StrategicIncentivesWhenSupplyingtoRivals

SergeMoresiandMariusSchwartz*

June26,2015

GeorgetownUniversity,DepartmentofEconomics,WorkingPaper15‐15‐05

AbstractWeconsideranunregulated,verticallyintegratedinputmonopolistthatsuppliestoa

differentiateddownstreamrival.Withlinearinputpricing,theintegratedfirm

unambiguouslywantstoinduceexpansionbytherival—theoppositeincentivefromthatin

standardoligopolysettingswithnosupplyrelationship,eventhoughthedownstream

competitioneffectisstillpresenthere.Thisresultholdswhetherdownstreamcompetition

involvespricesorquantitiesandstrategicsubstitutesorcomplements.Ifthefirmchargesa

two‐parttarifffortheinput,theresultcontinuestoholdunderBertrandcompetitioninthe

“normal”caseofpricesasstrategiccomplements,butisreversedforCournotandstrategic

substitutes.Weanalyzeonemechanismforinfluencingtheindependentdownstreamfirm,

verticaldelegation,wherebytheintegratedfirmchargesitsdownstreamunitanobservable

inputprice,andthedownstreamunitdoesnottreatthatpriceasapureinternaltransfer.

Verticaldelegationisshowntodominatecentralizedbehaviorbytheintegratedfirm,andwe

characterizehowtheinputpriceshouldbesetinordertoaltertheindependentfirm’schoice

dependingonthespecificsofdownstreamcompetition.

JELCodes: L13,D43,L14,L22

Keywords: StrategicCompetitionAgainstCustomers,VerticalDelegation

*Moresi:CharlesRiverAssociates,Inc.,WashingtonDC2004<smoresi@crai.com>.Schwartz:DepartmentofEconomics,GeorgetownUniversity,WashingtonDC20057<mariusschwartz@mac.com>.WethankAxelAnderson,YongminChen,JayEzrielev,JustinJohnson,PatrickRey,BillRogerson,SteveSalop,YossiSpiegel,andJohnVickers.Anyerrorsareoursalone.

1

1. Introduction

Avastliteraturestudiesafirm’sincentivestoelicitsoftercompetitivebehaviorfrom

oligopolisticrivals—contraction,ahigherpriceorloweroutput—bytakingobservableand

irreversibleactionsthatalterthefirm’sownstrategicpostureintheensuingcompetition

(Shapiro,1989).Mechanismstoachievesuchstrategiccommitmentincludeinvestment

thatlowersthefirm’smarginalcost(Spence,1977;Dixit,1980),advertising(Schmalensee,

1983),subsidiestodomesticfirmsforexportsorR&D(SpencerandBrander,1983),capital

structure(BranderandLewis,1986),managerialincentiveschemes(Vickers,1985;

FershtmanandJudd,1987),andverticalcontractualarrangements(BonannoandVickers,

1988;ReyandStiglitz,1995).Toinfluenceitsrivals,thefirmmayadoptatoughorsoft

strategicposture,dependingonwhetheritwishestodeterentryoraccommodaterivals

and,inthelattercase,whetherthecompetitivechoicevariablesarestrategicsubstitutesor

complements(FudenbergandTirole,1984;Bulow,GeanakoplosandKlemperer,1985).But

throughout,thegoalistoinducesoftercompetitionfromrivals.

Howdotheseincentiveschangewhenafirmnotonlycompeteswithrivals

downstreambutalsosuppliesthemwithinputs?Suchsituationsarefairlycommon.Tocite

justafewexamples,Qualcommmakeschipsusedinsmartphonesandlicenseskeypatents

torivalchipmanufacturers(BenoitandClark,2015);Samsungsuppliescomponentsfor

theiPhoneandcompetesagainsttheiPhone;andComcast‐NBCUsuppliesprogrammingto

videodistributorsandcompeteswiththeminvideodistribution(Rogerson,2013).

Weconsideranunregulated,verticallyintegratedinputmonopolistthatfaces

differentiated‐productscompetitiondownstream.Unsurprisingly,theintegratedfirmnow

perceivesatradeoff:softerbehaviorbyarival/customerincreasesthefirm’sdownstream

profitbutreducesitsprofitableinputsales.Moresurprisingly,weshowthatwithlinear

inputpricing,theeffectoninputsalesnecessarilydominates.Theverticallyintegrated

supplierunambiguouslywantstoinduceexpansionbyarival/customer—theopposite

incentivetothatinstandardtwo‐stagegameswithnosupplyrelationship.Thissharp

resultholdswhetherdownstreamcompetitionisinpricesorquantitiesandwhetherthese

variablesarestrategicsubstitutesorcomplements.

2

Theanalysisbecomesmoreintricatewhentheinputsupplierchargestherivala

two‐parttariffinsteadoflinearpricing.Thepreviousresultcontinuestoholdunder

Bertrandcompetitioninthe“normal”casewherepricesarestrategiccomplements,butthe

incentiveisreversedforCournotandstrategicsubstitutes:theintegratedfirmthenwants

toinducecontractionbyitsrival/customer.Weexplainwhytheincentivesdependonthe

specificsofdownstreamcompetitiononlywhentheinputissoldunderatwo‐parttariff.

Asnoted,theliteratureidentifiesnumerouspracticesafirmmayusetoalterits

strategicposture—shiftitsbest‐responsefunctionintheensuingcompetitiontosignala

changeinitsdownstreamchoice—soastoinfluencerivals’choices.Weapplyouranalysis

toonesuchmechanism,verticaldelegation:thefirmestablishesadownstreamunitwhose

objectivefunctiontreatsaportion(nomatterhowsmall)oftheinputpricechargedtoitby

theupstreamunitasacostratherthanapurelyinternaltransfer;andcommitstocharge

thedownstreamunitapubliclyobservableinputprice.1Echoingfamiliarideas(e.g.Vickers,

1985;Katz,2006;Heifetz,ShannonandSpiegel,2007),weshowthatwithdelegationthe

integratedfirmcanreplicatethedownstreamoutcomearisingwhenthefirmactsasa

centralizeddecisionmaker(e.g.asinChen,2001)andgenerallydoesbetter,evenifitcan

chargeatwo‐parttarifffortheinputunderbothregimes.Specifically,itattainsthe

outcomeitwouldachieveundercentralizationifitweretheStackelbergleaderin

downstreaminteraction.Wealsocharacterizethefirm’sincentivesregardingtheinput

pricetoitsdownstreamunit,dependingonthedownstreamcompetition—Bertrandor

Cournotandstrategicsubstitutesorcomplements.Forexample,withpricesasstrategic

complements,theintegratedfirmwantstoinduceareductioninthedownstreampriceof

itsrival/customer,whichrequiresloweringtheinputpricetoitsdownstreamunit.2

Thenextsectionofthepaperpresentsthemodelandmainresults.Section3applies

theanalysistoverticaldelegation,andSection4offersbriefconcludingremarks.

1Wethereforeabstractfromissuesofunobservabilityandrenegotiation(e.g.Katz,1991;CaillaudandRey,1994),butwilldiscusstheseissuesbrieflyinSection3.

2FordifferentiatedCournotcompetitionwithlineardemandsandlinearpricingoftheinput,Arya,MittendorfandYoon(2008)showthatunderdelegationtheintegratedfirmwouldpricetheinputtoitsdownstreamdivisionsoastoinduceexpansionbytherival/customer,consistentwithourgeneralresultdiscussedearlier.

3

2. TheSettingandMainResults

2.1 LinearPricingoftheInput

Aninputmonopolist,firm1,suppliestoitsdownstreamunitandtoanindependent

downstreamrival,firm2,asettingoftendescribedaspartialintegration.Thedownstream

choicevariables and areeitherper‐unitprices( and )orquantities( and ),

therebyallowingBertrandorCournotcompetitiondownstream.3Thetimingisasfollows.

First,firm1setsaper‐unitinputprice tofirm2.Then,firms1and2simultaneouslyset

downstreamvariables , ,consumerspurchase,andfirm2paysforfirm1’sinput.

Tosimplify,weassumeeachfirmrequiresoneunitofinputperunitofoutput,anduse

forbothk’soutputandinputamountsconditionalonthedownstreamvariables.

Firm2chooses tomaximizeitsprofit ; ,andfirm1chooses to

maximizeitstotalprofits:

; ; . (1)

Here, denotesfirm1’smarginalcostofproducingtheinput,assumedconstantoverthe

relevantrange, ; isfirm1’sprofitfromitsoutputsales,and isits

profitfrominputsalestofirm2.Thisisastandardrepresentationofbehaviorbyan

integratedfirmthatalsosuppliestoarival.4

Observethatiffirm1didnotsupplyinputstofirm2,orwasaregulatedmonopolist

thatmustsupplytofirm2atcost(i.e. ),thenfirm1’sprofitwouldcomesolelyfrom

itssales,i.e. ; ; .Inthatcase,firm1wouldwantfirm2toreduceoutput—i.e.

∂⁄ 0whendownstreamcompetitionisCournot,and ∂⁄ 0ifitisBertrand.5

3Theintegratedinputmonopolistpreferstomaintainfirm2asanactivebuyeroftheinputratherthanforeclosefirm2(underbothBertrandandCournotdownstreamcompetition)iffirm2isasufficientlyefficientcompetitor.SeeArya,Mittendorf,andSappington(2008).

4Underthisrepresentation,Chen(2001)comparespartialintegrationtonointegration.Arya,Mittendorf,andSappington(2008)comparetheoutcomesunderpartialintegrationwhendownstreamcompetitionisBertrandorCournot.

5Weassumethatalltherelevantfunctionsaredifferentiable.Implicitly,therearenobindingcapacityconstraintsandthedownstreamproductsaredifferentiated(i.e.imperfectsubstitutes).

4

Thisisbecauseacontractionbyfirm2wouldincreasethedemandfacedbyfirm1.

Inoursetting,firm1sellsinputstofirm2atamarkup(i.e. ),sofirm2’s

downstreamchoice, ,hastwoopposingeffectsonfirm1’sprofits:

Π12 2

2. (2)

where / ∂ ,isthe“downstreamcompetition”effectdiscussedpreviouslyand

∂⁄ isthe“inputsupply”effectwhichtypicallyrunsintheoppositedirection.

Forexample,underBertrandcompetitiondownstream,apriceincreasebyfirm2raises

firm1’sprofitintheoutputmarket,butlowersitsprofitintheinputmarketbecausefirm2

willproduceless,andhencepurchasefewerinputs,whenitraisesitsoutputprice.(Under

Cournot,anoutputreductionbyfirm2increasesfirm1’sprofitintheoutputmarket,but

decreasesitsprofitintheinputmarket.)Proposition1belowwillshowthat,inequilibrium,

theinputsaleseffectalwaysdominates,andthustheintegratedfirmwantsthe

rival/customertoexpandoutput.

Foranyinputprice ,weassumethereexistsauniquedownstreamNash

equilibriuminpurestrategies,withthechoiceofdownstreamrivalk(k=1,2)denoted∗ anditscorrespondingoutputleveldenoted ∗ .6,7Finally,weassumethatan

increaseintheinputpricetofirm2leadsittoreduceoutputunderCournotcompetition

(i.e.d ∗ d 0⁄ )andtoincreasepriceunderBertrandcompetition(i.e.d ∗ d 0⁄ ).8

Firm1sets tomaximize ∗ ≡ ∗ , ∗ ; .Let ∗denotethe

profit‐maximizingchoice,and ∗, ∗ denotetheresultingequilibriumdownstream

choices.Recallingthat underBertrandcompetitionand underCournot

( 1,2),wenowstateourmainresult:

6Weassumethefirms’reactionfunctionsdownstreamareeitherstrictlyincreasing(strategiccomplements)orstrictlydecreasing(strategicsubstitutes),andhaveslopessmallerthanunityinabsolutevalue.Theseconditionsareassumedtoholdovertherelevantrangeofthevariables.

7Wethushave ∗ ≡ ∗ , ∗ .UnderCournot, and ∗ ∗ .

8Weassumethatthedirectcosteffectdominatestheindirectstrategiceffect.SeeAppendixA.

5

Proposition1.Assumefirm1sellstheinputtofirm2underlinearpricing.Atthe

equilibriumoutcome , , ∗, ∗, ∗ ,firm1wantsfirm2toexpandoutput,i.e.

⁄ 0ifdownstreamcompetitionisCournot,and ⁄ 0ifitisBertrand.

Proof.Thefirst‐ordercondition(FOC)withrespectto is

∗ ≡

∗

∗ 0. (3)

TheFOCwithrespectto implies ⁄ 0,and(1)implies ⁄ ∗,sothat(3)

canberewrittenas

∗ ∗. (4)

Itthenfollowsfrom(4)and ∗ 0that

∗ 0. (5)

IfdownstreamcompetitionisBertrand,thend ∗ d 0⁄ and(5)implies 0.⁄ If

downstreamcompetitionisCournot,thend ∗ d 0⁄ and(5)implies 0⁄ .

Proposition1identifiesoppositeincentivestothoseinastandardduopolysetting

wherefirm1doesnotsupplyfirm2.There,holding constant,firm1wouldgainfroma

risein2’spriceunderBertrandcompetitionorafallin2’soutputunderCournot.Thisis

theusual“softeningdownstreamcompetition”effect.Itispresentalsohere,butdominated

byanopposing“inputsupply”effect.Thelogicisshownin(4).Giventhatfirm1sets at

theprofit‐maximizinglevel,areductionin wouldhavethefollowingequalbutopposite

effects:profitfrominframarginalinputsaleswouldfall,implyingthatprofitmustrisefrom

increasedinputsales—afterincorporatingthelossinfirm1’sdownstreamprofitcaused

byfirm2’soutputexpansion(see(2)).Thus,holding constantat ∗and constantat∗,theintegratedfirmwouldgainonbalance,despitethelossinitsdownstreamprofit,if

firm2wereexogenouslytoincreaseitsinputpurchasesanddownstreamsales.(Weshow

inSection3thatverticaldelegationcaninduceachangein bysignalingachangein .)

6

2.2 Two‐PartTarifffortheInput

Whentheintegratedfirmsellstheinputtofirm2usingatwo‐parttariff—i.e.apair , ,

where isafixedupfrontfee—itstotalprofitsare ; , ; ,where

; isgivenin(1).Aswewillexplain,theintegratedfirmcannotachievethevertically

integratedmonopolyprofitdespitetheuseofatwo‐parttariffand,therefore,atthe

equilibrium,itstillwantsfirm2tochangeitsoutput.Theanalysisismoreintricatethan

withlinearpricingandtheresultswilldependonthespecificsofdownstreamcompetition.

Inthatregard,let ; denotefirm1’s“reactionfunction”foritsdownstreamchoice,

i.e.thevalueof thatmaximizes ; .Notethat 0⁄ ifdownstream

competitionisBertrand,and 0⁄ ifitisCournot.9Bydefinition, 0⁄ if

downstreamchoicesarestrategiccomplements,and 0⁄ forstrategicsubstitutes.

Iffirm2acceptsthetwo‐parttariffoffer , (asitwillinequilibrium),the

downstreamoutcomeisgivenbythefunctions ∗ and ∗ aswithlinearpricing.

Denotefirm2’sprofitgrossofthefixedfeeas ∗ ∗ , ∗ ; .In

equilibrium,firm1willextractfirm2’sprofitbysetting ∗ .Therefore,itsets

tomaximize ∗ ∗ ∗ ,where ∗ isthesameaswithlinearpricing.

Let and ≡ ∗ denotetheprofit‐maximizingtwo‐parttariff,and , denote

theresultingequilibriumdownstreamchoices.

Proposition2.Assumefirm1sellstheinputtofirm2underatwo‐parttariff.Atthe

equilibriumoutcome , , , , , , ,firm1wantsfirm2toexpand

outputifdownstreamchoicesarestrategiccomplements,andwantsfirm2tocontract

outputifdownstreamcompetitionisCournotandquantitiesarestrategicsubstitutes:

(i) 0⁄ ifdownstreamcompetitionisCournotandquantitiesarestrategic

substitutes,and 0⁄ ifquantitiesarestrategiccomplements.

(ii) 0⁄ ifdownstreamcompetitionisBertrandandpricesarestrategic

complements,and ⁄ isambiguousifpricesarestrategicsubstitutes.

9DifferentiatingtheFOC ⁄ 0andusingtheSOC 0⁄ ,onefinds:sign ⁄sign ⁄ .From(1),wehave ⁄⁄ .UnderBertrand ⁄ 0while,underCournot, ⁄ 0.WeexplaintheintuitioninSection3,after(9).

7

Proof.Theintegratedfirmsets tomaximize ∗ ∗ ∗ .TheFOCis

∗ ≡

∗

∗

∗ ∗ ∗ 2

1

d 1∗

d 20, (6)

where ⁄ 0bytheEnvelopeTheorem.Decomposingd ∗ d ⁄ andrearranging,

∗ 2

1

1

2

d 2∗

d 2

1

2. (7)

(i)ForCournotcompetition:d ∗ d⁄ 0, ⁄ 0,and ⁄ 0;(7)implies

⁄ 0andhence ⁄ 0ifquantitiesarestrategicsubstitutes( ⁄ 0),

and ⁄ 0ifinstead ⁄ 0.

(ii)ForBertrandcompetition:d ∗ d⁄ 0, ⁄ 0,and ⁄ 0;(7)implies

⁄ 0ifpricesarestrategiccomplements( ⁄ 0).Ifinstead ⁄ 0,

theterminparenthesesin(7)hasanambiguoussign,hencesodoes ⁄ .

Interestingly,forCournotcompetitionandthe“normal”caseofstrategicsubstitutes,

thepatternisreversedrelativetoProposition1withnofixedfee(i.e. ≡ 0):startingat

theequilibriumoutcome,theintegratedfirmwantstoinduceadecreasein .ForBertrand

competitionandthe“normal”caseofstrategiccomplements,wehavethesamepatternas

inProposition1sincetheintegratedfirmwantstoinduceadecreasein .

ThesepatternscanbeunderstoodbycomparingtheFOCswithlinearinputpricing

versusatwo‐parttariff.Withlinearinputpricing,adecreaseinw2iscostlytofirm1since

revenuefallsfrominframarginalinputsales,term– ∗in(4);thus,attheoptimalw2,the

othereffectofadecreaseinw2—expansionbyfirm2—mustincreasetheintegratedfirm’s

profit,V.Therefore,theintegratedfirmwantsfirm2toexpand(Proposition1).Withatwo‐

parttarifffortheinput,adecreaseinw2doesnotreducefirm1’sprofitfrominframarginal

inputsales,duetothecompensatingincreaseinfirm2’sfixedfee(term ∗cancelsin(6)).

Instead,adecreaseinw2—besidesinducingexpansionbyfirm2—affectsfirm1’sprofitby

alteringitsdownstreamchoice,whichchangesfirm2’sprofitand,hence,thefixedfee

(terms ⁄ d ∗ d⁄ in(6)).

8

Inturn,d ∗ d⁄ isthesumoftwoeffectsshowninparenthesesin(7).With

Cournotcompetition,anincreasein willreducefirm2’soutput d ∗ d⁄ 0 and

increasefirm1’soutputinthe“normal”caseofstrategicsubstitutes ⁄ 0 .10Firm

1’sincreasedoutputreducesfirm2’sgrossprofitandthusthefixedfee,whichbyitself

reducesfirm1’stotalprofit.Fromtheoptimalityof ,theothereffectsoffirm2’s

contraction(inducedbyanincreasein )necessarilyincreasefirm1’sprofit.Thus,

startingattheoptimaltwo‐parttariff,firm1wantsfirm2tocontractifquantitiesare

strategicsubstitutes.11

WithBertrandcompetition,adecreasein willreducefirm1’sdownstreamprice

inthe“normal”caseofstrategiccomplementsduetothefallinfirm2’sprice,andfora

secondreason:byloweringtheprofitfrominputsalestofirm2and,hence,theopportunity

costofexpandingfirm1’soutput.12Sincethisreductioninfirm1’spricereducesfirm2’s

grossprofitandthusthefixedfee,thepreviouslogicimpliesthatattheoptimaltwo‐part

tariff,firm1wouldbenefitfromexpansionbyfirm2,aswithlinearinputpricing.13

Whywouldtheintegratedfirmbenefitfromanexogenouschangeinfirm2’schoice,

,notwithstandingtheassumedabilitytoextractfirm2’sprofitthroughthefixedfee?The

(perhapsobvious)answeristhatinourcontractingenvironmenttheintegratedfirmlacks

sufficientinstrumentstomaximizeoverallindustryprofit,downstreamplusupstream.

Denotetherequireddownstreamchoicesasthe“monopolysolution,” and .The

integratedfirmcaninduce byappropriatelysetting ∗ .However,sincewe

assumedthatfirm2’sfixedfee cannotbecontingenton ,theintegratedfirm’sbest

responsewillnotbe but ; ∗ ,thevalueof thatmaximizesfirm1’s

profitwhileignoringtheeffectonfirm2(sincethefixedfeeis“sunk”whenfirm1sets ).

At ; ∗ ,wehave ⁄ 0,implying Π ⁄ Π ⁄

10In(7), ⁄ 0since ⁄ 0underCournotcompetition.Seefootnote9.

11Ifquantitiesarestrategiccomplements,adecreasein leadsfirm1toexpandandthusreducethefixedfee(ceterisparibus),whichimpliesthatfirm1wantsfirm2toexpand.

12In(7), ⁄ >0since ⁄ 0underBertrandcompetition.Seefootnote9.

13Ifpricesarestrategicsubstitutes,adecreaseinw2hasopposingeffectsonfirm1’sdownstreamprice(thetermsinparenthesesin(7)runinoppositedirections),andtheneteffectisambiguous.

9

whichispositiveunderBertrandcompetitionandnegativeunderCournot.Thus,industry

profitwouldincreaseiftheintegratedfirmactedlessaggressivelydownstream(raised

priceorreducedoutput).Intuitively,giventheinabilitytocondition on ,firm2expects

firm1tofavoritsowndownstreamunitatfirm2’sexpense,andreducesaccordinglythe

fixedfeeitiswillingtopay.Thus,thetwo‐parttariffsolutionunderourinformational

assumptionsfailstomaximizeindustryprofit,leavingroomfortheintegratedfirmtodo

betterwithadditionalinstrumentseventhoughitalreadyextractsfirm2’sprofit.

Indeed,(7)showsthatunderatwo‐parttariff,theoptimalinputprice does

“doubleduty”bybalancingtwoeffects:how affectstheintegratedfirm’svariableprofit

byalteringfirm2’sdownstreamchoice ⁄ d ∗ d⁄ ;andhow affectsfirm2’s

profit(and,hence, )byalteringfirm1’sdownstreamchoice ⁄ d ∗ d⁄ .Thus,

isusedpartlytoalterfirm2’schoiceandpartlyasasignaltofirm2aboutfirm1’s

downstreamchoice.Thislogicsuggeststhattheintegratedfirmwouldbenefitifitfounda

waytosignalcrediblyaboutthelevelof beyondrelyingsolelyontheinputprice to

firm2.Thenextsectionconsidersthisissue.

3. Application:AlteringRival’sChoicethroughVerticalDelegation

AsnotedintheIntroduction,theliteratureonstrategiccompetitionintwo‐stagegames

identifiesvariouscommitmentmechanismsafirmmayusetovisiblyalteritsincentivesin

thesubsequentcompetition(shiftitsbest‐responsefunction)andsignalachangeinits

downstreamchoice,soastoalterarival’schoice.Inourcontext,theintegratedinput

supplierpotentiallycoulduseanyofthosemechanisms—dependingontheircost—to

inducethedesiredchangebyitsrival/customer.Forexample,firm1couldmakean

investmentthatchangesitsmarginalcostofsellingoutputinthedownstreammarket.

Here,wefocusonamechanismthatdoesnotrequireaninvestmentanddoesnot

haveanydirectcost(atleastnotexplicitlymodeled),verticalDelegation:theintegrated

firmestablishesadownstreamunit,division1,andcommitstochargeitapublicly

observableinputprice;andthedivisiontreatsaportion(nomatterhowsmall)oftheinput

pricechargedtoitbytheupstreamunitasacostratherthanapurelyinternaltransfer.To

simplifytheexposition,weassumethatdivision1maximizessolelyitsownprofit,andwill

10

explainlaterhowtheresultsextendtootherobjectivefunctions.Webeginwithlinear

pricingoftheinput,andthendiscussatwo‐parttariff.

UnderDelegation,thegameisasfollows.First,firm1publiclycommitstoapairof

inputprices , where denotesthepricetoitsdivision1.Giventhese

observedprices,division1andfirm2makedownstreamchoices simultaneously,

consumerspurchase,andfirm1receivesinputpayments.Division1chooses to

maximizeonlyitsprofit ; ≡ ; ,butinthepriorstagefirm1

nowsetsboth andthenewinstrument tomaximizeintegratedprofit, ; .14For

anygiven ,assumethereexistsauniquedownstreamequilibriuminpurestrategies,with

thechoiceofdownstreamrivalkdenoted anditsoutputleveldenoted .We

assume ⁄ 0ifdownstreamcompetitionisBertrand(Cournot).15Underthis

Delegationregime,firm1sets and tomaximizethecontinuationequilibriumprofit

function ≡ , ; .Let , denotetheoptimalchoice.

WewillcomparethisDelegationstructuretothestructurefromSection2,thatwe

nowlabelCentralization.16Forlateruse,observethatthebestresponsefunctionoffirm2,

; ,isthesameunderbothregimes.Also,let denotetheintegratedfirm’s

optimalchoicefor given ,i.e. maximizes .

Itisusefultointroducetheconceptofshadowmarginalcostoftheinput.Whenan

integratedfirmsellsinputstorivals,itsrelevantmarginalcostforexpandingits

downstreamoutputincludesboththeresourcemarginalcostcandanopportunitycostof

lostprofitsfromreducedinputsalestoitsdownstreamrival(s)(e.g.Sappington,2005).In

(1),wedecomposefirm1’sprofitfromitsdownstreamoutputsalesas ; ≡

,where isfirm1’sprofitonitsdownstreamsalesgrossofthecostofthe

internallyprovidedinput,andwrite:

14Firm1’stotalprofit, ,doesnotdependdirectlyon sinceinputpaymentsfromdivision1areapuretransfer,but willaffectVindirectlybychangingtheequilibriumvaluesof .

15SeeAppendixA.

16Arya,Mittendorf,andYoon(2008)establishresultssimilartoPropositions3through5belowforthecaseoflinear(differentiated)demandsandCournotcompetition.Theyfurthershowthatdecentralization(ourDelegation)increasestheintegratedfirm’sprofitiftheinputpricetothedownstreamdivisionisdeterminedbybargaining.

11

1

1

11

122

11 /

/)(

x

Q

xQ

xQcwc

x

G

x

V

. (8)

Theterminsquarebracketscanbeinterpretedastheintegratedfirm’sshadowmarginal

costforexpandingitsdownstreamoutput.Wewriteitmorecompactlyas

, ≡ , (9)

where ≡⁄

⁄istheinputdiversionratio—i.e.decreasedinputsalestofirm2per

extraunitofinputtodivision1.Notethat 0ifdownstreamcompetitionisBertrand,

sincedivision1’sexpansion(inducedbycuttingprice)displacessomesalesoffirm2;but

0ifcompetitionisCournot,sincedivision1thentakesfirm2’soutput,andthusalso

firm2’sinputpurchases,asgiven.17Let ∗ ≡ ∗ , ∗ , ∗ denotethe

shadowmarginalcostatthecontinuationequilibriumunderCentralization.

Proposition3.(i)UnderDelegation,theintegratedfirmcanachievethesameprofitas

underCentralizationbysetting ∗and ∗ ∗ ,i.e. ∗ ∗ ∗ ∗ , ∗ .

(ii)Therefore,theintegratedfirm’sequilibriumprofitis(weakly)higherwithDelegation:∗ ∗ ∗ ∗ , ∗ ∗ , ∗ , .

Proof.(i)UnderDelegation,supposetheintegratedfirmset ∗and ∗ ∗ ,i.e.

aninputpricetodivision1equaltotheshadowmarginalcostunderCentralization.This

wouldinducethesamedownstreamequilibriumchoicesaswithCentralization:

∗ ∗ , ∗ ∗and ∗ ∗ , ∗ ∗. (10)

Toseethis,observefirstthatunderCentralization,firm1’sbestresponsefunction

maximizes ; and,recalling(1),isimplicitlydeterminedby:

17Arya,Mittendorf,andSappington(2008)invokethisdistinctiontoshowthat,unlikeinstandardduopoly,BertrandcompetitiondownstreamcanyieldhigherpricesthanCournotwhenapartiallyintegratedinputmonopolistsellsalsotoadownstreamrival,becausethemonopolistinternalizesahigheropportunitycostunderBertrandthanunderCournot.

12

≡ 2 0, (11)

whereaswithDelegation,division1’sbestresponsefunctionmaximizes ; and,

hence,using ; ≡ ; ,isimplicitlydeterminedby

≡ 1 0. (12)

Setting , from(9)makes(12)identicalto(11),sothebestresponsefor

willbethesamefunctionof and inbothregimes.Sincefirm2’sbestresponseto

andtheexpected isalsothesamefunctioninbothregimes,setting ∗and∗ ∗ underDelegationwouldreplicatetheCentralizationoutcome.

(ii)UnderDelegation,theintegratedfirm’soptimalchoicefor conditionalon∗istypicallynot ∗ ∗ .Profitcouldbe(weakly)increasedbysetting atthe

optimallevel ∗ .Theactualoptimumwillgenerallyinvolvechangingboth and ,

therebyfurtherincreasingprofitcomparedtoCentralization.

ObservethattheabilityunderDelegationtochargedivision1anobservableinput

price ∗ ∗ benefitstheintegratedfirmsolelybecauseitaltersfirm2’schoiceby

signalingachangeindivision1’sdownstreamchoice.18InsteadofDelegation,thesame

outcomecouldbeachievedunderCentralizationiftheintegratedfirmcouldactasa

Stackelbergleaderinthedownstreamcompetition,asinthefollowinggame.19

18Thedirecteffectofchanging (holding constant)onfirm1’sprofitiszero;andthechangeininducedbyasmallchangein alsohasnofirst‐ordereffectbecause,given ∗and

∗,setting ∗ ∗ inducedthelevelof thatmaximizesfirm1’sprofit.

19Lu,Moresi,andSalop(2007)showthesameresultastheensuingProposition4assumingBertrandcompetition.OnthegeneralconnectionbetweenstrategicdelegationandStackelbergleadershipinthecompetitiongame,seeVickers(1985,sectionsIandII).Adaptedtooursetting,hisagentappointmentgame—thatprecedesdownstreamcompetition—correspondstowhethertheintegratedfirminitiallyadoptstheCentralizationstructureandsetsonlyaninputprice ortheDelegationstructureandsetsanadditionalinputprice .Theagentappointmentgamemaximizestheintegratedfirm’sprofitifandonlyifitimplementsthesamedownstreamoutcomeasStackelbergleadershipbytheintegratedfirm.SeealsoHeifetz,Shannon,andSpiegel(2007).

13

Leadership:AswithCentralization,firm1publiclycommitsonlyto .Butnow

downstreamchoicesoccursequentially,withfirstfirm1choosing tomaximize

, ; , andfirm2thenchoosingitsbestresponse, ; .

Proposition4.Theintegratedfirm’sprofitisthesameunderDelegationorLeadership.

Proof.UnderLeadership,theintegratedfirmsets and tomaximize:

, ; ; . (13)

UnderDelegation,let ; betheinversefunctionof , .Setting

and tomaximize , ; isequivalenttosetting and to

maximize:

, ; , ; . (14)

Since ; , ≡ ; ,thetwomaximizationproblemsareidentical.

Thenextresultidentifiestheintegratedfirm’sincentivetochange under

Delegation,relativetotheshadowmarginalcostunderCentralization, ∗ ∗ .

Proposition5.Holding constantat ∗,underDelegation:

(i)WithBertrandcompetition,theintegratedfirmwantstoinduceafallin2’sprice,

implying ∗ ∗ ∗ ifpricesarestrategiccomplements(substitutes).

(ii)WithCournotcompetition,theintegratedfirmwantstoinducearisein2’squantity,

implying ∗ ∗ ifquantitiesarestrategicsubstitutes(complements).

Proof.Startingat ∗and ∗ ∗ ,underDelegationtheintegratedfirm’s

desiredchangein isdeterminedbythesignof(from ≡ , ; ):

. (15)

Observethat ⁄ ⁄ ⁄ ,andweassumed ⁄ 0if

downstreamcompetitionisBertrand(Cournot),whilebydefinition ⁄ 0if

downstreamvariablesarestrategiccomplements(substitutes).

14

Setting hereisakintoan“investment”thataffectsdivision1’smarginalcostina

standardtwo‐stagegameandtherebyshiftsitsbest‐responsefunctionindownstream

competition.Thechangein neededtoinducethedesiredchangein willtherefore

dependonfamiliarissues—whetherdownstreamcompetitionisinpricesorquantities

andwhetherthesechoicevariablesarestrategiccomplementsorsubstitutes.Forbrevity,

wewillthereforeexplainonlypart(i)ofProposition5;thelogicfor(ii)issimilar.

WithBertrandcompetition,Proposition1showsthatfirm1’sprofitwouldincrease

iffirm2weretoloweritsprice .UnderDelegation,firm1caninducefirm2tocut by

lowering tosignalareductionindivision1’sprice ,inthe“normal”casewhereprices

arestrategiccomplements(orraising ifpricesarestrategicsubstitutes).Lowering is

animperfectwaytoshrinkfirm2’smarginandreducedoublemarginalizationininputsales

tofirm2.However,becausethereductionin isinducedbylowering ,firm2’soutputis

likelytodeclineonbalancerelativetoCentralization.20

Interestingly,whenmovingfromCentralizationtoDelegation,firm1likelywillraise

above ∗underBertrandcompetition.Formally,startingat ∗and ∗ ∗

underDelegation,andusing ⁄ 0,firm1’sdesiredchangein isdeterminedby

thesignof

2

2

2

2. (16)

Since ⁄ ∗ ∗ 0,(16)wouldbezeroiffirm2’s“pass‐throughrate”under

Delegation, ,⁄ wasequaltothatunderCentralization,d ∗ d⁄ (see(4)).However,

withBertrandcompetitionandstrategiccomplements, ⁄ likelyissmallerthan

d ∗ d⁄ ,asdiscussedinAppendixA,inwhichcase(16)ispositiveandfirm1hasan

incentivetoraise whenholding constantattheshadowmarginalcost ∗ ∗ .

Intuitively,raisingw2increasesfirm1’smarginalprofitabilityofsellingtofirm2,which

inducesadownstreampriceincreasebyfirm1underCentralizationbutnotunder

20ThelineardemandexampleinAppendixBsupportsthisintuition.Inequilibrium,theintegratedfirmtypicallywilladjustalso ,asdiscussedmomentarily.However,theequilibriumchangein

and canbeexpectedtotrackthedirectionfromPropositions5and1,respectively,asillustratedbytheexampleinAppendixB.

15

Delegation;thisexpecteddifferentialresponseleadsfirm2toraiseitspricebyless,hence

reduceitsinputpurchasesbyless,underDelegationfollowingagivenincreaseinw2.

Nowsupposefirm1chargesfirm2atwo‐parttarifffortheinput.Proposition3

continuestohold—firm1earns(weakly)higherprofitunderDelegation.WithDelegation,

firm1canbothmaximizeindustryprofitandcollectit,bysettingthemarginalinputprices

atthelevels and thatinducedivision1andfirm2tochoose(unilaterally)the

“monopolysolution” and ,andsetting toextractfirm2’sprofits.21With

Centralization,firm1generallycannotimplementthissolutionevenwithtwo‐parttariffs

(seeSection2.2).

Proposition4alsocontinuestohold.UnderLeadership,theintegratedfirmcan

achievethemonopolyoutcomebysetting and ,whichwillinducefirm2

toset .Andbydefinition,itcannotearnmorethanthemonopolyprofit.Thus,

DelegationandLeadershipyieldthesamepayoff,themonopolyprofit.Proposition5,

characterizingtheincentivesregarding underDelegation,mustbemodifiedforthecase

ofatwo‐parttariff,butforbrevityweforgothatexercise.

Weendthissectionwithabriefdiscussionofadditionalrelatedwork,andthe

plausibilityofDelegationbyaverticallyintegratedfirm.BonannoandVickers(1988)noted

thestrategicadvantageofcommitmenttoobservableinputprices.Theyconsider

differentiatedBertrandcompetitionbetweentwosupplierswithpricesasstrategic

complements.Ifbothsuppliersareverticallyintegrated,eachprovidestheinputtoits

downstreamretailunitatmarginalcost.Ifbothareverticallyseparated,eachsellsthrough

adifferentsingleretailerandcapturesthelatter’sprofitwithatwo‐parttariff.Forany

inputpricesetbyonesupplier(includingmarginalcostifintegrated),theothersupplier

preferstoverticallyseparateandraisetheinputpricesomewhatabovemarginalcost,to

coaxanincreaseintherival’sdownstreamprice.Oursettingfeaturesaninputmonopolist

21Whendivision1andfirm2arecompetitors,toimplement and theintegratedfirmtypicallymustsetthemarginalinputpricesabovethemarginalcost .Thequalitativeresultsdonotchangeiffirm2hasanalternativesourceofsupplyand,asaresult,theintegratedfirmcannotextractalltheprofitsoffirm2.

16

engagedindownstreamcompetition,andweshowedthatbecausetherivalisalsoaninput

customer,theincentiveistoinduceareductioninitsdownstreamprice.

OurassumptionthatunderDelegationthedownstreamdivisionacts(atleastpartly)

initsowninterestisreminiscentoftheliteratureonstrategicadvantagesofcreating

autonomouscompetingdivisions(e.g.SchwartzandThompson,1986;Baye,Crocker,andJu,

1996).There,divisionalautonomymakesthefirmatoughercompetitor,todeterentryor

induceoutputcontractionbyoligopolyrivalswhenoutputsarestrategicsubstitutes.Here,

autonomyinvolvesaverticaldivisionandthestrategicgaindoesnothingeonpresentinga

toughposture,e.g.underCournotcompetitionandlinearinputpricing,theintegratedfirm

usesDelegationtomakeitselfasoftercompetitordownstream.

RegardingthefeasibilityofverticalDelegation,therearetworequirements:the

verticallyintegratedsuppliercancommittochargeitsdownstreamdivisionaninputprice

observabletodownstreamrivals;andthedivisiondoesnot“undo”thestrategiceffectsof

thatinputpricebyfullyinternalizinghowitschoiceaffectstheupstreamprofitsofthe

affiliatedsupplier.Forexpositionalconvenienceweassumedthedivisionmaximizessolely

itsowndownstreamprofit,butallourresultsextendtoanyobjectivefunctionforwhich

theinputpriceaffectsthedivision’sdownstreamchoice.Forexample,ifthedivision

maximizesaweightedaverageofitsprofitandtheintegratedfirm’sprofit,thesuppliercan

implementitspreferreddownstreamoutcomebysuitablyraisingtheinputpricetoits

divisiontocompensateforthedivisiontreatingafractionofthatpriceasapuretransfer

(Arya,Mittendorf,andYoon,2008,Proposition3).Inpractice,integratedfirmsmayfindit

feasibleandbeneficialtoestablishdivisionsasprofitcentersforinternalincentivereasons.

Anothermechanismthatmaypreventthedownstreamunitfromactingasapassivearmof

anintegratedfirmistoallowminorityshareholdersinthedownstreamunit,whichmay

limittheabilitytotreatinputpaymentstotheupstreamaffiliateasapuretransfer.22

22Forexample,O’BrienandSalop(2000)state:“inmakingdecisionsthataffecttheacquiredfirm[theanalogueofourdownstreamdivision],theBoardofDirectorsoftheacquiredfirmisconstrainedtoignoretheimpactofitsactionsontheacquiringfirm,eveniftheacquiringfirmhasalargefinancialinterestintheacquiredfirm.Instead,theBoardmustmanagetheacquiredfirmtoactlikeanindependent,stand‐aloneentity.”(p.580)

17

Ouranalysissuggeststhatmechanismsthatyielddivisionalautonomycanhave

strategicbenefitsinpartialintegrationsettings,providedacommitmentcanbemadeto

chargethedivisionanobservableinputprice.23Thelatterassumptionismorequestionable

forunregulatedfirms.24Oneconsiderationthatmayaidsuchcommitmentisthesupplier’s

interestinmaintainingareputationfornotactingopportunisticallytodisadvantage

independentcustomersbysecretlyfavoringitsdivision(e.g.McAfeeandSchwartz,1994).

Topreservesuchareputation,itmayadoptapolicyofpublicandtransparentpricing.

4. ConcludingRemarks

Weconsideredaverticallyintegratedinputmonopolistthatbothsellsoutputinthe

downstreammarketandsuppliesinputstoadifferentiateddownstreamrival.Expansion

bytherivalharmstheintegratedfirmdownstreambutbenefitsitupstreamfromincreased

inputsales.Ourmainresultshowsthatunderlinearpricingoftheinput,theintegrated

firmwouldunambiguouslygainfromexogenousexpansionbyitsrival/customer.The

optimalinputpricebalancestherevenuelossoninframarginalinputsalesfromasmallcut

intheinputpriceagainsttheinducedexpansionbytherival/customer,implyingthatsuch

expansionmustincreasetheintegratedfirm’sprofit,despitethedownstreamloss.Ifthe

inputissold,instead,underatwo‐parttariff,theinframarginalsaleseffectisabsent,

becauseofthecompensatingchangeinthefixedfee,andisreplacedbyadifferenteffect:

howchangingtheinputpricealtersthesupplier’sownoptimaldownstreamchoiceand

howthat,inturn,affectstheattainablefixedfee.Theprofiteffectofexpansionbythe

rival/customerthendependsonthespecificsofdownstreamcompetition.WithBertrand

competitionandpricesasstrategiccomplements,theintegratedfirmwouldstillbenefit

fromexogenousexpansionbytherival/customer,butwouldloseunderCournot

competitionwithstrategicsubstitutes.

23Forageneraldiscussiononfirms’abilitytomakecommitments,seeShapiro(1989,p.382).

24Regulatedfirmsmaybesubjecttorulesgoverninginputpricingtosubsidiariesaswellasthesubsidiaries’behaviorthroughimputationrequirements(e.g.LaffontandTirole,2000).

18

Weanalyzedoneofpotentiallymanymechanismstoinduceexpansionor

contractionbytherival/customer,beyondrelyingsolelyontheinputpricetothatfirm.

Thatmechanismisverticaldelegation:establishingadownstreamdivisionthattreatsits

inputpricefromtheupstreamaffiliatepartlyasacost(ratherthanapurelyinternal

transfer),andchargingthatdivisionaninputpriceobservablebytherival/customer.

Verticaldelegationwasshowntodominatecentralizedbehaviorbytheintegratedinput

supplier,duetotheincreasedstrategicabilitytoalterthebehavioroftherival/customer.

Tofocusonthenewincentivescreatedwhenthedownstreamrivalisalsoaninput

customeroftheverticallyintegratedsupplier,weabstractedthroughoutfromupstream

competition.Animportantextensionwouldbetoexplorehowtheresultsareaffectedby

thepresenceofupstreamcompetition.

19

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21

Appendix

A. Owncosteffectandtheindirectstrategiceffect

InProposition1,weassumethatanincreaseintheinputpriceleadsfirm2toincrease

priceunderBertrandcompetition,i.e.d ∗ d 0⁄ ,andreduceoutputunderCournot

competition,i.e.d ∗ d 0⁄ .Thesederivativescanbeanalyzedbydifferentiatingthe

downstreamequilibriumconditions

∗ ∗; and ∗ ∗; (A1)

where denotesthereactionfunctionofdownstreamrivalk(k=1,2).Thus:

∗

. (A2)

In(A2),thedenominatorispositivesinceweassumethereactionfunctionshave

slopessmallerthanunityinabsolutevalue.Thefirstterminthenumerator, ⁄ ,is

the“directcosteffect”andweassumeitisstrictlypositiveunderBertrandcompetition—

i.e.holding constant,anincreasein leadsfirm2toraise —andstrictlynegative

underCournotcompetition—i.e.holding constant,anincreasein leadsfirm2to

decrease .Thesecondtermisthe“indirectstrategiceffect”anditiszerowithCournot

competition(sincethereisnoinputdiversion,firm1’sshadowmarginalcostin(9)is

unaffectedby ,implying ⁄ 0).WithBertrandcompetition,theindirect

strategiceffectispositive(likethedirectcosteffect)ifpricesarestrategiccomplements.If

insteadpricesarestrategicsubstitutes,weassumethatthedirectcosteffectdominatesthe

indirectstrategiceffect.

InProposition5,weassumedivision1’soutputfallsif increases(holding

constant ),i.e. ⁄ 0ifdownstreamcompetitionisBertrand(Cournot).

Thesederivativescanbeanalyzedbydifferentiating

; and ; . (A3)

22

Notethatdivision1’sreactionfunctionunderDelegation, ,isdifferentfromfirm1’s

reactionfunctionunderCentralization, .Inparticular,underDelegationthereisno

indirectstrategiceffectsince ⁄ 0(and ⁄ 0).Thus:

(A4)

(andasimilarexpressionfor ⁄ ).

UnderBertrandcompetition,strategiccomplementsandlineardemand,wehave

d ∗ d ⁄ 0⁄ because 0⁄⁄ , ⁄ 0and

⁄ ⁄ .25Intuitively,anincreasein willraisefirm1’sshadowmarginal

costunderCentralization(whichis , from(9))butwillnotaffectdivision1’s

perceivedmarginalcostunderDelegation(whichissimply ).Thus,followinganincrease

in ,firm2expectsalargerpriceincreasebyfirm1underCentralizationthanunder

Delegation,leadingtoagreaterequilibriumpass‐throughrate,d ∗ d ⁄ 0⁄ .

Inturn,firm2’slowerpass‐throughrateunderDelegationencouragesfirm1toraise

above ∗.26

ThisincentivehelpsexplainaseemingpuzzleinAppendixB,Table1,wherethe

equilibriuminputpricechargedtofirm2underDelegation(3.00)ishigherthanunder

Centralization(2.97),despitetwoforcesthatpushintheoppositedirection:firm1’s

outputpriceunderDelegation(3.00)islowerthanunderCentralization(3.12),whichby

itselfincreasestheprofitabilitytotheintegratedfirmofdivertingsalestofirm2by

reducing ;andfirm2’sequilibriumoutputunderDelegation(100)islowerthanunder

Centralization(109),whichalsobyitselfcallsforreducing .

25Withlineardemand,thederivativesofthereactionfunctionsarescalarsandhencedonotdependonprice,quantityorinputpricelevels.

26ThisincentivedoesnotariseunderCournotcompetitionandlineardemand.Thatis,d ∗ d ⁄⁄ since ⁄ 0⁄ and ⁄ ⁄ .

23

B. Example

Assumethatthemarginalcostoftheinputisconstant, 1,andthattherearenoother

downstreamcosts.Consumerdemandfortheproductsoffirms1and2isgiven

by 500 200 100 ,where denotestheoutputoffirm ,and and denote

thepricesoffirms and ,respectively( , ∈ 1,2 , ).27

WebeginwiththecaseofBertrandcompetition.Table1belowshowsthe

equilibriumprofits,outputsandpricesunderCentralizationandDelegation.

Table1:BertrandCompetitionDownstream

Centralization Delegation

ProfitofIntegratedFirm 696.97 700.00

ProfitofDownstreamRival 59.50 50.00

OutputofIntegratedFirm, 227.27 250.00

OutputofDownstreamRival, 109.09 100.00

OutputPriceofIntegratedFirm, 3.12 3.00

OutputPriceofDownstreamRival, 3.52 3.50

InputPriceChargedtoDownstreamRival, 2.97 3.00

ShadowCostofSupplyingInputtoDivision1, ∗ ∗ 1.98 NA

InputPriceChargedtoDivision1, NA 1.75∗ NA 1.74

Table1illustratesProposition3:Delegationallowstheintegratedfirmtoincreaseits

profitrelativetoCentralization.ItalsoillustratesProposition5(i):startingfromthe

Centralizationsolution,theintegratedfirmwantstoinduceareductioninfirm2’s

downstreamprice ;andsincelineardemandsimplythatpricesarestrategic

27Inthisexamplewherefirms1and2aresymmetric,verticalintegrationoffirm1andtheinputmonopolistdoesnotleadtoforeclosureoffirm2underCentralization.SeeArya,Mittendorf,andSappington(2008).OurresultsimplythatthisalsoistrueunderDelegation.

24

complements,inducingareductionin requiressignalingareductionin bylowering

: ∗ ∗ ∗ .Thatis,attheinputpricechargedtofirm2underCentralization

(2.97),theprofit‐maximizinginputpricetodivision1underDelegationislowerthanthe

shadowmarginalcostunderCentralization,1.74<1.98.IntheactualDelegation

equilibrium,theintegratedfirmraises slightly,from2.97to3.00,andreduces to1.75

(insteadof1.74)fromtheshadowmarginalcostof1.98.Thereductionin andincrease

in leadtoa(substantial)decreaseintheprofitofthedownstreamrival.

WenowturntothecaseofCournotcompetition.

Table2:CournotCompetitionDownstream

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 3.00

OutputPriceofDownstreamRival, 3.52 3.57

InputPriceChargedtoDownstreamRival, 2.97 3.00

ShadowCostofSupplyingInputtoDivision1, ∗ ∗ 1.00 NM

InputPriceChargedtoDivision1, NM 1.29∗ NM 1.28

Table2alsoillustratesProposition3—theintegratedfirm’sprofitagainishigherunder

Delegation.ItalsoillustratesProposition5(ii):startingfromtheCentralizationsolution,the

integratedfirmwantstoinduceanincreaseinfirm2’squantity ;andsincelinear

demandsimplythatquantitiesarestrategicsubstitutes,thisrequiressignalingareduction

in byraising .Thus,movingtoDelegationtheintegratedfirmwantstoraisetheinput

pricetodivision1abovetheshadowmarginalcostunderCentralization(which,under

Cournotcompetition,simplyequalstheresourcemarginalcost, 1,independentof

: ∗ 1.28 1.00 .IntheactualDelegationequilibrium, rises

25

substantially,to1.29(from 1),while risesslightly,from2.97to3.00,causingthe

integratedfirm’soutput tofallandtherival’soutput torise,consistentwiththe

incentivesdescribedabove.Thistime,therival’sprofitincreases,unlikeintheBertrand

case.