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Abstract

This is a successive oligopoly model with two varieties of a final product. Downstream firms choose one variety to sell on a final market. Upstream firms specialize in the production of one input specifically designed for one variety, but they also produce the input for the other variety at an extra cost. We show that as more downstream firms choose one particular variety, more upstream firms specialize in the input specific to that variety, and vice-versa. Multiple equilibria may result, and the softening effect of product differentiation on competition might not be strong enough to induce maximal differentiation.
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 Belleamme¤EricToulemondey
October2000
Abstract
Thisisasuccessiveoligopoly modelwithtwobrands.Each down-
streamrmchoosesonebrand tosell ona…nalmarket.Theupstream
rms specialize intheproductionofoneinputspecically designed
fortheproductionofonebrand,but theyalsoproduce theinputfor
theotherbrand atanextracost. Weshowthatwhenmoredown-
streamrmschooseonebrand,moreupstreamrmswill specialize in
theinputspecictothatbrand,and vice versa.Hence,multiple equi-
libria arepossibleand thesofteninge¤ectof brand di¤erentiationon
competitionmightnotbestrongenoughtoinduce maximaldi¤eren-
tiation.The existence ofequilibria and theirwelfareperformance are
alsoexamined.
JELclassicationcodes: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 forScienticResearch(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,dAspremontetal. (1979)establishthatrmschoosemaximal
di¤erentiationinthelinearcitymodelwithquadratictransportationcosts.
Thispredictionisechoedinmostmarketingtextsconcerningmarketseg-
mentation,whichrecommend …rmstodi¤erentiatetheirproducts.
There exist,however,forcesthatopposemaximal(orany)productdif-
ferentiation.FollowingTirole(1988),theseforcescan beputintothree
categories.Arst,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 byconsumersmayencouragermsto gather.Finally, thethird
categoryreferstopositive externalitiesthatinduce rmstolocatenearone
another.ThenowstandardclassicationofMarshallianexternalitiesisbe-
tweenlocalizationeconomies(whichrefertothebenetsgenerated bythe
proximityof …rmsproducingsimilargoods),and urbanizationeconomies
(whichaccountforall theadvantagesassociatedwiththeoverall levelof
activityprevailinginaparticulararea).Thegeneral ideaisthat,forgiven
inputs,theoutputofanindividualrmislargerthelargeristheaggregate
outputofotherrmsproducingthesamegoodinthesamelocale.
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 manydownstreamrms.Inadditiontothis
demand linkagebetweenindustries,thereisalso a costlinkage.Firms
inthedownstreamindustrywill havelowercostsifthey locatewhere
therearerelatively manyupstreamrmstheysavetrade 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 costlinkagesidentied byVenables.
Specically, we considertwovertically relatedindustrieswiththefollow-
ingfeatures.Thedownstreamindustryproducesa…nalproduct thatcan be
marketed undertwopossibledi¤erentiated brands.Intuitively, oneunder-
standsthat,absentanyotherconsideration,thisindustry isdriventowards
asituationwhererms splitequally (oralmostequally’ ifthereisanodd
numberofthem)betweenthetwobrandsinordertosoftencompetition
(thisisthetranslationofmaximaldi¤erentiationinoursetting).
However,thisintuitionmayprovewrongwhenwealsotakeinto account
an upstreamindustrythatproducesanessential inputforthedownstream
industry, and thatalsohastotakeits stand onthetwobrands.Thereasonis
thefollowing.Althoughtheuseofso-calledexiblemanufacturingsystems
(FMS)becomesincreasingly widespread,1economiesofscalearestill present
inmany industries.Itisthusreasonableto assumethat theinputismore
1Asexplained byNormanand Thisse(1999),the essence ofaFMSisthatitallows
rmstocustomize theirproductstotherequirementsofheterogeneouscustomersatlittle
ornocostpenalty.
3
costly toproduce whenithastot twodi¤erentiated brandsratherthan
asingleone.Thisinduces supplierstoselect thebrand fromwhichthey
want to “getcloser.Specically, supplierswill incurspecic coststo adapt
theintermediateproduct tothespecicneedsofasinglebrand ofthe…nal
product.Nevertheless,suchspecicinvestmentdoesnotcompletely prevent
themto alsoconformtheirinput totheotherbrand:depending onthe
demand expressed bythebuyersofthisotherbrand,theymightindeed …nd
itprotabletoincurthe extracost toservethem.Adoptingtheterminology
ofEatonand Schmitt (1994),we cansayequivalently thatupstreamrms
choosetodeveloponebasicproduct(thattsoneparticularbrand ofthe
…nalgood),and then produce,withanextracost,onevariationofthat
basicproduct (inordertot theotherbrand ofthe…nalgood).2
Reconsideringbrand choicesbydownstreamrmsinsuchabroadercon-
text,we canconjecturethat these choiceswill bedriven notonly bycom-
petitiononthe…nalmarket,butalsobycostconsiderationslinkedtothe
choicesmade byrmsintheupstreamindustry.In particular,downstream
rmswill bemoreattracted byabrand thatalargenumberof upstream
rmshave chosentoconformwith.Since, inturn,upstreamrmsaremore
likely toconformwithabrand thathasbeenselected byalargenumber
of downstreamrms,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 downstreamrmsthatchooseone
particularbrand increaseswiththenumberof upstreamrmsthatspecialize
intheinput thatis specictothatbrand,and vice versa.Thisraisesthe
possibilityofmultiple equilibria.Forinstance, itispossibletohaveanequi-
libriumwithoutanydi¤erentiation(upstreamand downstreamrms select
thesamebrand),oranequilibriumwithmaximaldi¤erentiation(rms split
equally between both brands)oranintermediate equilibrium(mostbutnot
allupstreamand downstreamrmschoosethesamebrand).Depending on
parameters,someofthese equilibria arepreferred byrmswhilethe 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 bytheupstreamrmsmightfollowfromothersourcesthanthe
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
enablethemtofulll therequirementsofmoretightly connectedand inte-
gratedinformation networks.Similarly, casestudiesinMarcussen(1996)
showthatEDIcapability in‡uencesthebuyingdecisionaboutwhich up-
streamrmswill be‘insuppliers(currently chosensuppliers)— and which
ofthe‘insupplierswill getordersforspecicitems.Iftwoleadingcompet-
ingsuppliersboth developEDIcapability, competitiveparitybetweenthe
twormsismaintained.However,theEDIcapable‘insuppliersstand to
winat the expenseof boththeinsupplierswithoutEDIcapabilityand 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
papersconsiderinsteadthatrms specialize intheproductionofasingle
component,anassumptionthatcomesclosertotheissueatstakehere(we
3EDIisthedirectcomputer-to-computerexchangeofinformationstoredinstandard
formatted business documents,suchasinvoices,billsoflading,purchaseorders,etc.,
amongrms.
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-
bilitycosts),theyshareourconcern foranalyzingthelocationdecisions
ofthesoftwarerms(theupstreamrmsinourterminology).5However,
itmustbestressedthat they leaveasidethelocationdecisionsofthe
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,asetNofnrmsproduce someproduct tobesold
ona…nalmarket.Theyall havetochooseunderwhich brand tomarket
thisproduct.Twohorizontally di¤erentiated brands,notedaand b,are
available. Once thermshavemadetheirchoice,theindustry ispartitioned
intothetwosubsetsNaand Nb(Na\Nb=;;Na[Nb=N). Weadopt
thefollowingnotation.Letyikdenotethequantityof brand k(k=a;b)
produced bysomedownstreamrmi(i=1;::: ;n),Yk´Pi2Nkyik,the
total quantityproducedof brand k,and Y¡i
k´Pj2Nk;j6=iyjk.
Withinthe existingtheory, therearetwobasicapproachestomodelhor-
izontaldi¤erentiation. Ontheonehand,thespatialmodelssuchasthe
5Forinstance,Matutesand Régibeau(1989)analyze thesituationwheretworms
sell one component (e.g., software)ofatwo-componentsystem,and where consumers
havealreadybought therstcomponent (e.g., thepersonalcomputer)sothatseveral
independentmarketsforsoftwareare created.Firmshavethustochoosethemarket(s)
that theywill serve.Theyalsodecidewhethertoproduce di¤erentsoftwareforeach
marketortosell softwarethatcan beusedwith bothtypesofcomputers.Churchand
Gandal(1992)developamodelwheretherearetwoincompatiblehardwaretechnologies,
anendogenouslydetermined numberofsoftwarerms,and consumerswhovaluesoftware
variety;theyinvestigatethedecisionofasoftwarermconcerningwhich networktojoin.
6
Hotellingslinearcityand Salopscircularcitymodels–generatemarketde-
mandsby integrating overconsumerswith di¤erent tastes. Ontheother
hand,themodelsintheSpence-Dixit-Stiglitztradition deriveademand
systemfordi¤erentiated productsfromtheutilityfunctionofarepresen-
tative consumerwithatasteforvariety.6Becausetherstapproachis
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
²whenthendownstreamrmschoosetoproduce thesamebrand k
(meaningthatNk=Nand that theotherbrand isnotproduced),
price onthemarketforbrand kisgiven bypk=1¡Yk;
²whenthedownstreamrmschoosetoproduce di¤erentbrands,prices
onthemarketaregiven bypa=1¡Ya¡°Yband pb=1¡Yb¡°Ya
intheregionofquantitieswherepricesarepositive,where0·°<1
isthedegree of di¤erentiation between brandsaand b.
Weassumethat thedownstreamrmsproduce the…nalproductby
transforming a singleintermediategoodona one-for-onebasis.Themarginal
costofthedownstreamrmsonly consistsoftheprice paid fortheinterme-
diategood(otherspecic costsareassumedtobe zero).
Theintermediategoodis supplied bytheupstreamindustry,whichcon-
sistsofasetMofmrms.Theproductiontechnologyfortheintermediate
goodassumes somedegree ofbrand specicity’. Thatis,each upstream
rmhastodesignitsproduction process inconformitywithaspecicbrand
ofthe…nalproduct (wesay inthesequelthat theychooseto “produce for
brand aorforbrand b).Asaresult, iftheintermediategoodistobetrans-
formedintothatspecicbrand,themarginalcostof productionisequalto
6Forarecentdiscussionofthesetwo approaches(and thepresentationofanovel
integrativeapproach),see Andersonand dePalma(2000).
7Theinversedemand scheduleisderivedfromthequadraticutilityfunctionofarepre-
sentative consumerwhichexhibitsloveforvariety;forthe exactderivation,see theproof
ofProposition7intheappendix.
7
c,whereasifithastobetransformedintotheotherbrand,anextraadap-
tationcosttistobeincurred,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
upstreamrmsdecidebeforedownstreamrms,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.Typicalrms
i2Naand j2Nbrespectively face thefollowingmaximization programs:
8Aconcrete examplethattsourassumptionsiswhatLevitt (1980)reportsabouta
specictypeofsteel, theso-calledNo.302,72-inch,hot-rolledstrip’. Notall generic
productsarethesame.(...)Becauseofslightdi¤erencesamong automobile company
manufacturingprocesses,onesuppliers“302” may, infact,bebetterthananothers.
Onemill’s302 maytake certaincoatingsmore easilyorquicklythananothers.One
suppliermayll 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.
Therst-orderconditionsforprotmaximizationyieldthefollowing
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,weidentifyapartitionofthermsto¢n´na¡nbwhere
nk=#Nk;k=a;b.Fora given¢ninherited fromtherststage,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).9Equilibriumprots
aresimply equaltothesquareofthe equilibriumquantities:¦d
a(¢n)=
[ya(¢n)]2and ¦d
b(¢n)= [yb(¢n)]2:
Letusnowturntotherststageofthegamewhererms simultaneously
choosewhich brand theywant toproduce.Inthisgame,a Nashequilibrium
with¢n6=§nischaracterized bytwoconditions,ensuringthatnorm…nds
itprotabletoswitch 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)satises(Da)(resp.(Db))only. Wedevelopthese conditions
intwospecic examples.
3.1Constantand identical costs
Intherstexample,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 downstreamrmsface thesame costofproduction,the
unique brandchoice equilibriumiswhererms splitequallybetweenthetwo
brands ( ¢n=0).10 Theintuitionis simple:whenthesplitisunequal, rms
onthe‘largermarkethaveanincentivetomovetothesmallermarket
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
1forjusticationsofsuchassumption).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
analmostequal’ splitofthesetofrms(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 ofthesmallermarket
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 downstreamrms(saybrand
k),we can usetheaboveanalysistoexpress theinversedemand forthe
intermediategoodtobetransformedin brand kaswk=1¡n+1
nYk:
4.1 Upstreamrms’ decisions
Letusconsiderthe casewherethetwobrandshavebeenselected bya
positivenumberof downstreamand upstreamrms.Since upstreamrms
havetheopportunitytoproduce theintermediateinputforeitherbrand,we
needaslightly di¤erentnotation.Let theupstreamindustrybepartitioned
intothetwosubsetsMaand Mb(Ma\Mb=;;Ma[Mb=M)according
tothermsbrand choices.Letxikdenotethequantityoftheintermediate
good produced bytheupstreamrmi(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
rmsmaximization programforatypicalupstreamrmi2Maasfollows:
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:
Asimilarexpressionobtainsfortheprot¦u
jofatypicalrmj2Mb.
Maximizing¦u
iwithrespect toxiaand xib,and ¦u
jwithrespect to
xjaand xjb,wegetasystemof fourrst-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
upstreamrmswill only …nd itprotabletoproduce 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 itprotabletoproduce theintermediategood forthetwobrands)
whateverthebrand choicesmadeinthetwoindustriesat therststageof
thegame.Asu¢cientcondition forthistobetrueis
(INT)t·n(1¡°)+°
n+m[n(1+°)¡°]:
12
Weassumethat thisconditionismet throughout therestofthepaperwhich
provesparticularly usefulforspecicationofequilibriaintherststage(we
discuss theraison dêtreofthisassumptionin Subsection4.4).
4.2 Equilibriumprots
Becauseofthemarketclearingconditionsand ofthesymmetryofthedown-
streamrms,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 equilibriumquantitiesforthedownstreamrmsas:
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 equilibriumprotsin
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 comebacktotherstspecialcasewherewa=wb.Second,
exogenousMarshallianexternalitiescan beobtainedinthegeneralsetting
bymakingthefollowing assumptions:(i)m=xn,wherexisaninteger
largerthanorequalto 1,(ii)each downstreamrmcan dictateitschoice of
brand to a separatesetofxupstreamrms.Theseassumptionsimply that
¢m=x¢n.Asaresult,somemanipulationsallowtorewritewa(¢m)as
w¡µna,withw=(1+xnt)=(xn+1)and µ=(xt)=(xn+1).
Inthesetwospecialcases,downstreamrmsare eithersplittingequally
orconcentrating onasinglebrand. Wewantnowtoinvestigatewhether
thesetofequilibriaexpandswhen upstreamrmsarefree tochoosethe
brand theywant toproduce for,and whentheadaptationcosttispositive
(althoughsmall enough forCondition(INT) tohold).Theanswertothis
questionwill beshowntodepend onthebalance betweenthefollowingtwo
conictingforces. Ontheonehand,downstreamrmshaveanincentiveto
choosedi¤erentbrandsinordertosoftencompetitiononthe…nalmarket;
stabilitythenrequiresthat theysplitequally betweenthetwobrands. On
theotherhand,brand choicesalsodeterminetherelativeprice oftheinput;
this second force drivesdownstreamrmstomodeltheirbehavioronthat
oftheupstreamrms.Since upstreamrmstend todoexactly thesame,
thisforce drivesthemarket towardsunanimousadoptionofasinglebrand.
Inwhatfollows,welookforNashequilibriainthebrand choice game,
i.e., forpartitionsofthetwosetsof …rms suchthatnormeitherupstream
ordownstreamhasanincentivetounilaterally switch brands. We establish
the existence ofa Nashequilibriumand drawinstructiveresultsabout the
characterizationofthesetofNashequilibria.
4.3 Existenceand characterizationofequilibria
Thebrand choice gamemightleadtotwotypical industrypatterns:eitherno
industry isconcentratedonasinglebrand (apatternreferredtohereafter
asdisper