Endogenizing the cost parameter in Cournot oligopoly

0.28 MB·

by  Stefanos Leonardos

Endogenizing the cost parameter in Cournot oligopoly
∗
6 Stefanos Leonardos 1 and CostisMelolidakis 1
† ‡
1
0 1SectionofStatisticsandOperationsResearch, Departmentof Mathematics,
2
NationalandKapodistrianUniversityof Athens,PanepistimioupolisGR-15784, Athens,
g
u Greece
A

0 August11,2016
1

]
C
O Abstract
. Westudyasingleproductmarketwithaffineinversedemandfunctioninwhichtwoproducers/retailers,
h
producing under capacity constraints, may order additional quantities from a single profit maximizing
t
a supplier. WeexpressthischainasatwostagegameandshowthatithasauniquesubgameperfectNash
m
equilibrium, whichweexplicitlycalculate. Ifthesupplier facesuncertainty aboutthedemandintercept
[ andwhenever aBayesianNash equilibriumexists, thesupplier’sprofit marginat equilibriumisafixed
pointofatranslationoftheMRLfunctionofhisbeliefundertheassumptionthathisbeliefisnon-atomic
5
andtheretailersareidentical. Ifthesupplier’sbeliefisofDecreasingMeanResidualLifetime(DMRL),
v
thegamehasauniquesubgameperfectBayesianNashequilibrium. Variouspropertiesoftheequilibrium
5
solutionareestablished,inefficienciesatequilibriumgeneratedbythelackofinformationofthesupplier
6
are investigated, and examples are provided for various interesting cases. Finally, the main results are
3
7 generalizedformorethantwoidenticalproducers/retailers.
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Keywords:CournotNash,NashEquilibrium,Duopoly,IncompleteInformation,Existence,Uniqueness
.
1 MathematicsSubjectClassification(2000):91A10,91A40
0
6
1 1 Introduction
:
v
i The main objective of this paper is to examine equilibria in a classic Cournot market with affine inverse
X
demand function, when the competing firms’ cost is affected exogenously by a supplier, who may even
r
a beuncertainaboutthe demandparameterofthemarket. AsMarxandSchaffer[2015]pointout,although
Cournot’smodelhasbecometheworkhorsemodelofoligopolytheory,therehasbeenverylittlediscussion
about the origin of the competing firms costs in the quite vast Cournot literature. As they underline, if
∗Thisresearchhasbeenco-financedbytheEuropeanUnion(EuropeanSocialFund ESF)andGreeknationalfundsthroughthe
OperationalProgram”EducationandLifelongLearning”oftheNationalStrategicReferenceFramework(NSRF)-ResearchFunding
Program”ARISTEIAII:OptimizationofStochasticSystemsUnderPartialInformationandApplication,Investinginknowledgesociety
throughtheEuropeanSocialFund”.
StefanosLeonardosgratefullyacknowledgessupportbyascholarshipoftheAlexanderS.OnassisPublicBenefitFoundation.
†[email protected]
‡[email protected]
1
these costs representpurchasesfrom a third party, then importantstrategic considerationscome into play,
whicharenotaddressedbytheprevailingapproach.MarxandSchafferquestiontherobustnessoftheresults
obtainedthusfarintheCournotoligopolytheoryduetothisnegligence.
ThekeyinstudyingtheeffectsofanexogenoussourceofsupplyforCournotoligopolistsistherelation
betweendemandandthevariouscosts. Ifthedemandishighenough,thecompetingfirmsmayhaveincen-
tivetoplaceorderstoanexternalsupplier,dependingofcourseonthepriceheasks. Intheclassicoriginal
Cournot model, the crucial parameter concerning demand is the y-intercept of the affine inverse demand
function, which implies its important role in studying Cournot markets with exogenous supply. Next, in
suchastudy,variousquestionshavetobeaddressed: Dothefirmshaveproductioncapacitiesoftheirown?
Ifyes,thenthesecapacitieshavetobeassumedbounded(astheyareinreality),sothatneedmayarisefor
additionalquantities. Doesthesupplierknowtheactualdemandtheoligopolistsface? Ifno,thenhemay
askforapricethatthecompetingfirmswillnotaccept,evenifitistohisadvantagenottodoso. Thelatter
questionimpliesanincompleteinformationgamesetup.
Itisthe aimofthispapertoaddressthequestionsraisedaboveandcontributetoextendingthe classic
Cournottheorytooligopoliststhatmaypurchaseadditionalquantitiesfromanexternalsupplier.Byviewing
the interaction of the competing Cournot oligopolists with their supplier as a two stage game, the cost
parameteroftheclassicCournotmodelisendogenizedandanswerscanbeworkedout.
Asweshallsee,obtainingtheequilibriumstrategiesfortheCournotfirmsinsuchamarketmaybevery
tedious, but nevertheless manageable after one formulates the setup. However, the case for the supplier,
whoisalsoaplayerinthisgame,issomehowmoredemanding. Intheduopolycase,ifthesupplierknows
the true value of the demandparameter, a uniquesubgameperfectequilibriumalways exists, althoughits
descriptionisquitecomplicated(seePropositionsA.1andA.2intheAppendix).Inallothercases(threeor
moreoligopolistsand/orincompleteinformation),thestudyofequilibriaispossibleonlyifweassumethat
thecompetingfirmsareidentical.So,assumingthattheoligopolistsareidentical,weobtainconcreteresults
concerningthepricethesupplierwillaskinboththecompleteandtheincompleteinformationcaseunder
equilibrium. Inthe latter, whenthe supplierdoesnotknowthe demandparameter,we givenecessaryand
sufficientconditionsfortheexistenceofasubgameperfectequilibriumviaafixedpointequationinvolving
atranslationoftheMeanResidualLifetime(MRL)ofhisbelief,providedthemeasurethisbeliefinduceson
thedemandparameterspaceisnon-atomic. Iftheyexist,theevaluationofallsubgameperfectequilibriais
possiblebyusingthisequation,i.e. ourapproachisconstructive. TheMRLfunctioniswellknownamong
actuarials,reliabilityengineers,andappliedprobabilistsbut,toourknowledge,hasneverarisenbeforeina
Cournotcontext.
Indetail,inspiredbytheclassicCournotoligopolywhereproducers/retailerscompeteoverquantity,we
study the market of a homogenousgood differing from the classic model in the following aspects. Each
producer/retailermayproduceonlya limitedquantityofthe gooduptoa specifiedandcommonlyknown
productioncapacity.Ifneeded,theretailersmayrefertoasinglesuppliertoorderadditionalquantities.The
suppliermayproduceunlimitedquantitiesbutatahighercostthantheretailers,makingitbestfortheretail-
erstoexhausttheirproductioncapacitiesbeforeplacingadditionalorders.Themarketclearsatapricethatis
determinedbyanaffineinversedemandfunction.Thedemandparameterordemandinterceptisconsidered
tobearandomvariablewithacommonlyknownnon-atomicdistributionhavingafiniteexpectation.
Dependingonthetimeofthedemandrealization,twovariationsofthismarketstructureareexamined,
which correspondto a completeand an incompleteinformationtwo-stage game playedas follows. In the
firststagethesupplierfixesapricebydecidinghisprofitmarginandtheninthesecondstage,theretailers,
knowing this price and the true value of the demand parameter, decide their production quantity (up to
capacity)andalsoplacetheirorders(ifany). Thedecisionsoftheproducers/retailersaresimultaneous. If
the supplier knows the true value of the demand parameter before making his decision as well, this is an
2
ordinarytwo stage game (and the demand random variable is just a formality). On the other hand, if the
randomdemandparameteris realized after the supplier hasset his price, then this is a two stage game of
incompleteinformationwherethedemanddistributionisthebeliefofthesupplieraboutthisvariable.So,in
bothvariationsthedemanduncertaintyisresolvedpriortothedecisionoftheretailers,whoalwaysknowthe
truevalueofthedemandintercept. Inthisrespect,oursetupdiffersfromtheusualincompleteinformation
modelsofCournotmarkets, where demanduncertaintyinvolvesthe producers(e.g. see Einyetal. [2010]
andreferencesgiventherein).
The limited production capacities of the producers/retailers may equivalently be viewed as inventory
quantities that may be drawn at a fixed cost per unit. The latter setting corresponds to the operation
of a retailer’s cooperative (hence the characterization of the ‘producers’ as retailers’ also). In case the
production capacities are 0, this is a classic Cournot model with the cost input determined exogenously.
MarxandSchaffer[2015]pointoutthescarcenessofresearchalongthislineandtheneedofendogenizing
thecostparameterintheCournotmodel.Todoso,theystudythegamethatariseswhencompetingCournot
firmspurchasetheirinputsfromacommonsupplierbasedoncontracts,withthecompetingfirmshavingthe
bargainingpower.
We study the equilibriumbehavior of the two-stage game by first determiningthe unique equilibrium
solution of the second stage game, which concernsthe quantitiesthat the producers/retailerswill produce
andtheadditionalquantitiestheywillorderfromthesupplier. Thisstageisthesameinboththecomplete
andtheincompleteinformationcaseandissolvedcompletely.Then,weexaminethefirststagegame,which
concernsthe price the supplier will ask for the product. In the case of the complete informationduopoly
(twoproducers/retailers-supplierknowsthedemandparameter),auniquesubgame-perfectNashequilibrium
always exists, which we give explicitly in the Appendix. Its descriptionis quite complicatedas it is very
sensitivetotherelationshipofallparametersoftheproblem.Otherwise,weproceedbyconsideringidentical
producers/retailers.Underthisassumption,thecompleteinformationcasesimplifiessignificantly.However,
ourfocusismainly onthe incompleteinformationcase. Forthatcase, we show thatif a subgameperfect
Bayesian-Nash equilibrium exists, then it will necessarily be a fixed point of a translation of the Mean
ResidualLifetime(MRL)functionofthesupplier’sbelief.Ifthesupportofthesupplier’sbeliefisbounded,
thensuchanequilibriumexists.IftheMRLfunctionisdecreasing(asinmost“wellbehaved”distributions),
thenasubgameperfectequilibriumalwaysexistsandisunique,irrespectiveofthesupplier’sbeliefsupport.
Inaddition,weinvestigatethebehaviorofthissubgameperfectequilibriuminvariousinterestingcases.
Awellknownbyproductofincompleteinformationistheappearanceofinefficienciesinthesensethat
transactionsthat would have been beneficialfor all marketparticipantsare excludedat equilibriumunder
incompleteinformation,whilethiswouldn’thavebeenthecaseundercompleteinformation. Inthepresent
paper,anupperboundonameasureofinefficiencyoverallDMRLdistributionsisderived,itisshownthat
it is tight, and the implicationsof the structure of the supplier’sbelief on the efficiencyof the incomplete
informationequilibriumareinvestigated.
Under incomplete information, the DMRL propertyis sufficient for the existence of a unique optimal
strategyforthesupplier,butnotnecessary. Possiblerelaxationsofthisconditionareexamined,basicallyin
termsoftheincreasinggeneralizedfailurerate(IGFR)condition,seeLariviere[2006]. Asfarasnecessary
conditionsare concerned,we don’thave a characterizationof the distributionwhen an equilibriumexists,
althoughwe havea characterizationofthe equilibriumitself throughthe MRL function. Finally, asfar as
the inversedemandfunctionis concerned,its affinityassumptionmaynotbe droppedif onestill wantsto
expressthe supplier’spayoff in terms of the MRL function. Hence, a generalizationin models with non-
affineinversedemandfunctionisopen.
Asadynamictwo-stagemodel,thecurrentworkcombineselementsfromvariousactiveresearchareas.
Cournotcompetitionwithanexternalsupplierhavinguncertaintyaboutthedemandinterceptmaybeviewed
3
as an applicationof gametheoretictoolsand conceptsin supplychainmanagement. However,we are in-
terested in the equilibriumbehaviorof the incompleteinformationtwo stage gamerather thanto building
coordinationmechanisms.Theincompleteinformationofthesupplierrelatesourworktoincompleteinfor-
mationCournotmodels,althoughthelackofinformationreferstotheproducersinmostpapersinthisarea.
Havingcapacityconstraintsfortheproducers/consumers,asitisnecessaryforourmodel,hasalsoattracted
attention in the Cournot literature, see eg. Bischietal. [2009]. Finally, the application of the concept of
themeanresiduallifetime,whichoriginatesfromreliabilitytheory,toapurelyeconomicsettingisanother
researchareatowhichourpaperisrelated.
CachonandNetessine[2004]surveytheapplicationsofgametheoreticconceptsinsupplychainanalysis
in detail and provide a thorough literature review up to 2004. They state that the concept that is mostly
used is that of static non-cooperativegamesand highlightthe need for a game theoretic analysis of more
dynamicsettings. Atthetimeoftheirsurvey,theyreportofonlytwopapersthatapplythesolutionconcept
of the subgame perfect equilibrium (in settings quite different to ours). BernsteinandFedergruen [2005]
investigatetheequilibriumbehaviorofadecentralizedsupplychainwithcompetingretailersunderdemand
uncertainty.Theirmodelaccountsfordemanduncertaintyfromtheretailers’pointofviewalsoandfocuses
mainlyonthedesignofcontractsthatwillcoordinatethesupplychain.Basedonwhethertheuncertaintyof
thedemandisresolvedpriortooraftertheretailers’decisions,theyidentifytwomainstreamsofliterature.
Forthe firststream, in whichthe presentpapermaybeplaced, theyreferto Vives[2001] foranextensive
survey.
Einyetal.[2010]examineCournotcompetitionunderincompleteproducers’informationaboutthede-
mandandproductioncosts. TheyprovideexamplesofsuchgameswithoutaBayesianCournotequilibrium
in purestrategies, theydiscuss the implicationof notallowing negativepricesand provideadditionalsuf-
ficient conditions that will guarantee existence and uniqueness of equilibrium. Richter [2013] discusses
Cournotcompetitionunderincompleteproducers’informationabouttheirproductioncapacitiesandproves
the existence of equilibrium under the assumption of stochastic independenceof the unknowncapacities.
He also discusses simplifications of the inverse demand function that result to symmetric equilibria and
implicationsofinformationsharingamongtheproducers.
Capacityconstrainedduopoliesaremostlystudiedintheviewofpriceratherthanquantitycompetition.
OsborneandPitchik[1986]andthereferencesthereinareamongtheclassicsinthisfield. Equallycommon
is the study of models where the capacity constraints are viewed as inventories kept by the retailers at a
lowercost. Papersinthisdirectionfocusmainlyondeterminingoptimalpoliciesinbuildingtheinventory
overmorethan oneperiods. Hartwigetal. [2015] and the referencesthereinare indicativeof this field of
research.
Theconditionofdecreasingmeanresiduallifetimethatarisesinourtreatment,iscloselyrelatedtothe
more general concept of log-concave probability. In an inspiring survey, BagnoliandBergstrom [2005]
examinea series of theoremsrelating log-concavityand/orlog-convexityof probabilitydensity functions,
distributionfunctions,reliabilityfunctions,andtheirintegralsandpointoutnumerousapplicationsoflog-
concaveprobabilitythatarise,amongotherareas,inmechanismdesign,monopolytheoryandintheanalysis
of auctions. Lagerlo¨f [2006] examines a Cournot duopoly where the producersdo not know the demand
interceptbuthaveacommonbeliefaboutit. Undertheconditionthatthedistributionofthedemandintercept
hasmonotonehazardrateandanadditionalmildtechnicalassumption,heprovesuniquenessofequilibrium.
GuessandProschan[1988]surveyearlieradvancesaboutthemeanresiduallifetimefunction,itsproperties
anditsapplications.Amongmanyotherfields,theymentionasingleeconomicapplicationinasettingabout
optimalinventorypoliciesfor perishableitemswith variablesupplywhich ishoweverratherdistinctfrom
ours.
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1.1 Outline
The rest of the paper is structured as follows. In Section 2 we build up the formal setting for a duopoly.
In Section 3 we present some preliminary results and treat the complete information case. In Section 4
weanalyzethemodelofincompleteinformation,stateourmainresultsandhighlightthespecialcasewith
no productioncapacities for the retailers. In Section 5 we investigateinefficiencies caused by the lack of
informationonthetruevalueofthedemandparameterbythesupplier. Theuseofresultsexhibitedin the
previoussectionsishighlightedbyvariousexamplesandcalculationsofthesubgameperfectequilibriumin
Section6. Finally,inSection7wegeneralizeourresultstothecaseofn>2-identicalretailers.Theproofs
ofSection3andSection7proceedbyanextensivecasediscriminationandarepresentedinAppendixA.
Throughout the rest of this paper we drop the double name “producers/retailers” and keep the term
“retailers”only.
2 The Model
2.1 Notationandpreliminaries
We consider the market of a homogenous good that consists of two producers/retailers (R and R ) that
1 2
competeoverquantity(R placesquantityQ)andasupplier(orwholesaler)underthefollowingassumptions
i i
1. Theretailersmayproducequantitiest andt uptoacapacitiesT andT respectivelyatacommonfixed
1 2 1 2
costhperunitnormalizedtozero1.
2. Additionally,theymayorderquantitiesq ,q fromthesupplieratapricewsetpriortoandindependently
1 2
oftheirorders.ThetotalquantityQ (w),i=1,2thateachretailerreleasestothemarketisequaltothesum
i
Q (w):=t (w)+q (w)
i i i
orshortly Q := t +q, where, fori = 1,2,thevariablet T isthequantitythatretailerR producesby
i i i i i i
≤
himselfordrawsfromhisinventory(atnormalizedzerocost)andq isthequantitythattheretailerR orders
i i
fromthesupplieratpricew.
3. Thesuppliermayproduceunlimitedquantityofthegoodatacostcperunit.Weassumethattheretailers
are moreefficientin the productionofthe goodorequivalentlythatc > h. After thenormalizationof the
retailers’productioncosthto0,therestoftheparametersi.e.wandcarealsonormalized.Thus,wrepresents
anormalizedprice,i.e. theinitialpricethatwassetbythesupplierminustheretailers’productioncostand
canormalizedcost,i.e. thesupplier’sinitialcostminustheretailers’productioncost. Thesupplier’sprofit
marginrisnotaffectedbythenormalizationandisequalto
r:=w c
−
4. AfterthetotalquantityQ= Q +Q thatwillbereleasedtothemarketissetbytheretailers,themarket
1 2
clearsataprice pthatisdeterminedbyaninversedemandfunction,whichweassumetobeaffine2
p=α Q (1)
−
1SeeSection1foranalternativeinterpretationofthesequantitiesasinventories.
2Afternormalizationoftheslopeparameter(initiallydenotedwithβ)oftheinversedemandfunctionto1,allvariablesin(1)are
expressedintheunitsofquantityandnotinmonetaryunitsandthereforeanyinterpretationsorcomparisonsofthesubsequentresults
shouldbedonewithcaution.
5
5. The demandparameterα is a non-negativerandomvariable with finite expectationE(α) < + and a
∞
continuous cumulative distribution function (cdf) F (i.e. the measure induced on the space of α is non-
atomic–singulardistributionfunctions,liketheCantorfunction,areacceptable).Wewillassumethatα h
≥
forallvaluesofα, i.e. thatthedemandparameterisgreaterorequaltotheretailers’productioncost. The
latter assumptionisconsistentwith theclassic Cournotduopolymodel,whichis resembledbythesecond
stage of the game (however, the second stage game is not a classic Cournot duopoly due to the capacity
constraintsT andT andtothepossibilityofw > α). Afternormalization,inwhatfollowswewillusethe
1 2
termα (resp. α )todenotethelowerupperbound(resp.theupperlowerbound)ofthesupportofα.
H L
6. The capacitiesT ,T and the distribution of the random demandparameterα are commonknowledge
1 2
amongthethreeparticipantsofthemarket(theretailersandthesupplier).
Based on these assumptions, a strategy s (w) for retailer R,i = 1,2 is a vector valued function s (w) =
i i i
(t(w),q (w))orshortlya pair s = (t,q) fori = 1,2. Equation(1) impliesthat Q maynotexceedαand
i i i i i i
hencethestrategysetS˜iofR willsatisfy
i
S˜i (t,q):0 t T and0 q +t α (2)
i i i i i i
⊂{ ≤ ≤ ≤ ≤ }
Denotingbys:=(s ,s )astrategyprofile,thepayoffui(s w)ofretailerR,i=1,2willbegivenby
1 2 i
|
ui(s w)= Q (α Q) wq =Q (α w Q)+wt (3)
i i i i
| − − − −
Wheneverconfusionmaynotarisewewillwriteui(s)insteadofui(s w). Astrategyforthesupplieristhe
|
pricewhechargestotheretailersorequivalentlyhisprofitmarginr. From(3)weseethatwmaynotexceed
a ,otherwisetheretailerswillnotorderforsure. Additionally,itmaynotbelowerthanc,sinceinthatcase
H
his payoff will becomenegative. Hence, in termsof his profitmarginr, the strategy set R of the supplier
satisfies
R r :0 r α c (4)
H
⊂{ ≤ ≤ − }
Consequently, a reasonable assumption is that c < α , otherwise the problem becomes trivial from the
H
supplier’sperspective. Foragivenvalueofα,thesupplier’spayofffunction,statedintermsofrratherthan
w=r+c,isgivenby
us(r α)=r(q (w)+q (w)) (5)
1 2
|
Ontheotherhand,fortheretailersitisnotnecessarytoknowtheexactvaluesofcandr andhence,from
their point of view (2nd stage game), we keep the notation w = r + c. If the supplier doesn’t know α
(incompleteinformationcase),thenhispayofffunctionwillbe
us(r):=Eus(r α) (6)
|
theexpectationtakenwithrespecttothedistributionofα.
Toproceedwiththeformaltwo-stagegamemodel,werecallthatboththeproduction/inventorycapaci-
tiesT andT oftheretailersandthedistributionofthedemandparameterαarecommonknowledgeofthe
1 2
threemarketparticipants.Wethenhave,
CompleteInformationCase: The demandparameterαis realizedandobservedbyboththesupplierand
theretailers3. Then,atstage1,thesupplierfixeshisprofitmarginrandhencehispricew. Hisstrategyset
3Ofcourse,thismeansthatthereisnorandomnessinα,sothedescriptionofαasarandomvariableisredundantinthecomplete
informationcase.Weuseitjusttogiveacommonformaldescriptionofboththecompleteandtheincompleteinformationcase.
6
andpayofffunctionare givenby(4)and(5). Atstage 2, basedonthevalueofw, eachcompetingretailer
choosesthequantityQ,i = 1,2thathewillreleasetothemarketbydetermininghowmuchquantityt he
i i
willdraw(atzerocost)fromhisinventoryT andhowmuchadditionalquantityq hewillorder(atpricew)
i i
fromthesupplier.Thestrategysetsandpayofffunctionsoftheretailersaregivenby(2)and(3)respectively.
IncompleteInformationCase: Atstage1,thesupplierchoosesrwithoutknowingthetruevalueofα. His
strategyset remainsthe same buthispayofffunctionisnow givenby (6). Afterr andhencew = r+c is
fixed,thedemandparameterαisrealizedandalongwiththepricewisobservedbytheretailers. Thenwe
proceedtostage2,whichisidenticaltothatoftheCompleteInformationCase.
Alltheaboveareassumedtobecommonknowledgeofthethreeplayers.
3 Subgame-perfect equilibria under complete information
Atfirstwetreatthecasewithnouncertaintyonthesideofthesupplieraboutthedemandparameterα. In
subsections3.1and3.2wedeterminethesubgameperfectequilibriaofthistwostagegame.
3.1 Equilibrium strategies ofthesecond stage
We beginwith the intuitiveobservationthatit is best forthe retailersto produceup to their capacitycon-
straintsorequivalentlytoexhausttheirinventoriesbeforeorderingadditionalquantititesfromthesupplier
atunitpricew. ForsimplicityinthenotationofLemma3.1andLemma3.2fixi 1,2 andletT :=T. As
i
∈{ }
above,Q =t +q.
i i i
Lemma 3.1. Any strategy s = (t,q) S˜i with t < T and q > 0 is strictly dominated by a strategy
i i i i i
∈
s = t ,q with
′i i′ ′i
(cid:0) (cid:1)
(T,Q T), ifQ T
ti′,q′i =(Q,0i)−, ifQi ≥<T
(cid:0) (cid:1)  i i
orequivalentlyby t ,q = min Q,T ,(Q T)+ .
i′ ′i { i } i−
(cid:0) (cid:1) (cid:0) (cid:1)
Proof. LetQ T. Thenforany s S˜j,(3)impliesthatui s,s = ui s,s +w(t T). Sincet < T,
i ≥ j ∈ i j ′i j i− i
theresultfollows.Similarly,ifQ <T,thenforanys S˜j,(3(cid:16))imp(cid:17)liesth(cid:16)atui (cid:17)s,s =ui s,s wq. (cid:3)
i j ∈ i j ′i j − i
(cid:16) (cid:17) (cid:16) (cid:17)
Accordingly,werestrictattentiontothestrategiesin
Si = (t,q):0 t <min α,T ,q =0 (t,q):T =min α,T ,0 q (α T)+
i i i i i i i
(cid:26) ≤ { } (cid:27)[(cid:26) { } ≤ ≤ − (cid:27)
Figure 1a depicts the set Si when T < α and Figure 1b when T α. As shown below, Lemma 3.1
≥
considerablysimplifiesthemaximizationofui ,s ,sinceforanystrategys ofretailerR ,themaximumof
j j j
·
ui ,s willbeattainedatthebottomorrighth(cid:16)ands(cid:17)ideboundaryoftheregion[0,min α,T ] 0,(α T)+ .
j
· { } × −
Mo(cid:16)reov(cid:17)er, Lemma 3.1 implies that when α T, retailer Ri will order no additional quan(cid:2)tity from th(cid:3)e
supplier4. Althoughtrivial,wemaynotexclud≤ethiscaseingeneral,sinceinSection4weconsiderαtobe
varying.
4InProposition3.3wewillseethatinequilibirumretailerRiwillordernoadditionalquantityifα 3T.
≤
7
qi qi
α T α
−
Qi=qi+ti:constant Qi=qi+ti:constant
Dominantstrategies Dominantstrategies
0 T ti 0 α T ti
(a)T <α (b)T α
≥
Figure1:RetailersstrategysetSi.
RestrictingattentiontoSifori=1,2,weobtainthebestreplycorrespondencesBR1andBR2ofretailers
R and R respectively. To proceed, we notice that the payoff of retailer R,i = 1,2 dependson the total
1 2 i
quantityQ thatretailerR , j=3 ireleasestothemarketandnotontheexplicitvaluesoft ,q ,cf. (3).
j j j j
−
Lemma3.2. ThebestreplycorrespondenceBRiofretailerR fori=1,2isgivenby
i
α w Q
j
t =T, q = − − T, if 0 Q <α w 2T (1)
i i j
 2 − ≤ − −
BRi(Qj)=ti =Tα, Qj qi =0, if α−w−2T ≤ Qj <α−2T (2)
The enumeratiotni =(1),(−22),(3,)qoif=th0e,different parts ofifthαe−be2sTt r≤epQlyjcorrespondencewill b(3e)used for a
moreclearcasediscriminationinthesubsequentequilibriumanalysis,seeAppendixA.
Proof. SeeAppendixA. (cid:3)
AgenericgraphofthebestreplycorrespondenceBR1 ofretailerR tothetotalquantityQ thatretailer
1 2
R releases to the market is given in Figure 2. The equilibrium analysis of the second stage of the game
2
proceedsinthestandardway, i.e. with theidentificationoftheNashequilibriathroughthe intersectionof
thebestreplycorrespondencesBR1andBR2.Forabetterexpositionoftheresultswerestrictattentionfrom
nowontothecaseofidenticalretailers,i.e. T =T = T. ThegeneralcaseT T istreatedinAppendix
1 2 1 2
≥
A,wherethecompletestatementsandtheproofsofProposition3.3andProposition3.4areprovided.
Iftheretailersareidentical,i.e. T = T = T onlysymmetricequilibriamayoccurinthesecondstage.
1 2
Theequilibriumstrategiesdependonthevalueofαanditsrelativepositionto3T.
Proposition3.3. IfT = T = T, thenforallvaluesofαthesecondstageequilibriumstrategiesbetween
1 2
retailers R and R are symmetric and for i = 1,2 they are given by s (w) = t (w),q (w) , or shortly
1 2 ∗i i∗ ∗i
s∗i = ti∗,q∗i ,with (cid:0) (cid:1)
(cid:0) (cid:1)
1 1
t (w)=T (3T α)+, q (w)= (α 3T w)+.
i∗ − 3 − ∗i 3 − −
8
Q2
α
(t1,q1)=(cid:18)α−2Q2,0(cid:19)
(3)
α 2T
−
(t1,q1)=(T,0)
(2)
α w 2T
− −
(t1,q1)=(cid:18)T,α−w2−Q2 −T(cid:19)
(1)
0 T α w Q1
2 − 2
Figure2:BestreplycorrespondenceofRetailerR .
1
Proof. SeeAppendixA,whereProposition3.3isstatedandprovedforthegeneralcaseT T . (cid:3)
1 2
≥
The quantity Q = t +q is the total quantityof the good that each retailer releases to the marketin
∗i i∗ ∗i
equilibrium.Ifα 3T thenQ isequaltotheequilibriumquantityofaclassicCournotduopolistwhofaces
≤ ∗i
linear inverse demand with intercept equal to α and has 0 cost per unit. If α > 3T, then the equilibrium
quantitiesoftheretailersdependonthesupplier’spricew. Ifwislowenough,i.e. ifw< α 3T,theywill
1 −
bewillingtoorderadditionalquantitiesfromthesupplier.Inthiscase,Q =t +q =T+ (α 3T w)=
∗i i∗ ∗i 3 − −
1
(α w),whichisequaltotheequilibriumquantityofaCournotduopolistwhofaceslineardemandwith
3 −
interceptequaltoαandhaswcostperunitforallproductunits,despitethefactthatinourmodeltheretailers
faceacostof0forthefirstT unitsandwfortherest. Finally,forw α 3T theretailerswillavoidordering
≥ −
andwillreleasetheirinventoriestothemarket.
Thetotalquantitythatisreleasedtothemarkethasadirectimpactontheconsumerswelfareasexpressed
intermsoftheconsumerssurplus5. Forlowervaluesofthedemandparametertheconsumersbenefitfrom
the “low cost” inventoriesor productioncapacities of the retailers, since otherwise no goods would have
beenreleasedtothemarket. Ontheotherhand,forhighervaluesofαthetotalquantitythatisreleasedto
themarketdoesnotdependdirectlyonT andthusthebenefitsfromkeepinginventoryareeliminatedforthe
consumers.
3.2 Equilibrium strategies ofthefirst stage
Thegeneralformofthesupplier’spayofffunctionus(r α) is givenin(5) andhisstrategysetin (4). Ob-
|
viously,thesupplierwillnotbewilling tochargepriceslowerthanhiscostc. Basedonthe discussionof
5Weremindthattheconsumerssurplusisproportionaltothesquareofthetotalquantitythatisreleasedtothemarket.
9
Proposition3.3aboveandtheconstraintw c(i.e.r 0),weconcludethatatransactionwilltakeplacefor
≥ ≥
valuesofα > 3T +candforr [0,α 3T c). Asweshallsee,inthatcase,theoptimalprofitmarginof
∈ − −
thesupplierisatthemidpointofther-interval.
Toseethis,letq (w):=q (w)+q (w)denotethetotalquantitythatthesupplierwillreceiveasanorder
∗ ∗1 ∗2
2
fromthe retailerswhentheyrespondoptimally. By Proposition3.3, q (w) = (α 3T w)+. Hence, on
∗
3 − −
theequilibriumpathandforr 0,thepayoffofthesupplieris
≥
2
us(r α)=rq (w)= r(α 3T c r)+ (7)
∗
| 3 − − −
Wethenhave
Proposition3.4. ForT = T = T andforallvaluesofα,thesubgameperfectequilibriumstrategyr (α)
1 2 ∗
ofthesupplierisgivenby
1
r (α)= (α 3T c)+
∗
2 − −
Proof. SeeAppendixA,whereProposition3.4isstatedandprovedforthegeneralcaseT T . (cid:3)
1 2
≥
Proposition3.4impliesthatifα<3T+c,thentheoptimalprofitofthesupplierisequalto0,i.e.hewill
setapriceequaltohiscost. Actually,heisindifferentbetweenanypricew csinceinthatcaseheknows
≥
thattheretailerswillordernoadditionalquantity.
In sum, Proposition3.3 and Proposition3.4 providethe subgameperfectequilibriumof the two stage
gameinthecaseofidentical(i.e. T =T =T)retailers.
1 2
Theorem 3.5. If the capacities of the producers (retailers) are identical, i.e. if T = T = T, then the
1 2
completeinformationtwostagegamehasauniquesubgameperfectNashequilibrium,underwhichthesup-
1
pliersellswithprofitmarginr (α) = (α 3T c)+ andeachoftheproducers(retailers)ordersquantity
∗
2 − −
1 1
q (w)= (α 3T w)+andproduces(releasesfromhisinventory)quantityt (w)=T (3T α)+.
∗ ∗
3 − − − 3 −
4 Subgame-perfect equilibrium under demand uncertainty
We investigatetheequilibriumbehaviorofthesupplychainassumingthatthesupplierhasnoinformation
aboutthetruevalueofthedemandparameterαwhenhesetshisprice,whiletheretailersknowitwhenthey
placetheirorders(ifany). Inthetwostage gamecontext,weassumethatthedemandisrealizedafterthe
first stage (i.e. after the suppliersets hisprice)butpriorto the second stage ofthe game(i.e. priorto the
decisionoftheretailersaboutthequantitythattheywillreleasetothemarket). ThecaseT T exhibits
1 2
≥
significantcomputationaldifficultiesandwewillrestrictourattentiontothesymmetriccaseT =T =T.
1 2
Underthese assumptionsthe equilibriumanalysis of the second stage (as presentedin subsection 3.1)
remainsunaffected: theretailersobservetheactualdemandparameterαandthepricewsetbythesupplier
andchoosethequantitiest andq fori = 1,2. However,inthefirststage,theactualpayoffofthesupplier
i i
dependsontheunknownparameterαforwhichhehasabelief:thedistributionFofα,whichinducesanon-
atomicmeasureon[0, )withfinite expectation,asassumed. Giventhevalueofα andtakingasgranted
∞
thatthe retailersrespondto the supplier’schoicew with theiruniqueequilibriumstrategies, the supplier’s
10