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Department of Economics
Product Differentiation in Successive Vertical Oligopolies
Paul Belleflamme and Eric Toulemonde
Working Paper No. 421 October 2000 ISSN 1473-0278
Productdi¤erentiationinsuccessivevertical
oligopolies
Paul Belle‡amme¤EricToulemondey
October2000
Abstract
Thisisasuccessiveoligopoly modelwithtwobrands.Each down-
stream…rmchoosesonebrand tosell ona…nalmarket.Theupstream
…rms specialize intheproductionofoneinputspeci…cally designed
fortheproductionofonebrand,but theyalsoproduce theinputfor
theotherbrand atanextracost. Weshowthatwhenmoredown-
stream…rmschooseonebrand,moreupstream…rmswill specialize in
theinputspeci…ctothatbrand,and vice versa.Hence,multiple equi-
libria arepossibleand thesofteninge¤ectof brand di¤erentiationon
competitionmightnotbestrongenoughtoinduce maximaldi¤eren-
tiation.The existence ofequilibria and theirwelfareperformance are
alsoexamined.
JELclassi…cationcodes:L11,L13,L23
Keywords:productdi¤erentiation, verticalrelationships,oligopoly
¤DepartmentofEconomics,QueenMary,Universityof London,MileEnd Road,Lon-
donE1 4NS,UnitedKingdom(e-mail: p.belleflamme@qmw.ac.uk,phone:+44 20 7882
5587,fax:+44 20 8983 3580,homepage:http://www.qmw.ac.uk/~ugte186/)
yChargéderechercheat theNationalFund forScienti…cResearch(Belgium),C.R.E.W.,
G.R.E.B.E., DepartmentofEconomics,UniversityofNamur,8RempartdelaVierge,5000
Namur,Belgium(e-mail: eric.toulemonde@fundp.ac.be,phone:+32 81 724810,fax:
+32 81 724840,homepage:http://www.fundp.ac.be/~etoulemo).
1
1Introduction
Since thepioneeringworkofHotelling(1929),thesofteninge¤ectof prod-
uctdi¤erentiationon price competition hasbeenabundantly studied.For
instance,d’Aspremontetal. (1979)establishthat…rmschoosemaximal
di¤erentiationinthelinearcitymodelwithquadratictransportationcosts.
Thispredictionisechoedinmostmarketingtextsconcerningmarketseg-
mentation,whichrecommend …rmstodi¤erentiatetheirproducts.
There exist,however,forcesthatopposemaximal(orany)productdif-
ferentiation.FollowingTirole(1988),theseforcescan beputintothree
categories.A…rst,obvious,categoryhastodowiththeabsence of price
competition: insomeinstances,theremayexistlegalortechnicalreasonwhy
thescopeof price competitionislimited.Thesecond categorycorresponds
tothemotto “Bewherethedemand is”.Although …rmsliketodi¤erentiate
forstrategicpurposes,theyalso all want tolocatewherethedemand is.
If,forexample,demand isconcentratedaround afewpoles,one caneasily
constructexamplesinwhich …rmsdi¤erentiatebutnotfully. Inasimilar
vein,search byconsumersmayencourage…rmsto gather.Finally, thethird
categoryreferstopositive externalitiesthatinduce …rmstolocatenearone
another.Thenowstandardclassi…cationofMarshallianexternalitiesisbe-
tweenlocalizationeconomies(whichrefertothebene…tsgenerated bythe
proximityof …rmsproducingsimilargoods),and urbanizationeconomies
(whichaccountforall theadvantagesassociatedwiththeoverall levelof
activityprevailinginaparticulararea).Thegeneral ideaisthat,forgiven
inputs,theoutputofanindividual…rmislargerthelargeristheaggregate
outputofother…rmsproducingthesamegoodinthesamelocale.
Regardingthelattercategory, Helpmanand Krugman(1985)pointout
thatMarshall (1920)’sexplanationisincomplete:externaleconomiescan
arisefromproximitytospecializedinputsonly ifthereisanaturalcom-
parativeadvantagefortheproductionoftheseinputsintheregion.As
Rotemberg and Saloner(2000)nicely putit,“Thepuzzleis simply rolled
backtothepreviousproductionstage: Whydotheproducersofinputs
locateintheregion?”.Theydevelopthereforethetheorythat the equi-
libriumlocationsof …rmsand theirinputsuppliersareinterdependent.So
doesVenables(1996)who argues:
“Ifindustriesarevertically linkedthroughaninput-outputstructure,
2
thenthedownstreamindustryformsthemarketforupstream.Market
access considerationsthen drawtheupstreamindustrytolocations
wheretherearerelatively manydownstream…rms.Inadditiontothis
demand linkagebetweenindustries,thereisalso a costlinkage.Firms
inthedownstreamindustrywill havelowercostsifthey locatewhere
therearerelatively manyupstream…rms–theysavetrade costsontheir
intermediateinputs.Puttingthedemand linkageand the costlinkage
togethercreatesaforce fortheagglomerationofactivity inasingle
location.”
ThepresentpapersharestheintuitionofVenables(1996)butdeparts
fromhisanalysisinoneimportantaspect.Venablesisconcernedwithre-
gionalagglomerationand trade,and considersthusphysical locationof …rms
in di¤erentregions.Bycontrast,weaddress theissueof productdi¤eren-
tiation,and considerthuslocationof …rmsinaproductratherthanina
geographicalspace. WhereVenablesconsidersexogenously di¤erentiated
productswhichcan besoldontwodi¤erentgeographicalmarkets,wefocus
onasinglemarketonwhichtwoendogenously di¤erentiated productscan
besold. Ourmainobjectiveisthusto assess howthesofteninge¤ectof
productdi¤erentiationoncompetitionmightbemitigated bythedemand
and costlinkagesidenti…ed byVenables.
Speci…cally, we considertwovertically relatedindustrieswiththefollow-
ingfeatures.Thedownstreamindustryproducesa…nalproduct thatcan be
marketed undertwopossibledi¤erentiated brands.Intuitively, oneunder-
standsthat,absentanyotherconsideration,thisindustry isdriventowards
asituationwhere…rms splitequally (or‘almostequally’ ifthereisanodd
numberofthem)betweenthetwobrandsinordertosoftencompetition
(thisisthetranslationofmaximaldi¤erentiationinoursetting).
However,thisintuitionmayprovewrongwhenwealsotakeinto account
an upstreamindustrythatproducesanessential inputforthedownstream
industry, and thatalsohastotakeits stand onthetwobrands.Thereasonis
thefollowing.Althoughtheuseofso-called‡exiblemanufacturingsystems
(FMS)becomesincreasingly widespread,1economiesofscalearestill present
inmany industries.Itisthusreasonableto assumethat theinputismore
1Asexplained byNormanand Thisse(1999),“the essence ofaFMSisthatitallows
…rmstocustomize theirproductstotherequirementsofheterogeneouscustomersatlittle
ornocostpenalty”.
3
costly toproduce whenithasto…t twodi¤erentiated brandsratherthan
asingleone.Thisinduces supplierstoselect thebrand fromwhichthey
want to “getcloser”.Speci…cally, supplierswill incurspeci…c coststo adapt
theintermediateproduct tothespeci…cneedsofasinglebrand ofthe…nal
product.Nevertheless,suchspeci…cinvestmentdoesnotcompletely prevent
themto alsoconformtheirinput totheotherbrand:depending onthe
demand expressed bythebuyersofthisotherbrand,theymightindeed …nd
itpro…tabletoincurthe extracost toservethem.Adoptingtheterminology
ofEatonand Schmitt (1994),we cansayequivalently thatupstream…rms
choosetodeveloponebasicproduct(that…tsoneparticularbrand ofthe
…nalgood),and then produce,withanextracost,onevariationofthat
basicproduct (inorderto…t theotherbrand ofthe…nalgood).2
Reconsideringbrand choicesbydownstream…rmsinsuchabroadercon-
text,we canconjecturethat these choiceswill bedriven notonly bycom-
petitiononthe…nalmarket,butalsobycostconsiderationslinkedtothe
choicesmade by…rmsintheupstreamindustry.In particular,downstream
…rmswill bemoreattracted byabrand thatalargenumberof upstream
…rmshave chosentoconformwith.Since, inturn,upstream…rmsaremore
likely toconformwithabrand thathasbeenselected byalargenumber
of downstream…rms,weareinthepresence ofmutually re-enforcingforces
thatdrivetheindustriestowardanabsence of productdi¤erentiation.
Ourgoal isto analyze thebalance betweenthesevariousforcesinasimple
statictwo-stagemodelwhere,…rst,…rmsin bothindustrieschoosesimulta-
neously betweenthetwobrandsand,second,competeontheir respective
markets. Weshowthat thenumberof downstream…rmsthatchooseone
particularbrand increaseswiththenumberof upstream…rmsthatspecialize
intheinput thatis speci…ctothatbrand,and vice versa.Thisraisesthe
possibilityofmultiple equilibria.Forinstance, itispossibletohaveanequi-
libriumwithoutanydi¤erentiation(upstreamand downstream…rms select
thesamebrand),oranequilibriumwithmaximaldi¤erentiation(…rms split
equally between both brands)oranintermediate equilibrium(most–butnot
all–upstreamand downstream…rmschoosethesamebrand).Depending on
parameters,someofthese equilibria arepreferred by…rmswhilethe con-
sumerpreferanotherequilibrium.Hence,thereisnoclear-cutconclusionto
2Thisextracostmightbeduetothefact thatmorerawmaterial isneededforeach
unitoftheinput tobetransformedintotheotherbrand.Itmightalsobethe consequence
ofadegradationofperformance oftheinputwhenitistransformedintotheotherbrand.
4
drawforsocietyasawhole,whichmightend up withexcessiveorinsu¢cient
di¤erentiation.
Wesee two otherpossibleapplicationsofoursetting.First,the extra
costincurred bytheupstream…rmsmightfollowfromothersourcesthanthe
necessitytophysically alterthebasicproduct.Itmightwell bethe casethat
suppliersface di¤erentcostsforservingdi¤erentbuyers,eventhoughthey
sell themexactly thesamephysicalgood.Forinstance,thedevelopment
ofEDI(ElectronicDataInterchange)ofteninduces supplierstoconduct
transactionsin di¤erentwayswith di¤erentbuyers.3Asexplained byAnge-
lesand Nath(2000),“withthemovement towardsmoreintegratedand agile
‘extendedenterprises’, hub …rms(i.e.…rmsthatinitiateEDIlinkages)have
been forcedtoprovideincentivestotheirsupplierstomakenon-contractible
investmentsininformationsharing, quality initiatives,and innovationto
enablethemtoful…ll therequirementsofmoretightly connectedand inte-
gratedinformation networks.”Similarly, casestudiesinMarcussen(1996)
showthatEDIcapability in‡uencesthebuyingdecisionaboutwhich up-
stream…rmswill be‘insuppliers’(currently chosensuppliers)— and which
ofthe‘insuppliers’will getordersforspeci…citems.“Iftwoleadingcompet-
ingsuppliersboth developEDIcapability, competitiveparitybetweenthe
two…rmsismaintained.However,theEDIcapable‘insuppliers’stand to
winat the expenseof boththe‘insuppliers’withoutEDIcapabilityand the
‘outsuppliers”’. Since EDIrequirementsoften di¤erfromonedownstream
…rmto another,supplierswillingtotradewith di¤erentbuyersface thesame
typeofcostsastheonesdescribedabove.
Second,theissueweaddress inthispapercanalsoberelatedtothe
literature consideringmix-and-matchand systemgoods.Asde…ned byEin-
horn(1992),“mix-and-matchcompatibletechnologiesaresystemsofinter-
connectedgoodsthatmay incorporatedi¤erentcomponentsfromdi¤erent
manufacturers”.Examplesabound,e.g., inthe computerand inthe con-
sumerelectronicsindustries.Somepapersinthisliterature considerindus-
trieswhere each …rmsellseverycomponentofacompletesystem.4Other
papersconsiderinsteadthat…rms specialize intheproductionofasingle
component,anassumptionthatcomesclosertotheissueatstakehere(we
3EDIisthedirectcomputer-to-computerexchangeofinformationstoredinstandard
formatted business documents,suchasinvoices,billsoflading,purchaseorders,etc.,
among…rms.
4See,e.g., Matutesand Régibeau(1988,1992),Economides(1989).
5
canindeedinterpretourintermediateand …nalproductsascomplementary
componentsformingasystemgood).Althoughthepapersinthis second
strand oftheliteraturefocuson di¤erentissuesthanours(and makequite
di¤erentassumptionsabout,notably, consumerpreferencesand ‘compati-
bility’ costs),theyshareourconcern foranalyzingthe“location”decisions
ofthe‘software’…rms(theupstream…rmsinourterminology).5However,
itmustbestressedthat they leaveasidethe“location”decisionsofthe
‘hardware’(ordownstream)…rmsand,therefore,abstractawayourmain
questionabouthowthedecisionsof bothtypesof …rmsa¤ecteachother.
Therestofthepaperisorganizedasfollows.Section2describesthe
modellingframework. Section3draws someintuition fromtwosimplemod-
elswhere costsaretakenasexogenous.Section4 analyzesthegeneraltwo-
stagegameand presentsourmainresults.Section5collects somewelfare
considerations.Section6concludesand proposes somedirectionsforfurther
research.
2 Themodel
Themodel isasuccessiveoligopoly modelwithtwo adjacentindustries.In
thedownstreamindustry,asetNofn…rmsproduce someproduct tobesold
ona…nalmarket.Theyall havetochooseunderwhich brand tomarket
thisproduct.Twohorizontally di¤erentiated brands,notedaand b,are
available. Once the…rmshavemadetheirchoice,theindustry ispartitioned
intothetwosubsetsNaand Nb(Na\Nb=;;Na[Nb=N). Weadopt
thefollowingnotation.Letyikdenotethequantityof brand k(k=a;b)
produced bysomedownstream…rmi(i=1;::: ;n),Yk´Pi2Nkyik,the
total quantityproducedof brand k,and Y¡i
k´Pj2Nk;j6=iyjk.
Withinthe existingtheory, therearetwobasicapproachestomodelhor-
izontaldi¤erentiation. Ontheonehand,thespatialmodels–suchasthe
5Forinstance,Matutesand Régibeau(1989)analyze thesituationwheretwo…rms
sell one component (e.g., software)ofatwo-componentsystem,and where consumers
havealreadybought the…rstcomponent (e.g., thepersonalcomputer)sothatseveral
independentmarketsforsoftwareare created.Firmshavethustochoosethemarket(s)
that theywill serve.Theyalsodecidewhethertoproduce di¤erentsoftwareforeach
marketortosell softwarethatcan beusedwith bothtypesofcomputers.Churchand
Gandal(1992)developamodelwheretherearetwoincompatiblehardwaretechnologies,
anendogenouslydetermined numberofsoftware…rms,and consumerswhovaluesoftware
variety;theyinvestigatethedecisionofasoftware…rmconcerningwhich networktojoin.
6
Hotelling’slinearcityand Salop’scircularcitymodels–generatemarketde-
mandsby integrating overconsumerswith di¤erent tastes. Ontheother
hand,themodelsintheSpence-Dixit-Stiglitztradition deriveademand
systemfordi¤erentiated productsfromtheutilityfunctionofarepresen-
tative consumerwithatasteforvariety.6Becausethe…rstapproachis
basedonarathersimplisticdescriptionofindividualdemand (consumers
areassumedtobuyatmostoneunitoftheproduct), itprovesill-suitedto
incorporatedemand and costlinkages. Wethusfavorthesecond approach
and useasystemof demand functionsfordi¤erentiated productsderived
fromthequadratic,separableutilityfunctionofarepresentative consumer,
asin Shubik (1980,Chapter7)orin Singhand Vives(1984).Accordingly,
wewritetheinversedemand scheduleasfollows:7
²whenthendownstream…rmschoosetoproduce thesamebrand k
(meaningthatNk=Nand that theotherbrand isnotproduced),
price onthemarketforbrand kisgiven bypk=1¡Yk;
²whenthedownstream…rmschoosetoproduce di¤erentbrands,prices
onthemarketaregiven bypa=1¡Ya¡°Yband pb=1¡Yb¡°Ya
intheregionofquantitieswherepricesarepositive,where0·°<1
isthedegree of di¤erentiation between brandsaand b.
Weassumethat thedownstream…rmsproduce the…nalproductby
transforming a singleintermediategoodona one-for-onebasis.Themarginal
costofthedownstream…rmsonly consistsoftheprice paid fortheinterme-
diategood(otherspeci…c costsareassumedtobe zero).
Theintermediategoodis supplied bytheupstreamindustry,whichcon-
sistsofasetMofm…rms.Theproductiontechnologyfortheintermediate
goodassumes somedegree of‘brand speci…city’. Thatis,each upstream
…rmhastodesignitsproduction process inconformitywithaspeci…cbrand
ofthe…nalproduct (wesay inthesequelthat theychooseto “produce for”
brand aorforbrand b).Asaresult, iftheintermediategoodistobetrans-
formedintothatspeci…cbrand,themarginalcostof productionisequalto
6Forarecentdiscussionofthesetwo approaches(and thepresentationofanovel
integrativeapproach),see Andersonand dePalma(2000).
7Theinversedemand scheduleisderivedfromthequadraticutilityfunctionofarepre-
sentative consumerwhichexhibitsloveforvariety;forthe exactderivation,see theproof
ofProposition7intheappendix.
7
c,whereasifithastobetransformedintotheotherbrand,anextra‘adap-
tation’costtistobeincurred,and thetotalmarginalcostof production
amountstoc+t(naturally, weassumec+t<1to avoid notradesitu-
ations).8Inthesequel, withoutany loss ofgenerality, wesetc=0:For
thesakeofsimplicity, wealso assumethatbothindustriescountaneven
numberof …rms(in Subsection4.4,wediscuss howanodd numberof …rms
ineitherindustrya¤ectsourmainresults).
Westudyadynamicgamewherelocation decisions(which brand topro-
duce,ortodesigntheintermediategood for?)precedeproduction decisions
(howmuchoftheintermediateand the…nalgoodstoproduce?).Regarding
production decisions,weassumeasequentialCournotcompetitioninwhich
upstream…rmsdecidebeforedownstream…rms,theprice oftheintermedi-
ategood being obtained byequatingsupply todemand.Regardingbrand
choices,weassumesimultaneousdecisionsbyupstreamand downstream
…rms.Thegameis solved forits subgame-perfectequilibriabythemethod
of backwardinduction.
Althoughthemodellingframeworkhasbeenkeptas simpleaspossible,
thedynamicgameisratherintricatetosolve.Therefore,toshedsomelight
onthevariousforcesatplay, weshall …rstfocusonthedownstreamindustry
and analyze twosimplemodelswhere costsaretakenasexogenous.This
will allowustobettergrasp howdownstreamdecisionsarea¤ected bythe
endogeneizationofcoststhroughtheupstreambrand choices.
3Brand choiceswithexogenouscosts
Westartwiththesecond-stageproduction decisions.Supposethat thetwo
brandshavebeenadopted byapositivenumberof …rms.Typical…rms
i2Naand j2Nbrespectively face thefollowingmaximization programs:
8Aconcrete examplethat…tsourassumptionsiswhatLevitt (1980)reportsabouta
speci…ctypeofsteel, theso-called‘No.302,72-inch,hot-rolledstrip’. “Notall generic
productsarethesame.(...)Becauseofslightdi¤erencesamong automobile company
manufacturingprocesses,onesupplier’s“302” may, infact,be“better”thananother’s.
Onemill’s302 maytake certaincoatingsmore easilyorquicklythananother’s.One
suppliermay…ll ordersfromasinglemill, and anotherfromseveral. Inthelattercase,
thesheenorhueofthegenericproductmay varyslightlyfrom mill tomill, whichmakes
considerabledi¤erence inthe caseofstainless steelthatisusedfordecorativetrim.”
8
max
yia
¦d
i=(1¡yia¡Y¡i
a¡°Yb)yia¡wayia,
max
yjb
¦d
j=(1¡yjb¡Y¡j
b¡°Ya)yjb¡wbyjb,
wherewaand wbarethepricesoftheintermediateinput tobetransformed
ineitherbrand aorb.Forthemoment,weassumethat thesepricesare
constant.
The…rst-orderconditionsforpro…tmaximizationyieldthefollowing
reaction functions:yia=(1=2)(1¡wa¡Y¡i
a¡°Yb)and yjb=(1=2)(1¡
wb¡Y¡j
b¡°Ya):Solvingthis systemand usingthesymmetricpositionsof
…rmswithineachmarket,wederivethe equilibriumquantities.Tosimplify
the exposition,weidentifyapartitionofthe…rmsto¢n´na¡nbwhere
nk=#Nk;k=a;b.Fora given¢ninherited fromthe…rststage,every
i2Naproduce aquantityya(¢n);and everyj2Nbaquantityyb(¢n):
ya(¢n)=2(n¡¢n+2)(1¡wa)¡°(n¡¢n)(1¡wb)
4(n+1)+(1¡°2)(n2¡¢2
n);(1)
yb(¢n)=2(n+¢n+2)(1¡wb)¡°(n+¢n)(1¡wa)
4(n+1)+(1¡°2)(n2¡¢2
n):(2)
Inthe casewhereall …rmsadopt thesamebrand (saybrand a),weset
¢n=ninexpression(1)and disregardexpression(2).9Equilibriumpro…ts
aresimply equaltothesquareofthe equilibriumquantities:¦d
a(¢n)=
[ya(¢n)]2and ¦d
b(¢n)= [yb(¢n)]2:
Letusnowturntothe…rststageofthegamewhere…rms simultaneously
choosewhich brand theywant toproduce.Inthisgame,a Nashequilibrium
with¢n6=§nischaracterized bytwoconditions,ensuringthatno…rm…nds
itpro…tabletoswitch brandsunilaterally: (Da)¦d
a(¢n)¸¦d
b(¢n¡2)and
(Db)¦d
b(¢n)¸¦d
a(¢n+2).A Nashequilibriumwith¢n=n(resp.
¢n=¡n)satis…es(Da)(resp.(Db))only. Wedevelopthese conditions
intwospeci…c examples.
3.1Constantand identical costs
Inthe…rstexample,weassumethatall …rmsface thesamemarginalcost,
whatevertheirchoice of brand.Forwa=wb=w;conditions(Da)and (Db)
9Wesupposethatparametervaluesaresuchthatequilibriumquantitiesarepositive
forall ¢n(thisissueisanalyzedrigorouslyinthegeneralsetting).
9
canrespectively berewrittenas
((Da)¡(¢n¡1)£4(n¡°)+(1¡°2)(n¡¢n)(n+¢n¡2)¤¸0;
(Db)¡(¡¢n¡1)£4(n¡°)+(1¡°2)(n+¢n)(n¡¢n¡2)¤¸0:
Because condition(Da)[resp.(Db)] isvalid for¡n<¢n·n[resp.
¡n·¢n<n], itisclearthat thetwobracketedtermsarestrictly pos-
itive,meaningthat thetwoconditionsboil downto¡1·¢n·1.In
words,whenall downstream…rmsface thesame costofproduction,the
unique brandchoice equilibriumiswhere…rms splitequallybetweenthetwo
brands ( ¢n=0).10 Theintuitionis simple:whenthesplitisunequal, …rms
onthe‘larger’markethaveanincentivetomovetothe‘smaller’market
where competitionis softer.
3.2 ExogenousMarshallian externalities
Inthesecond example,weintroduce exogenousMarshallianexternalitiesof
thefollowinglinearform:wk=w¡µnk,with0<µ<w=n,and k=a;b:We
assumethusthat themarginalcostof producingsomebrand decreaseslin-
early withthenumberof …rmsproducingthatparticularbrand (see Section
1forjusti…cationsofsuchassumption).Inthiscase,we can use expressions
(1)and (2) todevelopconditions(Da)and (Db)asfollows:
((Da)(¢n¡1)[µ(1+°n)¡(1¡°)(1¡w)]-(¢n)¸0
(Db)(¢n+1)[µ(1+°n)¡(1¡°)(1¡w)]-(¢n+2)·0;
where-(¢n)=£(n(1¡°)+2)(n(1+°)¡2°)¡(1¡°2)¢n(¢n¡2)¤:It
can beshownthat-(¢n)>0,forall ¡n+2·¢n·n.11 Thereare
thereforetwocasestoconsideraccordingtotheintensityoftheMarshallian
externalities. Wede…nethefollowingthreshold:¹
µ´(1¡°)(1¡w)=(1+°n):
1.ForweakMarshallianexternalities(µ·¹
µ),conditions(Da)and (Db)
boil downto¡1·¢n·1,meaningthat thepreviousconclusion
carriesover:equalsplitistheunique brandchoice equilibrium.
10 Whennisodd,therearetwoequilibria,¢n=1and ¢n=¡1,whichcorrespond to
an‘almostequal’ splitofthesetof…rms(see Subsection4.4formoreonthisissue).
11 See theproofof Lemma1intheAppendix.
10
2.ForstrongMarshallianexternalities(µ>¹
µ),conditions(Da)and (Db)
cannotholdat thesametime.Thereisthusnobrand choice equilib-
riumwherethetwobrandsarebothchosen byapositivenumberof
…rms.Theonlyequilibria arewhereall …rmschoosethesame brand.
Again,theintuitionis simple:whenMarshallianexternalitiesarestrong
enough,theyovercomethe competitiveattractiveness ofthe‘smaller’market
and drivetheindustrytowardsconcentrationonasinglebrand.
4Brand choiceswithendogenouscosts
Wenowexplicitly considertheupstreamindustry inordertoendogenize
thedownstreaminputcosts.Usingthedownstreamreaction functionsde-
rivedabove,weobtaintheinversedemandsfortheintermediategoodtobe
transformedineach brand:
wa=1¡n+¢n+2
n+¢n
Ya¡°Yb,(3)
wb=1¡n¡¢n+2
n¡¢n
Yb¡°Ya.(4)
Whenasinglebrand isadopted byall downstream…rms(saybrand
k),we can usetheaboveanalysistoexpress theinversedemand forthe
intermediategoodtobetransformedin brand kaswk=1¡n+1
nYk:
4.1 Upstream…rms’ decisions
Letusconsiderthe casewherethetwobrandshavebeenselected bya
positivenumberof downstreamand upstream…rms.Since upstream…rms
havetheopportunitytoproduce theintermediateinputforeitherbrand,we
needaslightly di¤erentnotation.Let theupstreamindustrybepartitioned
intothetwosubsetsMaand Mb(Ma\Mb=;;Ma[Mb=M)according
tothe…rms’brand choices.Letxikdenotethequantityoftheintermediate
good produced bytheupstream…rmi(i=1;::: ;m)tobetransformedinto
brand k(k=a;b),and let thetotal quantity, Xk,oftheintermediategood
tobetransformedintobrand kbede…nedas
Xk´Xak+Xbk´X
i2Ma
xik+X
j2Mb
xjk.
11
Becauseofthelineartechnology inthedownstreamindustry, wehave
thefollowingmarketclearingconditions:Ya=Xaand Yb=Xb:Usingthe
latterconditionsand expressions(3)and (4),we canwritetheupstream
…rms’maximization programforatypicalupstream…rmi2Maasfollows:
max
xia;xib
¦u
i=waxia+(wb¡t)xib
=·1¡n+¢n+2
n+¢n¡xia+X¡i
a¢¡°¡xib+X¡i
b¢¸xia+
·1¡t¡n¡¢n+2
n¡¢n¡xib+X¡i
b¢¡°¡xia+X¡i
a¢¸xib:
Asimilarexpressionobtainsforthepro…t¦u
jofatypical…rmj2Mb.
Maximizing¦u
iwithrespect toxiaand xib,and ¦u
jwithrespect to
xjaand xjb,wegetasystemof four…rst-orderconditionsthatwesolve
usingthesymmetryofthemodel(at thesymmetric equilibrium,wehave
xia=xaa and xib=xab,8i2Ma,and xja=xbaand xjb=xbb,8j2Mb).
Accordingly, weget thefollowinginteriorsolutions(with¢m´ma¡mb):
xaa(¢n;¢m)=(n+¢n) (n¡¢n+2)[2+(m¡¢m)t]
2(m+1)[4(n+1)+(1¡°2)(n2¡¢2
n)]
¡°(n2¡¢2
n)[2¡(m¡¢m+2)t]
2(m+1)[4(n+1)+(1¡°2)(n2¡¢2
n)];
xab(¢n;¢m)=(n¡¢n) (n+¢n+2)[2¡(m¡¢m+2)t]
2(m+1)[4(n+1)+(1¡°2)(n2¡¢2
n)]
¡°(n2¡¢2
n)[2+(m¡¢m)t]
2(m+1)[4(n+1)+(1¡°2)(n2¡¢2
n)];
xba(¢n;¢m)=xab(¡¢n;¡¢m);and xbb(¢n;¢m)=xaa(¡¢n;¡¢m):
Thisinteriorsolution holdsprovidedthatxaband xbaareboth non-
negative, i.e., providedthattisnot too large.Itisindeedclearthat the
upstream…rmswill only …nd itpro…tabletoproduce fortheotherbrand
thantheonetheyhave chosen providedthattislowenough.Formally, xab¸
0,t·^
tab(¢n;¢m);and xba¸0,t·^
tba(¢n;¢m)=^
tab(¡¢n;¡¢m):
Wehavethat^
tabdecreaseswith¢nand increaseswith¢m. Wewishto
focusonsituationswheretheinteriorsolution holds(i.e., whereall upstream
…rms…nd itpro…tabletoproduce theintermediategood forthetwobrands)
whateverthebrand choicesmadeinthetwoindustriesat the…rststageof
thegame.Asu¢cientcondition forthistobetrueis
(INT)t·n(1¡°)+°
n+m[n(1+°)¡°]:
12
Weassumethat thisconditionismet throughout therestofthepaperwhich
provesparticularly usefulforspeci…cationofequilibriainthe…rststage(we
discuss theraison d’êtreofthisassumptionin Subsection4.4).
4.2 Equilibriumpro…ts
Becauseofthemarketclearingconditionsand ofthesymmetryofthedown-
stream…rms,wehaveya=(1=na)(maxaa +mbxba)and yb=(1=nb)(maxab+
mbxbb):Pluggingthelatterexpressionsinto(3)and (4),weget the equilib-
riumpricesfortheintermediateinput tobetransformedinthetwobrand:
wa(¢m)=2+(m¡¢m)t
2(m+1)and wb(¢m)=wa(¡¢m):
Notsurprisingly, weobservethatwaTwb,¢mS0,and that thedif-
ference (wa¡wb)decreaseswith¢m.Inotherwords,themoreupstream
…rms selectsomebrand,the cheapertheinput tobetransformedintothat
brand.Substitutingtheinputpricesintoexpressions(1)and (2),weobtain
the equilibriumquantitiesforthedownstream…rmsas:
ya(¢n;¢m)=(n¡¢n+2)[m(2¡t)+¢mt]
2(m+1)[4(n+1)+(1¡°2)(n2¡¢2
n)]
¡°(n¡¢n)[m(2¡t)¡¢mt]
2(m+1)[4(n+1)+(1¡°2)(n2¡¢2
n)];(5)
yb(¢n;¢m)=ya(¡¢n;¡¢m):(6)
Asabove,¦d
a(¢n;¢m)= [ya(¢n;¢m)]2and ¦d
b(¢n;¢m)= [yb(¢n;¢m)]2:
Usingthepreviousresults,we canalsotoderivethe equilibriumpro…tsin
theupstreamindustry:
¦u
a(¢n;¢m)=(n+¢n) (n¡¢n+2)[2¡(m¡¢m)t]2
2(m+1)2[4(n+1)+(1¡°2)(n2¡¢2
n)]
+(n¡¢n) (n+¢n+2)[2¡(m¡¢m+1)t]2
2(m+1)2[4(n+1)+(1¡°2)(n2¡¢2
n)]
¡°(n2¡¢2
n)[2¡(m¡¢m)t] [2¡(m¡¢m+1)t]
2(m+1)2[4(n+1)+(1¡°2)(n2¡¢2
n)];
¦u
b(¢n;¢m)=¦u
a(¡¢n;¡¢m)
AsfarasCondition(INT)holds,thepreviousexpressionsare easily
usedtodescribelimitcaseswhereall …rmsinsomeindustryselect thesame
13
brand. Wehavethusall necessaryelementsatourdisposalto analyze the
…rst-stagebrand choicesinthetwoindustries.
Before consideringthegeneralframework in depth, letusquickly relateit
tothetwospecialcasesanalyzedabove.First, ifwesett=0inthegeneral
setting,we comebacktothe…rstspecialcasewherewa=wb.Second,
exogenousMarshallianexternalitiescan beobtainedinthegeneralsetting
bymakingthefollowing assumptions:(i)m=xn,wherexisaninteger
largerthanorequalto 1,(ii)each downstream…rmcan dictateitschoice of
brand to a separatesetofxupstream…rms.Theseassumptionsimply that
¢m=x¢n.Asaresult,somemanipulationsallowtorewritewa(¢m)as
w¡µna,withw=(1+xnt)=(xn+1)and µ=(xt)=(xn+1).
Inthesetwospecialcases,downstream…rmsare eithersplittingequally
orconcentrating onasinglebrand. Wewantnowtoinvestigatewhether
thesetofequilibriaexpandswhen upstream…rmsarefree tochoosethe
brand theywant toproduce for,and whentheadaptationcosttispositive
(althoughsmall enough forCondition(INT) tohold).Theanswertothis
questionwill beshowntodepend onthebalance betweenthefollowingtwo
con‡ictingforces. Ontheonehand,downstream…rmshaveanincentiveto
choosedi¤erentbrandsinordertosoftencompetitiononthe…nalmarket;
stabilitythenrequiresthat theysplitequally betweenthetwobrands. On
theotherhand,brand choicesalsodeterminetherelativeprice oftheinput;
this second force drivesdownstream…rmstomodeltheirbehavioronthat
oftheupstream…rms.Since upstream…rmstend todoexactly thesame,
thisforce drivesthemarket towardsunanimousadoptionofasinglebrand.
Inwhatfollows,welookforNashequilibriainthebrand choice game,
i.e., forpartitionsofthetwosetsof …rms suchthatno…rm–eitherupstream
ordownstream–hasanincentivetounilaterally switch brands. We establish
the existence ofa Nashequilibriumand drawinstructiveresultsabout the
characterizationofthesetofNashequilibria.
4.3 Existenceand characterizationofequilibria
Thebrand choice gamemightleadtotwotypical industrypatterns:eitherno
industry isconcentratedonasinglebrand (apatternreferredtohereafter
asdisper