Commitment
NolanH.Miller¤
AmitPazgaly
September7,1998
Abstract
Weconsideratwostagedi¤erentiatedproductsduopolymodel(withlineardemandandconstantmarginalcost).Inthe…rststagepro…tmaxi-mizingownerschooseincentiveschemesinordertoinducetheirmanagerstoexhibitacertaintypeofbehavior.Inthesecondstagethemanagerscompeteeitherinpricesorinquantities.IncontrasttoSinghandVives(1984),weshowthatiftheownershavesu¢cientpowertomanipulatetheincentivesoftheirmanagers,theequilibriumoutcomeisthesameregard-lessofwhetherthe…rmscompeteinpricesorinquantities.Basingthemanager’sobjectivefunctiononaconvexcombinationofownpro…tandthedi¤erencebetweenownpro…tandtherival…rm’spro…tissu¢cientfortheequivalenceresulttohold.
¤
ManagerialEconomicsandDecisionSciences,J.L.KelloggGraduateSchoolofManagement,
NorthwesternUniversity.y
DepartmentofMarketing,JohnM.OlinSchoolofBusiness,WashingtonUniversity.TheauthorsthankBillSandholm,DanielSpulber,andJeroenSwinkelsforhelpfulcomments.Allerrorsareourown.Sendcommentsto:pazgal@wuolin.wustl.edu
1.Introduction
SinghandVives(1984)provethesomewhatsurprisingresultthatinadi¤eren-tiatedproductsduopolywithconstantmarginalcosts,allelsebeingequal,theequilibriumunderpricecompetitiondi¤ersfromtheequilibriumunderquantitycompetition.Inparticular,theyshowthatinCournotcompetition1,quantitiesarelowerandpricesarehigherthaninBertrandcompetition,regardlessofwhetherthegoodsaresubstitutesorcomplements.Thisdi¤erencestemsfromthefactthattheperceivedelasticityofdemandwhena…rmtakesit’srival’spriceascon-stantislargerthantheperceivedelasticityofdemandwhena…rmtakesit’srival’squantityasconstant.Inotherwords,theoptimalreactionofthemanagerdi¤ersdependingonwhetherhebelieveshisopponenttobeplayingapricestrategy,inwhichcaseheimagineshisopponent’spricetobe…xed,oraquantitystrategy,inwhichcaseheimagineshisopponent’squantitytobe…xed.
Thedi¤erencebetweenCournotandBertrandequilibriaforthesamedemandsystemconstitutesamajorpuzzleforeconomistsseekingtounderstandthestrate-gicinteractionsof…rms.Thepolarcaseis,ofcourse,productsthatareperfectsubstitutes.Inthiscasewe…ndtheso-calledBertrandParadox,thatquantitycompetitionyieldspositivegrossmargins,whilepricecompetitionleadstopricesatthemarginalcostlevel.Fordi¤erentiatedproducts,pricecompetitionisgener-ally“moreaggressive”thanquantitycompetition,yieldinghigherquantitiesandlowerprices(SinghandVives1984).
TheproblemofunderstandingtheequilibriumbehaviorinaduopolymarketisapparentlyevenmorecomplicatedasdemonstratedbytheworkofFershtmanandJudd(1987),Sklivas(1987)andVickers(1985),whoshowinavarietyofmod-elsthattheownerofa…rmmaywanttodistortthepreferencesofitsmanagers(thepeopleresponsibleformakingthepriceorquantitydecision)awayfrompure
1
Throughoutthispaperweusetheterm“Cournotcompetition”whenthe…rmscompeteby
settingquantitiesand“Bertrandcompetition”whenthe…rmscompetebysettingprices.
2
pro…tmaximizationinordertocommittoacertaincompetitiveposture.Fer-shtmanandJudd(1987)andSklivas(1987)(hereafterFJS)considertwostageduopolymodelswhereinthe…rststagepro…tmaximizing…rmschoosecompen-sationschemesformanagersthatarealinearcombinationofpro…tsandsales.Inthesecondstage,themanagers,knowingbothcompensationschemes,competeinaduopoly.SklivasshowsthatinCournotcompetitionownerswillchoosetohavemanagersputlessthanfullweightontheircosts,whileinpricecompetitionwithdi¤erentiated,substitutegoods,…rmswillputmorethanunitaryweightontheircostsinevaluatingtheirmanagers.Thisisequivalenttoputtingpositiveweightonbothpro…tsandsalesinthe…rstcase,andputtingpositiveweightonpro…tsbutnegativeweightonsalesinthesecond.TheresultsofFershtmanandJuddaresimilarincharacter.
Fumas(1991)andMillerandPazgal(1997)considertheroleofrelativeper-formanceevaluationusingthenaturalvariationontheFJS-stylemodelwheremanagersarecompensatedbasedonalinearcombinationofthe…rm’sownpro…tandthepro…tofitsrival.Theystudyatwostagegamewhereinthe…rststagetheownersdecideontheweighttobeputonthepro…tsoftherival…rmandinthesecondstagethemanagersofthe…rmscompeteinaduopolygame.TheequilibriumofthisgameisshowntohavelowerpricesandhigherquantitiesinthecaseofCournotcompetitionandhigherpricesandlowerquantitiesinthecaseofBertrandcompetitionascomparedtotheequilibriumingameswithoutexantecommitmenttorelativeperformance.Hence,theequilibriaunderpriceandquantitycompetitionareshiftedclosertogetheroncetheroleofincentiveschemecommitmentisrecognized.
Inthispaperweconsiderageneralclassoftwo-stagegamesinwhichinthe…rststageownerschooseavectorofincentiveparametersfortheirmanagersandinthesecondstagethemanagerscompete(eitherinpricesorinquantities)inadi¤erentiatedproductsduopolyenvironmentwithlineardemandandconstantmarginalcost.Weshowthatiftheownershavesu¢cientpowertomanipulate
3
theincentivesoftheirmanagers,theequilibriaunderpriceandquantitycompeti-tionareequivalent.Speci…cally,foranypricecompetition(respectivelyquantitycompetition)equilibriumthereexistsaquantitycompetitionequilibrium(pricecompetitionequilibrium)thatinducesthesameequilibriumpricesandquantities,althoughtheequilibriumincentiveparameterswillalmostcertainlydi¤er.Section2describesthemodelandprovestheequivalenceresult.Section3showshowtheequivalenceresultholdsintherelativeperformancemodelofMillerandPazgal(1997)butdoesnotholdinamodelbasedonFershtmanandJudd(1987).Section4discussesissuesregardingtheingeneralizationoftheresulttononlineardemandandincentivesystems.Section5concludes.
2.TheEquivalenceofPriceandQuantityCompetition
WhiletheFJS,Fumas,andMillerandPazgalmodelsdi¤erinthespeci…cmech-anismbywhichthepro…t-maximizingownersofthe…rmscommittheirmanagerstoacertaincompetitivepostures,thetwostagegamestheyconsidershareabasicstructure.Inthesecondstagethemanagers,theirincentivessetbytheownersinthe…rststage,competeinaduopolygame.Themanager’soptimizationproblemthereforede…neshis2reactionfunctionsuchthatforanystrategysjplayedbyhisrival,themangerchoosestheoptimalstrategy,si:Graphically,ifweplotsiontheverticalaxisandsjonthehorizontalaxis,thereactionfunctionissuchthattheisoquantsofmanageri’sincentivefunctionhaveverticaltangentsthroughall(sj;si)onthereactionfunction.WewillrefertothisconditionastheManager’sOptimalityCondition.
Inthe…rststageofthecompetitiontheownerschooseanincentiveschemeforthemanagers.Thatis,giventhesecondstageequilibriuminducedbytheowners’choiceofincentiveparameters,theownerschoosetheincentiveparametersinorder
2
Throughoutthepaper,ownerswillbereferredtoas“she”andmanagersas“he”.
4
tomaximizetheirpro…t.Graphically,thisisequivalentto,wheneverpossible3,ownerichoosingtheincentiveparameterssuchthattheisopro…tcurveof…rmithroughthesecondstageequilibriumpointistangenttothereactionfunctionof…rmj.Thatis,inchoosingtheincentiveparameters,owneriischoosingitsmostpreferredpointo¤ofthereactioncurveofmanagerj.WewillrefertothisastheOwner’sOptimalityCondition.
Inthissection,weshowthatifownershavesu¢cientpowertomanipulatetheincentivesoftheirmanagers,thenforanyequilibriumofthetwo-stagepricesettinggame(respectively,quantitysettinggame),thereexistsanequilibriumofthetwo-stagequantitysettinggame(pricesettinggame)thatinducesthesameequilibriumpricesandquantities(althoughtheequilibriumvaluesoftheincentiveparametersmaydi¤er).
Finally,weconsideratwostagegame.Inthe…rststagetheownerofeach…rmchoosesavectorofincentiveparameters°i;fromafeasiblesetofincentiveparametervalues¡i,thatwillbeusedtodeterminethecompensationofherownmanager.FollowingFershtmanandJudd,whenwesay“owner”wemeananindividualoragroupwhosesolepurposeistomaximizethepro…tsofthe…rm.“Manager”referstoanagentthattheownerhirestomakerealtimeoperatingdecisions.
Atthebeginningofthesecondstagethecompensationschemesoftheman-agers(expressedby°1;°2)arerevealedandbecomecommonknowledge.Then,thetwomanagersengageinsomeformofduopolycompetition.
Inthispaper,weconsiderthelineardi¤erentiatedproductsdemandsystemgivenby
q1=a1¡b1p1+z1p2
Wesaywheneverpossiblesincetheownermaynothavesu¢cientpowertomanipulatethe
3
incentivesofhermanagerforthisconditiontohold.Inthiscase,theownerselectsthemostpreferred(highestpro…t)pointo¤ofheropponent’sreactioncurvepossible,giventhepowertosetincentivesthatsheposseses.
5
q2=a2¡b2p2+z2p1:
Firmsareassumedtohaveconstantmarginalcostequaltoci>04.Intheabsenceofincentiveschemecommitment,themanagers’reactionfunctionswouldbelinearinsuchagame,regardlessofwhetherthe…rmscompeteinpricesorinquantities.Wewishtoretainthisproperty,andsowerestricttheincentiveparametersavailabletotheownersaccordingtothefollowingcondition.Linearity:Theincentiveparametersavailabletotheownersaresuchthatthemanagers’incentivereactionfunctionsarelinearinthestrategyoftheother…rm.Thisconditionissatis…edbyallofthemodelsmentionedintheintroduction.Ageneralspeci…cationofamodelsatisfyingthelinearityconditionwouldbeonewhereownerichoosesweights(°i1;°i2;°i3;°i4)2¡i½<4suchthatthemanageriscompensatedaccordingtothefollowingfunction:
®i+¯i[°i1(piqi)¡°i2(ciqi)¡(°i3(pjqj)¡°i4(cjqj))]:
Theequilibriumconceptweemployissubgameperfection.Thatis,werequirethatinthesecondstagethemanagerschooseastrategythatmaximizestheirin-centivefunctionsforanygivenvectorofincentiveparameters,takingthestrategyoftheiropponentasgiven.Inthe…rststage,werequirethattheownerschoosetheincentiveparametersinordertomaximizetheirpro…t,giventhechoiceofincentiveparametersoftheiropponentandtheequilibriumthatwillresultwhenthemanagersplaythesecondstagegame.Thenextsubsectionsdescribesthegameingreaterdetail.2.1.TheSecondStage
Weassumethatthelinearityconditionholdsandthatinthe…rststagetheownershaveselectedtheincentiveparameters°i2¡i,i=1;2:Themanager’sincentive
4
Inordertoguaranteeproductionofpositivequantitiesinequilibriumweneedtoassume
bjai+zjajbibj¡zizj
thatthemarginalcostofproductionissmallenough.Speci…cally,weassumethatthemarginalcostof…rmiobeysci<
wherej=i.
6
functionMi(°i;si;sj)dependsontheincentiveparameterschosenbyhisownowner,thestrategythemanagerchooses,aswellasthestrategychosenbyhisopponent.Inpricecompetition,thestrategicvariablesareprices,whileinquan-titycompetition,theyarequantities.Themanager’sreactionfunctionRi(°i;sj)whichmapspairs(°i;sj)tostrategiessi,isthesolutionto
Ri(°i;sj)2argmaxMi(°i;si;sj):
si
SinceMi(°i;si;sj)isdi¤erentiable,thisisequivalenttoRi(°i;sj)beingsuchthat
@Mi(°i;Ri(°i;sj);sj)
@si
=0andtheappropriatesecondorderconditionsforamaximum
hold.
¤
i=1;2;j=i:Thatis,asecondstageequilibriumisapair(s¤1;s2)ofstrate-
¡¢¤¤¤
Asecondstageequilibriumisavector(s¤;s)suchthats=R°;siij;12i
giessuchthatthereactionfunctionsofthetwomanagersintersect.An(second
¤¤¤¤stage)equilibriumoutcomeunder(s¤1;s2)isapricep=(p1;p2)andquantity¤¤q¤=(q1;q2)suchthatp¤andq¤arethepriceandquantitythatoccurwhenthe¤equilibriumpair(s¤1;s2)areplayed.
2.2.TheFirstStage
Inthe…rststageofthegame,ownerschoosetheincentiveparametersinordertomaximizetheirpro…tsgiventhechoiceofincentiveparametersoftheiropponentsandtheequilibriumthesewillinduceinthesecondstage.Thatis,ownerichooses°¤iinordertosolvethefollowingprogram
¤
°¤2argmax(p¤ii¡ci)qi
°i¡¢j¤¤
s:t:s¤=R°;sj=1;2jj¡j
¤
andp¤;q¤istheequilibriumoutcomeunder(s¤1;s2).
Sincethepro…tfunctionisdi¤erentiable,theowner’schoiceof°¤iisequivalenttoherchoosing°¤iinordertoshiftherownreactionfunctionsothatitintersects
7
withtheopposingmanager’sreactionfunctionatthepointalongtheopponent’sreactionfunctionthatmaximizesitspro…t.Iftheownerhassu¢cientpowertomanipulatethereactionfunctionofitsmanager,thenthisamountstochoosingthepointalongitsopponent’sreactionfunctionsuchthatitspro…tisoquantistangenttotheopponent’sreactionfunctionatthatpoint.Withthisgraphicalintuitioninplace,wearereadytostatethemainresult.2.3.TheMainResult
Webeginbyconsideringconditionsunderwhichthereisacorrespondingequilib-riuminthetwostagequantitycompetitiongamethathasthesameequilibriumoutcomeasagivenequilibriumofthepricesettinggame.Asnotedabove,thetangencyconditioninthe…rststagewillonlyholdiftheownershavesu¢cientpowertochoosetheincentivesoftheirmanagers.Speci…cally,theownersmusthaveenoughpowertoshiftthereactioncurveoftheirownmanagerstothepointalongtheiropponent’sreactioncurvewheretheirownpro…tismaximized,i.e.theirpro…tisoquantistangenttothereactioncurveoftheiropponent.Forthereminderofthesectionweassumetheowner’shaveatleastthismuchpower.(TPE)TangencyinPriceEquilibrium:Assumethatinanypriceequi-librium,ownershaveenoughcontrolovertheirmanager’sincentivesinthepricesettinggamethatthepro…tisoquantofowneriistangenttothereactioncurveofmanagerjattheequilibriump¤foralli=j:
Forexample,powertomanipulateeithertheslopeortheinterceptofmanageri0sreactioncurveoverabroadenoughrangewillbesu¢cienttosatisfy(TPE).Anypointinthe(q1;q2)spacecanbemappedtoapointinthe(p1;p2)spaceusingtheequationscharacterizingthedemandsystem.Thisidenti…esthequantity(q1;q2)withthecorrespondingprices(p1;p2)thatwouldclearthemarketandcausethesequantitiestobesold.Sincethedemandsystemislinear,linesinthequantityspacemapintolinesinthepricespaceandviceversa.So,the
8
reactionfunctionresultingfromquantitycompetitiontransformationgivenbythedemandsystem.Considerthefollowingcondition:
qRi
inthequantityspacecanbetranslatedintothepricespacebyapplyingthe
¡
¤¤°qi;q¡i
¢
whichresides
(CQI)ControlofQuantityIncentives:Owners1and2havesu¢cientpowertomanipulatetheincentivesoftheirmanagerssothatifp¤istheequilib-riumoutcomepriceunderpricecompetition,thenthereexistincentiveparameters
q¤q¡q¤¤¢°isuchthatthequantitycompetitionreactionfunctionRi°i;q¡i,whentrans-formedintothepricespace,isidenticaltothepricecompetitionreactionfunction¡p¤¤¢p
Ri°i;p¡ithatinducesthep¤equilibriumoutcome.result.
Theorem1:Let(p¤;q¤)bethesecondstageequilibriumoutcomethatresultsfrompricecompetition.If(TPE)and(CQI)hold,thenthereisacorrespondingequilibriuminthetwostagequantitycompetitiongamethatinduces(p¤;q¤)asanequilibriumoutcome.
Proof:Considerthetwostagepricecompetitiongame.Attheequilibrium,
¤p¤
theownerschooseincentiveparameters(°p1;°2)suchthatthemanagers’reaction
Assumptions(TPE)and(CQI)aresu¢cientforthe…rsthalfoftheequivalence
functionsintersectatp¤.Since(TPE)holds,atthispointthereactioncurveofmanageriistangenttothepro…tisoquantofownerj,i=j:If(CQI)holds,there
¤q¤exist°qsuchthatthequantityreactioncurvesde…nedby°ii,whenprojected
intothepricespace,intersectatp¤andcoincidewiththepricereactioncurves.
¤
Considerthequantityreactioncurvesde…nedby°qi,i=1;2,inthequantity
space.Sincetheycoincidewiththeequilibriumpricereactioncurvesinthepricespace,andtheequilibriumpricereactioncurvesintersectatp¤inthepricespace,
¤¤thequantityreactioncurvesde…nedby°qi,i=1;2,intersectatqinthequan-
tityspace.Thus,theManager’sOptimalityConditionissatis…edatq¤forbothmanagers.Furthermore,thefactthatthepro…tisoquantofowneri;i=1;2,istangenttothereactioncurveofmanagerj=iinthepricespaceimpliesthatthe
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tangencystillholdsinthequantityspace,sincethetangentisjusttheslopeofthequantityreactionfunctioninthepricespaceandbotharetranslatedtothe
¤
quantityspaceviathedemandsystem.Thus°qiisanoptimalincentivepara-
meterchoiceforowneri;i=1;2,implyingtheOwner’sOptimalityConditionis
¤satis…ed.Therefore,°qiareequilibriumchoicesoftheincentiveparameterswhich
inducep¤;q¤astheequilibriumoutcome.¥
TheintuitiontheproofofTheorem1isrelativelystraightforwardandisillus-¤tratedinFigures1aand1b.Theproofshowsthatifthereexistsa°qisuchthat
thequantityreactioncurve,whentranslatedintothepricespace,corresponds
¤withthepricereactioncurveattheequilibrium°pi;thenthegeometricrelation-¤shipsthatmake°pisatisfytheManager’sandOwner’sOptimalityConditions
translateintothepricespaceinamannerthatpreservestheserelationships.IftheManager’sOptimalityConditionandOwner’sOptimalityConditionholdin
¤thequantityspace;then°qiinducesanequilibriumofthequantitysettinggame.¤Hence°qiinducesanequilibriuminthequantitygamethathasthesameequi-¤libriumoutcomeasthe°piequilibriuminthepricegame.
p2 Π1(p*)=k* q2 R2q(γ2q*)=R2p(γ2p*) γ’ varies
q2p* R2q(γ’2q) R2p(γ2p*)=R2q(γ2q*) p1 Π1(q*)=k* R2q(γ’2q) γ’2q varies q1 Theorem1showsthat(TPE)and(CQI)holdingsimultaneouslyaresu¢cientfortheretobeaquantityequilibriumoutcomecorrespondingtoanypriceequi-libriumoutcome.Wecanstatetheanalogousassumptionsabouttangencyofthe
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quantityequilibrium,(TQE)andcontrolofthepriceincentives,(CPI).Theseas-sumptionswillbesu¢cientforthesymmetricresultaboutequilibriumoutcomesofthequantitycompetition.
Theorem2:Let(p¤;q¤)bethesecondstageequilibriumoutcomethatresultsfromquantitycompetition.If(TQE)and(CPI)hold,thenthereisacorrespondingequilibriuminthetwostagepricecompetitiongamethatinduces(p¤;q¤)asanequilibriumoutcome.
Proof:MutatisMutandisastheorem1.¥
Theorems1and2,takentogether,providesu¢cientconditionsfortheequiv-alenceofthesetofequilibriumoutcomesunderpriceandquantitycompetition.Corollary1:Suppose(TQE),(TPE),(CQI),and(CPI)hold.Thenthesetofequilibriumoutcomesunderthetwostagepriceandquantitycompetitionareidentical.
Proof:FollowsdirectlyfromTheorems1and2.
Fortheremainderofthepaper,wewilluse(TE)torefertothecasewhen(TQE)and(TPE)hold,and(CI)torefertothecasewhen(CQI)and(CPI)hold.Theassumptions(TE)and(CI)explicitlystateconditionsunderwhichownershavesu¢cientpowertocommittheirmanagerstoacertaintypeofbehavior.The(TE)assumptionsrequirethatownershavesu¢cientcontroloverincentivesforagiventypeofcompetition,andthattheycanchoosetheirfavoritepointo¤oftheiropponent’sreactioncurvethroughtheirchoiceofincentiveparameters.The(CI)assumptionsrequirethatownershavesu¢cientcontrolovertheincentivesoftheirmanagersinonetypeofcompetition(sayquantity)togetthemtomimicthebehaviortheyexhibitinanequilibriumoftheothertypeofcompetition(sayprice).Thusthe(CI)assumptions,wouldgenerallyrequireownerstoexercisecontroloverboththeslopeandtheinterceptofthemanager’sreactioncurves,whilethe(TE)assumptionsseemtorequirecontrolonlyovertheslopeortheintercept.
Theorems1-2andCorollary1statethatiftheownershavesu¢cientcontrol
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overincentivesinthesenseof(TE)and(CI),thentheequilibriaareequivalent.Thisissomewhatparadoxical,sinceone’sinitialreactionistosay“iftheyhavethatmuchcontrol,can’ttheydoevenbetter?”However,thelogicoftheproofshowswherethisintuitionfails.Westartbytakinganequilibriuminonetypeoftwo-stagecompetitionasgiven.Wethenshowthatifassumptions(TE)and(CI)hold,thentheownerscanreplicatetheequilibriumconditions(tangencyoftheisopro…ttotheothermanager’sreactioncurve,andtheequilibriumpricebeingattheintersectionofthereactioncurves)intheothertypeofcompetitionbymanipulatingtheincentivesofthemanagers.Bystartingwithanequilibriuminonetypeofcompetitionandreplicatingitintheothertypeofcompetition,wearenottryingtoidentifytheabsolutebestoutcomefortheowners.Rather,weareaskingwhethertheconditionsthatmaketheoriginalequilibriumanequilibriumcanbereplicatedintheothertypeofcompetition.If(TE)and(CI)hold,theanswertothisquestionisyes.
Foranotherapproach,considerthefollowing.Thereasonwhysimplepriceandquantitycompetition(withnoincentiveschemecommitment)failtohavethesameequilibriumisthattheincentivesgiventoamanagerwhenhetakeshisopponent’spriceas…xedinchoosingastrategyaredi¤erentthantheincentivesgiventoamanagerwhenhetakeshisopponent’squantityas…xed.However,weassumethatownersherehavesu¢cientpoweroverincentives(reactionfunctions)tomitigatethesedi¤erences.Whenthereasonwhytheequilibriadi¤erisremovedbyadmittingincentiveschemecommitment,theycoincide.
3.Applications
Inthissectionweconsidertwodi¤erenttypesofincentiveschemecommitmentmechanisms.First,weconsiderthemodelofMillerandPazgal(1997),whereinthe…rststageownerscommittheirmanagerstoaspeci…cattitudetowardrelativeperformanceandinthesecondstagethemanagerscompeteineitherpricesor
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quantities.Weshowthatownershavingpowertocontroltheweightmanagersgivetotheirperformancerelativetothecompetitionissu¢cientfor(TE)and(CI)tohold.Consequentlytheequilibriaunderpriceandquantitycompetitionareidentical.
Second,weconsideramodelbasedonFershtmanandJudd(1987),whereownerschoosetheweightmanagersputonrevenues.Weshowthatinthiscase(CI)failstohold,andthustheequilibriadonotcoincide.3.1.CommitmenttoRelativePerformance
Inthemodelofcommitmenttoanincentiveschemebasedonrelativeperfor-mancesuggestedbyMillerandPazgal(1997),ownersgivemanagerstheincentivetomaximizenotonlythe…rm’sownpro…tinisolation.butmanagersarealsocompensatedbasedontheirperformancerelativetothecompetitors.Thisisequivalenttohavingincentivestomaximizeaconvexcombinationofownpro…tandthedi¤erencebetweenownpro…tandtherival’spro…t.
Thedemandstructurewewilluseisthefollowingsymmetricone:
q1=1¡p1+zp2q2=1¡p2+zp1:
(3.1)(3.2)
Inthisdemandstructurewehavesuppressedalloftheparametersexceptforthedi¤erentiationparameterz:Furthermore,wewillassumethat…rmshavezeromarginalcost5.
Webeginbyderivingtheequilibriumwherethemanagerscompeteinpricesinthesecondstage.Weletµpibetheweightowneriplacesonthepro…toftheother…rminthemanager’scompensationscheme.Inthisenvironment,themanagers’objectivefunctionsaregivenby
5
Wemaketheseassumptionsforexpositionalpurposes.Alloftheresultsholdforageneral
lineardemandstructurewith(possiblydi¤erent)constantmarginalcosts.
13
p
mpi=(1¡pi+zpj)pi¡µi(1¡pj+zpi)pj;i=1;2;j=i:
Thecorrespondingreactionfunctionsare
pi(pj)=
1
(1+z(1¡µpi)pj);i=1;2;j=i:2
andthesecondstageequilibriumpricesarethus
¡z+µpiz¡2;i=1;2;j=i:pi=2¡µpz2+µpz2µp¡4+z2¡µpzijji
Lettingxibethepro…tearnedby…rmi,theowner’sobjectivefunctionsaregivenby
xi=(¡z+µpiz¡2)¡¡
¢
¡4+
p2pp2¡µpiz+µjzµi¡µiz¡z¡2
z2¡
2µpiz
¡
2µpjz
+
Thisyieldstheowners’reactionfunctions
¢;p2p2µjzµi
i=1;2;j=i:(3.3)
z
;i=1;2;j=i:(3.4)2¡2µpz+2z+4¡z2+µpzjj
©pz+2pz+2ª
Thesystemofequationsde…nedby(3.4)hastwosolutions,µ1=z;µ2=z;
ª©pz+2pz+2ª©p
pzz
andµ1=z¡2;µ2=z¡2:However,theµ1=z;µ2=zsolutiondoesnot
©pªpzzsatisfythesecondorderconditions,andwediscardit.Theµ1=z¡;µ=22z¡2µpi
=¡1+
µpj
(z+2)
solutionyieldsthefollowingequilibrium:
11pp
q1=q2=z+;
421z¡2p
pp=p=;12
4z¡11z2¡4pp
x1=x2=;
16z¡1zpp
µ1=µ2=:
z¡214
Directcomputationoftheequilibriumwhenthemanagerscompeteinquanti-tiesshowsthatthesameequilibriumpricesandquantitiesobtain,withtheownersputtingweightµqi=
z,z+2i=1;2,onthepro…tsoftherival…rmintheobjective
functionofthemanagers.WenowshowhowthelogicoftheproofofTheorem1canbeusedtodemonstratethisequivalence.
Invertingthedemandsystemgivenby(3.1)and(3.2)yieldstheinversedemandsystem
p1=p2
q1¡1+zq2¡zz2¡1q2¡1+zq1¡z=:
z2¡1
Lettingµqibetheweightassignedtothepro…tof…rmjby…rmi,themanager’sobjectivefunctionswhenthe…rmscompeteinquantitiesaregivenby:
µ¶µ¶qi¡1+zqj¡zqj¡1+zqi¡zq
mq=q¡µqj;i=1;2;j=i:iii
z2¡1z2¡1Di¤erentiatingyieldsreactionfunctions
qi=
1
[1+z+z(µqi¡1)qj];i=1;2;j=i:2
Next,weusethedemandsystemtotranslatethequantityreactioncurvestothepricespace.Manageri’sreactioncurvemapsto:
q1+zpj¡µqiz+zµipj
;i=1;2;j=i:pi=2¡z2z2+µqi
(3.5)
1z¡2
:4z¡1zµq2=z+2:
p
Inpricecompetition,theequilibriumpricesaregivenbypp=p12=qq
andµyieldsµSubstitutingthesevaluesinto(3.5)andsolvingforµq1=12q
Hence,whenµq=µ12=
z,z+2thetranslatedquantityreactioncurvesintersectat
1z¡2:4z¡1q
thepriceequilibriumpricespq1=p2=qqatq1=q2=
1
z4Inthequantityspace,theyintersect
+1:Sincethesequantitieslieonthereactioncurvesofthe2managers,theyformasecondstageequilibriumofthegame.Thusinquantity
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q
competitionwhenµq=µ12=
zz+2theManager’sOptimalityConditionholdsat
zz+2theequilibriumpricesandquantitiesfromthepricecompetition.
q
Itremainstoshowthatµq=µ12=
formanequilibriumofthe…rststage
ofthegame.Todothis,wemustverifythattheOwner’sOptimalityCondition
1+µqholds.Theslopeof(3.5)isgivenbyz2+µqz21:Theslopeofmanageri’sequilib-¡z2i¡¢z1
riumpricereactioncurveisgivenby2z1¡z¡2:Wesettheseequalandsolvefor
q
µqiinorderto…ndthevalueofµithatmakestheslopeofthetranslatedquantity
zreactioncurvethesameastheoptimalpricereactioncurve.Thisyieldsµqi=z+2:
qzHencetheOwner’sOptimalityConditionholdsaswell,andµq1=µ2=z+2form
anequilibriumforthetwostagequantitycompetitiongamethatinducesthesamepricesandquantitiesasthetwostagepricecompetitiongame.Thustheabilityofownerstocontroltheweighttheirmanagersputonrelativeperformanceissu¢cienttosatisfy(TE)and(CI)andasaresultthereisaquantitycompetitionequilibriumthatinducesthesameoutcomeasthepricecompetitionequilibrium.Sincethesameargumentshowsthereexistsapricecompetitionequilibriumcor-respondingtoanyquantitycompetitionequilibrium,theequilibriaunderpriceandquantitycompetitioncoincide.
Itisinterestingtonotethatwhileingeneral(CI)willrequirecontrolovertheslopeandinterceptofthereactioncurves,thisexampleshowsthatitisnotnecessarytohaveindependentcontroloverbothslopeandinterceptofthereactioncurves.
3.2.AFershtmanandJuddStyleModel
Intheexampleoftheprevioussection,exantecommitmenttocompensateman-agersbasedonrelativeperformanceprovidedsu¢cientpowertocontrolincentivesfortheequivalenceresulttohold.OnemaywonderifasimilarresultholdsinaFershtmanandJuddstylemodel,whereownerscommittheirmanagerstobehav-iorthatmaximizesalinearcombinationofpro…tsandsales.Theanswertothis
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questionisno.
FershtmanandJuddconsideramodelwheremanagersmaximize
ai¼i+(1¡ai)Ri
where¼iisthepro…tearnedby…rmiandRiisitsrevenue.Each…rmisassumedtohaveconstantmarginalcostequaltoci,andweassumethecostissmallenoughtoallowfortheexistenceofequilibria.Considerthesimpli…edlineardemandsystemoftheprevioussection,andfocuson…rm1.(Thesymmetricargumentsholdfor…rm2.)Inpricecompetitionmanagersseektomaximize:
(1¡p1+zp2)(p1¡ap1c1)
Di¤erentiatingthiswithrespecttopiyieldspricereactioncurve
1
p1=(1+ap1c1+zp2);2
whichhasslope1z:2Thereactioncurveformanager1inthequantitygameisgivenby:
¢1¡qq2
q1=1¡a1c1+z+a1c1z¡zq2:
2
Transformingtheaboveresulttothepricespaceyields:
1q1q
+1zp+ac¡acz221122211
:p1=¡12
¡1+2z
12z:¡2+z2(3.6)
TheslopeofthiscurveisalwaysequaltoSincethisisindependentof
q1
aq;thereisnovalueofaiithatwillmaketheslopeof(3.6)equal2z,theslope
ofthepricereactioncurve.Sincechangingtheincentiveparameterscanonlyshiftthereactioncurveinaparallelmannerbutcannota¤ecttheslope,(CQI)failstohold,andthereisnoequilibriuminthequantitygamethatinducesthesameequilibriumoutcomeastheequilibriumofthepricegame(see…gure2).Furthercomputationveri…esthatthesameproblemcauses(CPI)tofail,anddirectcomputationcon…rmsthattheuniqueequilibriainthetwogamesare,infact,di¤erent.
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P2 Π1(p*)=k* P* R2p(γ2p*) R2q(γ2q) P1 4.Generalizations
Themodelpresentedhereconsidersonlythespecialcaseoflineardemandandconstantmarginalcost.However,theintuitiveargumentpresentedabovesuggeststhattheresultsholdinabroaderclassofdemandstructures.Sincethecrucialstepintheargumentinvolveshaving“su¢cientpowertomanipulatethereactioncurvesofthemanagers”suchthatthequantityreactioncurves(inthepricespace)passthroughtheequilibriumpricesfromthepricecompetitionandaretangenttothepro…tisoquantsoftheother…rmsatthispoint,itsuggeststhatallthatisneededfortheargumenttogothroughisforthepro…tisoquantsandreactionfunctionstobedi¤erentiableattheequilibriumprices.
However,whenwedepartfromtheworldofreactioncurvesthatarelinearunderallpossiblevaluesoftheincentiveparameters,theresultpresentedheremightfailtohold.Theintuitionisillustratedin…gure3.Itmaybepossibleforthequantityisoquants,whentranslatedtothepricespace,tointersectatthepriceequilibriumandbetangenttothepro…tisoquantoftheother…rmattheequilibriumprice,buttoalsocrossthepro…tisoquantawayfromtheequilibrium.Inthiscase,therewillbeapointalongthequantityreactioncurve
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thatyieldsahigherpro…tthanthepriceequilibrium.Hencethispricevectorandthecorrespondingquantityvectorwillnotgenerallybeanequilibriumoutcomeinthequantitygame.
Ontheotherhand,theresultspresentedhereeasilygeneralizetooligopolieswithmorethantwo…rms.Thesecondstageoptimalityconditionand…rststagetangencyconditiondonotdependonthecompetitiontakingplaceintwodi-mensionsfortheirvalidity.Thustheargumentspresentedhereholdinlineardi¤erentiatedproductsoligopolieswithanynumberof…rms.
p2 Π1(p*)=k* R2q(γ2q*)
p* R2p(γ2p*) R1q(γ’1q) p1 R1(γ1)=Rpp*q*1(γ1) q5.Discussion
Inthispaperweprovethatforlineardemanddi¤erentiatedproductsduopolieswithconstantmarginalcost,theequilibriumoutcomesunderpriceandquantitycompetitioncoincideifownershavesu¢cientpowertocontroltheincentivesoftheirmanagers.Weshowhowatwostagemodelwhereinthe…rststageowners
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choosetheweightthattheirmanagerswillgivetotheirperformancerelativetothatoftheirrivalsandinthesecondstagemanagerscompetewiththeseincentivesissu¢cientfortheequivalenceresulttohold.Whileamodelwhereownerscontrolonlytherelativeweightmanagersgivetopro…tsandsalesisnotsu¢cient,sinceownersdonothavethepowertomanipulatetheslopeoftheirmanagers’reactioncurves.
Whatdotheseresultsmeanfortheownersandmanagersinanindustryandforeconomiststryingtounderstandthem?Foreconomists,theseresultsmeanthatifownershavepowertocontroltheincentivesoftheirmanagers,weshouldexpectthattheywilluseittomitigatethedi¤erencesbetweenpriceandquantitycom-petition,makingpricecompetitionlessaggressiveandquantitycompetitionmoreaggressive.Thus,fromthepointofviewofeconomists,onceincentiveschemecommitmentisrecognized,thedi¤erencesbetweenpriceandquantitycompeti-tionequilibriaareexpectedtobelesspronouncedthantheoriginaleconomicmodelspredict.
Fromthepointofviewofownersandmanagersinanindustry,theresultspresentedheredonotimplythattheyneednotbeconcernedwithwhethertheyarecompetinginpricesorquantities.Astherelativeperformanceexampleshows,thesignoftheoptimalweightputonrelativeperformancewillgenerallydependonwhetherthe…rmscompeteinpricesorquantities.Thus,itiscrucialforownerstoknowwhatkindofstrategiestheirmanagersfundamentallyemploy,sincetheycannotdesignanoptimalincentiveschemewithoutsuchknowledge.
However,whileownersandmanagersstillneedtoknowwhethertheycompeteinpricesorquantities,incentiveschemecommitmentimpliesthatintermsofprof-its,the…rmsshouldbeindi¤erentbetweenanindustrywherethe…rmscompeteinpricesandoneinwhichtheycompeteinquantities,sinceoncethisisknown…rmswilladjusttheincentivesofthemanagersaccordingly,andtheultimateequilibriumoutcomewillbethesameregardlessofthetypeofcompetition.
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References
[1]Cheng,L.“ComparingBertrandandCournotEquilibria:aGeometricAp-proach.”RandJournalofEconomics,Vol.16(1985),pp.146-152.
[2]Fershtman,C.andJudd,K.“EquilibriumIncentivesinOligopoly.”American
EconomicReview,Vol.77(1987),pp.927-940.
[3]Fumas,V.S.,”RelativePerformanceevaluationofmanagement”International
JournalofIndustrialOrganizations,Vol.10(1992),pp.473-489.
[4]Miller,N.andPazgal,A.“RelativePerformanceasaStrategicCommitment
Mechanism”mimeo,NorthwesternUniversity,November1997.
[5]Singh,N.andVives,X.“PriceandQuantityCompetitioninaDi¤erentiated
Duopoly.”RandJournalofEconomics,Vol.15(1984),pp.546-554.
[6]Sklivas,S.D.“TheStrategicChoiceofManagerialIncentives.”RandJournal
ofEconomics,Vol.18(1987),pp.452-458.
[7]Vickers,J.“DelegationandtheTheoryoftheFirm.”EconomicJournal,Sup-plement,Vol.95(1985),pp.138-147.
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