganofferis
𝑝 𝐵 Offer =𝑝 𝐵 j 𝑝(Offer|𝐵)
𝑝(Offer)
=𝑝 𝐵 j 𝑝(Offer|𝐵)
𝑝 𝐵 j 𝑝 Offer 𝐵 + 𝑝 𝑁𝐵 j 𝑝(Offer|𝑁𝐵)
AdvancedMicroeconomicTheory 68
Sequential-MoveGamesunderIncompleteInformation
• Denoting𝜇 = 𝑝 𝐵 Offer ,𝛼$ = 𝑝(Offer|𝑖),where𝑖 =𝐵,𝑁𝐵 ,𝑝 = 𝑝 𝐵 ,and1 − 𝑝 = 𝑝 𝑁𝐵 ,player2’sbeliefcanbeexpressedas
𝜇 =𝑝 j 𝛼C
𝑝 j 𝛼C + (1 − 𝑝) j 𝛼]C• Ifplayer2assignsprobabilities𝛼C = 1/8 and𝛼]C =1/16,then
𝜇 =1/2 j 1/8
1/2 j 1/8 + 1/2 j 1/16 =23
• Wereferto𝜇 asoff-the-equilibriumbeliefs– Theprobabilityofbeinginanodeofaninformationsetthatisactuallynotreachedinequilibrium.
AdvancedMicroeconomicTheory 69
Sequential-MoveGamesunderIncompleteInformation
• ProceduretoFindPBEs:1. Specifyastrategyprofilefortheprivatelyinformed
player.– Intheinvestmentexample,therearefourpossiblestrategyprofilesfortheprivatelyinformedplayer1.
– Twoseparatingstrategyprofiles:𝑂C𝑁]C,𝑁C𝑂]C.– Twopoolingstrategyprofiles:𝑂C𝑂]C,𝑁C𝑁]C.
2. Updatetheuninformed player’sbeliefsusingBayes’sruleatallinformationsets,wheneverpossible.
AdvancedMicroeconomicTheory 70
Sequential-MoveGamesunderIncompleteInformation
• ProceduretoFindPBEs:(continued)3. Giventheuninformed player’supdatedbeliefs,findhis
optimalresponse– Intheinvestmentexample,weneedtodeterminetheoptimalresponseofplayer2uponreceivinganofferfromplayer1givenhisupdatedbelief.
4. Giventheoptimalresponseoftheuninformedplayerobtainedinstep3,findtheoptimalaction(message)foreachtypeofinformed player.
– Intheinvestmentexample,firstcheckifplayer1makesanofferwhentheinvestmentisbeneficial.
– Thencheckwhetherplayer1preferstomakeanoffer,whentheinvestmentisnotbeneficial.
AdvancedMicroeconomicTheory 71
Sequential-MoveGamesunderIncompleteInformation
• ProceduretoFindPBEs:(continued)5. Checkifthestrategyprofilefortheinformedplayer
foundinstep4coincideswiththeprofilesuggestedinstep1.
– Ifitcoincides,thenthisstrategyprofile,updatedbeliefs,andoptimalresponsescanbesupportedasaPBEoftheincompleteinformationgame.
– Otherwise,wesaythatthisstrategyprofilecannotbesustainedasaPBEofthegame.
AdvancedMicroeconomicTheory 72
Sequential-MoveGamesunderIncompleteInformation
• Example (Labormarketsignalinggame):– Thesequentialgamewithincompleteinformation.– Aworkerprivatelyobserveswhetherhehasahighproductivityoralowproductivity.
– Theworkerthendecideswhethertopursuemoreeducation(e.g.,anMBA)thathemightuseasasignalabouthisproductivity.
– Thefirmcaneitherhirehimasamanager(M)orasacashier(C).
AdvancedMicroeconomicTheory 73
Sequential-MoveGamesunderIncompleteInformation
AdvancedMicroeconomicTheory 74
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Wefocuson:
• Separatingstrategyprofiles: 𝐸g,𝑁𝐸h
• Poolingstrategyprofile:(𝑁𝐸g,𝑁𝐸h)– Exercise:
• Separatingstrategyprofiles: 𝑁𝐸g, 𝐸h
• Poolingstrategyprofile:(𝐸g, 𝐸h)
AdvancedMicroeconomicTheory 75
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):1. SeparatingPBE 𝑬𝑯,𝑵𝑬𝑳 :– Step1:Specifytheseparatingstrategyprofile𝐸g,𝑁𝐸h fortheinformedplayer.
AdvancedMicroeconomicTheory 76
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step2:UseBayes’s ruletoupdatetheuninformedplayer’s(firm)beliefs.• Takingintoaccount𝛼� = 1 while𝛼]� = 0,thefirmupdateditsbeliefsforaneducatedapplicantas
𝜇 =1/3 j 𝛼�
1/3 j 𝛼� + 2/3 j 𝛼]�= 1
• Intuitively,afterobservingthattheapplicantacquirededucation,thefirmassignsfullprobabilitytotheapplicantbeingofhighproductivity.
AdvancedMicroeconomicTheory 77
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):• Takingintoaccount𝛼� = 0 while𝛼]� = 1,thefirmupdateditsbeliefsforalessapplicantas
𝛾 =1/3 j 𝛼�
1/3 j 𝛼� + 2/3 j 𝛼]�= 0
1 − 𝛾 = 1• Intuitively,thefirmthatobservesthelesseducatedapplicantbelievesthatsuchanapplicantmustbeoflowproductivity.
AdvancedMicroeconomicTheory 78
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step3:Giventhefirm’sbeliefs,determinethefirm’soptimalresponse,afterobservingtheeducationleveloftheworker.
AdvancedMicroeconomicTheory 79
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step4:Giventhesestrategyprofiles,examinetheworker’soptimalaction.• High-productivitytype:Doesnothaveanincentivetodeviatefromthestrategyprofile(acquiringmoreeducation).
• Low-productivitytype:Thecostofacquiringeducationistoohighforthelow-productivityworker;andthusthatworkerchoosestonotpursueit.
AdvancedMicroeconomicTheory 80
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step5:Theseparatingstrategyprofile(𝐸g,𝑁𝐸h) canbesustainedasthePBEofthisincompleteinformationgame.• Neithertypeofworkerhastheincentivetodeviatefromtheprescribedseparatingstrategyprofile(𝐸g, 𝑁𝐸h).
AdvancedMicroeconomicTheory 81
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):2. PoolingPBE 𝑵𝑬𝑯,𝑵𝑬𝑳 :– Step1:Specifytheseparatingstrategyprofile𝑁𝐸g,𝑁𝐸hfortheinformedplayer.
AdvancedMicroeconomicTheory 82
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step2:UseBayes’s ruletoupdatetheuninformedplayer’s(firm)beliefs.• Takingintoaccount𝛼� = 1 while𝛼]� = 1,thefirmupdateditsbeliefsforaneducatedapplicantas
𝛾 =1/3 j 𝛼�
1/3 j 𝛼� + 2/3 j 𝛼]�= 1/3
• Intuitively,sinceneithertypeofapplicantobtainseducationinthisstrategyprofile,thefirm’sobservationofanuneducatedapplicantdoesnotallowthefirmtofurtherrestrictitsposteriorbeliefsabouttheapplicant’stype.
AdvancedMicroeconomicTheory 83
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):• Takingintoaccount𝛼� = 0 while𝛼]� = 0,thefirmupdateditsbeliefsforalessapplicantas
𝜇 =1/3 j 𝛼�
1/3 j 𝛼� + 2/3 j 𝛼]�= 0
• Thisplayer’soff-the-equilibriumbeliefsareleftunrestrictedat𝜇 ∈[0,1].
AdvancedMicroeconomicTheory 84
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step3:Giventhefirm’sbeliefs,determinethefirm’soptimalresponse,afterobservingtheeducationleveloftheworker.• Uponobservingalesseducatedapplicant:
𝐸𝑈nopq 𝑀4 Noeducation =13 j 10 +
23 j 0 =
103
𝐸𝑈nopq 𝐶4 Noeducation =13 j 4 +
23 j 4 = 4
• Hence,thefirmoptimallyrespondsbyofferingtheapplicantacashierposition.
AdvancedMicroeconomicTheory 85
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):• Uponobservingamoreeducatedapplicant:
𝐸𝑈nopq 𝑀 Education = 𝜇 j 10 + (1 − 𝜇) j 0 = 10𝜇𝐸𝑈nopq 𝐶 Education = 𝜇 j 4 + (1 − 𝜇) j 4 = 4
• Thefirmrespondsbyofferingtheapplicantamanagerpositionifandonlyif
10𝜇 > 4 ⇒ 𝜇 > 2/5• Wethusneedtodividethefifthsstep(theoptimalactionsoftheworker)intotwocases:1. 𝜇 > 2/5,wherethefirmrespondswith𝑀2. 𝜇 ≤ 2/5,wherethefirmrespondswith𝐶
AdvancedMicroeconomicTheory 86
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step4:Giventhesestrategyprofiles,examinetheworker’soptimalaction.
– Case1:𝜇 > 2/5• High-productivitytype:Hasanincentivetodeviatefromtheprescribedstrategyprofile.ThusitcannotbesupportedasaPBE.
• Low-productivitytype:Doesnothaveincentivestodeviatefromtheprescribedstrategyprofile.
AdvancedMicroeconomicTheory 87
Sequential-MoveGamesunderIncompleteInformation
AdvancedMicroeconomicTheory 88
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Case2:𝜇 ≤ 2/5
• High-productivitytype:Doesnothaveincentivestodeviatefromtheprescribedstrategyprofile.
• Low-productivitytype:Doesnothaveincentivestodeviatefromtheprescribedstrategyprofile.
AdvancedMicroeconomicTheory 89
Sequential-MoveGamesunderIncompleteInformation
AdvancedMicroeconomicTheory 90
Sequential-MoveGamesunderIncompleteInformation
• Example (continued):– Step5:Thepoolingstrategyprofile(𝑁𝐸g,𝑁𝐸h) canbesupportedasthePBEwhenoff-the-equilibriumbeliefssatisfy𝜇 ≤ 2/5.• Neithertypeofworkerhastheincentivetodeviatefromtheprescribedseparatingstrategyprofile(𝐸g, 𝑁𝐸h).
AdvancedMicroeconomicTheory 91
BertrandModelofPriceCompetition
AdvancedMicroeconomicTheory 92
BertrandModelofPriceCompetition
• Consider:– Anindustrywithtwofirms,1and2,sellingahomogeneousproduct
– Firmsfacemarketdemand𝑥(𝑝),where𝑥(𝑝) iscontinuousandstrictlydecreasingin𝑝
– Thereexistsahighenoughprice(chokeprice) �̅� < ∞ suchthat𝑥(𝑝) = 0 forall𝑝 > �̅�
– Bothfirmsaresymmetricintheirconstantmarginalcost𝑐 > 0,where𝑥 𝑐 ∈ (0,∞)
– Everyfirm𝑗 simultaneously setsaprice𝑝°
AdvancedMicroeconomicTheory 93
BertrandModelofPriceCompetition
• Firm𝑗’sdemandis
𝑥°(𝑝°, 𝑝±) =
𝑥(𝑝°)if𝑝° < 𝑝±12 𝑥(𝑝°)if𝑝° = 𝑝±0if𝑝° > 𝑝±
• Intuition:Firm𝑗 captures– allmarketifitspriceisthelowest,𝑝° < 𝑝±– nomarketifitspriceisthehighest,𝑝° > 𝑝±– sharesthemarketwithfirm𝑘 ifthepriceofbothfirmscoincide,𝑝° = 𝑝±
AdvancedMicroeconomicTheory 94
BertrandModelofPriceCompetition
• Givenprices𝑝° and𝑝±,firm𝑗’sprofitsaretherefore
(𝑝° − 𝑐) j 𝑥° (𝑝°, 𝑝±)
• WearenowreadytofindequilibriumpricesintheBertrandduopolymodel.– ThereisauniqueNE(𝑝°∗, 𝑝±∗) intheBertrandduopolymodel.Inthisequilibrium,bothfirmssetpricesequaltomarginalcost,𝑝°∗ = 𝑝±∗ = 𝑐.
AdvancedMicroeconomicTheory 95
BertrandModelofPriceCompetition
• Let’susdescribethebestresponsefunctionoffirm𝑗.• If𝑝± < 𝑐,firm𝑗 setsitspriceat𝑝° = 𝑐.– Firm𝑗 doesnotundercutfirm𝑘 sincethatwouldentailnegativeprofits.
• If𝑐 < 𝑝± < 𝑝°,firm𝑗 slightlyundercutsfirm𝑘,i.e.,𝑝° = 𝑝± − 𝜀.– Thisallowsfirm𝑗 tocaptureallsalesandstillmakeapositivemarginoneachunit.
• If𝑝± > 𝑝�, where𝑝� isamonopolyprice,firm𝑗 doesnotneedtochargemorethan𝑝�,i.e.,𝑝° = 𝑝�.– 𝑝� allowsfirm𝑗 tocaptureallsalesandmaximizeprofitsastheonlyfirmsellingapositiveoutput.
AdvancedMicroeconomicTheory 96
pj
pk
pm
pm
c
c
pj(pk)
45°-line(pj=pk)
BertrandModelofPriceCompetition
• Firm𝑗’sbestresponsehas:– aflatsegmentforall𝑝± < 𝑐,where𝑝°(𝑝±) = 𝑐
– apositiveslopeforall𝑐 < 𝑝± < 𝑝°,wherefirm𝑗 chargesapriceslightlybelowfirm𝑘
– aflatsegmentforall𝑝± > 𝑝�,where𝑝°(𝑝±) = 𝑝�
AdvancedMicroeconomicTheory 97
pj
pk
c
c
pj(pk)
pk(pj)
pm
pm
45°-line(pj=pk)
BertrandModelofPriceCompetition
• Asymmetricargumentappliestotheconstructionofthebestresponsefunctionoffirm𝑘.
• Amutualbestresponseforbothfirmsis
(𝑝/∗, 𝑝H∗) = (𝑐, 𝑐)wherethetwobestresponsefunctionscrosseachother.
• ThisistheNEoftheBertrandmodel– Firmsmakenoeconomicprofits.
AdvancedMicroeconomicTheory 98
BertrandModelofPriceCompetition
• Withonlytwofirmscompetinginpricesweobtaintheperfectlycompetitiveoutcome,wherefirmssetpricesequaltomarginalcost.
• Pricecompetitionmakeseachfirm𝑗 faceaninfinitelyelasticdemandcurveatitsrival’sprice,𝑝±.– Anyincrease(decrease)from𝑝± infinitelyreduces(increases,respectively)firm𝑗’sdemand.
AdvancedMicroeconomicTheory 99
BertrandModelofPriceCompetition
• HowmuchdoesBertrandequilibriumhingeintoourassumptions?– Quitealot
• ThecompetitivepressureintheBertrandmodelwithhomogenousproductsisamelioratedifweinsteadconsider:– Pricecompetition(butallowingforheterogeneousproducts)
– Quantitycompetition(stillwithhomogenousproducts)
– CapacityconstraintsAdvancedMicroeconomicTheory 100
BertrandModelofPriceCompetition
• Remark:– Howourresultswouldbeaffectediffirmsfacedifferentproductioncosts,i.e.,0 < 𝑐/ < 𝑐H?
– Themostefficientfirmsetsapriceequaltothemarginalcostoftheleastefficientfirm,𝑝/ = 𝑐H.
– Otherfirmswillsetarandompriceintheuniforminterval
[𝑐/, 𝑐/ + 𝜂]where𝜂 > 0 issomesmallrandomincrementwithprobabilitydistribution𝑓 𝑝, 𝜂 > 0 forall𝑝.
AdvancedMicroeconomicTheory 101
CournotModelofQuantityCompetition
AdvancedMicroeconomicTheory 102
CournotModelofQuantityCompetition
• Letusnowconsiderthatfirmscompeteinquantities.
• Assumethat:– Firmsbringtheiroutput𝑞/ and𝑞H toamarket,themarketclears,andthepriceisdeterminedfromtheinversedemandfunction𝑝(𝑞),where𝑞 = 𝑞/ + 𝑞H.
– 𝑝(𝑞) satisfies𝑝’(𝑞) < 0 atalloutputlevels𝑞 ≥ 0,– Bothfirmsfaceacommonmarginalcost𝑐 > 0– 𝑝(0) > 𝑐 inordertoguarantee thattheinversedemandcurvecrossestheconstantmarginalcostcurveataninteriorpoint.
AdvancedMicroeconomicTheory 103
CournotModelofQuantityCompetition
• Letusfirstidentifyeveryfirm’sbestresponsefunction
• Firm1’sPMP,foragivenoutputlevelofitsrival,𝑞·H,
maxº»¼½
𝑝 𝑞/ + 𝑞·HPrice
𝑞/ − 𝑐𝑞/
• WhensolvingthisPMP,firm1treatsfirm2’sproduction,𝑞·H, asaparameter,sincefirm1cannotvaryitslevel.
AdvancedMicroeconomicTheory 104
CournotModelofQuantityCompetition
• FOCs:𝑝4(𝑞/ + 𝑞·H)𝑞/ + 𝑝(𝑞/ + 𝑞·H) − 𝑐 ≤ 0
withequalityif𝑞/ > 0• Solvingthisexpressionfor𝑞/,weobtainfirm1’sbestresponsefunction(BRF),𝑞/(𝑞·H).
• Asimilarargumentappliestofirm2’sPMPanditsbestresponsefunction𝑞H(𝑞·/).
• Therefore,apairofoutputlevels(𝑞/∗, 𝑞H∗) isNEoftheCournotmodelifandonlyif
𝑞/∗ ∈ 𝑞/(𝑞·H) forfirm1’soutput𝑞H∗ ∈ 𝑞H(𝑞·/) forfirm2’soutput
AdvancedMicroeconomicTheory 105
CournotModelofQuantityCompetition
• Toshowthat𝑞/∗, 𝑞H∗ > 0,letusworkbycontradiction,assuming𝑞/∗ = 0.– Firm2becomesamonopolistsinceitistheonlyfirmproducingapositiveoutput.
• UsingtheFOCforfirm1,weobtain𝑝4(0 + 𝑞H∗)0 + 𝑝(0 + 𝑞H∗) ≤ 𝑐
or𝑝(𝑞H∗) ≤ 𝑐• AndusingtheFOCforfirm2,wehave
𝑝4(𝑞H∗ + 0)𝑞H∗ + 𝑝(𝑞H∗ + 0) ≤ 𝑐or𝑝4(𝑞H∗)𝑞H∗ + 𝑝(𝑞H∗) ≤ 𝑐
• Thisimpliesfirm2’sMRundermonopolyislowerthanitsMC.Thus,𝑞H∗ = 0.
AdvancedMicroeconomicTheory 106
CournotModelofQuantityCompetition
• Hence,if𝑞/∗ = 0, firm2’soutputwouldalsobezero,𝑞H∗ = 0.
• Butthisimpliesthat𝑝(0) < 𝑐 sincenofirmproducesapositiveoutput,thusviolatingourinitialassumption𝑝(0) > 𝑐.– Contradiction!
• Asaresult,wemusthavethatboth𝑞/∗ > 0 and𝑞H∗ > 0.
• Note:Thisresultdoesnotnecessarilyholdwhenbothfirmsareasymmetricintheirproductioncost.
AdvancedMicroeconomicTheory 107
CournotModelofQuantityCompetition
• Example (symmetriccosts):– Consideraninversedemandcurve𝑝(𝑞) = 𝑎 −𝑏𝑞,andtwofirmscompetingàlaCournotbothfacingamarginalcost𝑐 > 0.
– Firm1’sPMPis𝑎 − 𝑏(𝑞/ + 𝑞·H) 𝑞/ − 𝑐𝑞/
– FOCwrt 𝑞/:𝑎 − 2𝑏𝑞/ − 𝑏𝑞·H − 𝑐 ≤ 0withequalityif𝑞/ > 0
AdvancedMicroeconomicTheory 108
CournotModelofQuantityCompetition
• Example (continue):– Solvingfor𝑞/, weobtainfirm1’sBRF
𝑞/(𝑞·H) =¿*ÀHÁ
− º·ÂH
– Analogously,firm2’sBRF
𝑞H(𝑞·/) =¿*ÀHÁ
− º·»H
AdvancedMicroeconomicTheory 109
CournotModelofQuantityCompetition
AdvancedMicroeconomicTheory 110
• Firm1’sBRF:– When𝑞H = 0,then𝑞/ =
¿*ÀHÁ
,whichcoincideswithitsoutputundermonopoly.
– As𝑞H increases,𝑞/decreases (i.e.,firm1’sand2’soutputarestrategicsubstitutes)
– When𝑞H =¿*ÀÁ, then
𝑞/ = 0.
CournotModelofQuantityCompetition
AdvancedMicroeconomicTheory 111
• Asimilarargumentappliesforfirm2’sBRF.
• Superimposingbothfirms’BRFs,weobtaintheCournotequilibriumoutputpair(𝑞/∗, 𝑞H∗).
CournotModelofQuantityCompetition
AdvancedMicroeconomicTheory 112
q1
q2
a–c
q1(q2)
2bq2(q1)
a–cb
a–cb
a–c2b
a–c3b
a–c3b
(q1,q2)**
45°-line(q1=q2)q1+q2=qc= a–cb
q1+q2=qm= a–c2b
Perfectcompetition
Monopoly
45°
CournotModelofQuantityCompetition
• Cournotequilibriumoutputpair(𝑞/∗, 𝑞H∗) occursattheintersectionofthetwoBRFs,i.e.,
(𝑞/∗, 𝑞H∗) =¿*ÀÃÁ, ¿*ÀÃÁ
• Aggregateoutputbecomes
𝑞∗ = 𝑞/∗ + 𝑞H∗ =¿*ÀÃÁ
+ ¿*ÀÃÁ
= H(¿*À)ÃÁ
whichislargerthanundermonopoly,𝑞� = ¿*ÀHÁ
,butsmallerthanunderperfectcompetition,𝑞À =¿*ÀÁ.
AdvancedMicroeconomicTheory 113
CournotModelofQuantityCompetition
• Theequilibriumpricebecomes
𝑝 𝑞∗ = 𝑎 − 𝑏𝑞∗ = 𝑎 − 𝑏 H ¿*ÀÃÁ
= ¿1HÀÃ
whichislowerthanundermonopoly,𝑝� = ¿1ÀH,but
higherthanunderperfectcompetition,𝑝À = 𝑐.
• Finally,theequilibriumprofitsofeveryfirm𝑗
𝜋°∗ = 𝑝 𝑞∗ 𝑞°∗ − 𝑐𝑞°∗ =¿1HÀÃ
¿*ÀÃÁ
− 𝑐 ¿*ÀÃÁ
= ¿*À Â
ÄÁ
whicharelowerthanundermonopoly,𝜋� = ¿*À Â
ÅÁ,
buthigherthanunderperfectcompetition,𝜋À = 0.AdvancedMicroeconomicTheory 114
CournotModelofQuantityCompetition
• Quantitycompetition(Cournotmodel)yieldslesscompetitiveoutcomesthanpricecompetition(Bertrandmodel),wherebyfirms’behaviormimicsthatinperfectlycompetitivemarkets– That’sbecause,thedemandthateveryfirmfacesintheCournotgameisnotinfinitelyelastic.
– Areductioninoutputdoesnotproduceaninfiniteincreaseinmarketprice,butinsteadanincreaseof− 𝑝′(𝑞/ + 𝑞H).
– Hence,iffirmsproducethesameoutputasundermarginalcostpricing,i.e.,halfof¿*À
H,eachfirmwould
haveincentivestodeviatefromsuchahighoutputlevelbymarginallyreducingitsoutput.
AdvancedMicroeconomicTheory 115
CournotModelofQuantityCompetition
• EquilibriumoutputunderCournotdoesnotcoincidewiththemonopolyoutputeither.– That’sbecause,everyfirm𝑖,individuallyincreasingitsoutputlevel𝑞$,takesintoaccounthowthereductioninmarketpriceaffectsitsownprofits,butignorestheprofitloss(i.e., anegativeexternaleffect)thatitsrivalsuffersfromsuchalowerprice.
– Sinceeveryfirmdoesnottakeintoaccountthisexternaleffect,aggregateoutputistoolarge,relativetotheoutputthatwouldmaximizefirms’jointprofits.
AdvancedMicroeconomicTheory 116
CournotModelofQuantityCompetition
• Example (Cournotvs.Cartel):– Letusdemonstratethatfirms’Cournotoutputislargerthanthatunderthecartel.
– PMPofthecartelismaxº»,ºÂ
(𝑎 − 𝑏(𝑞/+𝑞H))𝑞/ − 𝑐𝑞/+ (𝑎 − 𝑏(𝑞/+𝑞H))𝑞H − 𝑐𝑞H
– Since𝑄 = 𝑞/ + 𝑞H,thePMPcanbewrittenasmaxº»,ºÂ
𝑎 − 𝑏(𝑞/+𝑞H) (𝑞/+𝑞H) − 𝑐(𝑞/+𝑞H)
= maxÈ
𝑎 − 𝑏𝑄 𝑄 − 𝑐𝑄 = 𝑎𝑄 − 𝑏𝑄H − 𝑐𝑄AdvancedMicroeconomicTheory 117
CournotModelofQuantityCompetition
• Example (continued):– FOCwithrespectto𝑄
𝑎 − 2𝑏𝑄 − 𝑐 ≤ 0– Solvingfor𝑄,weobtaintheaggregateoutput
𝑄∗ = ¿*ÀHÁ
whichispositivesince𝑎 > 𝑐,i.e.,𝑝(0) = 𝑎 > 𝑐.– Sincefirmsaresymmetricincosts,eachproduces
𝑞$ =ÈH= ¿*À
ÅÁ
AdvancedMicroeconomicTheory 118
CournotModelofQuantityCompetition
• Example (continued):– Theequilibriumpriceis
𝑝 = 𝑎 − 𝑏𝑄 = 𝑎 − 𝑏 ¿*ÀHÁ
= ¿1ÀH
– Finally,theequilibriumprofitsare𝜋$ = 𝑝 ⋅ 𝑞$ − 𝑐𝑞$
= ¿1ÀH⋅ ¿*ÀÅÁ
− 𝑐 ¿*ÀÅÁ
= ¿*À Â
ÊÁwhichislargerthanfirmswouldobtainunder
Cournotcompetition, ¿*ÀÂ
ÄÁ.
AdvancedMicroeconomicTheory 119
CournotModelofQuantityCompetition:CournotPricingRule
• Firms’marketpowercanbeexpressedusingavariationoftheLernerindex.– Considerfirm𝑗’sprofitmaximizationproblem
𝜋° = 𝑝(𝑞)𝑞° − 𝑐°(𝑞°)– FOCforeveryfirm𝑗
𝑝′ 𝑞 𝑞° + 𝑝 𝑞 − 𝑐° = 0or𝑝(𝑞) − 𝑐° = −𝑝′ 𝑞 𝑞°
– Multiplyingbothsidesby𝑞 anddividingthemby𝑝(𝑞)yield
𝑞𝑝 𝑞 − 𝑐°𝑝(𝑞) =
−𝑝4 𝑞 𝑞°𝑝(𝑞) 𝑞
AdvancedMicroeconomicTheory 120
CournotModelofQuantityCompetition:CournotPricingRule
– Recalling/Ë= −𝑝4 𝑞 ⋅ º
� º,wehave
𝑞 � º *ÀÌ�(º)
= /Ë𝑞°
or� º *ÀÌ�(º)
= /˺̺
– Defining𝛼° ≡ºÌº
asfirm𝑗’smarketshare,weobtain𝑝 𝑞 − 𝑐°𝑝(𝑞) =
𝛼°𝜀
whichisreferredtoastheCournotpricingrule.AdvancedMicroeconomicTheory 121
CournotModelofQuantityCompetition:CournotPricingRule
– Note:§When𝛼° = 1,implyingthatfirm𝑗 isamonopoly,theIEPRbecomesaspecialcaseoftheCournotpricerule.
§ Thelargerthemarketshare𝛼° ofagivenfirm,thelargerthepricemarkupoffirm𝑗.
§ Themoreinelasticdemand𝜀 is,thelargerthepricemarkupoffirm𝑗.
AdvancedMicroeconomicTheory 122
CournotModelofQuantityCompetition:CournotPricingRule
• Example (MergereffectsonCournotPrices):– Consideranindustrywith𝑛 firms andaconstant-elasticitydemandfunction𝑞(𝑝) = 𝑎𝑝*/, where𝑎 > 0 and𝜀 = 1.
– Beforemerger,wehave𝑝C − 𝑐𝑝C =
1𝑛 ⟹ 𝑝C =
𝑛𝑐𝑛 − 1
– Afterthemergerof𝑘 < 𝑛 firms𝑛 − 𝑘 + 1 firmsremainintheindustry,andthus
𝑝A − 𝑐𝑝A =
1𝑛 − 𝑘 + 1 ⟹ 𝑝A =
𝑛 − 𝑘 + 1 𝑐𝑛 − 𝑘
AdvancedMicroeconomicTheory 123
CournotModelofQuantityCompetition:CournotPricingRule
• Example (continued):– Thepercentagechangeinpricesis
%Δ𝑝 =𝑝A − 𝑝C
𝑝C =𝑛 − 𝑘 + 1 𝑐𝑛 − 𝑘 − 𝑛𝑐
𝑛 − 1𝑛𝑐𝑛 − 1
=𝑘 − 1
𝑛(𝑛 − 𝑘) > 0
– Hence,pricesincreaseafterthemerger.– Also,%Δ𝑝increasesasthenumberofmergingfirms𝑘 increases
𝜕%Δ𝑝𝜕𝑘 =
𝑛 − 1𝑛 𝑛 − 𝑘 H > 0
AdvancedMicroeconomicTheory 124
%Δp
k20 40 60 80 100
0.10
0.20 %Δp(k)
CournotModelofQuantityCompetition:CournotPricingRule
• Example (continued):– Thepercentageincreaseinpriceafterthemerger,%Δ𝑝,asafunctionofthenumberofmergingfirms,𝑘.
– Forsimplicity,𝑛 =100.
AdvancedMicroeconomicTheory 125
CournotModelofQuantityCompetition:SOC
• Letuscheckifthefirstorder(necessary)conditionsarealsosufficient.
• RecallthatFOCsare𝑝4 𝑞 𝑞° + 𝑝 𝑞 − 𝑐°4(𝑞°) ≤ 0
• DifferentiatingFOCswrt 𝑞° yields𝑝44 𝑞 𝑞° + 𝑝4 𝑞 + 𝑝4 𝑞 − 𝑐°44(𝑞°) ≤ 0
– 𝑝4 𝑞 < 0:bydefinition(negativelyslopedinversedemandcurve)
– 𝑐°44(𝑞°) ≥ 0: byassumption(constantorincreasingmarginalcosts)
– 𝑝44 𝑞 𝑞° ≤ 0:aslongasthedemandcurvedecreasesataconstantordecreasingrate
AdvancedMicroeconomicTheory 126
CournotModelofQuantityCompetition:SOC
• Example (lineardemand):– Thelinearinversedemandcurveis𝑝(𝑞) = 𝑎 − 𝑏𝑞andconstantmarginalcostis𝑐 > 0.
– Since𝑝4 𝑞 = −𝑏 < 0, 𝑝44 𝑞 = 0, 𝑐4 𝑞 = 𝑐 and𝑐44(𝑞) = 0,theSOCreducesto
0 − 2𝑏 − 0 = −2𝑏 < 0
where𝑏 > 0 bydefinition.– Hencetheequilibriumoutputisindeedprofitmaximizing.
AdvancedMicroeconomicTheory 127
CournotModelofQuantityCompetition:SOC
• NotethatSOCscoincideswiththecross-derivative
𝜕H𝜋°𝜕𝑞°𝜕𝑞±
=𝜕𝜕𝑞±
𝑝4 𝑞 𝑞° + 𝑝 𝑞 − 𝑐4(𝑞°)
= 𝑝44 𝑞 𝑞° − 𝑝4 𝑞 forall𝑘 ≠ 𝑗.• Hence,thefirm𝑗’sBRFdecreasesin𝑞± aslongas𝑝44 𝑞 𝑞° − 𝑝4 𝑞 < 0– Thatis,firm𝑗’sBRFisnegativelysloped.
AdvancedMicroeconomicTheory 128
CournotModelofQuantityCompetition:AsymmetricCosts
• Assumethatfirm1and2’sconstantmarginalcostsofproductiondiffer,i.e.,𝑐/ > 𝑐H,sofirm2ismoreefficientthanfirm1.Assumealsothattheinversedemandfunctionis𝑝 𝑄 = 𝑎 − 𝑏𝑄,and𝑄 = 𝑞/ + 𝑞H.
• Firm𝑖’sPMPismaxºÒ
𝑎 − 𝑏(𝑞$ + 𝑞°) 𝑞$ − 𝑐$𝑞$• FOC:
𝑎 − 2𝑏𝑞$ − 𝑏𝑞° − 𝑐$ = 0
AdvancedMicroeconomicTheory 129
CournotModelofQuantityCompetition:AsymmetricCosts
• Solvingfor𝑞$ (assuminganinteriorsolution)yieldsfirm𝑖’sBRF
𝑞$(𝑞°) =𝑎 − 𝑐$2𝑏 −
𝑞°2
• Firm1’soptimaloutputlevelcanbefoundbypluggingfirm2’sBRFintofirm1’s
𝑞/∗ =𝑎 − 𝑐/2𝑏 −
12𝑎 − 𝑐H2𝑏 −
𝑞/∗
2 ⟺ 𝑞/∗ =𝑎 − 2𝑐/ + 𝑐H
3𝑏• Similarly,firm2’soptimaloutputlevelis
𝑞H∗ =𝑎 − 𝑐H2𝑏 −
𝑞/∗
2 =𝑎 + 𝑐/ − 2𝑐H
3𝑏AdvancedMicroeconomicTheory 130
CournotModelofQuantityCompetition:AsymmetricCosts
• Iffirm𝑖’s costsaresufficientlyhighitwillnotproduceatall.– Firm1: 𝑞/∗ ≤ 0 if¿1ÀÂ
H≤ 𝑐/
– Firm2: 𝑞H∗ ≤ 0 if¿1À»H
≤ 𝑐H• Thus,wecanidentifythreedifferentcases:– If𝑐$ ≥
¿1ÀÌH
forallfirms𝑖 = {1,2},nofirmproducesapositiveoutput
– If𝑐$ ≥¿1ÀÌH
but𝑐° <¿1ÀÒH
,thenonlyfirm𝑗 producespositiveoutput
– If𝑐$ <¿1ÀÌH
forallfirms𝑖 = {1,2},bothfirmsproducepositiveoutput
AdvancedMicroeconomicTheory 131
c2
c1
a
a
45°(c1=c2)a+c22c1=
a+c12c2=
a2
a2
Nofirmsproduce
Bothfirmsproduce
Onlyfirm1produces
Onlyfirm2produces
CournotModelofQuantityCompetition:AsymmetricCosts
AdvancedMicroeconomicTheory 132
CournotModelofQuantityCompetition:AsymmetricCosts
• Theoutputlevels(𝑞/∗, 𝑞H∗) alsovarywhen(𝑐/, 𝑐H)changes
𝜕𝑞/∗
𝜕𝑐/= −
23𝑏 < 0and
𝜕𝑞/∗
𝜕𝑐H=13𝑏 > 0
𝜕𝑞H∗
𝜕𝑐/=13𝑏 > 0and
𝜕𝑞H∗
𝜕𝑐H= −
23𝑏 < 0
• Intuition:Eachfirm’soutputdecreasesinitsowncosts,butincreasesinitsrival’scosts.
AdvancedMicroeconomicTheory 133
q1
q2a–c22b
a–c12b
a–c1b
a–c2b
(q1,q2)**q1(q2)
q2(q1)
CournotModelofQuantityCompetition:AsymmetricCosts
• BRFsforfirms1and2when𝑐/ >
¿1ÀÂH
(i.e.,onlyfirm2produces).
• BRFscrossattheverticalaxiswhere𝑞/∗ = 0 and𝑞H∗ > 0 (i.e.,acornersolution)
AdvancedMicroeconomicTheory 134
CournotModelofQuantityCompetition:𝐽 > 2 firms
• Consider𝐽 > 2firms,allfacingthesameconstantmarginalcost𝑐 > 0.Thelinearinversedemandcurveis𝑝 𝑄 = 𝑎 − 𝑏𝑄,where𝑄 =∑ 𝑞±�Ù .
• Firm𝑖’sPMPis
maxºÒ
𝑎 − 𝑏 𝑞$ +Ú𝑞±
�
±Û$
𝑞$ − 𝑐𝑞$
• FOC:
𝑎 − 2𝑏𝑞$∗ − 𝑏Ú𝑞±∗�
±Û$
− 𝑐 ≤ 0
AdvancedMicroeconomicTheory 135
CournotModelofQuantityCompetition:𝐽 > 2 firms
• Solvingfor𝑞$∗,weobtainfirm𝑖’sBRF
𝑞$∗ =𝑎 − 𝑐2𝑏 −
12Ú𝑞±∗
�
±Û$• Sinceallfirmsaresymmetric,theirBRFsarealsosymmetric,implying𝑞/∗ = 𝑞H∗ = ⋯ = 𝑞Ù∗.Thisimplies∑ 𝑞±∗�
±Û$ = 𝐽𝑞$∗ − 𝑞$∗ = 𝐽 − 1 𝑞$∗.• Hence,theBRFbecomes
𝑞$∗ =𝑎 − 𝑐2𝑏 −
12 𝐽 − 1 𝑞$∗
AdvancedMicroeconomicTheory 136
CournotModelofQuantityCompetition:𝐽 > 2 firms
• Solvingfor𝑞$∗
𝑞$∗ =𝑎 − 𝑐𝐽 + 1 𝑏
whichisalsotheequilibriumoutputforother𝐽 − 1firms.
• Therefore,aggregateoutputis
𝑄∗ = 𝐽𝑞$∗ =𝐽
𝐽 + 1𝑎 − 𝑐𝑏
andthecorrespondingequilibriumpriceis
𝑝∗ = 𝑎 − 𝑏𝑄∗ =𝑎 + 𝐽𝑐𝐽 + 1
AdvancedMicroeconomicTheory 137
CournotModelofQuantityCompetition:𝐽 > 2 firms
• Firm𝑖’sequilibriumprofitsare𝜋$∗ = 𝑎 − 𝑏𝑄∗ 𝑞$∗ − 𝑐𝑞$∗
= 𝑎 − 𝑏𝐽
𝐽 + 1𝑎 − 𝑐𝑏
𝑎 − 𝑐𝐽 + 1 𝑏 − 𝑐
𝑎 − 𝑐𝐽 + 1 𝑏
=𝑎 − 𝑐𝐽 + 1 𝑏
H= 𝑞$∗ H
AdvancedMicroeconomicTheory 138
CournotModelofQuantityCompetition:𝐽 > 2 firms
• Wecanshowthat
limÙ→H
𝑞$∗ =𝑎 − 𝑐2 + 1 𝑏 =
𝑎 − 𝑐3𝑏
limÙ→H
𝑄∗ =2(𝑎 − 𝑐)2 + 1 𝑏 =
2(𝑎 − 𝑐)3𝑏
limÙ→H
𝑝∗ =𝑎 + 2𝑐2 + 1 =
𝑎 + 2𝑐3
whichexactlycoincidewithourresultsintheCournotduopolymodel.
AdvancedMicroeconomicTheory 139
CournotModelofQuantityCompetition:𝐽 > 2 firms
• Wecanshowthat
limÙ→/
𝑞$∗ =𝑎 − 𝑐1 + 1 𝑏 =
𝑎 − 𝑐2𝑏
limÙ→/
𝑄∗ =1(𝑎 − 𝑐)1 + 1 𝑏 =
𝑎 − 𝑐2𝑏
limÙ→/
𝑝∗ =𝑎 + 1𝑐1 + 1 =
𝑎 + 𝑐2
whichexactlycoincidewithourfindingsinthemonopoly.
AdvancedMicroeconomicTheory 140
CournotModelofQuantityCompetition:𝐽 > 2 firms
• WecanshowthatlimÙ→ß
𝑞$∗ = 0
limÙ→ß
𝑄∗ =𝑎 − 𝑐𝑏
limÙ→ß
𝑝∗ = 𝑐
whichcoincideswiththesolutioninaperfectlycompetitivemarket.
AdvancedMicroeconomicTheory 141
ProductDifferentiation
AdvancedMicroeconomicTheory 142
ProductDifferentiation
• Sofarweassumedthatfirmssellhomogenous(undifferentiated)products.
• Whatifthegoodsfirmssellaredifferentiated?– Forsimplicity,wewillassumethatproductattributesareexogenous(notchosenbythefirm)
AdvancedMicroeconomicTheory 143
ProductDifferentiation:BertrandModel
• Considerthecasewhereeveryfirm𝑖,for𝑖 ={1,2},facesdemandcurve
𝑞$(𝑝$, 𝑝°) = 𝑎 − 𝑏𝑝$ + 𝑐𝑝°where𝑎, 𝑏, 𝑐 > 0 and𝑗 ≠ 𝑖.
• Hence,anincreasein𝑝° increasesfirm𝑖’ssales.• Firm𝑖’sPMP:
max�Ò¼½
(𝑎 − 𝑏𝑝$ + 𝑐𝑝°)𝑝$
• FOC:𝑎 − 2𝑏𝑝$ + 𝑐𝑝° = 0
AdvancedMicroeconomicTheory 144
ProductDifferentiation:BertrandModel
• Solvingfor𝑝$,wefindfirm𝑖’sBRF
𝑝$(𝑝°) =𝑎 + 𝑐𝑝°2𝑏
• Firm𝑗 alsohasasymmetricBRF.• Note:– BRFsarenowpositivelysloped– Anincreaseinfirm𝑗’spriceleadsfirm𝑖 toincreasehis,andviceversa
– Inthiscase,firms’choices(i.e.,prices)arestrategiccomplements
AdvancedMicroeconomicTheory 145
ProductDifferentiation:BertrandModel
AdvancedMicroeconomicTheory 146
p1
p2
a2b
p2*
p1(p2)
p2(p1)
p1*
a2b
ProductDifferentiation:BertrandModel
• SimultaneouslysolvingthetwoBRSyields𝑝$∗ =
𝑎2𝑏 − 𝑐
withcorrespondingequilibriumsalesof
𝑞$∗(𝑝$∗, 𝑝°∗) = 𝑎 − 𝑏𝑝$∗ + 𝑐𝑝°∗ =𝑎𝑏
2𝑏 − 𝑐andequilibriumprofitsof
𝜋$∗ = 𝑝$∗ j 𝑞$∗ 𝑝$∗, 𝑝°∗ =𝑎
2𝑏 − 𝑐𝑎𝑏
2𝑏 − 𝑐
=𝑎H𝑏
2𝑏 − 𝑐 HAdvancedMicroeconomicTheory 147
ProductDifferentiation:CournotModel
• Considertwofirmswiththefollowinglinearinversedemandcurves
𝑝/(𝑞/, 𝑞H) = 𝛼 − 𝛽𝑞/ − 𝛾𝑞Hforfirm1𝑝H(𝑞/, 𝑞H) = 𝛼 − 𝛾𝑞/ − 𝛽𝑞Hforfirm2
• Weassumethat𝛽 > 0 and𝛽 > 𝛾– Thatis,theeffectofincreasing𝑞/ on𝑝/ islargerthantheeffectofincreasing𝑞/ on𝑝H
– Intuitively,thepriceofaparticularbrandismoresensitivetochangesinitsownoutputthantochangesinitsrival’soutput
– Inotherwords,own-priceeffectsdominatethecross-priceeffects.
AdvancedMicroeconomicTheory 148
ProductDifferentiation:CournotModel
• Firm𝑖’sPMPis(assumingnocosts)maxºÒ¼½
(𝛼 − 𝛽𝑞$ − 𝛾𝑞°)𝑞$
• FOC:𝛼 − 2𝛽𝑞$ − 𝛾𝑞° = 0
• Solvingfor𝑞$ wefindfirm𝑖’sBRF𝑞$(𝑞°) =
𝛼2𝛽 −
𝛾2𝛽 𝑞°
• Firm𝑗 alsohasasymmetricBRF
AdvancedMicroeconomicTheory 149
ProductDifferentiation:CournotModel
AdvancedMicroeconomicTheory 150
q1
q2
(q1,q2)**
q1(q2)
q2(q1)α2β
αγ
α2β
αγ
ProductDifferentiation:CournotModel
• Comparativestaticsoffirm𝑖’sBRF– As𝛽 → 𝛾 (productsbecomemorehomogeneous),BRFbecomessteeper.Thatis,theprofit-maximizingchoiceof𝑞$ ismoresensitivetochangesin𝑞° (toughercompetition)
– As𝛾 → 0 (productsbecomeverydifferentiated),firm𝑖’sBRFnolongerdependson𝑞° andbecomesflat(mildercompetition)
AdvancedMicroeconomicTheory 151
ProductDifferentiation:CournotModel
• SimultaneouslysolvingthetwoBRFyields
𝑞$∗ =𝛼
2𝛽 + 𝛾 forall𝑖 = {1,2}
withacorrespondingequilibriumpriceof
𝑝$∗ = 𝛼 − 𝛽𝑞$∗ − 𝛾𝑞°∗ =𝛼𝛽
2𝛽 + 𝛾andequilibriumprofitsof
𝜋$∗ = 𝑝$∗𝑞$∗ =𝛼𝛽
2𝛽 + 𝛾𝛼
2𝛽 + 𝛾 =𝛼H𝛽
2𝛽 + 𝛾 H
AdvancedMicroeconomicTheory 152
ProductDifferentiation:CournotModel
• Note:– As𝛾 increases(productsbecomemorehomogeneous),individualandaggregateoutputdecrease,andindividualprofitsdecreaseaswell.
– If𝛾 → 𝛽 (indicatingundifferentiatedproducts),then𝑞$∗ =
àHá1á
= àÃá
asinstandardCournotmodelsofhomogeneousproducts.
– If𝛾 → 0 (extremelydifferentiatedproducts),then𝑞$∗ =
àHá1½
= àHá
asinmonopoly.
AdvancedMicroeconomicTheory 153
DynamicCompetition
AdvancedMicroeconomicTheory 154
DynamicCompetition:SequentialBertrandModelwithHomogeneousProducts
• Assumethatfirm1choosesitsprice𝑝/ first,whereasfirm2observesthatpriceandrespondswithitsownprice𝑝H.
• Sincethegameisasequential-movegame(ratherthanasimultaneous-movegame),weshouldusebackwardinduction.
AdvancedMicroeconomicTheory 155
DynamicCompetition:SequentialBertrandModelwithHomogeneousProducts
• Firm2(thefollower)hasaBRFgivenby
𝑝H(𝑝/) = â𝑝/ − 𝜀if𝑝/ > 𝑐𝑐if𝑝/ ≤ 𝑐
whilefirm1’s(theleader’s)BRFis𝑝/ = 𝑐
• Intuition:thefollowerundercutstheleader’sprice𝑝/ byasmall𝜀 > 0 if𝑝/ > 𝑐,orkeepsitat𝑝H = 𝑐 iftheleadersets𝑝/ = 𝑐.
AdvancedMicroeconomicTheory 156
DynamicCompetition:SequentialBertrandModelwithHomogeneousProducts
• Theleaderexpectsthatitspricewillbe:– undercutbythefollowerwhen𝑝/ > 𝑐 (thusyieldingnosales)
– mimickedbythefollowerwhen𝑝/ = 𝑐 (thusentailinghalfofthemarketshare)
• Hence,theleaderhas(weak)incentivestosetaprice𝑝/ = 𝑐.
• Asaconsequence,theequilibriumpricepairremainsat(𝑝/∗, 𝑝H∗) = (𝑐, 𝑐),asinthesimultaneous-moveversionoftheBertrandmodel.
AdvancedMicroeconomicTheory 157
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• Assumethatfirmsselldifferentiatedproducts,wherefirm𝑗’sdemandis
𝑞° = 𝐷°(𝑝°, 𝑝±)– Example: 𝑞°(𝑝°, 𝑝±) = 𝑎 − 𝑏𝑝° + 𝑐𝑝±,where𝑎, 𝑏, 𝑐 >0 and𝑏 > 𝑐
• Inthesecondstage,firm2(thefollower)solvesfollowingPMP
max�¼½
𝜋H = 𝑝H𝑞H − 𝑇𝐶(𝑞H)
= 𝑝H𝐷H(𝑝H, 𝑝/) − 𝑇𝐶(𝐷H(𝑝H, 𝑝/)ºÂ
)
AdvancedMicroeconomicTheory 158
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• FOCswrt 𝑝H yield
𝐷H(𝑝H, 𝑝/) + 𝑝H𝜕𝐷H(𝑝H, 𝑝/)
𝜕𝑝H−𝜕𝑇𝐶 𝐷H(𝑝H, 𝑝/)𝜕𝐷H(𝑝H, 𝑝/)
𝜕𝐷H(𝑝H, 𝑝/)
𝜕𝑝HUsingthechainrule
= 0
• Solvingfor𝑝H producesthefollower’sBRFforeverypricesetbytheleader,𝑝/,i.e.,𝑝H(𝑝/).
AdvancedMicroeconomicTheory 159
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• Inthefirststage,firm1(leader)anticipatesthatthefollowerwilluseBRF𝑝H(𝑝/) torespondtoeachpossibleprice𝑝/,hencesolvesfollowingPMP
max�»¼½
𝜋/ = 𝑝/𝑞/ − 𝑇𝐶 𝑞/
= 𝑝/𝐷/ 𝑝/, 𝑝H 𝑝/Cä~Â
− 𝑇𝐶 𝐷/ 𝑝/, 𝑝H(𝑝/)º»
AdvancedMicroeconomicTheory 160
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• FOCswrt 𝑝/ yield𝐷/(𝑝/, 𝑝H) + 𝑝/
𝜕𝐷/(𝑝/, 𝑝H)𝜕𝑝/
+𝜕𝐷/(𝑝/, 𝑝H)𝜕𝑝H(𝑝/)
𝜕𝑝H(𝑝/)𝜕𝑝/
NewStrategicEffect
−𝜕𝑇𝐶 𝐷/(𝑝/, 𝑝H)𝜕𝐷/(𝑝/, 𝑝H)
𝜕𝐷/(𝑝/, 𝑝H)𝜕𝑝/
+𝜕𝐷/(𝑝/, 𝑝H)𝜕𝑝H(𝑝/)
𝜕𝑝H(𝑝/)𝜕𝑝/
NewStrategicEffect
= 0
• Ormorecompactlyas𝐷/(𝑝/, 𝑝H)
+ 𝑝/ −𝜕𝑇𝐶 𝐷/(𝑝/, 𝑝H)𝜕𝐷/(𝑝/, 𝑝H)
𝜕𝐷/(𝑝/, 𝑝H)𝜕𝑝/
1 +𝜕𝑝H(𝑝/)𝜕𝑝/]��
= 0
AdvancedMicroeconomicTheory 161
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• IncontrasttotheBertrandmodelwithsimultaneouspricecompetition,anincreaseinfirm1’spricenowproducesanincreaseinfirm2’spriceinthesecondstage.
• Hence,theleaderhasmoreincentivestoraiseitsprice,ultimatelysofteningthepricecompetition.
• Whileasoftenedcompetitionbenefitsboththeleaderandthefollower,therealbeneficiaryisthefollower,asitsprofitsincreasemorethantheleader’s.
AdvancedMicroeconomicTheory 162
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• Example:– Consideralineardemand𝑞$ = 1 − 2𝑝$ + 𝑝°,withnomarginalcosts,i.e.,𝑐 = 0.
– SimultaneousBertrandmodel:thePMPismax�̼½
𝜋° = 𝑝° j (1 − 2𝑝° + 𝑝±)forany𝑘 ≠ 𝑗
whereFOCwrt 𝑝° producesfirm𝑗’sBRF
𝑝°(𝑝±) =14 +
14𝑝±
– SimultaneouslysolvingthetwoBRFsyields𝑝°∗ =/Ã≃
0.33,entailingequilibriumprofitsof𝜋°∗ =HÄ≃ 0.222.
AdvancedMicroeconomicTheory 163
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• Example (continued):– SequentialBertrandmodel: inthesecondstage,firm2’s(thefollower’s)PMPis
max�¼½
𝜋H = 𝑝H j 1 − 2𝑝H + 𝑝/
whereFOCwrt 𝑝H producesfirm2’sBRF
𝑝H(𝑝/) =14 +
14𝑝/
– Inthefirststage,firm1’s(theleader’s)PMPis
max�»¼½
𝜋/ = 𝑝/ j 1 − 2𝑝/ +14 +
14𝑝/
Cä~Â
= 𝑝/ j14 (5 − 7𝑝/)
AdvancedMicroeconomicTheory 164
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• Example (continued):– FOCwrt 𝑝/,andsolvingfor𝑝/,producesfirm1’sequilibriumprice𝑝/∗ =
ç/Å= 0.36.
– Substituting𝑝/∗ intotheBRFoffirm2yields𝑝H∗ 0.36 = /
Å+ /
Å0.36 = 0.34.
– Equilibriumprofitsarehence
𝜋/∗ = 0.3614 5 − 7 0.36 = 0.223forfirm1
𝜋H∗ = 0.34 1 − 2 0.34 + 0.36 = 0.230forfirm2
AdvancedMicroeconomicTheory 165
p1
p2
p1(p2)
p2(p1)
⅓
¼
¼
Priceswithsequentialpricecompetition
0.36
0.34⅓
Priceswithsimultaneouspricecompetition
DynamicCompetition:SequentialBertrandModelwithHeterogeneousProducts
• Example (continued):– Bothfirms’pricesandprofitsarehigherinthesequentialthaninthesimultaneousgame.
– However,thefollowerearnsmorethantheleaderinthesequentialgame!(secondmover’sadvantage)
AdvancedMicroeconomicTheory 166
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
• Stackelberg model:firm1(theleader)choosesoutputlevel𝑞/,andfirm2(thefollower)observingtheoutputdecisionoftheleader,respondswithitsownoutput𝑞H(𝑞/).
• Bybackwardinduction,thefollower’sBRFis𝑞H(𝑞/) forany𝑞/.
• Sincetheleaderanticipates𝑞H(𝑞/) fromthefollower,theleader’sPMPis
maxº»¼½
𝑝 𝑞/ + 𝑞H(𝑞/)Cä~Â
𝑞/ − 𝑇𝐶/(𝑞/)
AdvancedMicroeconomicTheory 167
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
• FOCswrt 𝑞/ yields
𝑝 𝑞/ + 𝑞H(𝑞/) + 𝑝4 𝑞/ + 𝑞H(𝑞/) 𝑞/ +𝜕𝑞H(𝑞/)𝜕𝑞/
𝑞/
−𝜕𝑇𝐶/(𝑞/)𝜕𝑞/
= 0
ormorecompactly
𝑝 𝑄 + 𝑝4 𝑄 𝑞/ + 𝑝4 𝑄𝜕𝑞H(𝑞/)𝜕𝑞/
𝑞/
StrategicEffect
−𝜕𝑇𝐶/ 𝑞/𝜕𝑞/
= 0
• ThisFOCcoincideswiththatforstandardCournotmodelwithsimultaneousoutputdecisions,exceptforthestrategiceffect.
AdvancedMicroeconomicTheory 168
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
• Thestrategiceffectispositivesince𝑝4(𝑄) < 0andèºÂ(º»)
躻< 0.
• Firm1(theleader)hasmoreincentivetoraise𝑞/relativetotheCournotmodelwithsimultaneousoutputdecision.
• Intuition (first-moveradvantage):– Byoverproducing,theleaderforcesthefollowertoreduceitsoutput𝑞H bytheamountèºÂ(º»)
躻.
– Thishelpstheleadersellitsproductionatahigherprice,asreflectedby𝑝′(𝑄);ultimatelyearningalargerprofitthaninthestandardCournotmodel.
AdvancedMicroeconomicTheory 169
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
• Example:– Considerlinearinversedemand𝑝 = 𝑎 − 𝑄,where𝑄 = 𝑞/ + 𝑞H,andaconstantmarginalcostof𝑐.
– Firm2’s(thefollower’s)PMPismaxºÂ
(𝑎 − 𝑞/ − 𝑞H)𝑞H − 𝑐𝑞H– FOC:
𝑎 − 𝑞/ − 2𝑞H − 𝑐 = 0– Solvingfor𝑞H yieldsthefollower’sBRF
𝑞H 𝑞/ = ¿*º»*ÀH
AdvancedMicroeconomicTheory 170
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
• Example (continued):– Plugging𝑞H 𝑞/ intotheleader’sPMP,weget
maxº»
𝑎 − 𝑞/ −¿*º»*À
H𝑞/ − 𝑐𝑞/ =
/H(𝑎 − 𝑞/ − 𝑐)
– FOC:/H𝑎 − 2𝑞/ − 𝑐 = 0
– Solvingfor𝑞/,weobtaintheleader’sequilibriumoutputlevel𝑞/∗ =
¿*ÀH.
– Substituting𝑞/∗ intothefollower’sBRFyieldsthefollower’sequilibriumoutput𝑞H∗ =
¿*ÀÅ.
AdvancedMicroeconomicTheory 171
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
AdvancedMicroeconomicTheory 172
q1
q2
a–c
q1(q2)
2
q2(q1)
a–c2
CournotQuantities
StackelbergQuantities
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
• Example (continued):– Theequilibriumpriceis
𝑝 = 𝑎 − 𝑞/∗ − 𝑞H∗ =𝑎 + 3𝑐4
– Andtheresultingequilibriumprofitsare
𝜋/∗ =¿1ÃÀÅ
¿*ÀH
− 𝑐 ¿*ÀH
= ¿*À Â
Ê
𝜋H∗ =¿1ÃÀÅ
¿*ÀÅ
− 𝑐 ¿*ÀÅ
= ¿*À Â
/é
AdvancedMicroeconomicTheory 173
Price
a
a+c2
a–c2b
Monopoly
Units
a+2c3
a+3c4
2(a–c)3b
3(a–c)4b
a–cb
ab
pm=
pCournot=
pStackelberg=
pP.C.=pBertrand=c
Cournot
Stackelberg
BertrandandPerfectCompetition
DynamicCompetition:SequentialCournotModelwithHomogenousProducts
• Linearinversedemand𝑝 𝑄 = 𝑎 − 𝑄
• Symmetricmarginalcosts𝑐 > 0
AdvancedMicroeconomicTheory 174
DynamicCompetition:SequentialCournotModelwithHeterogeneousProducts
• Assumethatfirmsselldifferentiatedproducts,withinversedemandcurvesforfirms1and2
𝑝/(𝑞/, 𝑞H) = 𝛼 − 𝛽𝑞/ − 𝛾𝑞Hforfirm1𝑝H(𝑞/, 𝑞H) = 𝛼 − 𝛾𝑞/ − 𝛽𝑞Hforfirm2
• Firm2’s(thefollower’s)PMPismaxºÂ
(𝛼 − 𝛾𝑞/ − 𝛽𝑞H) j 𝑞H
where,forsimplicity,weassumenomarginalcosts.• FOC:
𝛼 − 𝛾𝑞/ − 2𝛽𝑞H = 0AdvancedMicroeconomicTheory 175
DynamicCompetition:SequentialCournotModelwithHeterogeneousProducts
• Solvingfor𝑞H yieldsfirm2’sBRF𝑞H(𝑞/) =
à*꺻Há
• Plugging𝑞H 𝑞/ intotheleader’sfirm1’s(theleader’s)PMP,weget
maxº»
𝛼 − 𝛽𝑞/ − 𝛾à*꺻Há
𝑞/ =
maxº»
𝛼 Há*êHá
− HáÂ*êÂ
Há𝑞/ 𝑞/
• FOC:
𝛼 Há*êHá
− HáÂ*êÂ
á𝑞/ = 0
AdvancedMicroeconomicTheory 176
DynamicCompetition:SequentialCournotModelwithHeterogeneousProducts
• Solvingfor𝑞/,weobtaintheleader’sequilibriumoutputlevel𝑞/∗ =
à(Há*ê)H(HáÂ*êÂ)
• Substituting𝑞/∗ intothefollower’sBRFyieldsthefollower’sequilibriumoutput
𝑞H∗ =à*꺻∗
Há= à(ÅáÂ*Háê*êÂ)
Åá(HáÂ*êÂ)• Note:– 𝑞/∗ > 𝑞H∗– If𝛾 → 𝛽 (i.e.,theproductsbecomemorehomogeneous),(𝑞/∗, 𝑞H∗) convegetothestandardStackelberg values.
– If𝛾 → 0 (i.e.,theproductsbecomeverydifferentiated),(𝑞/∗, 𝑞H∗) convergetothemonopolyoutput𝑞� = à
Há.
AdvancedMicroeconomicTheory 177
CapacityConstraints
AdvancedMicroeconomicTheory 178
CapacityConstraints• HowcomeareequilibriumoutcomesinthestandardBertrandandCournotmodelssodifferent?
• Dofirmsreallycompeteinpriceswithoutfacingcapacityconstraints?– Bertrandmodelassumesafirmcansupplyinfinitelylargeamountifitspriceislowerthanitsrivals.
• ExtensionoftheBertrandmodel:– Firststage:firmssetcapacities,𝑞·/ and𝑞·H,withacostofcapacity𝑐 > 0
– Secondstage:firmsobserveeachother’scapacitiesandcompeteinprices,simultaneouslysetting𝑝/ and𝑝H
AdvancedMicroeconomicTheory 179
CapacityConstraints• Whatistheroleofcapacityconstraint?– Whenafirm’spriceislowerthanitscapacity,notallconsumerscanbeserved.
– Hence,salesmustberationedthroughefficientrationing:thecustomerswiththehighestwillingnesstopaygettheproductfirst.
• Intuitively,if𝑝/ < 𝑝H andthequantitydemandedat𝑝/ issolargethat𝑄(𝑝/) > 𝑞·/,thenthefirst𝑞·/unitsareservedtothecustomerswiththehighestwillingnesstopay(i.e.,theuppersegmentofthedemandcurve),whilesomecustomersareleftintheformofresidualdemandtofirm2.
AdvancedMicroeconomicTheory 180
p
q
p2
p1
Q(p2) Q(p1)
Q(p)
q1,firm1'scapacity
q1 Unservedcustomersbyfirm1
Theseunitsbecomeresidualdemandforfirm2.
Q2(p2)–q1
1st
2nd3rd
4th
5th
6th
CapacityConstraints
AdvancedMicroeconomicTheory 181
• At𝑝/ thequantitydemandedis𝑄(𝑝/),butonly𝑞·/ unitscanbeserved.
• Hence,theresidualdemandis𝑄(𝑝/) −𝑞·/.
• Sincefirm2setsapriceof𝑝H,itsdemandwillbe𝑄(𝑝H).
• Thus,aportionoftheresidualdemand,i.e.,𝑄(𝑝H) − 𝑞·/,iscaptured.
CapacityConstraints
• Hence,firm2’sresidualdemandcanbeexpressedas
â𝑄 𝑝H − 𝑞·/if𝑄 𝑝H − 𝑞·/ ≥ 00otherwise
• Shouldwerestrict𝑞·/ and𝑞·H somewhat?– Yes.Afirmwillneversetahugecapacityifsuchcapacityentailsnegativeprofits,independentlyofthedecisionofitscompetitor.
AdvancedMicroeconomicTheory 182
CapacityConstraints
• Howtoexpressthisratherobviousstatementwithasimplemathematicalcondition?– Themaximalrevenueofafirmundermonopolyismaxº(𝑎 − 𝑞)𝑞,whichismaximizedat𝑞 = ¿
H,yielding
profitsof¿Â
Å.
– Maximalrevenuesarelargerthancostsif¿Â
Å≥ 𝑐𝑞·°,or
solvingfor𝑞·°,¿Â
ÅÀ≥ 𝑞·°.
– Intuitively,thecapacitycannotbetoohigh,asotherwisethefirmwouldnotobtainpositiveprofitsregardlessoftheopponent’sdecision.
AdvancedMicroeconomicTheory 183
CapacityConstraints:SecondStage
• Bybackwardinduction,westartwiththesecondstage(pricinggame),wherefirmssimultaneouslychooseprices𝑝/ and𝑝H asafunctionofthecapacitychoices𝑞·/ and𝑞·H.
• Wewanttoshowthatinthissecondstage,bothfirmssetacommonprice
𝑝/ = 𝑝H = 𝑝∗ = 𝑎 − 𝑞·/ − 𝑞·Hwheredemandequalssupply,i.e.,totalcapacity,
𝑝∗ = 𝑎 − 𝑄·,where𝑄· ≡ 𝑞·/ + 𝑞·HAdvancedMicroeconomicTheory 184
CapacityConstraints:SecondStage
• Inordertoprovethisresult,westartbyassumingthatfirm1sets𝑝/ = 𝑝∗.Wenowneedtoshowthatfirm2alsosets𝑝H = 𝑝∗,i.e.,itdoesnothaveincentivestodeviatefrom𝑝∗.
• Iffirm2doesnotdeviate,𝑝/ = 𝑝H = 𝑝∗,thenitsellsuptoitscapacity𝑞·H.
• Iffirm2reducesitspricebelow𝑝∗,demandwouldexceeditscapacity𝑞·H.Asaresult,firm2wouldsellthesameunitsasbefore,𝑞·H,butatalowerprice.
AdvancedMicroeconomicTheory 185
CapacityConstraints:SecondStage• If,instead,firm2chargesapriceabove𝑝∗,then𝑝/ = 𝑝∗ < 𝑝H anditsrevenuesbecome
𝑝H𝑄ë(𝑝H) = â𝑝H(𝑎 − 𝑝H − 𝑞·/)if𝑎 − 𝑝H − 𝑞·/ ≥ 00otherwise
• Note:– ThisisfundamentallydifferentfromthestandardBertrandmodelwithoutcapacityconstraints,whereanincreaseinpricebyafirmreducesitssalestozero.
– Whencapacityconstraintsarepresent,thefirmcanstillcapturearesidualdemand,ultimatelyraisingitsrevenuesafterincreasingitsprice.
AdvancedMicroeconomicTheory 186
CapacityConstraints:SecondStage• Wenowfindthemaximumofthisrevenuefunction.FOCwrt 𝑝H yields:
𝑎 − 2𝑝H − 𝑞·/ = 0 ⟺ 𝑝H =𝑎 − 𝑞·/2
• Thenon-deviatingprice𝑝∗ = 𝑎 − 𝑞·/ − 𝑞·H liesabovethemaximum-revenueprice𝑝H =
¿*º·»H
when
𝑎 − 𝑞·/ − 𝑞·H >𝑎 − 𝑞·/2 ⟺ 𝑎 > 𝑞·/ + 2𝑞·H
• Since¿Â
ÅÀ≥ 𝑞·° (capacityconstraint),wecanobtain
𝑎H
4𝑐 + 2𝑎H
4𝑐 > 𝑞·/ + 2𝑞·H ⇔ 3𝑎H
4𝑐 > 𝑞·/ + 2𝑞·HAdvancedMicroeconomicTheory 187
CapacityConstraints:SecondStage
• Therefore,𝑎 > 𝑞·/ + 2𝑞·H holdsif𝑎 >ÿÂ
ÅÀwhich,
solvingfor𝑎,isequivalenttoÅÀÃ> 𝑎.
AdvancedMicroeconomicTheory 188
CapacityConstraints:SecondStage
• WhenÅÀÃ> 𝑎 holds,
capacityconstraint¿Â
ÅÀ≥
𝑞·° transformsintoÿÂ
ÅÀ>
𝑞·/ + 2𝑞·H,implying𝑝∗ >𝑝H = 𝑎 − º·»
H.
• Thus,firm2doesnothaveincentivestoincreaseitsprice𝑝H from𝑝∗,sincethatwouldloweritsrevenues.
AdvancedMicroeconomicTheory 189
CapacityConstraints:SecondStage
• Inshort,firm2doesnothaveincentivestodeviatefromthecommonprice
𝑝∗ = 𝑎 − 𝑞·/ − 𝑞·H• Asimilarargumentappliestofirm1(bysymmetry).
• Hence,wehavefoundanequilibriuminthepricingstage.
AdvancedMicroeconomicTheory 190
CapacityConstraints:FirstStage
• Inthefirststage(capacitysetting),firmssimultaneouslyselecttheircapacities𝑞·/ and𝑞·H.
• Insertingstage2equilibriumprices,i.e.,𝑝/ = 𝑝H = 𝑝∗ = 𝑎 − 𝑞·/ − 𝑞·H,
intofirm𝑗’sprofitfunctionyields𝜋°(𝑞·/, 𝑞·H) = (𝑎 − 𝑞·/ − 𝑞·H)
�∗𝑞·° − 𝑐𝑞·°
• FOCwrt capacity𝑞·° yieldsfirm𝑗’sBRF
𝑞·°(𝑞·±) =𝑎 − 𝑐2 −
12𝑞·±
AdvancedMicroeconomicTheory 191
CapacityConstraints:FirstStage
• SolvingthetwoBRFssimultaneously,weobtainasymmetricsolution
𝑞·° = 𝑞·± =𝑎 − 𝑐3
• ThesearethesameequilibriumpredictionsasthoseinthestandardCournotmodel.
• Hence,capacitiesinthistwo-stagegamecoincidewithoutputdecisionsinthestandardCournotmodel,whilepricesaresetequaltototalcapacity.
AdvancedMicroeconomicTheory 192
EndogenousEntry
AdvancedMicroeconomicTheory 193
EndogenousEntry
• Sofarthenumberoffirmswasexogenous• Whatifthenumberoffirmsoperatinginamarketisendogenouslydetermined?
• Thatis,howmanyfirmswouldenteranindustrywhere– TheyknowthatcompetitionwillbealaCournot– Theymustincurafixedentrycost𝐹 > 0.
AdvancedMicroeconomicTheory 194
EndogenousEntry• Considerinversedemandfunction𝑝(𝑞),where𝑞denotesaggregateoutput
• Everyfirm𝑗 facesthesametotalcostfunction,𝑐(𝑞°),ofproducing𝑞° units
• Hence,theCournotequilibriummustbesymmetric– Everyfirmproducesthesameoutputlevel𝑞(𝑛),whichisafunctionofthenumberofentrants.
• Entryprofitsforfirm𝑗 are𝜋° 𝑛 = 𝑝 𝑛 j 𝑞 𝑛
È�(È)
𝑞 𝑛 − 𝑐 𝑞 𝑛ProductionCosts
− 𝐹⏟FixedEntryCost
AdvancedMicroeconomicTheory 195
EndogenousEntry
• Threeassumptions(validundermostdemandandcostfunctions):– individualequilibriumoutput𝑞(𝑛) isdecreasingin𝑛;
– aggregateoutput𝑞 ≡ 𝑛 j 𝑞(𝑛) increasesin𝑛;– equilibriumprice𝑝(𝑛 j 𝑞(𝑛)) remainsabovemarginalcostsregardlessofthenumberofentrants𝑛.
AdvancedMicroeconomicTheory 196
EndogenousEntry
• Equilibriumnumberoffirms:– Theequilibriumoccurswhennomorefirmshaveincentivestoenterorexitthemarket,i.e.,𝜋°(𝑛�) = 0.
– Notethatindividualprofitsdecreasein𝑛,i.e.,
𝜋4 𝑛 = 𝑝 𝑛𝑞 𝑛 − 𝑐4 𝑞 𝑛1
𝜕𝑞(𝑛)𝜕𝑛
*
+𝑞 𝑛 𝑝4 𝑛𝑞 𝑛*
𝜕[𝑛𝑞 𝑛 ]𝜕𝑛1
< 0
AdvancedMicroeconomicTheory 197
EndogenousEntry
• Socialoptimum:– Thesocialplannerchoosesthenumberofentrants𝑛� thatmaximizessocialwelfare
maxI𝑊 𝑛 ≡ ï 𝑝 𝑠 𝑑𝑠 − 𝑛 j 𝑐 𝑞 𝑛 − 𝑛 j 𝐹
Iº(I)
½
AdvancedMicroeconomicTheory 198
p
Q
p(n∙q(n))
p(Q)n∙c(q(n))
n∙c(q)A
B
C
D
n∙q(n)
EndogenousEntry
• ∫ 𝑝 𝑠 𝑑𝑠Iº(I)½ =𝐴 + 𝐵 + 𝐶 + 𝐷
• 𝑛 j 𝑐 𝑞 𝑛 =𝐶 + 𝐷
• Socialwelfareisthus𝐴 + 𝐵 minustotalentrycosts𝑛 j 𝐹
AdvancedMicroeconomicTheory 199
EndogenousEntry
– FOCwrt 𝑛 yields
𝑝 𝑛𝑞 𝑛 𝑛𝜕𝑞 𝑛𝜕𝑛 + 𝑞 𝑛 − 𝑐 𝑞 𝑛 − 𝑛𝑐4 𝑞 𝑛
𝜕𝑞 𝑛𝜕𝑛 − 𝐹 = 0
or,re-arranging,
𝜋 𝑛 + 𝑛 𝑝 𝑛𝑞 𝑛 − 𝑐4 𝑞 𝑛𝜕𝑞(𝑛)𝜕𝑛 = 0
– Hence,marginalincreasein𝑛 entailstwooppositeeffectsonsocialwelfare:a) theprofitsofthenewentrantincreasesocialwelfare(+,
appropriability effect)b) theentrantreducestheprofitsofallpreviousincumbentsin
theindustry astheindividualsalesofeachfirmdecreasesuponentry(-,businessstealing effect)
AdvancedMicroeconomicTheory 200
EndogenousEntry
• The“businessstealing”effectisrepresentedby:
𝑛 𝑝 𝑛𝑞 𝑛 − 𝑐4 𝑞 𝑛𝜕𝑞(𝑛)𝜕𝑛 < 0
whichisnegativesinceèº(I)èI
< 0 and𝑛 𝑝 𝑛𝑞 𝑛 − 𝑐4 𝑞 𝑛 > 0 bydefinition.
• Therefore,anadditionalentryinducesareductioninaggregateoutputby𝑛 èº(I)
èI,whichin
turnproducesanegativeeffectonsocialwelfare.
AdvancedMicroeconomicTheory 201
EndogenousEntry
• Giventhenegativesignofthebusinessstealingeffect,wecanconcludethat
𝑊4 𝑛 = 𝜋 𝑛 + 𝑛 𝑝 𝑛𝑞 𝑛 − 𝑐4 𝑞 𝑛𝜕𝑞 𝑛𝜕𝑛
*
< 𝜋(𝑛)
andthereforemorefirmsenterinequilibriumthaninthesocialoptimum,i.e.,𝑛� > 𝑛�.
AdvancedMicroeconomicTheory 202
EndogenousEntry
AdvancedMicroeconomicTheory 203
EndogenousEntry
• Example:– Consideralinearinversedemand𝑝 𝑄 = 1 − 𝑄andnomarginalcosts.
– Theequilibriumquantityinamarketwith𝑛 firmsthatcompetealaCournotis
𝑞 𝑛 = /I1/
– Let’scheckifthethreeassumptionsfromabovehold.
AdvancedMicroeconomicTheory 204
EndogenousEntry
• Example (continued):– First,individualoutputdecreaseswithentry
èº IèI
= − /I1/ Â < 0
– Second,aggregateoutput𝑛𝑞(𝑛) increaseswithentry
è Iº IèI
= /I1/ Â > 0
– Third,priceliesabovemarginalcostforanynumberoffirms
𝑝 𝑛 − 𝑐 = 1 − 𝑛 j /I1/
= /I1/
> 0forall𝑛AdvancedMicroeconomicTheory 205
EndogenousEntry
• Example (continued):– Everyfirmearnsequilibriumprofitsof
𝜋 𝑛 =1
𝑛 + 1�(I)
1𝑛 + 1º(I)
− 𝐹 =1
𝑛 + 1 H − 𝐹
– Sinceequilibriumprofitsafterentry, /I1/ Â,is
smallerthan1evenifonlyonefirmenterstheindustry,𝑛 = 1,weassumethatentrycostsarelowerthan1,i.e.,𝐹 < 1.
AdvancedMicroeconomicTheory 206
EndogenousEntry
• Example (continued):– Socialwelfareis
𝑊 𝑛 = ï (1 − 𝑠)𝑑𝑠 − 𝑛 j 𝐹II1/
½
= 𝑠 −𝑠2 ñ
½
II1/
− 𝑛 j 𝐹
=𝑛 𝑛 + 2
21
𝑛 + 1
H− 𝑛 j 𝐹
AdvancedMicroeconomicTheory 207
EndogenousEntry
• Example (continued):– Thenumberoffirmsenteringthemarketinequilibrium,𝑛�,isthatsolving𝜋 𝑛� = 0,
1𝑛� + 1 H − 𝐹 = 0 ⟺ 𝑛� =
1𝐹�− 1
whereasthenumberoffirmsmaximizingsocialwelfare,i.e.,𝑛� solving𝑊4 𝑛� = 0,
𝑊4 𝑛� =1
𝑛� + 1 Ã = 0 ⟺𝑛� =1𝐹ò − 1
where𝑛� < 𝑛� foralladmissiblevaluesof𝐹,i.e.,𝐹 ∈ 0,1 .
AdvancedMicroeconomicTheory 208
Entrycosts,F
ne=–1(Equilibrium)1F½
no=–1(Soc.Optimal)1F⅓
Numberoffirms
0
EndogenousEntry
• Example (continued):
AdvancedMicroeconomicTheory 209
RepeatedInteraction
AdvancedMicroeconomicTheory 210
RepeatedInteraction• Inallpreviousmodels,weconsideredfirmsinteractingduringoneperiod(i.e.,one-shotgame).
• However,firmscompeteduringseveralperiodsand,insomecases,duringmanygenerations.
• Inthesecases,afirm’sactionsduringoneperiodmightaffectitsrival’sbehaviorinfutureperiods.
• Moreimportantly,wecanshowthatundercertainconditions,thestrongcompetitiveresultsintheBertrand(and,tosomeextent,intheCournot)modelcanbeavoidedwhenfirmsinteractrepeatedlyalongtime.
• Thatis,collusioncanbesupportedintherepeatedgameevenifitcouldnotintheone-shotgame.
AdvancedMicroeconomicTheory 211
RepeatedInteraction:BertrandModel
• Considertwofirmssellinghomogeneousproducts.• Let𝑝°� denotefirm𝑗’spricingstrategyatperiod𝑡,whichisafunctionofthehistoryofallpricechoicesbythetwofirms,𝐻�*/ = 𝑝/�, 𝑝H� �ó/
�*/.• Conditioning𝑝°� onthefullhistoryofplayallowsforawiderangeofpricingstrategies:– settingthesamepriceregardlessofprevioushistory– retaliationiftherivallowersitspricebelowa“thresholdlevel”
– increasingcooperationiftherivalwascooperativeinpreviousperiods(untilreachingthemonopolyprice𝑝�)
AdvancedMicroeconomicTheory 212
RepeatedInteraction:BertrandModel
• Finitelyrepeatedgame:– CanwesupportcooperationiftheBertrandgameisrepeatedforafinitenumberof𝑇 rounds?§ No!
– Toseewhy,considerthelastperiodoftherepeatedgame(period𝑇):§ Regardlessofpreviouspricingstrategies,everyfirms’optimalpricingstrategyinthisstageistoset𝑝$,ô∗ = 𝑐,asintheone-shotBertrandgame.
AdvancedMicroeconomicTheory 213
RepeatedInteraction:BertrandModel
– Now,movetotheprevioustothelastperiod(𝑇 − 1):§ Bothfirmsanticipatethat,regardlessofwhattheychooseat𝑇 − 1,theywillbothselect𝑝$,ô∗ = 𝑐 inperiod𝑇.Hence,itisoptimalforbothtoselect 𝑝$,ô*/∗ = 𝑐 inperiod𝑇 − 1 aswell.
– Now,movetoperiod(𝑇 −2):§ Bothfirmsanticipatethat,regardlessofwhattheychooseat𝑇 − 2,theywillbothselect𝑝$,ô∗ = 𝑐 inperiod𝑇 and𝑝$,ô*/∗ = 𝑐inperiod𝑇 − 1.Thus,itisoptimalforbothtoselect𝑝$,ô*H∗ = 𝑐inperiod𝑇 − 2 aswell.
– Thesameargumentextendstoallpreviousperiods,includingthefirstroundofplay𝑡 = 1.
– Hence,bothfirmsbehaveasinone-shotBertrandgame.
AdvancedMicroeconomicTheory 214
RepeatedInteraction:BertrandModel
• Infinitelyrepeatedgame:– CanwesupportcooperationiftheBertrandgameisrepeatedforaninfiniteperiods?§ Yes!Cooperation(i.e.,selectingpricesabovemarginalcost)canindeedbesustainedusingdifferentpricingstrategies.
– Forsimplicity,considerthefollowingpricingstrategy
𝑝°� 𝐻�*/ = â𝑝� ifallelementsin𝐻�*/are 𝑝�, 𝑝� or𝑡 = 1
𝑐otherwise
§ Inwords,everyfirm𝑗 setsthemonopolyprice𝑝� inperiod1.Then,ineachsubsequentperiod𝑡 > 1,firm𝑗 sets𝑝� ifbothfirmscharged𝑝� inallpreviousperiods.Otherwise,firm𝑗 chargesapriceequaltomarginalcost.
AdvancedMicroeconomicTheory 215
RepeatedInteraction:BertrandModel
– ThistypeofstrategyisusuallyreferredtoasNashreversionstrategy (NRS):§ firmscooperateuntiloneofthemdeviates,inwhichcasefirmsthereafterreverttotheNashequilibriumoftheunrepeatedgame(i.e.,setpricesequaltomarginalcost)
– LetusshowthatNRScanbesustainedintheequilibriumoftheinfinitelyrepeatedgame.
–Weneedtodemonstratethatfirmsdonothaveincentivestodeviatefromit,duringanyperiod𝑡 > 1andregardlessoftheirprevioushistoryofplay.
AdvancedMicroeconomicTheory 216
RepeatedInteraction:BertrandModel
– Consideranyperiod𝑡 > 1,andahistoryofplayforwhichallfirmshavebeencooperativeuntil𝑡 − 1.
– Bycooperating,firm𝑗’sprofitswouldbe(𝑝� −𝑐) /
H𝑥(𝑝�),i.e.,halfofmonopolyprofitsõ
ö
H,inall
subsequentperiods𝜋�
2 + 𝛿𝜋�
2 + 𝛿H𝜋�
2 +⋯
= 1 + 𝛿 + 𝛿H +⋯𝜋�
2 =1
1 − 𝛿𝜋�
2where𝛿 ∈ (0,1) denotesfirms’discountfactor
AdvancedMicroeconomicTheory 217
RepeatedInteraction:BertrandModel
– If,incontrast,firm𝑗 deviatesinperiod𝑡,theoptimaldeviationis𝑝°,� = 𝑝� − 𝜀, where𝜀 > 0,givenitsrivalstillsetsaprice𝑝±,� = 𝑝�.
– Thisallowsfirm𝑗 tocaptureallmarket,andobtainmonopolyprofits𝜋� duringthedeviatingperiod.
– A deviationisdetectedinperiod𝑡 + 1,triggeringaNRSfromfirm𝑘 (i.e.,settingapriceequaltomarginalcost)thereafter,andentailingazeroprofitforbothfirms.
– Thediscountedstreamofprofitsforfirm𝑗 isthen𝜋� + 𝛿0 + 𝛿H0 +⋯ = 𝜋�
AdvancedMicroeconomicTheory 218
RepeatedInteraction:BertrandModel
– Hence,firm𝑗 preferstosticktotheNRSatperiod𝑡 if1
1 − 𝛿𝜋�
2 > 𝜋� ⟺ 𝛿 >12
– Thatis,cooperationcanbesustainedaslongasfirmsassignasufficientlyhighvaluetofutureprofits.
AdvancedMicroeconomicTheory 219
Profits
TimePeriodst t+1 t+2 ...
InstantaneousGain
Profitfromcooperating
πm2
πm
FutureLosses
πm2πm2
δ
δ2
RepeatedInteraction:BertrandModel
– Instantaneousgainsandlossesfromcooperationanddeviation
AdvancedMicroeconomicTheory 220
RepeatedInteraction:BertrandModel
• Whataboutfirm𝑗’sincentivestouseNRSafterahistoryofplayinwhichsomefirmsdeviated?– NRScallsforfirm𝑗 toreverttotheequilibriumoftheunrepeatedBertrandmodel.
– Thatis,toimplementthepunishmentembodiedinNRSafterdetectingadeviationfromanyplayer.
• BystickingtotheNRS,firm𝑗’sdiscountedstreamofpayoffsis
0 + 𝛿0 +⋯ = 0AdvancedMicroeconomicTheory 221
RepeatedInteraction:BertrandModel
• BydeviatingfromNRS(i.e.,settingaprice𝑝° =𝑝� whileitsopponentsetsapunishingprice𝑝± = 𝑐),profitsarealsozeroinallperiods.
• Hence,firm𝑗 hasincentivestocarryoutthethreat– Thatis,settingapunishingpriceof𝑝° = 𝑐,uponobservingadeviationinanypreviousperiod.
• Asaresult,theNRScanbesustainedinequilibrium,sincebothfirmshaveincentivestouseit,atanytimeperiod𝑡 > 1 andirrespectiveoftheprevioushistoryofplay.
AdvancedMicroeconomicTheory 222
RepeatedInteraction:BertrandModel
• Example:– Consideranindustrywithonly2firms,alineardemand𝑄 = 5000 − 100𝑝, andconstantandaveragemarginalcostsof𝑐 = $10.
– Ifone-shotBertrandgameisplayed,firmswould§ chargeapriceof𝑝 = 𝑐 = $10§ sellatotalquantityof4000unitsofaproduct§ earnzeroeconomicprofits
– If,incontrast,firmscolludetofixpricesatthemonopolyprice,cansuchcollusionbesustained?
AdvancedMicroeconomicTheory 223
RepeatedInteraction:BertrandModel
• Example (continued):–Monopolypriceisdeterminedbysolvingthefirms’jointPMP
max� 𝑝 − 10 ⋅ 𝑄 = 𝑝(5000
− 100𝑝) − 10(5000 − 100𝑝)– FOC:
5000 − 200𝑝 + 1000 = 0– Solvingfor𝑝 yieldsthemonopolyprice𝑝� = 30.– Theaggregateoutputis𝑄 = 2000 (i.e.,1000unitsperfirm)andthecorrespondingprofitsare𝜋� = $40,000 ($20,000perfirm).
AdvancedMicroeconomicTheory 224
RepeatedInteraction:BertrandModel
• Example (continued):– Collusionatthemonopolypriceissustainableif
𝜋�
21
1 − 𝛿 ≥ 𝜋� +𝛿
1 − 𝛿 ⋅ 0
– Since𝜋� = $40,000,theinequalityreducesto
200001
1 − 𝛿 ≥ 40000 ⟺ 𝛿 ≥12
AdvancedMicroeconomicTheory 225
RepeatedInteraction:BertrandModel
• Example (continued):–Whatwouldhappeniftherewere𝑛 firms?– Eachfirm’sshareofthemonopolyprofitstreamundercollusionwouldbeõ
ö
I= Ž½½½
I.
– CollusionatthemonopolypriceissustainableifŽ½½½I
//*ù
≥ 40000 ⟺ 𝛿 ≥ 1 − /I≡ 𝛿̅
– Hence,asthenumberoffirmsintheindustryincreases,itbecomesmoredifficulttosustaincooperation.
AdvancedMicroeconomicTheory 226
RepeatedInteraction:BertrandModel
• Example (continued):minimaldiscountfactorsustainingcollusion
AdvancedMicroeconomicTheory 227
RepeatedInteraction:CournotModel
• WecanextendasimilaranalysistotheCournotmodelofquantitycompetitionwithtwofirmssellinghomogeneousproducts.
• Forsimplicity,considerthefollowingNRSforeveryfirm𝑗
𝑞°� 𝐻�*/ =𝑞�
2 ifallelementsin𝐻�*/equal𝑞�
2 ,𝑞�
2 or𝑡 = 1
𝑞°ú�û�I��otherwise
§ Inwords,firm𝑗’sstrategyistoproducehalfofthemonopolyoutputº
ö
Hinperiod𝑡 = 1.Then,ineachsubsequentperiod𝑡 >
1,firm𝑗 continuesproducingºö
Hifbothfirmsproducedº
ö
Hinall
previousperiods.Otherwise,firm𝑗 revertstotheCournotequilibriumoutput.
AdvancedMicroeconomicTheory 228
RepeatedInteraction:CournotModel
• LetusshowthatNRScanbesustainedintheequilibriumoftheinfinitelyrepeatedgame.
• Iffirm𝑗 usestheNRSinperiod𝑡,itobtainshalfofmonopolyprofits,õ
ö
H, thereafter,witha
discountedstreamofprofitsofõö
H/
/*ù.
• But,whatiffirm𝑗 deviatesfromthisstrategy?Whatisitsoptimaldeviation?– Sincefirm𝑘 stickstotheNRS,andthusproducesº
ö
Hunits,wecanevaluatefirm𝑗’sBRF𝑞°(𝑞±) at𝑞± =
ºö
H,
or𝑞°ºö
H.
AdvancedMicroeconomicTheory 229
RepeatedInteraction:CournotModel
• Forcompactness,let𝑞°��� ≡ 𝑞°ºö
Hdenotefirm
𝑗’soptimaldeviation.• Thisyieldsprofitsof
𝜋°��� ≡ 𝑝 𝑞°���,𝑞�
2 ×𝑞°��� − 𝑐°×𝑞°���
• Bydeviatingfirm𝑗 obtainsfollowingstreamofprofits
𝜋°��� + 𝛿𝜋°ú�û�I�� + 𝛿H𝜋°ú�û�I�� + ⋯
= 𝜋°��� +𝛿
1 − 𝛿 𝜋°ú�û�I��
AdvancedMicroeconomicTheory 230
RepeatedInteraction:CournotModel
• Hence,firm𝑗 stickstotheNRSaslongas1
1 − 𝛿𝜋�
2 > 𝜋°��� +𝛿
1 − 𝛿 𝜋°ú�û�I��
• Multiplyingbothsidesby(1 − 𝛿) andsolvingfor𝛿 weobtain
𝛿 >𝜋°��� −
𝜋�2
𝜋°��� − 𝜋°ú�û�I��≡ 𝛿̅
• Intuitively,everyfirm𝑗 stickstotheNRSaslongasitassignsasufficientweighttofutureprofits.
AdvancedMicroeconomicTheory 231
RepeatedInteraction:CournotModel
• Instantaneousgainsandlossesfromdeviation
AdvancedMicroeconomicTheory 232
RepeatedInteraction:CournotModel
• Whataboutfirm𝑗’sincentivestouseNRSafterahistoryofplayinwhichsomefirmsdeviated?– NRScallsforfirm𝑗 toreverttotheequilibriumoftheunrepeatedCournotmodel.
– Thatis,toimplementthepunishmentembodiedinNRSafterdetectingadeviationfromanyplayer.
• BystickingtotheNRS,firm𝑗’sdiscountedstreamofpayoffsis /
/*ùõö
H.
• Bydeviatingfrom𝑞°ú�û�I��, whilefirm𝑘 produces𝑞±ú�û�I��,firm𝑗’sprofits,𝜋ý ,arelowerthan𝜋°ú�û�I��sincefirm𝑗’sbestresponsetoitsrivalproducing𝑞±ú�û�I�� is𝑞°ú�û�I��.
AdvancedMicroeconomicTheory 233
RepeatedInteraction:CournotModel
• Firm𝑗 stickstotheNRSafterahistoryofdeviationssince
𝜋ú�û�I�� + 𝛿𝜋ú�û�I�� + ⋯ > 𝜋ý + 𝛿𝜋ú�û�I�� + ⋯
whichholdsgiventhat𝜋ú�û�I�� > 𝜋ý .
• Hence,noneedtoimposeanyfurtherconditionsontheminimaldiscountfactorsustainingcooperation,𝛿̅.
AdvancedMicroeconomicTheory 234
RepeatedInteraction:CournotModel
• Example:– Consideranindustrywith2firms,alinearinversedemand𝑝(𝑞/, 𝑞H) = 𝑎 − 𝑏(𝑞/ + 𝑞H),andconstantandaveragemarginalcostsof𝑐 > 0.
– Firm𝑖’sPMPismaxºÒ
𝑎 − 𝑏(𝑞$ + 𝑞°) 𝑞$ − 𝑐𝑞$– FOCs:𝑎 − 2𝑏𝑞$ − 𝑏𝑞° − 𝑐 = 0– Solvingfor𝑞$ yieldsfirm𝑖’sBRF
𝑞$(𝑞°) =¿*ÀHÁ
− ºÌH
AdvancedMicroeconomicTheory 235
RepeatedInteraction:CournotModel
• Example (continued):– SolvingthetwoBRFssimultaneouslyyields
𝑞$ú�û�I�� =¿*ÀÃÁ
withcorrespondingpriceof
𝑝 = 𝑎 − 𝑏 ¿*ÀÃÁ
+ ¿*ÀÃÁ
= ¿1HÀÃ
andequilibriumprofitsof
𝜋$ú�û�I�� =¿1HÀÃ
¿*ÀÃÁ
− 𝑐 ¿*ÀÃÁ
= ¿*À Â
ÄÁ
AdvancedMicroeconomicTheory 236
RepeatedInteraction:CournotModel
• Example (continued):– If,instead,eachfirmproducedhalfofmonopolyoutput,𝑞$� = ºö
H= ¿*À
ÅÁ,theywouldfacea
correspondingpriceof𝑝� = ¿1ÀH
andreceivehalf
ofthemonopolyprofits𝜋$� = õö
H= ¿*À Â
ÊÁ.
– Inthissetting,theoptimaldeviationoffirm𝑖 isfoundbyplugging𝑞$� intoitsBRF
𝑞$þ�� = 𝑞$(𝑞°�) =¿*ÀHÁ
− /H¿*ÀÅÁ
= Ã ¿*ÀÊÁ
AdvancedMicroeconomicTheory 237
RepeatedInteraction:CournotModel
• Example (continued):– Thisyieldspriceof
𝑝 = 𝑎 − 𝑏 Ã(¿*À)ÊÁ
+ ¿*ÀÅÁ
= ÿ1çÀÊ
andprofitsof
𝜋$þ�� =ÿ1çÀÊ
Ã(¿*À)ÊÁ
− 𝑐 Ã ¿*ÀÊÁ
= Ä ¿*À Â
éÅÁforthedeviatingfirm,and
𝜋ý = ÿ1çÀÊ
(¿*À)ÅÁ
− 𝑐 ¿*ÀÅÁ
= Ã ¿*À Â
ÃHÁforthenon-deviatingfirm.
AdvancedMicroeconomicTheory 238
RepeatedInteraction:CournotModel
• Example (continued):– Cooperationissustainableif
//*ù
õö
H> 𝜋°��� +
ù/*ù
𝜋°ú�û�I��
or,inourcase,/
/*ù¿*À Â
ÊÁ> Ä ¿*À Â
éÅÁ+ ù
/*ù¿*À Â
ÄÁ⟺ 𝛿 > Ä
/ÿ
– Forthenon-deviatingfirm,wehave𝜋$ú�û�I�� > 𝜋ý§ Thatis,iftherivalfirmdefects,thenon-defectingfirmwillobtainalargerprofitbyrevertingtotheCournotoutputlevel.
AdvancedMicroeconomicTheory 239
RepeatedInteraction:CournotModel
• Extensions:– Temporaryreversionstotheequilibriumoftheunrepeatedgame
– (Temporary)punishmentsthatyieldevenlowerpayoffs
– Less“pure”formsofcooperations– Imperfectmonitoring
AdvancedMicroeconomicTheory 240