外文Word格式.docx
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外文Word格式.docx
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E½
Ue~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
i¼
1
2
piqi,wherepiandqidenotethepriceandamountofgoodi.The
presentstudyassumesUisstrictlyconcaveandtakesasimplifiedquadraticform:
q2T?
aeq1tq2T_
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?
½
ae1_gT_p1tgp2_
1_g2;
q2?
ae1_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<
0;
andindependentproductsbetweenthetwosupports~g12;
~g21?
0.Basedonthedirect
demandsystem(3)theprofitequationsfortheindustryaregivenby
~p
1?
ep1_c1Tq1?
ep1_c1T½
ae1_~g12T_p1t~g12p2_
e1_~g12
~g21T
;
e4T
~p2?
ep2_c2Tq2?
ep2_c2T½
ae1_~g21T_p2t~g21p1_
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
E絜p1_c1Tq1_?
ae1_g12T_p1tg12p2_;
Varep1T?
s2
p1;
e7T
p2?
E絜p2_c2Tq2_?
ae1_g21T_p2tg21p1_;
Varep2T?
p2e8T
wherethebarovertherandomproduct-typeparametersdenotestheirexpectedvalues.The
necessaryconditionsforvaluemaximizingpricesaregivenby
p12argmaxfCE1g:
@CE1
@p1?
0!
@p1
:
5r1
@s2
e9T
p22argmaxfCE2g:
@CE2
@p2?
@p2
5r2
e10T
yieldingthepricereactioncurves:
5½
ae1_g12Ttg12p2tc1__R1;
R1?
52r1
e11T
ae1_g21Ttg21p1tc2__R2;
R2?
52r2
e12T
ThepricereactionsdependonunitcostandCARAparametersandestimatesofthe
expectedproducttypeparameters.Notethattheinterceptandslopeofthebestreplycurves
dependontheexpectedproduct-typeparameter.TheBayes-Nashequilibrium(BNE)prices
areobtainedbysolving(11)and(12)simultaneously,yielding
e
4
4_g12g21T:
ae1_g12Ttc1_t
g12
4½
ae1_g21Ttc2__½
R1t
R2_
__
e13T
ae1_g21Ttc2_t
g21
½
ae1_g12Ttc1__½
R2t
R1_
e14T
TheindividualBNEpricesdependonthecostandCARAparametersofbothfirms,and
theexpectedproduct-typeparametersofbothfirms.
3Comparativestatics
Figures1a–billustratetheimpactofriskaversiononthefirms’pricereactioncurves
(11and12)andtheresultingBNEprices(13and14).Thecurveshavepositiveslopesfor
differentiatedsubstituteproductse0<
1T;
negativeslopesfordifferentiated
complementproducts(_1<
0);
andzeroslope(notshown)forindependent
monopolymarkets(~g12;
0).
Figure1aillustratestheBNEsolutionfordifferentiatedsubstitutesunderriskneutral
conditions(r1,r2=0)andrisk-averseconditions(r1,r2,>
0),denotedbypoints0and1.The
slopeandintercepttermsofthepricereactioncurvesreflecttheexpectationofsubstitutes.
Greaterriskaversionorlowerunitcostsshiftthecurvesinward,resultingintheBNE
solutionatpoint1wherethefirmssetlowerpricesp11
p12
.Forstrongersubstitutesthe
pricereactioncurvesshiftinwardandrotateoutward.
Figure1billustratestheBNEsolutionfordiff
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