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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.


GameTheoryTools


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.


FirmBLowprices Highprices

FirmALowprices 5,5 9,1Highprices 1,9 7,7

GameTheoryTools• Payoffmatrix(NormalFormGame)


• “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.


GameTheoryTools

• 𝐴𝑑𝑜𝑝𝑡 isbetterthan𝑁𝑜𝑡𝑎𝑑𝑜𝑝𝑡 forfirm𝐴 ifitsopponentselects𝐴𝑑𝑜𝑝𝑡,butbecomesworseotherwise.

• Similarly argumentappliesforfirm𝐵.• Hence,nostrictlydominatedstrategiesforeitherplayer.• Whatistheequilibriumofthegamethen?


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$∗.


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).


Mixed-StrategyNashEquilibrium

• Insofarwerestrictedplayerstouseoneoftheiravailablestrategies100%ofthetime(commonlyknownas“purestrategies”)

• Generally,playerscouldrandomize(mix)theirchoices.– Example:Choosestrategy𝐴 withprobability𝑝 andstrategy𝐵 withprobability1 − 𝑝.



• 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.



• ThreepointsaboutmsNE:1. Playersmustbeindifferentamongall(oratleastsome)

oftheirpurestrategies.2. Sinceplayersneverusestrictlydominatedstrategies,a

NEassignsazeroprobabilitytodominatedstrategies.3. Ingameswithafinitesetofplayersandafinitesetof

availableactions,thereisgenerallyanoddnumberofequilibria.



• Example (noNEinpurestrategies):


FirmBAdopt Notadopt

FirmAAdopt 3,-3 -4,0

Not adopt -3,1 2,-2

– Thereisnocellofthematrixinwhichplayersselectmutualbestresponses.

– ThuswecannotfindaNEinpurestrategies.– FirmA(B)seekstocoordinate(miscoordinate)itsdecisionwiththatoffirmB(A,respectively).


• Example (continued):– Giventheiropposedincentives,firmAwouldliketomakeitschoicedifficulttoanticipate.

– IffirmAchoosesaspecificactionwithcertainty,firmBwillbedriventoselecttheoppositeaction.

– AnanalogousargumentappliestofirmB.– Asaconsequence,playershaveincentivestorandomizetheiractions.



• 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.



• 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



• Example (Technologyadoptiongame):


FirmBAdopt Notadopt

FirmAAdopt 3,1 0,0

Not adopt 0,0 1,3

– ThegamehastwoNashequilibriainpurestrategies:(𝐴𝑑𝑜𝑝𝑡, 𝐴𝑑𝑜𝑝𝑡)and(𝑁𝑜𝑡𝐴𝑑𝑜𝑝𝑡, 𝑁𝑜𝑡𝐴𝑑𝑜𝑝𝑡).

– Thereis,however,athirdNashequilibriainwhichbothfirmsuseamixedstrategy.


• 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.



• 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



• Example (continued):best-response


Sequential-MoveGames


Sequential-MoveGames• Whenplayerschoosetheirstrategiessequentially,ratherthan

simultaneously,thedefinitionofstrategybecomesmoreinvolved.

• Strategyisnowacompletecontingentplandescribingwhatactionplayer𝑖 choosesateachpointatwhichheiscalledontomove,giventheprevioushistoryofplay.

• Suchhistorycanbeobservableornotobservablebyplayer𝑖.



• Sequential-movegamesarerepresentedusinggametreesratherthanwithmatrices.– The“root”ofthetree,wherethegamestarts,isreferredtoastheinitialnode.

– Thelastnodesofthetree,wherenomorebranchesoriginate,aretheterminalnodes.



• Basicrules:1. Atreemusthaveonlyoneinitialnode.2. Everynodeofthetreehasexactlyoneimmediate

predecessorexcepttheinitialnode,whichhasnopredecessor.

3. Multiplebranchesextendingfromthesamenodemusthavedifferentactionlabels.

4. Everyinformationsetcontainsdecisionnodesforonlyoneoftheplayersinthegame.

5. Allnodesinagiveninformationsethavethesameimmediatesuccessors



• Informationsetsareusedtodenoteagroupofnodesamongwhichaplayercannotdistinguish.

• Acommonfeatureoftreesrepresentinggamesofincompleteinformation.

• Informationsetsarisewhenaplayerdoesnotobservetheactionthathispredecessorchose.



• CanwesimplyusetheNEsolutionconceptinordertofindequilibriumpredictionsinsequential-movegames?

• Wecan,butsomeoftheNEpredictionsarenotverysensible(credible).



• Example (Entryandpredationgame):– Consideranentrant’sdecisiononwhethertoenterintoanindustrywhereanincumbentfirmoperatesortostayout.



• Example (continued):– InordertofindtheNEofthisgame,itisusefultorepresentthegameinmatrixform.


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.


Sequential-MoveGames• Asubgame canbeidentifiedbydrawingarectanglearound

asectionofthegametreewithout“breaking”anyinformationset


Sequential-MoveGames• Thebackwardinductionrequirestofindthestrategythateveryplayer𝑖 findsoptimalateverysubgamealongthegametree.– Startbyidentifyingtheoptimalbehavioroftheplayerwhoactslast(inthelastsubgameofthetree).

– Takingtheoptimalactionofthisplayerintoaccount,movetotheprevioustothelastplayerandidentifyhisoptimalbehavior.

– Repeatthisprocessuntiltheinitialnode.• SubgameperfectNashequilibrium (SPNE):Astrategyprofile(𝑠/∗, 𝑠H∗, … , 𝑠]∗ ) isaSPNEifitspecifiesaNEforeachsubgame.



• Example (Entryandpredationgame):– Identifythesubgamesofthegametree


– TheSPNEis(𝐼𝑛, 𝐴𝑐𝑐𝑜𝑚𝑚𝑜𝑑𝑎𝑡𝑒),whichcoincideswithoneoftheNEofthisgame.


• Example (backwardinductioninthreesteps):



• Example (backwardinductionininformationsets):



• Example (continued):– Thesmallestsubgameisisstrategicallyequivalenttooneinwhichplayer1and2choosetheiractionssimultaneously.


Player2X Y

Player1A 3,4 1,4B 2,1 2,0

– TheNEofthesubgameis(𝐴, 𝑋).


• Example (continued):– Oncewehaveareduced-formgametree,wecanmoveonestepbackward(theinitialnode)


– TheSPNEofthisgameis(𝑈𝑝|𝐴, 𝑋).– Player1’sstrategy:play𝑈𝑝 inthefirstnodeand𝐴 afterwards– Player2’sstrategy:play𝑋


• Example (continued):– Normal-formrepresentationofthesequentialgame


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



• Thestrategicsettingspreviouslyanalyzedassumethatallplayersareperfectlyinformedaboutallrelevantdetailsofthegame.

• Thereareoftenreal-lifesituationswhereplayersoperatewithoutsuchinformation.

• Playersactunder“incompleteinformation”ifatleastoneplayercannotobserveapieceofinformation.– Example:marginalcostsofrivalfirms



• 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).



• Example (technologyadoption):– Afirstmoveofnaturedeterminestheprecisetypeof𝜃$.– FirmAhastwopossibletypes,eitherhighorlowcosts,withassociatedprobabilities2/3and1/3.

– FirmAobservesitsowntype,butfirmBcannotobserveit.– Graphically,firmAknowswhichpayoffmatrixfirmsareplaying,whilefirmBcanonlyassignaprobability2/3(1/3)toplayingtheright-hand(left-hand)matrix.



• 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*$afterobservinghisowntype𝜃$.



• 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$ .



• Approach1:FourstepstofindallBNEsinsimultaneous-movegamesofincompleteinformation.

• Example (technologyadoption):1. Strategysets:Identifythestrategysetforeachplayer,

whichcanbeafunctionofhisprivatelyobservedtype𝑆/ = 𝐼g𝐼h, 𝐼g𝑁𝐼h, 𝑁𝐼g𝐼h, 𝑁𝐼g𝑁𝐼h𝑆H = 𝐼, 𝑁𝐼



• Example (continued):2. Bayesiannormal-formrepresentation:Usethe

strategysetsidentifiedinstep1toconstructthe“Bayesiannormal-form”representationoftheincompleteinformationgame.


FirmB𝐼 𝑁𝐼

Firm A

𝐼g𝐼h𝐼g𝑁𝐼h𝑁𝐼g𝐼h𝑁𝐼g𝑁𝐼h


• Example (continued):3. Expectedpayoffs:Findtheexpectedpayoffsthat

wouldgoineverycell.


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


• Example (continued):4. Findbestresponsesforeachplayer:Followan

approachsimilartothatinsimultaneous-movegamesofcompleteinformationtofindbest-responsepayoffs.

– TheBNEsare(𝐼g𝐼h, 𝐼) and(𝐼g𝑁𝐼h, 𝑁𝐼).



• Approach2:FindthesetofBNEsbyfirstanalyzingbestresponsesfortheprivatelyinformedplayer,andthenusethoseinouridentificationofbestresponsesfortheuninformedplayer.



• Example (technologyadoption):– Thetwopossiblegamesthatfirmscouldbeplaying.



• 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.



• 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



• 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)



• 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).


Sequential-MoveGamesunderIncompleteInformation



• TheBNEsolutionconcepthelpsusfindequilibriumoutcomesinsettingswhereplayersinteractunderincompleteinformation.

• Whiletheapplicationsintheprevioussectionconsideredthatplayersactsimultaneously,wecanalsofindtheBNEsofincompleteinformationgamesinwhichplayersactsequentially.




• Investmentgame:


• InordertofindthesetofBNEs,wefirstrepresenttheBayesiannormal-form representationofthegametree.

• Thematrixincludesexpectedpayoffsforeachplayer.


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


• TherearetwoBNEsinthisgame:(𝑂C𝑂]C, 𝐴)(𝑁C𝑁]C, 𝑅)

• ThefirstBNEisrathersensible– Player1makestheofferregardlessofhistype,andthustheuninformedplayer2choosestoaccepttheofferifhereceivesone.



• 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.



• Inordertoavoididentifyingequilibriumpredictionsthatarenotsequentiallyrational,weapplythePerfectBayesianEquilibrium(PBE)thatcandealwithsequentialmovegameswithincompleteinformation.

• ThePerfectBayesianEquilibrium(PBE):Astrategyprofit(𝑠/, 𝑠H, … , 𝑠]) andbeliefs𝜇 overthenodesatallinformationsetsareaPBEif:1. eachplayer’sstrategiesspecifyoptimalactions,giventhe

strategiesoftheotherplayers,andgivenhisbeliefs,and2. beliefsareconsistentwithBayes’s rule,whenever

possible.



• 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)



• 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.



• ProceduretoFindPBEs:1. Specifyastrategyprofilefortheprivatelyinformed

player.– Intheinvestmentexample,therearefourpossiblestrategyprofilesfortheprivatelyinformedplayer1.

– Twoseparatingstrategyprofiles:𝑂C𝑁]C,𝑁C𝑂]C.– Twopoolingstrategyprofiles:𝑂C𝑂]C,𝑁C𝑁]C.

2. Updatetheuninformed player’sbeliefsusingBayes’sruleatallinformationsets,wheneverpossible.



• ProceduretoFindPBEs:(continued)3. Giventheuninformed player’supdatedbeliefs,findhis

optimalresponse– Intheinvestmentexample,weneedtodeterminetheoptimalresponseofplayer2uponreceivinganofferfromplayer1givenhisupdatedbelief.

4. Giventheoptimalresponseoftheuninformedplayerobtainedinstep3,findtheoptimalaction(message)foreachtypeofinformed player.

– Intheinvestmentexample,firstcheckifplayer1makesanofferwhentheinvestmentisbeneficial.

– Thencheckwhetherplayer1preferstomakeanoffer,whentheinvestmentisnotbeneficial.



• ProceduretoFindPBEs:(continued)5. Checkifthestrategyprofilefortheinformedplayer

foundinstep4coincideswiththeprofilesuggestedinstep1.

– Ifitcoincides,thenthisstrategyprofile,updatedbeliefs,andoptimalresponsescanbesupportedasaPBEoftheincompleteinformationgame.

– Otherwise,wesaythatthisstrategyprofilecannotbesustainedasaPBEofthegame.



• Example (Labormarketsignalinggame):– Thesequentialgamewithincompleteinformation.– Aworkerprivatelyobserveswhetherhehasahighproductivityoralowproductivity.

– Theworkerthendecideswhethertopursuemoreeducation(e.g.,anMBA)thathemightuseasasignalabouthisproductivity.

– Thefirmcaneitherhirehimasamanager(M)orasacashier(C).



• Example (continued):– Wefocuson:

• Separatingstrategyprofiles: 𝐸g,𝑁𝐸h

• Poolingstrategyprofile:(𝑁𝐸g,𝑁𝐸h)– Exercise:

• Separatingstrategyprofiles: 𝑁𝐸g, 𝐸h

• Poolingstrategyprofile:(𝐸g, 𝐸h)



• Example (continued):1. SeparatingPBE 𝑬𝑯,𝑵𝑬𝑳 :– Step1:Specifytheseparatingstrategyprofile𝐸g,𝑁𝐸h fortheinformedplayer.



• 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.



• 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.



• Example (continued):– Step3:Giventhefirm’sbeliefs,determinethefirm’soptimalresponse,afterobservingtheeducationleveloftheworker.



• Example (continued):– Step4:Giventhesestrategyprofiles,examinetheworker’soptimalaction.• High-productivitytype:Doesnothaveanincentivetodeviatefromthestrategyprofile(acquiringmoreeducation).

• Low-productivitytype:Thecostofacquiringeducationistoohighforthelow-productivityworker;andthusthatworkerchoosestonotpursueit.



• Example (continued):– Step5:Theseparatingstrategyprofile(𝐸g,𝑁𝐸h) canbesustainedasthePBEofthisincompleteinformationgame.• Neithertypeofworkerhastheincentivetodeviatefromtheprescribedseparatingstrategyprofile(𝐸g, 𝑁𝐸h).



• Example (continued):2. PoolingPBE 𝑵𝑬𝑯,𝑵𝑬𝑳 :– Step1:Specifytheseparatingstrategyprofile𝑁𝐸g,𝑁𝐸hfortheinformedplayer.



• 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.



• 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].



• 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.



• 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



• Example (continued):– Step4:Giventhesestrategyprofiles,examinetheworker’soptimalaction.

– Case1:𝜇 > 2/5• High-productivitytype:Hasanincentivetodeviatefromtheprescribedstrategyprofile.ThusitcannotbesupportedasaPBE.

• Low-productivitytype:Doesnothaveincentivestodeviatefromtheprescribedstrategyprofile.



• Example (continued):– Case2:𝜇 ≤ 2/5

• High-productivitytype:Doesnothaveincentivestodeviatefromtheprescribedstrategyprofile.

• Low-productivitytype:Doesnothaveincentivestodeviatefromtheprescribedstrategyprofile.



• Example (continued):– Step5:Thepoolingstrategyprofile(𝑁𝐸g,𝑁𝐸h) canbesupportedasthePBEwhenoff-the-equilibriumbeliefssatisfy𝜇 ≤ 2/5.• Neithertypeofworkerhastheincentivetodeviatefromtheprescribedseparatingstrategyprofile(𝐸g, 𝑁𝐸h).


BertrandModelofPriceCompetition



• Consider:– Anindustrywithtwofirms,1and2,sellingahomogeneousproduct

– Firmsfacemarketdemand𝑥(𝑝),where𝑥(𝑝) iscontinuousandstrictlydecreasingin𝑝

– Thereexistsahighenoughprice(chokeprice) �̅� < ∞ suchthat𝑥(𝑝) = 0 forall𝑝 > �̅�

– Bothfirmsaresymmetricintheirconstantmarginalcost𝑐 > 0,where𝑥 𝑐 ∈ (0,∞)

– Everyfirm𝑗 simultaneously setsaprice𝑝°



• 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±



• Givenprices𝑝° and𝑝±,firm𝑗’sprofitsaretherefore

(𝑝° − 𝑐) j 𝑥° (𝑝°, 𝑝±)

• WearenowreadytofindequilibriumpricesintheBertrandduopolymodel.– ThereisauniqueNE(𝑝°∗, 𝑝±∗) intheBertrandduopolymodel.Inthisequilibrium,bothfirmssetpricesequaltomarginalcost,𝑝°∗ = 𝑝±∗ = 𝑐.



• 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.


pj

pk

pm

pm

c

c

pj(pk)

45°-line(pj=pk)


• Firm𝑗’sbestresponsehas:– aflatsegmentforall𝑝± < 𝑐,where𝑝°(𝑝±) = 𝑐

– apositiveslopeforall𝑐 < 𝑝± < 𝑝°,wherefirm𝑗 chargesapriceslightlybelowfirm𝑘

– aflatsegmentforall𝑝± > 𝑝�,where𝑝°(𝑝±) = 𝑝�


pj

pk

c

c

pj(pk)

pk(pj)

pm

pm

45°-line(pj=pk)


• Asymmetricargumentappliestotheconstructionofthebestresponsefunctionoffirm𝑘.

• Amutualbestresponseforbothfirmsis

(𝑝/∗, 𝑝H∗) = (𝑐, 𝑐)wherethetwobestresponsefunctionscrosseachother.

• ThisistheNEoftheBertrandmodel– Firmsmakenoeconomicprofits.



• Withonlytwofirmscompetinginpricesweobtaintheperfectlycompetitiveoutcome,wherefirmssetpricesequaltomarginalcost.

• Pricecompetitionmakeseachfirm𝑗 faceaninfinitelyelasticdemandcurveatitsrival’sprice,𝑝±.– Anyincrease(decrease)from𝑝± infinitelyreduces(increases,respectively)firm𝑗’sdemand.



• HowmuchdoesBertrandequilibriumhingeintoourassumptions?– Quitealot

• ThecompetitivepressureintheBertrandmodelwithhomogenousproductsisamelioratedifweinsteadconsider:– Pricecompetition(butallowingforheterogeneousproducts)

– Quantitycompetition(stillwithhomogenousproducts)

– CapacityconstraintsAdvancedMicroeconomicTheory 100


• Remark:– Howourresultswouldbeaffectediffirmsfacedifferentproductioncosts,i.e.,0 < 𝑐/ < 𝑐H?

– Themostefficientfirmsetsapriceequaltothemarginalcostoftheleastefficientfirm,𝑝/ = 𝑐H.

– Otherfirmswillsetarandompriceintheuniforminterval

[𝑐/, 𝑐/ + 𝜂]where𝜂 > 0 issomesmallrandomincrementwithprobabilitydistribution𝑓 𝑝, 𝜂 > 0 forall𝑝.


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.



• Letusfirstidentifyeveryfirm’sbestresponsefunction

• Firm1’sPMP,foragivenoutputlevelofitsrival,𝑞·H,

maxº»¼½

𝑝 𝑞/ + 𝑞·HPrice

𝑞/ − 𝑐𝑞/

• WhensolvingthisPMP,firm1treatsfirm2’sproduction,𝑞·H, asaparameter,sincefirm1cannotvaryitslevel.



• 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



• 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.



• Hence,if𝑞/∗ = 0, firm2’soutputwouldalsobezero,𝑞H∗ = 0.

• Butthisimpliesthat𝑝(0) < 𝑐 sincenofirmproducesapositiveoutput,thusviolatingourinitialassumption𝑝(0) > 𝑐.– Contradiction!

• Asaresult,wemusthavethatboth𝑞/∗ > 0 and𝑞H∗ > 0.

• Note:Thisresultdoesnotnecessarilyholdwhenbothfirmsareasymmetricintheirproductioncost.



• 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



• Example (continue):– Solvingfor𝑞/, weobtainfirm1’sBRF

𝑞/(𝑞·H) =¿*ÀHÁ

− º·ÂH

– Analogously,firm2’sBRF

𝑞H(𝑞·/) =¿*ÀHÁ

− º·»H




• Firm1’sBRF:– When𝑞H = 0,then𝑞/ =

¿*ÀHÁ

,whichcoincideswithitsoutputundermonopoly.

– As𝑞H increases,𝑞/decreases (i.e.,firm1’sand2’soutputarestrategicsubstitutes)

– When𝑞H =¿*ÀÁ, then

𝑞/ = 0.



• Asimilarargumentappliesforfirm2’sBRF.

• Superimposingbothfirms’BRFs,weobtaintheCournotequilibriumoutputpair(𝑞/∗, 𝑞H∗).



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°


• 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.



• 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


• Quantitycompetition(Cournotmodel)yieldslesscompetitiveoutcomesthanpricecompetition(Bertrandmodel),wherebyfirms’behaviormimicsthatinperfectlycompetitivemarkets– That’sbecause,thedemandthateveryfirmfacesintheCournotgameisnotinfinitelyelastic.

– Areductioninoutputdoesnotproduceaninfiniteincreaseinmarketprice,butinsteadanincreaseof− 𝑝′(𝑞/ + 𝑞H).

– Hence,iffirmsproducethesameoutputasundermarginalcostpricing,i.e.,halfof¿*À

H,eachfirmwould

haveincentivestodeviatefromsuchahighoutputlevelbymarginallyreducingitsoutput.



• EquilibriumoutputunderCournotdoesnotcoincidewiththemonopolyoutputeither.– That’sbecause,everyfirm𝑖,individuallyincreasingitsoutputlevel𝑞$,takesintoaccounthowthereductioninmarketpriceaffectsitsownprofits,butignorestheprofitloss(i.e., anegativeexternaleffect)thatitsrivalsuffersfromsuchalowerprice.

– Sinceeveryfirmdoesnottakeintoaccountthisexternaleffect,aggregateoutputistoolarge,relativetotheoutputthatwouldmaximizefirms’jointprofits.



• 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


• 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



• 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.


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



– Recalling/Ë= −𝑝4 𝑞 ⋅ º

� º,wehave

𝑞 � º *ÀÌ�(º)

= /Ë𝑞°

or� º *ÀÌ�(º)

= /ËºÌº

– Defining𝛼° ≡ºÌº

asfirm𝑗’smarketshare,weobtain𝑝 𝑞 − 𝑐°𝑝(𝑞) =

𝛼°𝜀

whichisreferredtoastheCournotpricingrule.AdvancedMicroeconomicTheory 121


– Note:§When𝛼° = 1,implyingthatfirm𝑗 isamonopoly,theIEPRbecomesaspecialcaseoftheCournotpricerule.

§ Thelargerthemarketshare𝛼° ofagivenfirm,thelargerthepricemarkupoffirm𝑗.

§ Themoreinelasticdemand𝜀 is,thelargerthepricemarkupoffirm𝑗.



• 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



• 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


%Δp

k20 40 60 80 100

0.10

0.20 %Δp(k)


• Example (continued):– Thepercentageincreaseinpriceafterthemerger,%Δ𝑝,asafunctionofthenumberofmergingfirms,𝑘.

– Forsimplicity,𝑛 =100.


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



• 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.



• 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.


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



• 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


• 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


c2

c1

a

a

45°(c1=c2)a+c22c1=

a+c12c2=

a2

a2

Nofirmsproduce

Bothfirmsproduce

Onlyfirm1produces

Onlyfirm2produces




• 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.


q1

q2a–c22b

a–c12b

a–c1b

a–c2b

(q1,q2)**q1(q2)

q2(q1)


• BRFsforfirms1and2when𝑐/ >

¿1ÀÂH

(i.e.,onlyfirm2produces).

• BRFscrossattheverticalaxiswhere𝑞/∗ = 0 and𝑞H∗ > 0 (i.e.,acornersolution)


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



• 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$∗



• Solvingfor𝑞$∗

𝑞$∗ =𝑎 − 𝑐𝐽 + 1 𝑏

whichisalsotheequilibriumoutputforother𝐽 − 1firms.

• Therefore,aggregateoutputis

𝑄∗ = 𝐽𝑞$∗ =𝐽

𝐽 + 1𝑎 − 𝑐𝑏

andthecorrespondingequilibriumpriceis

𝑝∗ = 𝑎 − 𝑏𝑄∗ =𝑎 + 𝐽𝑐𝐽 + 1



• 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



• 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.



• 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.



• WecanshowthatlimÙ→ß

𝑞$∗ = 0

limÙ→ß

𝑄∗ =𝑎 − 𝑐𝑏

limÙ→ß

𝑝∗ = 𝑐

whichcoincideswiththesolutioninaperfectlycompetitivemarket.


ProductDifferentiation


ProductDifferentiation

• Sofarweassumedthatfirmssellhomogenous(undifferentiated)products.

• Whatifthegoodsfirmssellaredifferentiated?– Forsimplicity,wewillassumethatproductattributesareexogenous(notchosenbythefirm)


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



• Solvingfor𝑝$,wefindfirm𝑖’sBRF

𝑝$(𝑝°) =𝑎 + 𝑐𝑝°2𝑏

• Firm𝑗 alsohasasymmetricBRF.• Note:– BRFsarenowpositivelysloped– Anincreaseinfirm𝑗’spriceleadsfirm𝑖 toincreasehis,andviceversa

– Inthiscase,firms’choices(i.e.,prices)arestrategiccomplements




p1

p2

a2b

p2*

p1(p2)

p2(p1)

p1*

a2b


• 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.



• Firm𝑖’sPMPis(assumingnocosts)maxºÒ¼½

(𝛼 − 𝛽𝑞$ − 𝛾𝑞°)𝑞$

• FOC:𝛼 − 2𝛽𝑞$ − 𝛾𝑞° = 0

• Solvingfor𝑞$ wefindfirm𝑖’sBRF𝑞$(𝑞°) =

𝛼2𝛽 −

𝛾2𝛽 𝑞°

• Firm𝑗 alsohasasymmetricBRF




q1

q2

(q1,q2)**

q1(q2)

q2(q1)α2β

αγ

α2β

αγ


• Comparativestaticsoffirm𝑖’sBRF– As𝛽 → 𝛾 (productsbecomemorehomogeneous),BRFbecomessteeper.Thatis,theprofit-maximizingchoiceof𝑞$ ismoresensitivetochangesin𝑞° (toughercompetition)

– As𝛾 → 0 (productsbecomeverydifferentiated),firm𝑖’sBRFnolongerdependson𝑞° andbecomesflat(mildercompetition)



• 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



• Note:– As𝛾 increases(productsbecomemorehomogeneous),individualandaggregateoutputdecrease,andindividualprofitsdecreaseaswell.

– If𝛾 → 𝛽 (indicatingundifferentiatedproducts),then𝑞$∗ =

àHá1á

= àÃá

asinstandardCournotmodelsofhomogeneousproducts.

– If𝛾 → 0 (extremelydifferentiatedproducts),then𝑞$∗ =

àHá1½

= àHá

asinmonopoly.


DynamicCompetition


DynamicCompetition:SequentialBertrandModelwithHomogeneousProducts

• Assumethatfirm1choosesitsprice𝑝/ first,whereasfirm2observesthatpriceandrespondswithitsownprice𝑝H.

• Sincethegameisasequential-movegame(ratherthanasimultaneous-movegame),weshouldusebackwardinduction.



• 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.



• Theleaderexpectsthatitspricewillbe:– undercutbythefollowerwhen𝑝/ > 𝑐 (thusyieldingnosales)

– mimickedbythefollowerwhen𝑝/ = 𝑐 (thusentailinghalfofthemarketshare)

• Hence,theleaderhas(weak)incentivestosetaprice𝑝/ = 𝑐.

• Asaconsequence,theequilibriumpricepairremainsat(𝑝/∗, 𝑝H∗) = (𝑐, 𝑐),asinthesimultaneous-moveversionoftheBertrandmodel.


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�Â¼½

𝜋H = 𝑝H𝑞H − 𝑇𝐶(𝑞H)

= 𝑝H𝐷H(𝑝H, 𝑝/) − 𝑇𝐶(𝐷H(𝑝H, 𝑝/)ºÂ

)



• 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/).



• 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»



• 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



• IncontrasttotheBertrandmodelwithsimultaneouspricecompetition,anincreaseinfirm1’spricenowproducesanincreaseinfirm2’spriceinthesecondstage.

• Hence,theleaderhasmoreincentivestoraiseitsprice,ultimatelysofteningthepricecompetition.

• Whileasoftenedcompetitionbenefitsboththeleaderandthefollower,therealbeneficiaryisthefollower,asitsprofitsincreasemorethantheleader’s.



• Example:– Consideralineardemand𝑞$ = 1 − 2𝑝$ + 𝑝°,withnomarginalcosts,i.e.,𝑐 = 0.

– SimultaneousBertrandmodel:thePMPismax�Ì¼½

𝜋° = 𝑝° j (1 − 2𝑝° + 𝑝±)forany𝑘 ≠ 𝑗

whereFOCwrt 𝑝° producesfirm𝑗’sBRF

𝑝°(𝑝±) =14 +

14𝑝±

– SimultaneouslysolvingthetwoBRFsyields𝑝°∗ =/Ã≃

0.33,entailingequilibriumprofitsof𝜋°∗ =HÄ≃ 0.222.



• Example (continued):– SequentialBertrandmodel: inthesecondstage,firm2’s(thefollower’s)PMPis

max�Â¼½

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/)



• 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


p1

p2

p1(p2)

p2(p1)

⅓

¼

¼

Priceswithsequentialpricecompetition

0.36

0.34⅓

Priceswithsimultaneouspricecompetition


• Example (continued):– Bothfirms’pricesandprofitsarehigherinthesequentialthaninthesimultaneousgame.

– However,thefollowerearnsmorethantheleaderinthesequentialgame!(secondmover’sadvantage)


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/)



• 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.



• Thestrategiceffectispositivesince𝑝4(𝑄) < 0andèºÂ(º»)

èº»< 0.

• Firm1(theleader)hasmoreincentivetoraise𝑞/relativetotheCournotmodelwithsimultaneousoutputdecision.

• Intuition (first-moveradvantage):– Byoverproducing,theleaderforcesthefollowertoreduceitsoutput𝑞H bytheamountèºÂ(º»)

èº».

– Thishelpstheleadersellitsproductionatahigherprice,asreflectedby𝑝′(𝑄);ultimatelyearningalargerprofitthaninthestandardCournotmodel.



• 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



• 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.




q1

q2

a–c

q1(q2)

2

q2(q1)

a–c2

CournotQuantities

StackelbergQuantities


• 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


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


• Linearinversedemand𝑝 𝑄 = 𝑎 − 𝑄

• Symmetricmarginalcosts𝑐 > 0


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


• Solvingfor𝑞H yieldsfirm2’sBRF𝑞H(𝑞/) =

à*êº»Há

• Plugging𝑞H 𝑞/ intotheleader’sfirm1’s(theleader’s)PMP,weget

maxº»

𝛼 − 𝛽𝑞/ − 𝛾à*êº»Há

𝑞/ =

maxº»

𝛼 Há*êHá

− HáÂ*êÂ

Há𝑞/ 𝑞/

• FOC:

𝛼 Há*êHá

− HáÂ*êÂ

á𝑞/ = 0



• Solvingfor𝑞/,weobtaintheleader’sequilibriumoutputlevel𝑞/∗ =

à(Há*ê)H(HáÂ*êÂ)

• Substituting𝑞/∗ intothefollower’sBRFyieldsthefollower’sequilibriumoutput

𝑞H∗ =à*êº»∗

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.


CapacityConstraints


CapacityConstraints• HowcomeareequilibriumoutcomesinthestandardBertrandandCournotmodelssodifferent?

• Dofirmsreallycompeteinpriceswithoutfacingcapacityconstraints?– Bertrandmodelassumesafirmcansupplyinfinitelylargeamountifitspriceislowerthanitsrivals.

• ExtensionoftheBertrandmodel:– Firststage:firmssetcapacities,𝑞·/ and𝑞·H,withacostofcapacity𝑐 > 0

– Secondstage:firmsobserveeachother’scapacitiesandcompeteinprices,simultaneouslysetting𝑝/ and𝑝H


CapacityConstraints• Whatistheroleofcapacityconstraint?– Whenafirm’spriceislowerthanitscapacity,notallconsumerscanbeserved.

– Hence,salesmustberationedthroughefficientrationing:thecustomerswiththehighestwillingnesstopaygettheproductfirst.

• Intuitively,if𝑝/ < 𝑝H andthequantitydemandedat𝑝/ issolargethat𝑄(𝑝/) > 𝑞·/,thenthefirst𝑞·/unitsareservedtothecustomerswiththehighestwillingnesstopay(i.e.,theuppersegmentofthedemandcurve),whilesomecustomersareleftintheformofresidualdemandtofirm2.


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


• 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.


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.


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


• Inordertoprovethisresult,westartbyassumingthatfirm1sets𝑝/ = 𝑝∗.Wenowneedtoshowthatfirm2alsosets𝑝H = 𝑝∗,i.e.,itdoesnothaveincentivestodeviatefrom𝑝∗.

• Iffirm2doesnotdeviate,𝑝/ = 𝑝H = 𝑝∗,thenitsellsuptoitscapacity𝑞·H.

• Iffirm2reducesitspricebelow𝑝∗,demandwouldexceeditscapacity𝑞·H.Asaresult,firm2wouldsellthesameunitsasbefore,𝑞·H,butatalowerprice.


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.


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


• Therefore,𝑎 > 𝑞·/ + 2𝑞·H holdsif𝑎 >Ã¿Â

ÅÀwhich,

solvingfor𝑎,isequivalenttoÅÀÃ> 𝑎.



• WhenÅÀÃ> 𝑎 holds,

capacityconstraint¿Â

ÅÀ≥

𝑞·° transformsintoÃ¿Â

ÅÀ>

𝑞·/ + 2𝑞·H,implying𝑝∗ >𝑝H = 𝑎 − º·»

H.

• Thus,firm2doesnothaveincentivestoincreaseitsprice𝑝H from𝑝∗,sincethatwouldloweritsrevenues.



• Inshort,firm2doesnothaveincentivestodeviatefromthecommonprice

𝑝∗ = 𝑎 − 𝑞·/ − 𝑞·H• Asimilarargumentappliestofirm1(bysymmetry).

• Hence,wehavefoundanequilibriuminthepricingstage.


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·±


CapacityConstraints:FirstStage

• SolvingthetwoBRFssimultaneously,weobtainasymmetricsolution

𝑞·° = 𝑞·± =𝑎 − 𝑐3

• ThesearethesameequilibriumpredictionsasthoseinthestandardCournotmodel.

• Hence,capacitiesinthistwo-stagegamecoincidewithoutputdecisionsinthestandardCournotmodel,whilepricesaresetequaltototalcapacity.


EndogenousEntry


EndogenousEntry

• Sofarthenumberoffirmswasexogenous• Whatifthenumberoffirmsoperatinginamarketisendogenouslydetermined?

• Thatis,howmanyfirmswouldenteranindustrywhere– TheyknowthatcompetitionwillbealaCournot– Theymustincurafixedentrycost𝐹 > 0.


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


EndogenousEntry

• Threeassumptions(validundermostdemandandcostfunctions):– individualequilibriumoutput𝑞(𝑛) isdecreasingin𝑛;

– aggregateoutput𝑞 ≡ 𝑛 j 𝑞(𝑛) increasesin𝑛;– equilibriumprice𝑝(𝑛 j 𝑞(𝑛)) remainsabovemarginalcostsregardlessofthenumberofentrants𝑛.


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


EndogenousEntry

• Socialoptimum:– Thesocialplannerchoosesthenumberofentrants𝑛� thatmaximizessocialwelfare

maxI𝑊 𝑛 ≡ ï 𝑝 𝑠 𝑑𝑠 − 𝑛 j 𝑐 𝑞 𝑛 − 𝑛 j 𝐹

Iº(I)

½


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


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)


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.


EndogenousEntry

• Giventhenegativesignofthebusinessstealingeffect,wecanconcludethat

𝑊4 𝑛 = 𝜋 𝑛 + 𝑛 𝑝 𝑛𝑞 𝑛 − 𝑐4 𝑞 𝑛𝜕𝑞 𝑛𝜕𝑛

*

< 𝜋(𝑛)

andthereforemorefirmsenterinequilibriumthaninthesocialoptimum,i.e.,𝑛� > 𝑛�.


EndogenousEntry


EndogenousEntry

• Example:– Consideralinearinversedemand𝑝 𝑄 = 1 − 𝑄andnomarginalcosts.

– Theequilibriumquantityinamarketwith𝑛 firmsthatcompetealaCournotis

𝑞 𝑛 = /I1/

– Let’scheckifthethreeassumptionsfromabovehold.


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.


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


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 .


Entrycosts,F

ne=–1(Equilibrium)1F½

no=–1(Soc.Optimal)1F⅓

Numberoffirms

0

EndogenousEntry

• Example (continued):


RepeatedInteraction


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.


RepeatedInteraction:BertrandModel

• Considertwofirmssellinghomogeneousproducts.• Let𝑝°� denotefirm𝑗’spricingstrategyatperiod𝑡,whichisafunctionofthehistoryofallpricechoicesbythetwofirms,𝐻�*/ = 𝑝/�, 𝑝H� �ó/

�*/.• Conditioning𝑝°� onthefullhistoryofplayallowsforawiderangeofpricingstrategies:– settingthesamepriceregardlessofprevioushistory– retaliationiftherivallowersitspricebelowa“thresholdlevel”

– increasingcooperationiftherivalwascooperativeinpreviousperiods(untilreachingthemonopolyprice𝑝�)



• Finitelyrepeatedgame:– CanwesupportcooperationiftheBertrandgameisrepeatedforafinitenumberof𝑇 rounds?§ No!

– Toseewhy,considerthelastperiodoftherepeatedgame(period𝑇):§ Regardlessofpreviouspricingstrategies,everyfirms’optimalpricingstrategyinthisstageistoset𝑝$,ô∗ = 𝑐,asintheone-shotBertrandgame.



– 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.



• 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.



– ThistypeofstrategyisusuallyreferredtoasNashreversionstrategy (NRS):§ firmscooperateuntiloneofthemdeviates,inwhichcasefirmsthereafterreverttotheNashequilibriumoftheunrepeatedgame(i.e.,setpricesequaltomarginalcost)

– LetusshowthatNRScanbesustainedintheequilibriumoftheinfinitelyrepeatedgame.

–Weneedtodemonstratethatfirmsdonothaveincentivestodeviatefromit,duringanyperiod𝑡 > 1andregardlessoftheirprevioushistoryofplay.



– 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



– 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�



– Hence,firm𝑗 preferstosticktotheNRSatperiod𝑡 if1

1 − 𝛿𝜋�

2 > 𝜋� ⟺ 𝛿 >12

– Thatis,cooperationcanbesustainedaslongasfirmsassignasufficientlyhighvaluetofutureprofits.


Profits

TimePeriodst t+1 t+2 ...

InstantaneousGain

Profitfromcooperating

πm2

πm

FutureLosses

πm2πm2

δ

δ2


– Instantaneousgainsandlossesfromcooperationanddeviation



• Whataboutfirm𝑗’sincentivestouseNRSafterahistoryofplayinwhichsomefirmsdeviated?– NRScallsforfirm𝑗 toreverttotheequilibriumoftheunrepeatedBertrandmodel.

– Thatis,toimplementthepunishmentembodiedinNRSafterdetectingadeviationfromanyplayer.

• BystickingtotheNRS,firm𝑗’sdiscountedstreamofpayoffsis

0 + 𝛿0 +⋯ = 0AdvancedMicroeconomicTheory 221


• BydeviatingfromNRS(i.e.,settingaprice𝑝° =𝑝� whileitsopponentsetsapunishingprice𝑝± = 𝑐),profitsarealsozeroinallperiods.

• Hence,firm𝑗 hasincentivestocarryoutthethreat– Thatis,settingapunishingpriceof𝑝° = 𝑐,uponobservingadeviationinanypreviousperiod.

• Asaresult,theNRScanbesustainedinequilibrium,sincebothfirmshaveincentivestouseit,atanytimeperiod𝑡 > 1 andirrespectiveoftheprevioushistoryofplay.



• Example:– Consideranindustrywithonly2firms,alineardemand𝑄 = 5000 − 100𝑝, andconstantandaveragemarginalcostsof𝑐 = $10.

– Ifone-shotBertrandgameisplayed,firmswould§ chargeapriceof𝑝 = 𝑐 = $10§ sellatotalquantityof4000unitsofaproduct§ earnzeroeconomicprofits

– If,incontrast,firmscolludetofixpricesatthemonopolyprice,cansuchcollusionbesustained?



• 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).



• Example (continued):– Collusionatthemonopolypriceissustainableif

𝜋�

21

1 − 𝛿 ≥ 𝜋� +𝛿

1 − 𝛿 ⋅ 0

– Since𝜋� = $40,000,theinequalityreducesto

200001

1 − 𝛿 ≥ 40000 ⟺ 𝛿 ≥12



• Example (continued):–Whatwouldhappeniftherewere𝑛 firms?– Eachfirm’sshareofthemonopolyprofitstreamundercollusionwouldbeõ

ö

I= Å½½½½

I.

– CollusionatthemonopolypriceissustainableifÅ½½½½I

//*ù

≥ 40000 ⟺ 𝛿 ≥ 1 − /I≡ 𝛿̅

– Hence,asthenumberoffirmsintheindustryincreases,itbecomesmoredifficulttosustaincooperation.



• Example (continued):minimaldiscountfactorsustainingcollusion


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.



• 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.



• 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��



• 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.



• Instantaneousgainsandlossesfromdeviation



• 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��.



• Firm𝑗 stickstotheNRSafterahistoryofdeviationssince

𝜋ú�û�I�� + 𝛿𝜋ú�û�I�� + ⋯ > 𝜋ý + 𝛿𝜋ú�û�I�� + ⋯

whichholdsgiventhat𝜋ú�û�I�� > 𝜋ý .

• Hence,noneedtoimposeanyfurtherconditionsontheminimaldiscountfactorsustainingcooperation,𝛿̅.



• 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



• 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



• 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



• Example (continued):– Thisyieldspriceof

𝑝 = 𝑎 − 𝑏 Ã(¿*À)ÊÁ

+ ¿*ÀÅÁ

= Ã¿1çÀÊ

andprofitsof

𝜋$þ�� =Ã¿1çÀÊ

Ã(¿*À)ÊÁ

− 𝑐 Ã ¿*ÀÊÁ

= Ä ¿*À Â

éÅÁforthedeviatingfirm,and

𝜋ý = Ã¿1çÀÊ

(¿*À)ÅÁ

− 𝑐 ¿*ÀÅÁ

= Ã ¿*À Â

ÃHÁforthenon-deviatingfirm.



• 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.



• Extensions:– Temporaryreversionstotheequilibriumoftheunrepeatedgame

– (Temporary)punishmentsthatyieldevenlowerpayoffs

– Less“pure”formsofcooperations– Imperfectmonitoring


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