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外文

DuopolyPricingUnder‘PrivateKnowledge’

ofProductDifferentiation

CarlosA.Ulibarri

Received:

12July2010/Revised:

10January2011

Accepted:

4February2011

SpringerScience+BusinessMedia,LLC2011

AbstractThisnotestudiespricedecisionsinaduopolyindustrywherefirmshaveprivateinformationoverthedegreeofproductdifferentiation（product-type）.ABayesian-Nashpricesolutionisderivedassumingfirmsmaximizetheir‘certainty-equivalent’profitlevels.Thecomparative-staticsindicatethatincreasedriskaversionovertherival’sproducttriggerspricecompetition.Consequently,theresultsofthestudysuggestrevealinginformationisahigherrewardstrategythanconcealinginformationinsituationswhererivalshaveasymmetricinformationoverproducttype.ThesefindingscontributetotheindustrialeconomicliteraturebygeneralizingtheBertrandequilibriuminanasymmetricinformationgamemodel.

Keywordsduopolymarket.asymmetricinformationgamemodel.Bayes-Nashequilibrium

JELClassificationsD43.D41

1Introduction

TheBertrandmodelofpricecompetitionhasbeenextensivelyanalyzedingame-theoreticanalysisofproductdifferentiation.ThenecessaryandsufficientconditionsforNash（1950）andBayesian-NashequilibriaaredescribedinVives（1990,1999）.InbrieftheNashequilibriumisguaranteedinpurestrategiesundercompleteinformationbutnotunderasymmetricinformation.Forthelatter,a‘BayesianNashsolution’inpurestrategiesmaybeestablished,orpossiblymixedstrategieswithplayersrandomizingoveracertainrangeofprices.Thepresentstudyusesanasymmetricinformationgame（AIG）modeltoexaminetheBayesian-Nashpriceequilibrium（BNEprices）assumingfirmshaveprivateknowledgeofthedegreeofproductdifferentiation.Inthissettingweshowinformationsharingisa‘first-best’strategyfor‘valuemaximizing’duopolyfirms.Forexample,thefirmscouldagreetoshareproductdesigninformationthroughamarketingortradeassociation.

Industrialorganizationmodelsthatincorporateprivateinformationareparticularlyusefulinexaminingproblemsofoligopolyandimperfectcompetition,includingtheincentivestorevealcostordemandinformation（MilgromandRoberts1987）.Inparticular,theinformationsharingprobleminoligopoly/duopolysettingshasbeenexaminedextensivelyinthetradeassociationliterature.Notablecasesincludetwo-stagegameswithBertrandorCournotfirmscompetingunderuncertaintyabouttheindustry’sdemandparameter（commonknowledgetoallparticipants）,oracostparameterthatisfirm-specific（privatevalues）.1Thepresentstudyconsidersadifferentiatedproductmarketwhereduopolyfirmsareuncertainabouteachother’sdegreeofproductdifferentiation,andthusmakestrategicpricingdecisionsunderasymmetricinformationoverproducttype.

Thenormal-formrepresentationofBertrandcompetitionismodeledassumingrisk-aversefirmsmovesimultaneouslyinsettingpricesofdifferentiatedproducts.Theanalysisassumeseachfirmhasprivateinformationoverthedegreeofproductdifferentiation,andthatthisiscommonknowledge.Thefirmsthenchoosetheirfirst-bestpricesiftheyaresufficientlycompensatedforrisk-taking,asmeasuredbytheir‘certainty-equivalent’profitE½CEi_?

pi_ri;thatis,the

expectedprofitminustheriskpremium.Giventheutilityfromthecertainty-equivalentprofit

equalstheexpectedutilityfromtherandomprofit,Uepi_riT?

E½Ue~piT_;theriskpremiums

reflectcompensationforincompleteinformationoverproducttype;orinotherwords,the

maximumfirmsarewilling–to-paytoconvertrandomprofitsintodeterministicprofits.2

Thestudyproceedsasfollows.Section2constructsanormal-formrepresentationofthe

duopolyindustry,derivingBNEpriceswhichmaximize‘certainty-equivalent’profits.

Section3examinesthecomparativestaticsofthemodelforsubstituteandcomplement

products.Section4interpretsthepracticalimplicationsofthestudyrelativetotheliterature.

2Risk-averseBertrandcompetition

SinghandVives（1984）proposeaconvenientdemandsystemtostudydifferentiated

productmarketsinwhicharepresentativeconsumermaximizesutilityminusexpenditures:

Ueq1;q2T_

P

i¼1

2

piqi,wherepiandqidenotethepriceandamountofgoodi.The

presentstudyassumesUisstrictlyconcaveandtakesasimplifiedquadraticform:

Ueq1;q2T?

aeq1tq2T_

1

2

q1

2t2gq1q2tq2

2__

:

e1T

The‘gamma’parameteregTdescribesthenatureofproductdifferentiation:

thegoodsare

substitutesifγ>0,independentifγ=0andcomplementsifγ<0.ThestrictconcavityofUis

ensuredbyΔ=1−γ2>0.Theinverseanddirectdemandsystemsaregivenbyexpressions

（2）and（3）:

p1?

a_q1_gq2;p2?

a_q2_gq1;e2T

q1?

½ae1_gT_p1tgp2_

1_g2;q2?

½ae1_gT_p2tgp1_

1_g2:

e3T

ThestandardmodelofBertrandcompetitionassumescompleteinformationoverproduct

type—acommongammaparameterobservabletobothfirms.However,suppose

perceptionsoftherival’sproductarebasedonincompleteinformation,andthatthisis

commonknowledgetobothfirms.Thisbegsthequestion:

howdoesprivateknowledge

overproductdifferentiationaffectpricecompetitionbetweenriskaversefirms?

Toexaminethisquestionsupposeeachfirmhasprivateinformationandsetspriceunder

incompleteinformationoverthedegreeofproductdifferentiation.Thepricedecisionsare

madeassumingthecross-priceparameterse~g12;~g21Tinthedemandsystemarei.i.d.

randomvariables,drawnfromajointprobabilitydistributionfunctionFe~g12;~g21T;whichis

commonknowledgetobothfirms.Essentiallytherandomizedterm~g12describesfirm1’s

perceptionofits’degreeofproductdifferentiationand~g21describesfirm2’sperceptionof

its’degreeofproductdifferentiation.Hencesubstituteproductsaredefinedonthepositive

interval0<~g12;~g21<1;complementproductsonthenegativeinterval_1<~g12;~g21<0;

andindependentproductsbetweenthetwosupports~g12;~g21?

0.Basedonthedirect

demandsystem（3）theprofitequationsfortheindustryaregivenby

~p

1?

ep1_c1Tq1?

ep1_c1T½ae1_~g12T_p1t~g12p2_

e1_~g12

~g21T

;e4T

~p2?

ep2_c2Tq2?

ep2_c2T½ae1_~g21T_p2t~g21p1_

e1_~g12

~g21T

:

e5T

Thepresentstudyfocusesonduopolistswithconstantabsoluteriskaversion（CARA）

definedbyparametersr1andr2,payingriskpremiumstoconvertrandomprofits~pitotheir

certainty-equivalentCEi?

pi_:

5riVarepiT.3Thereforeincompactformthevalue

maximizationmodelfortheduopolymarketisspecifiedas:

CE1

CE2

"#

¼

p1

p2

"#

_:

5

r1

r2

"#

Varep1T

Varep2T

"#

:

e6T

TwomajorassumptionsareusedinspecifyingtheAIGmodel.Assumption（i）:

firm1

takesexpectationsoveritsprofitEq.4givenpriorinformation~g21?

o;andlikewise,firm2

takesexpectationsoveritsprofitEq.5given~g12?

o:

Thelogicbehindthisassumptionis

thatBayesianagentsinitiallytakeexpectationsofvariatesintheirowndemandcurves,not

theirrivals.Assumption（ii）:

firmshave‘commonknowledge’ofeachother’sdegreeofrisk

aversion.4Clearly‘commonknowledge’oftherival’sdegreeofriskaversionisastrong

assumptionincharacterizingstrategicchoicebehaviorintheAIGframework.Nonetheless,

ityieldstestableimplicationsregardingduopolisticreactionstouncertaintyinBertrand

markets.Inthissettingthemeansandvariancesoftheprofitequationsaredefinedby

p1?

E絜p1_c1Tq1_?

ep1_c1T½ae1_g12T_p1tg12p2_;Varep1T?

s2

p1;e7T

p2?

E絜p2_c2Tq2_?

ep2_c2T½ae1_g21T_p2tg21p1_;Varep2T?

s2

p2e8T

wherethebarovertherandomproduct-typeparametersdenotestheirexpectedvalues.The

necessaryconditionsforvaluemaximizingpricesaregivenby

p12argmaxfCE1g:

@CE1

@p1?

0!

@p1

@p1?

:

5r1

@s2

p1

@p1

;e9T

p22argmaxfCE2g:

@CE2

@p2?

0!

@p2

@p2?

:

5r2

@s2

p2

@p2

;e10T

yieldingthepricereactioncurves:

p1?

:

5½ae1_g12Ttg12p2tc1__R1;R1?

:

52r1

@s2

p1

@p1

;e11T

p2?

:

5½ae1_g21Ttg21p1tc2__R2;R2?

:

52r2

@s2

p2

@p2

:

e12T

ThepricereactionsdependonunitcostandCARAparametersandestimatesofthe

expectedproducttypeparameters.Notethattheinterceptandslopeofthebestreplycurves

dependontheexpectedproduct-typeparameter.TheBayes-Nashequilibrium（BNE）prices

areobtainedbysolving（11）and（12）simultaneously,yielding

p1?

e

4

4_g12g21T:

5½ae1_g12Ttc1_t

g12

4½ae1_g21Ttc2__½R1t

g12

2

R2_

__

;e13T

p2?

e

4

4_g12g21T:

5½ae1_g21Ttc2_t

g21

4

½ae1_g12Ttc1__½R2t

g21

2

R1_

__

:

e14T

TheindividualBNEpricesdependonthecostandCARAparametersofbothfirms,and

theexpectedproduct-typeparametersofbothfirms.

3Comparativestatics

Figures1a–billustratetheimpactofriskaversiononthefirms’pricereactioncurves

（11and12）andtheresultingBNEprices（13and14）.Thecurveshavepositiveslopesfor

differentiatedsubstituteproductse0<~g12;~g21<1T;negativeslopesfordifferentiated

complementproducts（_1<~g12;~g21<0）;andzeroslope（notshown）forindependent

monopolymarkets（~g12;~g21?

0）.

Figure1aillustratestheBNEsolutionfordifferentiatedsubstitutesunderriskneutral

conditions（r1,r2=0）andrisk-averseconditions（r1,r2,>0）,denotedbypoints0and1.The

slopeandintercepttermsofthepricereactioncurvesreflecttheexpectationofsubstitutes.

Greaterriskaversionorlowerunitcostsshiftthecurvesinward,resultingintheBNE

solutionatpoint1wherethefirmssetlowerpricesp11

;p12

__

.Forstrongersubstitutesthe

pricereactioncurvesshiftinwardandrotateoutward.

Figure1billustratestheBNEsolutionfordiff