StrategicIncentivesWhenSupplyingtoRivals
withanApplicationtoVerticalFirmStructure
SergeMoresiandMariusSchwartz*
November29,2015
Abstract
Weconsideraverticallyintegratedinputmonopolistsupplyingtoadifferentiated
downstreamrival.Withlinearinputpricing,atthemarginthefirmunambiguouslywants
therivaltoexpand—unlikestandardoligopolywithnosupplyrelationship—foreither
CournotorBertrandcompetition.Withatwo‐parttarifffortheinput,thesameresultholds
ifdownstreamchoicesarestrategiccomplements,butisreversedforCournotwith
strategicsubstitutes.Weanalyzeverticaldelegationasonemechanismforinducing
expansionorcontractionbytherival/customer.
JELCodes: L13,D43,L14,L22
Keywords: StrategicCompetitionagainstCustomers,VerticalDelegation
*Moresi:CharlesRiverAssociates,Inc.,WashingtonDC2004<[email protected]>.
Schwartz:DepartmentofEconomics,GeorgetownUniversity,WashingtonDC20057
<[email protected]>.WethankAxelAnderson,YongminChen,JayEzrielev,JustinJohnson,
NadavLevy,PatrickRey,BillRogerson,SteveSalop,YossiSpiegel,JohnVickers,AleksYankelevich,
andparticipantsattheIIOC2015conference,CRESSE2015conferenceandSearleCenter2015
conference.Anyerrorsareoursalone.
1.
Introduction
Verticallyintegratedfirmsoftensupplyinputstootherfirmswithwhomtheycompeteina
downstreammarket.Tocitejustafewexamples,Qualcommmakeschipsusedin
smartphonesandlicenseskeypatentstorivalchipmanufacturers(BenoitandClark,
2015);SamsungsuppliescomponentsforiPhonesandproducescompetingdevices;
Comcast‐NBCUsuppliesprogrammingtovideodistributorsandcompeteswiththemin
videodistribution(Rogerson,2013);andtheUSPostOfficesupplieslastmiledistribution
servicesofpackagestoprivatecompetitorssuchasFedExandUPS(Panzar,2015).
Tougherbehavioror“expansion”byarival/customer—anoutputincreaseorprice
decrease—thenhasopposingeffectsontheintegratedfirm’sprofit:downstreamprofits
fall,thecompetitioneffect,butinputsalesandupstreamprofitsrise,thesupplyeffect.Atthe
margin,wouldtheintegratedfirmgainorlosefromexpansionbyitsrival/customer?
Besidestheoreticalinterestthequestionhaspracticalrelevance.Forexample,the
firmmaybeabletoelicitthedesiredchangebymakingobservablepre‐commitmentsthat
signalitsownstrategicpostureindownstreamcompetition,asdiscussedintheliterature
onstrategiccommitmentsinoligopoly(seeShapiro,1989).There,afirmmayadoptatough
orsoftposturedependingonwhetherthecompetitivechoicevariablesarestrategic
substitutesorstrategiccomplements(FudenbergandTirole,1984;Bulow,Geanakoplos
andKlemperer,1985)—butthegoalthroughoutistoinducesofterbehaviorbytherival(s).
Inoursetting,however,thedownstreamrivalisalsoaninputcustomer.
Specifically,weconsideranunregulatedandverticallyintegratedinputmonopolist
thatpreferstosupplytheinputalsotoadownstreamfirmsellingadifferentiated
substituteproduct.DownstreamcompetitionmaybeCournotorBertrand,andwedonot
imposeafunctionalformondemand.Despitethetradeoffbetweendownstreamprofitsand
inputprofits,weareabletocharacterizetheintegratedfirm’sincentiveregardingits
rival/customerstartingattheequilibriumcontractunderfairlygeneralconditions.
Whentheinputissoldunderlinearpricing,atthemargintheintegratedfirm
necessarilybenefitsfromexpansionbytherival/customer(Proposition1).Thissharp
resultholdswhetherdownstreamcompetitionisinpricesorquantitiesandwhetherthese
1
variablesarestrategicsubstitutesorcomplements.Inhindsight,thelogicissimple.
Loweringtheinputpricewillalwayssacrificerevenuefrominframarginalinputsalesand
induceexpansionbytherival/customer,whoseeffectontheintegratedfirm’sprofitis
ambiguousaprioriduetotheopposinginputsupplyanddownstreamcompetitioneffects.
Attheprofit‐maximizinginputprice,however,expansionalonemustincreasethe
integratedfirm’sprofit,otherwisethefirmwouldgainbyraisingtheinputpriceand
boostingrevenuefrominframarginalinputsales.OurresultgeneralizesafindingbyArya,
MittendorfandYoon(2008)forCournotcompetitionwithlinear(differentiated)demands,
andtheproofrevealstheunderlyingforces.
Wealsoanalyzethecasewheretheinputissoldunderatwo‐parttariffandthe
integratedfirmextractstheotherfirm’sprofitviathefixedfee.Nonetheless,theintegrated
firmdoesnotmaximizeindustryprofit—becauseitcannotcommittotherival/customer
regardingitsowndownstreamchoice—andwillstillbenefitatthemarginfromexpansion
orcontractionbytheotherfirm.Butdeterminingwhichoftheseitprefersismoreintricate
thanunderlinearpricing,andthespecificsofcompetitionnowmatter.Thepreviousresult
persistswithBertrandorCournotcompetitionifdownstreamchoicesarestrategic
complements(Proposition2).However,forCournotandthe“normal”caseofstrategic
substitutes,theresultisreversed:expansionbytherival/customerwouldharmthe
integratedfirm;forBertrandandstrategicsubstitutestheeffectisambiguous.
Thelogicunderlyingthesepatternsissubtle,andthefollowingisonlyasketch.As
before,loweringtheinputpricewouldinduceexpansionbytherival/customer,whose
impactontheintegratedfirm’sprofitstartingattheequilibriumcontractmustbeinferred
fromtheothereffectofloweringtheinputprice.Buttheothereffectnolongerinvolves
reducedrevenuefrominframarginalinputsales,becauseofanoffsettingincreaseinthe
fixedfee.Instead,aneweffectarises:loweringtheinputpricemaysignaleithertougheror
softercompetitionbytheintegratedfirmdownstream(asdiscussedshortly)and,
therefore,alterthemaximalfixedfeethatcanbechargedtotheotherfirm.Ifitsignals
toughercompetition,thenloweringtheinputpricewouldbecostlytotheintegratedfirm
byreducingthemaximalfixedfee.Inthiscase,attheequilibriumcontract,theintegrated
firmmustgainfromexogenousexpansionbytherival/customer(sinceloweringtheinput
pricetoinduceexpansionwouldbecostly).Theoppositeholdsif,instead,loweringthe
2
inputpricesignalssoftercompetitionbytheintegratedfirm:itthenmustlosefrom
exogenousexpansionbytherival/customer.
Whetheralowerinputpricesignalstougherorsoftercompetitionfromthe
integratedfirmwilldependonwhetherdownstreamvariablesarestrategicsubstitutesor
complements,andwhethercompetitionisCournotorBertrand.Interestingly,thelatter
distinctionmattersindependently,becausetheinputpriceaffectstheintegratedfirm’s
shadowmarginalcostofitsdownstreamoutput—henceaffectstherival/customer’s
expectationofdownstreamaggressiveness—onlyunderBertrand.
Weanalyzeonepotentialmechanismtheintegratedfirmmayusetoelicitexpansion
orcontractionbyitsrival/customer:verticaldelegation.Thefirmcreatesanautonomous
downstreamunitthatsetsdownstreampriceorquantitytomaximizeitsobjectiverather
thanintegratedprofit(alternatively,controlofthedownstreamunitisspunofftooutside
investorswhilethesupplierretainsmajoritypassiveownership),andchargesitaninput
priceobservabletotherival/customer.1Becausetheinputpriceisnottreatedasapurely
internaltransfer,itcanservetosignalthedownstreamunit’scompetitivepostureand
(indirectly)altertherival/customer’sbehavior.
Focusingonlinearinputpricing(forreasonsweshallexplain),weapplyProposition
1toshowthattheintegratedfirmcangainbysuitablyadjustingtheinputpricetoits
downstreamunitrelativetoitsshadowmarginalcostundertheoriginalcentralization
regime.Itcangainbyraisingtheinputpriceifcompetitioninvolvesstrategicsubstitutes—
tosignalcontractionbyitsdownstreamunitandinduceexpansionbytherival/customer—
andloweringtheinputpricewithstrategiccomplements(Proposition3).Interestingly,
underBertrandcompetition,theintegratedfirmcangainbyreducingitspricetoinducea
pricereductionbytherival/customer,eventhoughtherival’soutputislikelytofall,andwe
explainwhythisisconsistentwiththelogicofProposition1.Anotherinterestingfeature
underBertrandandstrategiccomplementsisthatdelegationalsocreatesanincentiveto
raisetheinputpricetotherival.
1Wethereforeabstractfromissuesofunobservabilityandrenegotiation(e.g.Katz,1991;Caillaud
andRey,1995).WewilldiscusstheseissuesbrieflyinSection4,alongwiththeplausibilityofour
verticaldelegationandtherelationshiptorelatedliterature.
3
Thepaperisorganizedasfollows.Section2presentsthemodelandaddresses
linearinputpricing.Section3analyzestwo‐parttariffs.Section4appliestheanalysisto
verticaldelegation,andSection5concludes.TheAppendixprovidestechnicaldetailsanda
numericalexampleoftheapplicationtoverticaldelegation.
2.
TheSettingandLinearInputPricing
Aninputmonopolist,firm1,suppliestoitsdownstreamunitandtoanindependent
downstreamrival,firm2,asettingsometimesdescribedaspartialforwardintegrationor
dualdistribution.Thefirmsproduceimperfectsubstitutes,andwithenoughdifferentiation
firm1indeedwillprefertosupplyalsotofirm2ratherthanforecloseitentirely.
Downstreamchoicevariables and areeitherper‐unitquantities( and )orprices
( and ),therebyallowingBertrandorCournotcompetitiondownstream.Tostreamline
theexposition,wedefine  forCournotand 
forBertrand;thus,anincreasein
representsgreater“aggressiveness”or“toughness”becauseitreducesdownstream
profitoftheotherfirm.Thetimingisasfollows.First,firm1setsaper‐unitinputprice
tofirm2.Then,firms1and2simultaneouslysetdownstreamvariables
,
,
consumerspurchase,andfirm2paysforfirm1’sinput.
Eachfirmrequiresoneunitofinputperunitofoutput,andweuse
forbothk’s
outputandinputamountsconditionalonthedownstreamvariables.Firm2chooses to
maximizeitsprofit
;
;
,
,andfirm1chooses
;
tomaximizeitstotalprofits:
.
(1)
Here, isfirm1’smarginalcostofproducingtheinput,assumedconstantovertherelevant
range;
;
isfirm1’sprofitfromitsoutputsales;and
isitsprofitfrom
inputsalestofirm2.2Thisisastandardrepresentationofbehaviorbyanintegratedfirm
thatalsosuppliestoarival.3Alltherelevantfunctionsareassumeddifferentiable.
2Forconvenience,wewilloftensuppressthethirdargumentof
,theexogenousmarginalcost .
3Underthisrepresentation,Chen(2001)comparespartialintegrationtonointegration.Arya,
Mittendorf,andSappington(2008)comparetheoutcomesunderpartialintegrationwhen
downstreamcompetitionisBertrandorCournot.
4
Notethatiffirm1didnotsupplyinputstofirm2,orwasaregulatedmonopolist
thatmustsupplytofirm2atcost(i.e.
itsownsales,i.e.
aggressive,
;
⁄∂
;
),thenfirm1’sprofitwouldcomesolelyfrom
.Inthatcase,firm1wouldpreferthatfirm2beless
0,i.e.reduceoutputwhendownstreamcompetitionisCournot,or
raisepriceifcompetitionisBertrand,soastoincreasethedemandfor1’sproduct.
),sofirm2’s
Inoursetting,firm1sellsinputstofirm2atamarkup(i.e.
downstreamchoice, ,hastwoopposingeffectsonfirm1’sprofits:
where
Π1
2
2
2
,
(2)
/ ∂ ,isthe“downstreamcompetition”effectdiscussedpreviouslyand
⁄∂ isthe“inputsupply”effectwhichtypicallyrunsintheoppositedirection.
Forexample,underdownstreamBertrandcompetition,apricerisebyfirm2increasesfirm
1’sprofitintheoutputmarket,butlowersitsprofitintheinputmarketbecausefirm2will
produceless,andhencepurchasefewerinputs,whenitraisesitsoutputprice.Proposition
1belowwillshowthat,inequilibrium,theinputsaleseffectalwaysdominates,andthusthe
integratedfirmwantstherival/customertobecomemoreaggressive.
Wemakethefollowingassumptions:
Assumption1.Foranyinputprice
,thereexistsauniquedownstreamNashequilibrium
∗
inpurestrategies,withthechoiceofdownstreamrivalk(k=1,2)denoted
correspondingoutputlevel(andinputlevel)denoted
UnderCournot,
Assumption2.Let
and
;
∗
∗
≡
∗
,
∗
.
.
denotefirm1’s“reactionfunction”foritsdownstream
choice,i.e.thevalueof thatmaximizes
;
∗
andits
;
,anddenotefirm2’sreactionfunctionby
.Overtherelevantrange,afirm’sreactionfunctioniseitherstrictlydecreasingin
therival’schoiceorstrictlyincreasing:
⁄
⁄
<0(strategicsubstitute)or
(strategiccomplement),withslopesmallerthanunityinabsolutevalue,|
⁄
>0
|<1.
Assumption3.Anincreaseintheinputpricemakesfirm2lessaggressive—thatis,reduce
outputunderCournotcompetitionandincreasepriceunderBertrandcompetition(i.e.
5
d
∗⁄
d
0,recallingthat  forCournotand 
Firm1sets
tomaximize
∗
profit‐maximizingchoice,and
∗
∗
,
∗
≡
forBertrand).4
∗
,
;
.Let
∗
denotethe
theresultingequilibriumdownstreamchoices.We
nowstateasharpresult:
Proposition1.Assumethatfirm1sellstheinputtofirm2underlinearpricing.Atthe
equilibriumoutcome
,
⁄
aggressive,i.e.
∗
,
,
∗
,
∗
,firm1wantsfirm2tobecomemore
0.
Proof.Thefirst‐ordercondition(FOC)withrespectto
∗
∗
≡
∗
0.
TheFOCfor inthedownstreamcompetitionimplies
⁄
∗
is
(3)
⁄
0,and(1)implies
,sothat(3)canberewrittenas
∗
Theassumptions
∗
∗
0andd
.
(4)
∗⁄
d
0imply
⁄
0attheequilibriumpoint.
Proposition1identifiesoppositeincentivestothoseinastandardduopolysetting
wherefirm1doesnotsupplyfirm2.There,holding constant,firm1wouldgainfroma
riseinfirm2’spriceunderBertrandcompetitionorafallinfirm2’soutputunderCournot.
Thisistheusual“softeningdownstreamcompetition”effect.Itispresentalsohere,but
dominatedbyanopposing“inputsupply”effect.Thelogicisshownin(4).Giventhatfirm1
sets
attheprofit‐maximizinglevel,areductionin
wouldhavethefollowingequalbut
oppositeeffects:profitfrominframarginalinputsaleswouldfall,implyingthatprofitmust
4Anincreaseintheinputprice(
)hasa“directcosteffect”inthesensethatitmakesfirm2less
aggressiveholdingfirm1’sstrategicvariable( )constant.WithBertrandcompetition,itcanalso
havean“indirectstrategiceffect”thatworkseitherinthesamedirectionasthecosteffectorinthe
oppositedirection.Inthelattercase,weassumethatthecosteffectdominates.SeetheAppendix.
6
risefromincreasedinputsales—afterincorporatingthelossinfirm1’sdownstreamprofit
causedbyfirm2’soutputexpansion(see(2)).5Thus,holding
constantat
∗
and constantat ∗ ,theintegratedfirmwouldgainonbalance,despitethelossindownstream
profits,iffirm2wereexogenouslytoincreaseitsinputpurchasesanddownstreamsales.
(WeshowinSection4thatverticaldelegationcanbeusedtoinducethedesiredchangein
,andwealsodiscussinformallyotherpossiblemechanisms.)
3.
Two‐PartTarifffortheInput
Whentheintegratedfirmsellstheinputtofirm2usingatwo‐parttariff—i.e.apair
where isafixedupfrontfee—itstotalprofitsare
;
;
,
,
,
,where
;
isgivenin(1).Aswewillexplain,theintegratedfirmcannotachievethevertically
integratedmonopolyprofitdespitetheuseofatwo‐parttariffand,therefore,atthe
equilibriumoutcome,itwouldstillbenefitfromasuitablemarginalchangeinfirm2’s
outputorprice.
Iffirm2acceptsthetwo‐parttariffoffer
∗
downstreamoutcomeisgivenbythefunctions
Denotefirm2’sprofitgrossofthefixedfeeas
,
(asitwillinequilibrium),the
and
Let
and
∗
≡
∗
∗
∗
,where
aswithlinearpricing.
∗
∗
equilibrium,firm1willextractfirm2’sprofitbysetting
tomaximize
∗
∗
,
∗
∗
;
.In
.Therefore,itsets
isthesameaswithlinearpricing.
denotetheprofit‐maximizingtwo‐parttariff,and
,
denote
theresultingequilibriumdownstreamchoices.
Attheequilibrium,firm1mightbenefitfromfirm2beingmoreaggressive,aswith
linearpricing,orfromfirm2beinglessaggressive.Whichcaseprevailswilldependonthe
specificsofdownstreamcompetition:strategicsubstitutesorcomplements,andCournotor
Bertrandcompetition.Toseewhythelatterdistinctionalsomatters,notethatwhenafirm
sellsaninputtoarival,itsrelevantmarginalcostofsupplyingtheinputtoitsdownstream
5Equivalently,whatstopstheintegratedfirmfromraisingtheinputpriceabovetheoptimallevel—
despiteincreasedprofitoninframarginalinputsales—isthatreducedinputsaleswouldlowerthe
integratedfirm’sprofit,andbyenoughtooutweighthegaindownstream.
7
unitincludesboththeresourcemarginalcostcandanopportunitycostfromreducedinput
salestothedownstreamrival(e.g.Chen,2001;Sappington,2005).Thisopportunitycostis
positiveunderBertrandcompetition,butzerounderCournot,asshownnext.6
Define
≡
;
asfirm1’sprofitfromdownstreamsalesgrossof
theresourcecostoftheinternallyprovidedinput.Then,using(1),firm1’stotalprofitgross
ofthefixedfeecanbere‐writtenas
;
,
,hence:
 Q2 / x1  Q1
V G 

.

 c  ( w2  c) 
x1 x1 
 Q1 / x1  x1
(5)
Theterminsquarebracketscanbeinterpretedastheintegratedfirm’sshadowmarginal
costofprovidingtheinputtoitsdownstreamunit.Weexpressitmorecompactlyas where
,
≡
⁄
⁄
≡
,
istheinputdiversionratio—i.e.decreasedinputsalestofirm2per
extraunitofinputto1—andtheopportunitycostis
Notethat
(6)
multipliedbytheinputmargin.7
0withBertrandcompetition,sincefirm1’sdownstreamexpansion
inducedbycuttingpricewilldisplacesomesalesoffirm2;but
0withCournot,since
firm1thentakesfirm2’soutput,andthusalsoitsinputpurchases,asgivenwhensetting
itsowndownstreamoutput.Therefore,raisingtheinputpricetofirm2willincreasethe
shadowmarginalcostoffirm1’sownoutputunderBertrandcompetition,butnotaffectit
underCournot.Asaresult,firm1’sdownstream“reactionfunction”
;
),definedin
6Arya,Mittendorf,andSappington(2008)invokethisdistinctiontoshowthat,unlikeinstandard
duopoly,BertrandcompetitiondownstreamcanyieldhigherpricesthanCournotwhenapartially
integratedinputmonopolistsellsalsotoadownstreamrival,becausethemonopolistinternalizesa
higheropportunitycostunderBertrandthanunderCournot.
7MoresiandSalop(2013)analyzeunilateralpricingincentivesinverticalmergersandshowthata
verysimilaropportunitycostcreatesupwardpressureonthedownstreampricechargedbythe
mergedfirm.Asimilarconceptalsoarisesintheregulationliteratureonefficientaccesspricingtoa
downstreamcompetitor,wheretheopportunitycostisthereductioninprofittotheintegratedfirm
perunitofaccessprovidedtothecompetitor(e.g.Armstrong,Doyle,andVickers,1996).
8
⁄
Assumption2,exhibits
⁄
0underCournot,but
0underBertrand.8
Thenextresultshowsthat,holdingallelseconstant,anexpostmarginalincreasein
(firm2’saggressiveness)willbenefitfirm1ifitsdownstreamchoiceisastrategic
complementforfirm2’schoice;harmfirm1ifdownstreamcompetitionisCournotand
firm1’squantityisastrategicsubstituteforfirm2’squantity;andhaveanambiguous
effectifcompetitionisBertrandandfirm1’spriceisastrategicsubstituteforfirm2’sprice.
Proposition2.Assumethatfirm1sellstheinputtofirm2underatwo‐parttariff.Atthe
equilibriumoutcome
,
,
,
,
,
,asmallchangeinfirm2’schoice ,
affectsfirm1’sprofit excludingthefixedfeeasfollows:
(i)WithCournotcompetition,
⁄
⁄
0if
⁄
0if
0(strategiccomplement)and
0(strategicsubstitute).
(ii)WithBertrandcompetition,
ambiguousif
⁄
⁄
⁄
⁄
0if
0,andthesignof
⁄
is
0.
Proof.Firm1sets
∗
tomaximize
≡
∗
∗
∗
∗
wherethefirstequalityexploitsthat
⁄
*

R dX *
V dX 2
  2 { 1 2  x2 dw2
x1 x2 dw2



strategic
effect
∗
∗⁄
⁄
∗
2d 1
1d 2
0,
(7)
0attheprofit‐
andrearranginggives
d
R1
w2

.TheFOCis
∗
0and
maximizingdownstreamchoices.Decomposingd
∗
∗
}
(8)
opportunity
costeffect
wherethestrategicandopportunitycosteffectsareindicatedforfuturereference.In(8),
⁄
0andusingtheSOC
⁄
.From(1),wehave
⁄
0.
),whileunderCournot
8DifferentiatingtheFOC
⁄
sign
(recalling 
⁄
⁄
⁄
0,onefinds:sign
⁄
.UnderBertrand
0
9
d
∗⁄
d
⁄
0byAssumption3,and
theoppositesignoftheterminparentheses.Givend
⁄
(i)WithCournotcompetition,sign
⁄
(ii)WithBertrandcompetition,
⁄
⁄
0bydefinitionof .Thus,
∗⁄
0,(8)thenimplies:
⁄
sign
⁄
0if
d
takes
(since
⁄
0).
0(since
⁄
0).Ifinstead
0,theterminparenthesesin(8)hasanambiguoussign,hencesodoes
⁄
.
Thus,forCournotcompetitionandthe“normal”caseofstrategicsubstitutes,the
patternisreversedfromthecasewithnofixedfee(i.e.
≡ 0)addressedinProposition1:
startingattheequilibriumoutcome,theintegratedfirmnowwouldbenefitfroman
exogenousdecreasein .ForBertrandcompetitionandthe“normal”caseofstrategic
complements,wehavethesamepatternasinProposition1:theintegratedfirmwould
benefitfromareductionin ,whichconstitutesmoreaggressivebehaviorbyfirm2.
Thesepatterns,andtheirrelationshiptolinearinputpricing,canbeunderstoodby
comparingtheFOCsunderthetwopricingregimes.Withlinearinputpricing,adecreasein
w2willreducefirm1’srevenuefrominframarginalinputsales,term–
sustain
∗
∗
in(4);thus,to
asfirm1’soptimalpriceitmustbetruethatexpansionbyfirm2—inducedby
decreasingw2—mustbyitselfraisefirm1’sprofitV.Withatwo‐parttarifffortheinput,
however,w2nolongeraffectsfirm1’sprofitfrominframarginalinputsales,duetothe
compensatingchangeinfirm2’sfixedfee(term
∗
cancelsin(7)).Instead,w2nowaffects
firm1’sprofit(besidesthroughthedirectimpactoffirm2’sexpansiononV)bysignalinga
changeinfirm1’sdownstreamchoice,whichaltersfirm2’sprofitandthereforethefixed
feethatfirm1canextract(collectively,thelasttermin(7),
⁄
d
∗⁄
d
).
Signalingsofterbehaviorletsfirm1extractahigherfixedfee,byincreasingfirm2’s
expectedprofit.Whethersignalingsoftnessrequiresloweringorraisingw2dependsonthe
signofd
effectof
∗⁄
d
=
⁄
d
∗⁄
d
+
⁄
.Thelasttermistheopportunitycost
onfirm1’s(downstream)choice,whichiszerounderCournotcompetition.The
firsttermisthestrategiceffect—howtheforeseenchangeinfirm2’sequilibriumchoice
inducedbyaltering
altersfirm1’schoice.
WithCournotcompetition,onlythestrategiceffectispresent.Iffirm1’soutputisa
strategicsubstitutefor2’soutput
⁄
0 ,thenincreasingfirm2’soutput(through
10
cutting
)signalsaloweroutputbyfirm1andbooststhefixedfee,andtheequilibriumw2
reflectsthisincentive.Therefore,holdingfirm1’soutputconstant,firm1wouldgainfrom
anexogenouscontractionbyfirm2.If,instead,firm1’soutputisastrategiccomplement
⁄
0 ,thenfirm1wouldgainfromexogenousexpansionbyfirm2.
ForBertrandcompetitionandthe“normal”strategiccomplementcase,firm1
signalssoftnessbyraising
,whichincreasesfirm1’spricefortworeasons:theforeseen
riseinfirm2’sprice,andtheincreasedopportunitycostoffirm1’soutput.9Because
is
distortedup,thepreviouslogicimpliesthatfirm1wouldgainfromamarginalexogenous
expansionbyfirm2,aswithlinearinputpricing.Iffirm1’spriceisastrategicsubstitutefor
firm2’sprice,thestrategicandopportunitycosteffects(thetermsinparenthesesin(8))
workinoppositedirections,renderingthedirectionofdistortioninw2ambiguous.
Whywouldtheintegratedfirmbenefitfromanexogenouschangeinfirm2’schoice,
,notwithstandingtheassumedabilitytoextractfirm2’sprofitthroughthefixedfee?The
(perhapsobvious)answeristhatinourcontractingenvironmenttheintegratedfirmlacks
sufficientinstrumentstomaximizeoverallindustryprofit,downstreamplusupstream.
Denotetherequireddownstreamchoicesasthe“monopolysolution,”
couldinduce
byappropriatelysetting
∗
and
.Firm1
.However,sinceweassumedthat
firm2’sfixedfee cannotbecontingenton ,firm1’sbestresponsewouldnotbe
but
;
∗
,thevalueof thatmaximizesfirm1’sprofitwhileignoringthe
effectonfirm2(since is“sunk”whenfirm1sets ).At
⁄
0and
Π ⁄
Π ⁄
;
∗
,wehave
0.Thus,industryprofitwouldincreaseifthe
integratedfirmactedlessaggressivelydownstream(raisedpriceorreducedoutput).
Intuitively,giventheinabilitytocondition on ,firm2expectsfirm1toexpand
downstreamatfirm2’sexpense,andreducesaccordinglythefixedfeeitiswillingtopay.
Thisleavesroomfortheintegratedfirmtodobetterwithadditionalinstrumentseven
thoughitalreadyextractsfirm2’sprofit.10Ofcourse,additionalinstrumentsarevaluableto
⁄
<0,
regardlessofwhetherpricesarestrategiccomplementsorsubstitutes.Seefootnote8.
9In(8),theincreaseintheopportunitycostmakesfirm1lessaggressive,i.e.
10Indeed,(7)showsthatunderatwo‐parttariff,theequilibriummarginalinputpricedoes“double
duty”bybalancingtwoeffects:how
affectsfirm1’svariableprofitbyalteringfirm2’schoice,i.e.
11
theintegratedfirmalsowhenitsellstheinputunderlinearpricing,asintheapplication
discussednext.
4.
Application:VerticalDelegationtoSignalDownstreamBehavior
TheliteratureonstrategiccommitmentsinoligopolymentionedintheIntroduction
identifiesvariousmechanismsafirmmayusetosignalachangeinitsbehaviorinthe
subsequentcompetition(shiftitsbest‐responsecurve)soastoalterarival’sbehavior.11An
integratedfirmpotentiallycouldalsoemployseveralsuchmechanismstoinducethe
desiredchangebyitsrival/customer.
Weanalyzeonepossiblemechanism,verticaldelegation.Thefirmestablishesan
autonomousdownstreamunit,division1,andchargesitapubliclyobservableinputprice;
andthedivisiontreatsanincreaseintheinputpriceasraisingitsperceivedmarginalcost,
notasanirrelevantinternaltransfer.Forexpositionalsimplicity,weassumethatdivision1
maximizessolelyitsownprofit.Attheendofthissectionwewillexplainwhythelogic
extendstootherobjectivefunctions,discusstheplausibilityofverticaldelegation,and
compareourresultstorelatedliterature.Webeginwithlinearpricingoftheinput.
Underdelegationthegameisasfollows.First,firm1publiclycommitstoapairof
inputprices
,
where
isthepricetoitsdivision1.Thendivision1andfirm2
makedownstreamchoices simultaneously,consumerspurchase,andfirm1receives
inputpayments.Firm2’sbestresponsefunction,
;
,isunchangedfromtheoriginal
game;butnowdivision1chooses tomaximizeonlyitsprofit,
andthenewinstrument
;
.Firm1setsboth
tomaximizeitsintegratedprofit ,definedin(1).12
⁄
d
∗⁄
;andhow affectsfirm2’sprofit(and,hence, )byalteringfirm1’schoice,
d
⁄
i.e.
d ∗ ⁄d
.Thus, isusedpartlytoalterfirm2’schoiceandpartlyasasignalto
firm2aboutfirm1’sdownstreamchoice.
11Thesemechanismsinclude:investmentstoshiftdownstreammarginalcost(Spence,1977;Dixit,
1980);advertising(Schmalensee,1983);managerialincentivesschemes(Vickers,1985);capital
structure(BranderandLewis,1986);verticalcontracts(BonnanoandVickers,1988;Reyand
Stiglitz,1995).Foradditionalmechanismsandreferences,seeShapiro(1989).
12Although
upstream,
doesnotdependdirectlyon ,sincedivision1’sinputpaymentsaccrueasrevenue
willaffectVindirectlybychangingtheequilibriumvaluesof .
12
Wemakesimilarassumptionsasbefore:
Assumption4.Foranygiven ,thereexistsauniquedownstreamequilibriuminpure
strategies,withthechoiceofdownstreamrivalk(k=1,2)denoted
≡
inputleveldenoted
,
,and
⁄
,itsoutputand

0(where
for
Bertrandcompetition).13
Firm1sets
and
tomaximizethecontinuationequilibriumprofitfunction
≡
underdelegation,
,
;
.Let
,
denotetheoptimalchoice.
WewillcomparethisregimetotheonefromSection2,thatwenowlabelcentralization.
Let
∗
≡
∗
,
∗
,
∗
denotefirm1’sshadowmarginalcost,definedin(6),
evaluatedatthecontinuationequilibriumundercentralization.
Underdelegation,iftheintegratedfirmsets
∗
and
∗
∗
,i.e.aninput
pricetodivision1equaltotheshadowmarginalcostinthecentralizationequilibrium,it
willinducethesamedownstreamchoicesaswithcentralizationandearnthesameprofit.
Butitgenerallycandobetter,bysuitablychoosing
∗
∗
and
∗
.Specifically,
underdelegationfirm1achievestheprofititwouldearnundercentralizationifitwerethe
Stackelbergleaderindownstreamcompetition,ratherthanmakingitsdownstreamchoice
simultaneouslywithfirm2.14Intuitively,theStackelbergequivalencearisesbecause
choosing
servesasacommitmentregardingthelevelof .15
13SeeAppendix.
14SeeMoresiandSchwartz(2015,Propositions3and4).Lu,Moresi,andSalop(2007,Appendix4)
showtheStackelbergequivalenceresultforBertrandcompetition.Arya,Mittendorf,andYoon
(2008)establishtheseresults,andaresultanalogoustoourProposition3below,fortherestricted
caseoflinear(differentiated)demandsandCournotcompetition.Theyfurthershowthat
decentralization(ourdelegation)increasestheintegratedfirm’sprofitiftheinputpricetothe
downstreamdivisionisdeterminedbybargaining.
15OnthegeneralconnectionbetweenstrategicdelegationandStackelbergleadershipinthe
competitiongame,seeVickers(1985,SectionsIandII).Adaptedtooursetting,Vickers’sagent
appointmentgame—thatprecedesdownstreamcompetition—correspondstowhetherthe
integratedfirminitiallyadoptsthecentralizationstructureandsetsonlyaninputprice orthe
delegationstructureandsetsanadditionalinputprice .Theagentappointmentgamemaximizes
theintegratedfirm’sprofitifandonlyifitimplementsthesamedownstreamoutcomeas
Stackelbergleadershipbytheintegratedfirm.SeealsoHeifetz,Shannon,andSpiegel(2007).
13
Thenextresultcharacterizeshowtheintegratedfirmunderdelegationcan
profitablychange
locallyrelativeto
∗
∗
.Proposition1showedthatstartingatthe
centralizationequilibriumwithlinearinputpricing,firm1benefitsiffirm2becomes
marginallymoreaggressive,i.e.increasesitsquantityunderCournotcompetitionor
decreasesitspriceunderBertrand.Delegationenablesfirm1toinducethedesiredchange
byusing
tosignalasuitablechangeindivision1’sdownstreamchoice:
Proposition3.Assumethatfirm1sellstheinputtofirm2underlinearpricing.Holding
constantatthecentralizationequilibriumlevel
∗
,underdelegation:
(i)Iffirm2’schoiceisastrategicsubstitutefordivision1’schoice(
gainsfromasmallincreasein
∗
above
∗
⁄
,tosignalsofterbehaviorbydivision1.
(ii)Iffirm2’schoiceisastrategiccomplementfordivision1’schoice(
1gainsfromasmalldecreasein
below
∗
0 ,firm1
∗
⁄
0 ,firm
,tosignaltougherbehaviorbydivision1.
∗
Proof.Startingat
desiredchangein
and
wherethefirstequalityfollowsfrom
assumption,
⁄
∗
,underdelegationtheintegratedfirm’s
isdeterminedbythesignof(from
∗
≡
,
;
,
⁄
0.Thus,thesignof
(9)
0.FromProposition1,
⁄
):
⁄
istheoppositeofthatof
0and,by
⁄
.
Proposition3showsthat,withCournotcompetitionandstrategicsubstitutes,firm1
underverticaldelegationcangainbyraising
tosignalareductionindivision1’soutput
andtherebyinduceanincreaseinfirm2’soutput.16WithBertrandcompetitionand
strategiccomplements,firm1cangainbylowering
tosignaladecreaseindivision1’s
priceandinduceadecreaseinfirm2’sprice.
16Wepurposelysay“cangainbyraising
”insteadof“gainsbyraising ”becausewehavenot
characterizedtheequilibriuminputpricesunderdelegation,andareinsteaddescribingaprofitable
localdeviationfromcentralization.Weelaborateonthisshortly,whendiscussinganexample.
14
Interestingly,withBertrandandstrategiccomplements,firm2’sequilibriumoutput
islikelytofallunderdelegation.ThisdoesnotcontradictProposition1,whichstatesthat
firm1gainsfromamarginalexpansionbyfirm2.Startingatthecentralizationequilibrium,
asmalldecreasein underdelegation(signaledbysetting
below
∗
∗
)becomes
profitableonlybecauseitinducesadecreasein .Thisreductionin benefitsfirm1
solelybyincreasingfirm2’ssalesandinputpurchases,whichshowsthatunilateral
expansionbyfirm2—theexperimentofProposition1—indeedbenefitsfirm1.Inother
words,asmallreductionin from ∗ underdelegationisprofitableonlybecausethe
reductioninfirm2’spricemitigatesthedecreaseinfirm2’soutputandinputpurchases
causedbythedecreasein .
AnotherinterestingfeatureunderBertrandandstrategiccomplementsisthat
delegationcreatesanincentiveforfirm1notonlytolower
alsotoraise
⁄
above
∗
∗
.Startingat
and
0,underdelegationfirm1’sdesiredchangein
2
2
∗
⁄
Because
delegation,
⁄
2
∗
2
∗
∗
∗⁄
∗
)butlikely
,andrecallingthat
isgivenbythesignof
(10)
,(10)wouldbezeroiffirm2’s“pass‐throughrate”under
,wereequaltothatundercentralization,d
absolutevaluethand
∗
.
withBertrandcompetitionandstrategiccomplements,
gainsfromraising
below
d
⁄
∗⁄
d
(see(3)).However,
likelyissmallerin
(seeAppendix),inwhichcase(10)ispositiveandfirm1
whenholding
at
∗
∗
.Intuitively,undercentralization,raising
increasesfirm1’sopportunitycostofitsownoutput,17leadingittoincrease ;this
effectof
on isabsentunderdelegation,becausedivision1ignoresupstreamprofit.
Recognizingthisdifferentialresponseof ,firm2raisesitspricebyless—hencereduces
itsinputpurchasesbyless—underdelegationfollowingagivenincreasein
pass‐throughrateemboldensfirm1toraise
.Thislower
underdelegation.
17ThisopportunitycosteffectisabsentwithCournotcompetition,sincefirm1’ssalesdonotaffect
firm2’ssales(henceinputpurchases).Seethediscussionafter(6).
15
Example.Proposition3conductsalocalexperimentfor
theinputprice
,whileholdingconstant
totherival/customer,firm2.TheAppendixreportsanexampleshowing
theequilibriumvaluesofallthevariablesundercentralizationanddelegation,forCournot
orBertrandcompetition.Intheexample,thefirms’quantitiesarestrategicsubstitutes,
whilepricesarestrategiccomplements.Movingtodelegation,theequilibriumvaluesof
andfirm1’sdownstreamchoice shiftinthedirectionsuggestedbyProposition3.With
Cournotcompetition,theintegratedfirmraises
(above
∗
∗
))andlowersitsoutput,
toinduceanoutputexpansionbyfirm2.WithBertrandcompetition,theintegratedfirm
reduces
anditsdownstreamprice,toinduceapricereductionbyfirm2;however,firm
2’soutputfalls(andweexplainedwhythisisconsistentwithProposition1).
Thechangein
alsoisconsistentwiththeearlierdiscussion.WithBertrand
competition,theinputpricetofirm2risesunderdelegationdespitethedecreaseinfirm
1’sfinalprice ,whichbyitselfcallsforreducing
thelowerpass‐throughratefrom
.18Theincreasein
ismotivatedby
to underdelegationthanundercentralization,
whichoccursonlywithBetrandcompetitionandstrategiccomplements.19
Two‐PartTarifffortheInput.Supposethatinstageone,firm1publiclyoffersa
pair
,
tofirm2,andaper‐unitinputprice
todivision1.Downstreamcompetition
thenoccursasabove.Firm1cannowearnthemaximalindustryprofit,bysettinginput
pricesatthelevels
and
the“monopolysolution”
thatinducedivision1andfirm2tochoose(unilaterally)
and
,andsetting toextractfirm2’sprofits.20However,
withobservabletwo‐parttariffs,firm1couldachievethesamemaximalprofitbyvertically
18Decreasing
makesitmoreattractivetoreduce fortworeasons:thedecreasedmarginon
firm1’ssalesrendersoutputdiversionawayfromfirm1lesscostly(“margineffect”);andthe
reductioninfirm2’sinputdemandduetothedecreasein (“demandeffect”).
19Theequilibrium
increaseswithdelegationalsounderCournotcompetition,butthatis
explainedbythemargineffectanddemandeffectcausedbyraising .Holding constantatthe
shadowmarginalcostundercentralization( ∗ ∗ ,firm1’sconstrainedoptimal under
delegationwouldremainat ∗ withCournotcompetitionbutrisewithBertrand.
20Whendivision1andfirm2arecompetitors,toimplement
and theintegratedfirm
typicallymustsetthemarginalinputpricesabovethemarginalcost .
16
separatingentirely,settingtheinputprices(
,
),andchargingfixedfeesthatfully
extracteachdownstreamfirm’sprofits.Bycontrast,delegationandcompleteseparation
arenotequivalentwhenfirm1chargeslinearinputprices.21Thus,delegationforpricing
purposesisarguablyoflessinterestwhenobservabletwo‐parttariffsarefeasible.
Partlyforthisreason,weforgoaresultanalogoustoProposition2.22Asecond
reasonisthatalthoughthelocaldeviationsin identifiedinProposition2wouldincrease
firm1’sprofit,theygenerallywillnottrackthechangein whenfirm1implementsthe
“monopolysolution”underatwo‐parttariffanddelegation.Astheexampleinthe
Appendixsuggests,withcentralizationthereisabiastowardsfavoringfirm1’soutput,and
movingtothemonopolysolutioncorrectsthisbyreducingfirm1’soutputandincreasing
firm2’s;theprimeroleof
becomesto“reinin”firm1’saggressiveness.
Althoughverticaldelegationmaybeoflessinterestwhentwo‐parttariffsare
feasible,Proposition2canstillhavepracticalrelevance.Forexample,MilliouandPetrakis
(2015)consideraverticallyintegratedfirmthatcansharecost‐reducingknowledgewitha
downstreamrivaltowhomitsuppliesakeyinput.OurProposition2suggeststhatevenif
thescopeforsharingarisesunexpectedlyafterthetwo‐parttarifffortheinputhasbeen
set,somesharingwouldbeprofitableincertaincases,becauseonthemarginthe
integratedfirm’sgainfromexpandedinputsalesoutweighsitslossinthedownstream
market(andthisisalwaystrueunderlinearinputpricing,byProposition1).
DiscussionandRelatedWork.Ourverticaldelegationmechanismhastwo
requirements:thesuppliercancommittochargeitsdivisionaninputpriceobservableto
downstreamrivals;andthedivisiondoesnot“undo”thestrategiceffectsofthatpriceby
fullyinternalizinghowitschoiceaffectstheaffiliatedsupplier’supstreamprofits.To
streamlinetheexposition,weassumedthedivisionmaximizessolelyitsownprofit,butall
21Weshowedthatdelegationthendominatescentralization,andunderreasonableconditions
centralizationwilldominatecompleteseparation(byeliminatingdoublemarginalizationbetween
firm1’supstreamanddownstreamunits).
22Also,onecannolongeruseasimpleenvelopeargumenttoshowthatfirm1benefitsfromasmall
∗
changein todivision1from ∗
.Although ⁄
0at
,
,thechangein
inducedbychanging wouldalterfirm2’sprofitand,hence,theattainablefixedfee .
17
ourresultsextendtoanyobjectivefunctionwheretheinputpriceaffectsthedivision’s
downstreamchoice.23
Inpractice,integratedfirmsoftentreatdivisionsasprofitcenters.EcclesandWhite
(1988)examineindetailthemaintransferpricingpoliciesobservedintheintensivefield
studybyEccles(1985),andfindthatonesuchpolicy,exchangeautonomy,grantsdivisions
greatlatitudeintransactingwithoneother.Consistentwith(some)divisionalautonomy,
Crawford,Lee,Whinston,andYurukoglu(2015)presentempiricalevidencethatvideo
distributorswhoareverticallyintegratedintoprogrammingdonotfullyinternalizethe
effectsoftheirpricingandprogramcarriagedecisionsontheirprofitsfromprogramming.
Insteadofrelyingsolelyoninternalprotocols,anotherwaytolimitthedownstream
unitfromactingasapassivearmofitsupstreamsupplierisbyhavingminorityoutside
shareholders(e.g.O’BrienandSalop,2000).Inprinciple,aninitiallyintegratedfirmcould
retainanarbitrarilyhighshareofthedownstreamunit’sprofitswhilestillinducingitto
maximizesolelydownstreamprofitbyspinningoffthecontrolrights(votingstock)to
outsideshareholdersandbecomingapassivemajorityowner.24Alternatively,theinitially
integratedfirmcouldspinofftothesameoutsidershareholderstheupstreamunitand
non‐votingstockofthedownstreamunit,therebyretainingthecontrolrightsoverthe
downstreamunit.Ouranalysisillustratesastrategicbenefitofsuchspin‐offswhenthe
supplieralsosellstodownstreamrivals.
Besidesdownstreamautonomy,signalingthroughverticaldelegationalsorequires
thattheinputpricechargedtothedivision(orspun‐offunit)beobservablebythe
downstreamrival.Thisassumptionismorequestionableforunregulatedfirms,25andits
23Forexample,ifthedivisionmaximizesaweightedaverageofitsprofitandtheintegratedfirm’s
profit,thesuppliercanimplementitspreferreddownstreamoutcomebysuitablyraisingtheinput
pricetoitsdivisiontocompensateforthedivisiontreatingafractionofthatpriceasapuretransfer
(Arya,Mittendorf,andYoon,2008,Proposition3).
24Familyrunfirmssometimesretaincontrolrightsbutspinoffmajorityownershipforpurposesof
raisingcapital.Here,theinitiallyintegratedfirmspinsoffcontrolrightssolelytosignalthatthe
downstreamunitwillactindependently,nottoraisecapital.
25Regulatedfirmsmaybesubjecttorulesgoverninginputpricingtosubsidiariesaswellasthe
subsidiaries’behaviorthroughimputationrequirements(e.g.LaffontandTirole,2000).
18
validitywillbecontextspecific.Oneconsiderationthatmayaidobservabilityisthe
supplier’sincentivetomaintainareputationfornotactingopportunisticallyagainst
customersbyofferingsecretdiscountstotheirrivals(e.g.O’BrienandShaffer,1992;
McAfeeandSchwartz,1994).Topreservesuchareputation,itmayadoptapolicyof
committingtopublicandtransparentpricing.26
BonannoandVickers(1988)notedearlyonthestrategicadvantageofcommitment
toobservableinputprices.TheyconsidertwosupplychainsindifferentiatedBertrand
competitionwithpricesasstrategiccomplements.Byverticallyseparatingandchargingits
singleretaileraninputpricesomewhatabovemarginalcost,eachsuppliercanprofitably
coaxanincreaseintherival’sdownstreamprice.Oursettingdiffersbyhavingasingle
supplier,whoisverticallyintegratedandsuppliestoadownstreamrival.Consequently,
underlinearinputpricingthesupplierbenefitsfrominducingtougherratherthansofter
behaviorfromtherival/customer.Inthesamevein,ouranalysisdiffersfromtheliterature
onstrategicadvantagesfromcreatingautonomouscompetingdivisions(e.g.Schwartzand
Thompson,1986;Baye,Crocker,andJu,1996).There,theroleofdivisionalautonomyisto
signaltoughnessagainstrivals.Here,thedivisionisatadifferentverticalstageandthe
strategicgainmayinvolvepresentingatoughorsoftposture.
5.
ConcludingRemarks
Weconsideredaverticallyintegratedinputmonopolistthatbothsellsoutputinthe
downstreammarketandsuppliesinputstoadifferentiateddownstreamrival.Weshowed
thatunderlinearpricingoftheinput,theintegratedfirmwouldunambiguouslygainfrom
exogenousexpansionbyitsrival/customer.Underatwo‐parttarifffortheinput,the
integratedfirmwouldstillbenefitfromexogenousexpansionbytherival/customerif
downstreamcompetitionisBertrandwithstrategiccomplements,butwouldloseunder
Cournotcompetitionwithstrategicsubstitutes.
26Anotherconsiderationisthatpublicdelegationcontractscanbeusefulmechanismsevenwhen
theycanberenegotiatedsecretly(e.g.CaillaudandRey,1995).
19
Weanalyzedoneofpotentiallymanymechanismstoinduceexpansionor
contractionbytherival/customer,beyondrelyingsolelyontheinputpricetothatfirm.
Thatmechanismisverticaldelegation:establishingadownstreamdivisionthattreatsits
inputpricefromtheupstreamaffiliatepartlyasacost(ratherthanapurelyinternal
transfer),andchargingthatdivisionaninputpriceobservablebytherival/customer.
Alternatively,controlrightsinthedownstreamunit(butonlyaminorityofownership)
couldbespunofftooutsideshareholders.
Thisresearchpotentiallycouldbeextendedinseveraldirections.OurPropositions1
and2characterizethesupplier’sincentivesintwopolarcases,whenitcapturesnoneorall
oftheprofitoftherival/customer;theintermediatecaseofbargainingoverthetwo‐part
tariffmaybeworthexploring.Also,theresultsmaychangeunderotherformsofnon‐linear
contractsdifferentfromtwo‐parttariffs.27Anotherdirectioninvolvesalternativemarket
structures.Addressingthecasewheretheinputsuppliersellstomultipledownstream
rivalswouldnotraiseconceptualdifficultiesifrivalscanobserveeachother’sinputprices,
butwithsecretoffers,beliefsaboutoff‐equilibriumofferswillintroducefamiliar
complexities.Introducingupstreamcompetitionfromacompetitivefringealsowouldbe
fairlystraightforward;ourPropositions1and2wouldextendifthefringemarginalcost
capthesupplier’spricecloseenoughtoitsunconstrainedoptimum.Allowingforstrategic
competitionupstream,however,wouldsignificantlycomplicatetheanalysis.
Adifferentbutcomplementarydirectioninvolvesapplicationsoftheanalysis:
mechanismsthesuppliermightemploytoaltertherival/customer’schoice.Regarding
verticaldelegation,animportantempiricalissueishowthesuppliermaysetanobservable
inputprice.Alternativestoverticaldelegationcouldalsobeexplored.
27Forexample,considera“quantityforcingcontract”thatspecifiesafixedquantityofinputtobe
purchasedbyfirm2andafixedfeetobepaidbyfirm2.UnderCournotcompetition,firm1always
benefitsiffirm2exogenouslyreducesoutputinthecompetitionstage.Withatwo‐parttariff,
Proposition2showsthatfirm1benefitsfromacontractionbyfirm2onlywhenquantitiesare
strategicsubstitutes.
20
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Appendix
A.
DirectCostEffectandIndirectStrategicEffect
Assumption3saysthatanincreaseintheinputpricemakesfirm2lessaggressive,i.e.
d
∗⁄
d
0(recallingthat  forCournotand 
forBertrand).Thisderivative
canbeanalyzedbydifferentiating
∗
where
∗
;
∗
and
∗
;
(A1)
(A2)
denotesthereactionfunctionofdownstreamrivalk(k=1,2).Thus:
∗
⁄
Assumption2impliesthatthedenominatorin(A2)ispositive.Theterm
in
thenumeratoristhe“directcosteffect”andweassumeitisnegative—i.e.ahigherinput
pricemakesfirm2lessaggressive,holding constant.Thesecondtermisthe“indirect
strategiceffect”anditiszerowithCournotcompetition.(Asthereisnoinputdiversion,
firm1’sshadowmarginalcostin(6)isunaffectedby
⁄
,implying
0.)With
Bertrandcompetition,thereisinputdiversion,theshadowmarginalcostincreaseswith
andweassumethatahighershadowmarginalcostmakesfirm1lessaggressive,i.e.
⁄
0.Thus,theindirectstrategiceffectisnegative,likethedirectcosteffect,if
pricesarestrategiccomplements,i.e.if
⁄
0.Ifinsteadpricesarestrategic
substitutes,weassumethatthedirectcosteffectdominates.Itfollowsthatd
Assumption4saysthatdivision1islessaggressiveif
⁄
constant),i.e.
∗⁄
d
0.
increases(holding
0.Similarly,Assumption3nowsays
⁄
0.These
derivativescanbeanalyzedbydifferentiating
;
and
;
. (A3)
Thus:
24
and
0.Itfollowsthat
⁄
⁄
0.Similarly,
(A4)
Holding constant,weassumethatdivision1islessaggressiveif
⁄
increases,i.e.
0.
Finally,observethatunderBertrandcompetition,strategiccomplementsandlinear
demand,anincreaseintheinputpricemakesfirm2lessaggressiveandmoresounder
centralizationthanunderdelegation,i.e.d
and(A4),
⁄
⁄
∗⁄
⁄
d
⁄
0and
0.Thisfollowsfrom(A2)
⁄
.28Intuitively,anincreasein
raisesfirm1’sshadowmarginalcostoftheinputundercentralization(whichis
,
from(6))butwillnotaffectdivision1’sperceivedmarginalcostunderdelegation
(whichis
).Thus,followinganincreasein
,firm2expectsalargerpriceincreaseby
firm1undercentralizationthanunderdelegation,leadingtoagreaterequilibriumpass‐
throughrate,i.e.d
∗⁄
d
⁄
0.Inturn,firm2’slowerpass‐throughrateunder
delegationencouragesfirm1toraisew2.
B.
Example
Proposition3showshowfirm1canraiseitsprofitunderdelegationwithalocaldeviation:
maintaintheinputpricetofirm2atthecentralizationlevel
∗
andofferdivision1aninput
pricedifferentfromtheshadowmarginalcostundercentralization,
∗
∗
.Inthe
examplebelow,wefindtheactualequilibriumvaluesunderbothregimes.
Assumethatthemarginalcostoftheinputisconstant,
downstreamcosts.Consumerdemandisgivenby
500
200
1,andtherearenoother
100 ,where denotestheoutputoffirm ,and and denotethepricesoffirms and ,respectively
28Withlineardemand,thederivativesofthereactionfunctionsarescalarsandhencedonot
dependonprice,quantityorinputpricelevels.UnderCournotcompetitionandlineardemand,we
⁄
⁄
⁄
⁄
d ∗ ⁄d
0because
.
0and
have
25
( , ∈ 1,2 ,
).29Givenlineardemands,underCournotcompetitionfirms’quantitiesare
strategicsubstituteswhileunderBertrandcompetitionpricesarestrategiccomplements.
∗
IntheensuingTables1and2,underdelegation,
for
given
∗
∗
,and
∗
isfirm1’soptimalchoice
given
istheoptimalchoicefor
∗
∗
.
(NAmeansNotApplicable.)
Table1:CournotCompetitionDownstream(LinearInputPrice)
Centralization Delegation
ProfitofIntegratedFirm
682.76
685.71
ProfitofDownstreamRival
45.66
48.98
OutputofIntegratedFirm, 279.31
257.14
OutputofDownstreamRival, 82.76
85.71
OutputPriceofIntegratedFirm, 2.86
3
OutputPriceofDownstreamRival, 3.52
3.57
2.97
3
1
NA
NA
1.29
NA
1.28
NA
2.97
InputPriceChargedtoDownstreamRival,
ShadowCostofSupplyingInputInternally,
∗
InputPriceChargedtoDivision1,
∗
∗
∗
∗
Table1illustratesProposition3(i):startingfromthecentralizationoutcomeand
holding
constant,firm1wantstoinduceanincreaseinfirm2’squantity .This
requiressignalingareductionin (sincequantitiesherearestrategicsubstitutes)by
raising
todivision1abovetheshadowmarginalcostundercentralization(which,for
Cournotcompetition,equalstheresourcemarginalcost,
∗
1.28
1.00
∗
∗
1,independentof
.Intheactualdelegationequilibrium,
2.97to3.00,while risessubstantiallyto1.29(from
∗
:
risesslightly,from
1).Theintegratedfirm’s
29Inthisexample,firm1hasanincentivetosupplytheinputtofirm2undercentralization.See
Arya,Mittendorf,andSappington(2008).Ourresultsimplythatthisalsoistrueunderdelegation.
26
output fallsandtherival’soutput rises,consistentwiththelocalincentivesdescribed
inProposition3(i).(Therisein2’soutputandinputdemandinresponsetotheforeseen
largedecreasein explainstheincreasein
.)Profitsofbothfirmsincrease.
Table2:BertrandCompetitionDownstream(LinearInputPrice)
Centralization
Delegation
ProfitofIntegratedFirm
696.97
700
ProfitofDownstreamRival
59.50
50
OutputofIntegratedFirm, 227.27
250
OutputofDownstreamRival, 109.09
100
OutputPriceofIntegratedFirm, 3.12
3
OutputPriceofDownstreamRival, 3.52
3.5
2.97
3
1.98
NA
NA
1.75
NA
1.74
NA
3.04
InputPriceChargedtoDownstreamRival,
ShadowCostofSupplyingInputInternally,
∗
InputPriceChargedtoDivision1,
∗
∗
∗
∗
)
Table2illustratesProposition3(ii):startingfromthecentralizationoutcomeand
holding
constant,firm1wantstoinduceandecreaseinfirm2’sprice .Thisrequires
signalingareductionin (sincepricesherearestrategiccomplements)bylowering
belowtheshadowmarginalcostundercentralization:
∗
1.74
1.98
∗
∗
.
ThepatternsalsoillustratethediscussionaftertheproofofProposition3(ii).Under
delegation,if
gainbyraising
weresetatthecentralizationshadowmarginalcost,
:
∗
∗
3.04
∗
∗
,firm1would
2.97.Thisincentiveemergesbecausefirm2
raisespricebylessinresponsetoagivenincreasein
underdelegationthanunder
centralization(sinceitrecognizesthatdivision1willnotraise duetoanincreasein
,
whereasundercentralizationfirm1internalizestheincreasedopportunitycost).
27
Intheactualdelegationequilibrium,
risesbyless,from2.97to3.00,becausefirm
2’sinputdemandfallsduetothereductionin (from3.12to3.00)inducedbythe
decreasein
(to1.75fromtheshadowmarginalcostof1.98).Thereductionin
increasein
causefirm1’soutputtoriseandfirm2’stofall.(Butthefallin ismitigated
bythereductionin promptedbytheobservedreductionin
and
,whichiswhyfirm1
benefitsfromreducing underdelegation.)Delegationnowreducesfirm2’sprofit,unlike
theCournotcase,butasexpected,firm1’sprofitstillincreases.
ThefollowingTables3and4showtheresultswhenfirm1sellstheinputtofirm2
usingatwo‐parttariff.
Table3:CournotCompetitionDownstream(Two‐PartTariffInputPrice)
Centralization Delegation
ProfitofIntegratedFirm
784.62
800
FixedFeePaidbyDownstreamRival
227.22
266.67
OutputofIntegratedFirm, 253.85
200
OutputofDownstreamRival, 184.62
200
OutputPriceofIntegratedFirm, 2.69
3
OutputPriceofDownstreamRival, 2.92
3
1.69
1.667
1
NA
NA
1.667
NA
1.666
NA
1.69
InputPriceChargedtoDownstreamRival,
ShadowCostofSupplyingInputInternally,
∗
InputPriceChargedtoDivision1,
∗
Table3canbecomparedtoProposition2(i):startingfromthecentralization
outcome,firm1wouldbenefitiffirm2exogenouslyreduceditsoutput (sincequantities
herearestrategicsubstitutes).Thismightsuggestthat,underdelegation,firm1should
signalanincreasein bysetting
below1(theshadowmarginalcostunderCournot).
However,theoppositeistruesincefirm1insteadraisestheinputpricetoitsdivision
abovetheshadowmarginalcost(1.667
1 .Thereasonisexplainednext.
28
Undercentralization,whenfirm1sellstheinputtofirm2usingatwo‐parttariff,
firm1cannotachievetheintegratedmonopolysolutionbecauseitcannotcommitnotto
behaveopportunisticallyinthecompetitionstagebyselling“toomuch”output.Firm1
thuscanachieveonlyasecond‐bestsolution.Startingfromthatsecond‐best,firm1would
benefitiffirm2exogenouslyreducedoutput(sincehere
⁄
0 .However,firm1
wouldbenefitevenmoreifsomehowitcouldcommittobehavelessaggressivelyinthe
competitionstage.Thatisexactlywhatdelegationandatwo‐parttariffallowfirm1todo:
implementtheintegratedmonopolysolution(i.e.maximizeandobtaintheindustrytotal
profits).Itfollowsthatthemainroleof
becomestomakedivision1lessaggressiveso
astomaximizeindustryprofits,ratherthantoinducefirm2toreduceoutput.
Table4:BertrandCompetitionDownstream(Two‐PartTariffInputPrice)
Centralization
Delegation
ProfitofIntegratedFirm
790.48
800
FixedFeePaidbyDownstreamRival
146.94
200
OutputofIntegratedFirm, 242.86
200
OutputofDownstreamRival, 171.43
200
OutputPriceofIntegratedFirm, 2.81
3
OutputPriceofDownstreamRival, 3.05
3
2.19
2
1.60
NA
NA
2
NA
2.01
NA
1.98
InputPriceChargedtoDownstreamRival,
ShadowCostofSupplyingInputInternally,
∗
InputPriceChargedtoDivision1,
∗
)
Table4canbecomparedtoProposition2(ii):startingfromthecentralization
outcome,firm1wouldbenefitiffirm2exogenouslyreduceditsprice (sincepriceshere
arestrategiccomplements).Thismightsuggestthat,underdelegation,firm1shouldsignal
areductionin bysetting
below1.60(theshadowmarginalcost).However,the
29
oppositeistruesincefirm1insteadraisestheinputpricetoitsdivisionabovetheshadow
marginalcost(2
1.60 .ThereasonissimilarasthatgivenaboveforTable3.Inboth
cases,CournotorBertrandcompetition,themovetodelegationlowersfirm1’soutputand
raisesfirm2’s,withtheinputpricerisingsubstantiallytodivision1andfalling(byless)to
firm2.
30